vdoc.pub mathematics-in-fun-and-in-earnest

iiiatnematics
in Fun and in
Earnest
nathan fl. Court
I
I
MATHEMATICS
in Fun and in Earnest
Nathan A. Court
DOVER P U B L I C A T I O N S , INC.
Mineola, New York
T o DAVID, LOIS a n d ELLEN
Copyright
Copyright © 1935, 1945, 1948, 1953, 1954, 1955, 1956, 1958 by Nathan
Altshiller Court
All rights reserved.
Bibliographical
Note
This Dover edition, first published in 2006, is an unabridged republication of
the work originally published in 1958 by the Dial Press, New York.
"Sonnet XLV" by Edna St. Vincent Millay from Collected Poems, Harper &
Brothers, copyright 1920, 1948 by Edna St. Vincent Millay.
International
Standard Book Number:
0-486-44968-8
Manufactured in the United States of America
Dover Publications, Inc., 31 East 2nd Street, Mineola, N.Y. 11501
Preface
Toward the end of the last century ("fin de siecle") the
idea was afloat that the planet Mars was inhabited. Venturesome spirits were anxious to convince those Martians that their
neighboring planet Earth, too, is people with intelligent creatures. Those Earthians, or Terrestrians had the urge to tell
their hypothetical neighbors that they are not only interested
in the inhabitants of the red planet but would also like to
"hear" from them.
The surest way to bring about those desirable ends would
be to construct, say, in the Sahara desert a gigantic geometrical
figure of the kind used to prove the Pythagorean theorem. If
such a figure powerfully illuminated were to be flashed into
the sky at an appropriate time, it would surely attract the attention of the Martians and induce them to reciprocate in
kind. For mathematical propositions are universal and eternal
verities and therefore familiar to all intelligent creatures
everywhere.
The Martians did not receive that signal: the project
never got beyond the talking stage. But the story serves as
"documentary evidence" of the attitude of the educated public
toward mathematics as recently as half a century ago.
New conquests of science in general and of mathematics
in particular have created a different intellectual climate
which caused many preconceived notions to be abandoned,
many a cherished myth to be given up.
The mathematician came to the conclusion that his science
is a human enterprise, beset with all the foibles inherent in
man's handiwork, but also resplendent with his creative power,
his imaginative sweep, radiant with his love for beauty. The
mathematician still feels that he is justified in his claim that
mathematics is the brightest jewel in the intellectual crown of
mankind.
Most of the essays assembled in this book attempt to
mirror this "new look" or new outlook of mathematics. They
were written over a considerable period of time, mainly during
the last decade or so. The author was privileged to be able to
share his reflections with others, from the speaker's platform,
over the air, and through the printed page of the periodical
press. The more than favorable reception which was con5
6
PREFACE
sistently accorded those utterances provided the incentive for
collecting thei^ within the covers of the same book.
Although these writings met with approval of very competent judges, they were not composed for the benefit of the
experts in the field. The author had primarily in mind the
cultured lay reader whose intellectual curiosity impels him to
try to keep abreast of the times, and on the other hand the
professional whose field of special interest is more or less
removed from the domain of mathematics. On that account
care has been taken to avoid technical mathematics, and
where recourse has been had to it, its scope does not surpass
the high school level. And even that part may usually be
skipped without necessarily losing the trend of the argument
at hand.
By the nature of their origin, each of these essays is complete in itself. From the point of view of the reader this has the
advantage that the book may be read section by section in
any order that may be found interesting or convenient, not
necessarily in the order adopted in the book. On the other
hand this independence of the various sections from one another and also their restricted size account for the fact that
some topics are discussed in more than one place, although
usually from a different angle. This is particularly true about
the axiomatic method in mathematics. However, the fundamental importance of this subject, not only in mathematics,
makes it bear repetition quite gracefully. The reader who
would be interested in following up any given topic may be
helped by the cross references indicated in the text at the
appropriate places and stated explicitly at the end of each
chapter.
The lay leader would not be surprised to see that mathematicians are concerned with the history and philosophy of
their subject, with the relation of mathematics to social problems, etc. But he is rarely led to suspect that the practitioners
of the most "mysterious" of the sciences find within their
subject room for recreation, for play, to use a simpler word.
It seemed to this writer that, to be complete, the picture
of mathematics should also comprise something of the
"lighter" side of this discipline. Moreover, the reader may
perhaps be induced to try his own hand at playing some game
of the mathematician. Should he yield to such a temptation
he may be surprised to find how fascinating such a game may
turn out to be. It may even happen that he, the reader, may
be amused by this writer's clumsy, no doubt, and decidedly
unorthodox attempts at fictionalizing or dramatizing some
geometrical propositions.
7
PREFACE
If some serious minded reader would come to the conclusion that the author does not always treat the earnest topics
with quite the traditional dignity (synonym—stiffness) becoming such a subject, he may be perfectly right. But I do not
propose to apologize for this misdeed. I would rather hope
to win that reader over to my own credo:
Mathematics in earnest should be fun,
Mathematics in fun may be earnest.
The University of Oklahoma
Norman
June, 1958
N. A.
c.
CONTENTS
Chapter I
1
• S O M E PHILOSOPHICAL ASPECTS OF MATHEMATICS
A
B
C
D
E
F
G
H
1
2
• The nature of Mathematics
• The unity of form and number
• The dimensions of space
• Postulational Mathematics
• The question of consistency
• The empirical origin of the axioms
• The worth of deductive reasoning
• Imagination and imitation
• Conclusion
• G E O M E T R Y AND EXPERIENCE
A
B
C
D
E
3
MATHEMATICS AND PHILOSOPHY
• Origins of geometrical knowledge
• The sense of touch
• The sense of vision
• Metrical Geometry and Projective Geometry
• Conflicting testimony of the senses
• T H E M I G H T AND PLIGHT OF REASONING
A
3
C
D
E
F
G
• Reasoning and psychology
• The role of the body in the reasoning process
• A definition of reasoning
• Applications of the definition
• Pitfalls and merits of reasoning
• More checks on the definition
• Reasoning, memory, and invention
8
15
15
16
17
19
23
23
25
26
28
28
28
29
30
31
33
35
35
36
38
40
42
44
46
4
• PLANE G E O M E T R Y AND PLAIN LOGIC
47
A • The impact of non-Euclidean Geometry and of
Projective Geometry
B • The formalist approach to Geometry
C • Role of the knower
D • Axiomatic method
E • Meaning of intuition
F • Foundations of logic
G • Symbolic representation, or miniature realization
H • Rational theory of objective existence
I • Logic
Chapter II
1
SOME SOCIOLOGIC ASPECTS OF
MATHEMATICS
• MATHEMATICS AND CIVILIZATION
63
A • The early beginnings of counting and reckoning
B • Measuring • Beginnings of Geometry and
Chronology
C • The Renaissance period. The great voyages. The
invention of Analytic Geometry and of the
Calculus
D • Mathematics for the modern age
E ' Conclusion
2
• MATHEMATICS AND GENIUS
THE LURE OF THE
1 • T H E VAGARIES OF THE INFINITE
A
B
C
D
63
67
71
73
74
75
A • The "heroic" and the "objective" interpretations
of history
B • Are inventions inevitable?
C • Genius and environment
D • Genius and the "instinct of workmanship"
E • Mathematics—the patrimony of the race
Chapter III
47
49
50
52
53
55
57
58
59
75
77
79
81
82
INFINITE
84
• No largest number
84
• A part as big as the whole
85
• Arithmetical operations performed on the infinite 88
• No escape from the infinite
90
9
3
T H E INFINITE IN G E O M E T R Y
91
A • Parallelism in Euclid's Elements
B • The difference between Metric and Projective
geometry
C • The point at infinity of a line
D • The line at infinity of a plane and the plane at
infinity of space
E • Advantages and limitations of the elements at
infinity
F • Could Euclid find room in his Elements for
points at infinity
G • Do the elements at infinity "enrich" Projective
Geometry
91
• T H E MOTIONLESS A R R O W
92
93
94
96
97
98
99
A • Arrows
99
B • The arguments of Zeno and those of his imitators 99
C • Aristotle's arguments about the infinite divisibility
of both time and space
101
D • The potential infinity and the actual infinity
103
E • Motion and dynamics
104
F • Motion—an "undefined" term
106
G • Theory and observation
107
H • Instantaneous velocity
108
1 ° A modern answer to Zeno's paradoxes
110
Chapter IV
1
MATHESIS THE
• MATHEMATICS AND ESTHETICS
111
111
112
A • Beauty in Mathematics
B • Mathematics in beauty
2
BEAUTIFUL
• A R T AND MATHEMATICS
113
• Mathematics, logic, music
113
• Opinions of mathematicians about Mathematics 115
• The opinion of a poet
116
• Mathematics—a creation of the imagination
117
• Further analogies between Mathematics and the
arts: symbolism, condensation, care in execution, etc.
118
F • "Movements" in Mathematics
121
G • Conclusion
122
A
B
C
D
E
10
Chapter V
1
MATHEMATICS AND THE
MATHEMATICIAN
• I s MATHEMATICS AN EXACT SCIENCE?
A • Mathematicians Are Human
a • A definition of Mathematics
b • Is the mathematician "objective"?
c • Priority disputes
d • Withholding results
e • Mistakes of mathematicians
/ • Disputes over results obtained
B • Schools of Thought in Mathematics
g • The quest for rigor
h • Euclid and the "obvious" foundations of
Mathematics
i • Formalism
/ • Logicalism
k • Intuitionism
I • New logics
m - Conclusion
2
• PERPLEXITIES OF A POTATO-PUSHER
A
B
C
D
E
F
3
132
133
134
135
137
339
• GEOMETRICAL MAGIC
140
A • A point
3 • A square deal
fixation
MATHEMATICAL
•
•
•
•
•
•
140
143
ASIDES
1 • MATHEMATICAL ASIDES
A
B
C
D
E
F
127
127
129
130
131
131
132
• Winning a prize
• Gambling and statistics
• Tit-tat-toe ancient and modern
• New checker games for old
• Potato-pushing a la mode
• Conclusion
Chapter VI
123
123
123
123
124
125
125
125
126
126
147
"It is obvious that . „
Four examples
An explanation
Analogy as a useful guide to discovery
Limitations of that method
The altitudes of a triangle and of a tetrahedron
11
147
147
149
151
152
154
2
• " T H E FIGURE OF THE BRIDE"
156
A • Historical data
B • The theorem of Pythagoras in India
C • The story of Lilavati
3
• RUNNING AROUND IN CIRCLES
4
• Too
159
MANY?
Chapter VII
164
MATHEMATICS
AS
RECREATION
1 • MATHEMATICAL FOLKLORE
A
B
C
D
2
"River crossing" problems
Multiplication performed on the fingers
"Pouring" problems • The "Robot" method
The "false coin" problem
• FAMOUS PROBLEMS
A
B
C
D
E
F
G
3
•
•
•
•
•
•
•
•
•
•
•
156
157
158
168
168
171
173
178
182
Morley's problem
The problem of Appolonius
"Fermat's last theorem"
Goldbach's conjecture
The problem of the tangent
The recurrence of "famous" problems
Conclusion
• WITHOUT THE BENEFIT OF PAPER AND PENCIL
A • Mathematics and computation
B • Problems
C ' Solutions
12
182
183
183
184
185
185
186
187
187
188
189
Acknowledgments
It is a great pleasure for me to express my indebtedness to my friends and colleagues for all I
learned from them in our informal, private, and
after-dinner conversations, and for the opportunity I had to clarify my own ideas that find their
expression in the pages that follow. I also thank
those of them who read in manuscript some sections of this book or supplied me with references
in the field of their competence.
Dr. Duane H. D. Roller earned my sincere
gratitude for having read, carefully and attentively, a considerable part of the manuscript, and for
his willing and obliging help as curator of the DeGolyer Collection of History of Science and
Technology.
Sophie Court read intelligently and criticized
mercilessly the entire manuscript, from beginning
to end. She corrected all of the typescript and all
the printed proofs with skill and devotion. To
try to thank Sonia would be to belittle both her
and her help.
Last but not least I wish to record my thanks
to Clarkson N. Potter of The Dial Press, without
whose initiative and sustained interest this book
may perhaps not have seen the light of day.
N. A. C.
13
I
1
MATHEMATICS AND PHILOSOPHY
• Some Philosophical Aspects of Mathematics
Introduction. The historical relation of Philosophy and Mathematics is a matter that a mathematician may point out
with some legitimate pride. Philosophy, as it is understood in
our Western world, is the creation of the ancient Greeks.
The Greek "love of wisdom" included the study of nature as
well as the inquiry into the forms of human relations, that is
to say it embraced all learning. Only later did different
branches of philosophy break away from their parent and
form independent disciplines. Physics, for instance, did not
emancipate itself until sometime in the sixteenth century, and
psychology only during the nineteenth century. Mathematics,
however, was never a part of philosophy. It was recognized
by the Greek philosophers as an independent intellectual pursuit from the very start. The history of mathematics in
Greece runs parallel to the history of philosophy itself.
A ' The Nature of Mathematics
Mathematics has its roots
deep in the soil of everyday life and is basic in our highest
technological achievements. We use mathematics when we
count the lumps of sugar for our breakfast cup of coffee,
we use mathematics when we build our houses, erect our
lofty skyscrapers, when we construct our wonderful printing
presses and our imposing bridges, our mysterious radios, our
supersonic airplanes. At the same time mathematics is reputed to be, and actually is, the most abstract, the most
hypothetical of sciences. Let us illustrate this statement by
an example. The Greeks considered what they called "per15
16
MATHEMATICS IN FUN AND IN EARNEST
feet" numbers, that is, numbers which are equal to the sum
of their divisors. The number six is such a number, for
6 = 1 + 2 + 3 . Again 28 = 1 + 2 + 4 + 7 + 1 4 is a perfect number,
and 496 is another example of such a number. Notice
that all three examples offered are even numbers. Whether
there are odd perfect numbers remains an open question.
What is certain is that no such number has even been found.
However, this circumstance has not deterred the mathematician from studying odd perfect numbers and proving theorems concerning them. In other words, the mathematician is
ready to make the statement: "If there are odd perfect numbers, they will exhibit such and such characteristics."
B ' The Unity of Form and Number
The representation
of numerical data in graphical form has achieved wide acceptance. The daily press, and periodicals in general, often
have recourse to this device when discussing various phases
of our economic life, like the fluctuations of commodity
prices, population data, changes in the size of crops through
the years, etc. Graphs are used by governmental agencies, by
industrial and commercial concerns to render account of
their activities. Curves representing data obtained in scientific experiments abound in both professional journals and in
learned tomes.
This very efficacious union of numbers and picture, or
forms, is one of the applications of analytic geometry, or
Cartesian geometry as it is called after its inventor Rene
Descartes (1596-1650).
However, the ambitions of analytic geometry in the use
of number go much farther. This branch of mathematics replaces straight lines, circles, and other curves, by algebraic
equations, and does the same for cylinders, spheres, ellipsoids, and other surfaces. By manipulating those equations
analytic geometry obtains the geometrical properties of those
curves and surfaces. Thus geometrical reasoning is replaced
by algebraic operations, pretty much as the use of algebra
takes the place of arithmetical arguments in the solution of
problems which we in our earlier school days tried to puzzle
out by arithmetic.
How far can this union of algebra and geometry be pursued?
The invention of projective geometry, early in the nineteenth century, brought forward the idea that geometric
properties may be divided into two kinds, those of measure
and those of position. The length of the circumference of a
circle is a question of measure, but that this circumference
MATHEMATICS AND PHILOSOPHY
17
cannot have more than two points in common with a straight
line has nothing to do with measurements and is a property
of position.1 Now, in the middle of the nineteenth century
Karl Staudt (1789-1867) showed that the entire domain of
projective geometry, which domain deals with questions of
position exclusively, may be developed with complete independence from the notion of measure. On the other hand,
the classical analytic geometry essentially presupposes a unit
of length and therefore measurements. But this serious difficulty has not divorced number from geometry. Even before
the work of Staudt had appeared, a suitable apparatus for
the analytical treatment of projective properties was already
in use.
But algebra has no exclusive rights to the exploration of
geometry. The invention of the calculus was to a large degree inspired by geometrical problems. The calculus, in its
turn, applied its great powers to further the study of geometry, and during the nineteenth century created the vast domain of differential geometry.
Furthermore, during the present century projective concepts were introduced into differential geometry, and thus the
new doctrine of projective differential geometry came into
being. The union between number and space instead of growing weaker has become so intimate that E. J. Wilczynski
(1876-1932), one of the founders of Projective Differential geometry, could declare categorically: "Every problem of
mathematical analysis (i.e., of the study of number) has a
geometrical interpretation, and every problem of geometry
may be formulated analytically."2
The philosophical implications of this union of number
and space has not escaped the notice even of those who witnessed the birth of analytic geometry. The notion of number
and the idea of space seem so far apart, qualitatively so different that the correspondence between algebra and geometry revealed by Descartes' invention is philosophically as far
reaching as it is unexpected. This correspondence goes to
show that the various concepts which we elaborate starting
with different kinds of perception may not be as far apart
as their origins would imply. Furthermore this correspondence may be indicative of the "unity of knowledge" or of the
unity of the external world, although the aspects under which
we both perceive and conceive that world may seem to us to
be quite different.
C * The Dimensions of Space The union between analysis
and geometry sheds a brilliant light upon the question of di-
18
MATHEMATICS IN FUN AND IN EARNEST
mensionability of space. A solid having three of its points
fixed cannot move. If only two of its points are fixed, the
solid is free to rotate about the line joining the two points,
and we say that the solid has one degree of freedom. If only
one of its points is fixed, it has two degrees of freedom. A
solid none of whose points is fixed has three degrees of freedom. Such is, in brief, the intuitive origin of our belief that
space has three dimensions. On the other hand, in analytic
geometry a point in the plane is associated with two numbers, x and y, the coordinates of the point, and vice versa, a
pair of numbers is interpreted as a point in the plane. In
space of three dimensions a point has three coordinates x, y,
z, associated with it, and three numbers represent a point.
We thus obtain another intuitive corroboration that our
space has three dimensions.
However, analytic considerations lead us much further.
Analytic geometry also shows that four numbers, a, b, c, d,
determine a definite straight line in space, that is to say, the
four given numbers may be interpreted as a definite straight
line. If only three of these numbers are assigned, and the
fourth is allowed to vary, the corresponding straight line will
cover a surface; if only two of the four numbers are fixed,
the corresponding line will form what is called a congruence;
if just one of the numbers is fixed we obtain a complex of
lines, and finally, if all the four numbers vary, the straight
line fills space. A similar story may be told about the sphere,
for a sphere is also determined in analytic geometry by four
numbers. We are thus suddenly confronted with the fact that
the dimensionality of space is a relative question, namely
relative to the element with which we want to fill space. Our
ordinary intuitive space is three-dimensional with respect to
points, and four-dimensional with respect to lines, or spheres.
Under the impact of these ideas space loses its majestic
rigidity, its fixed and immutable form of a ready container,
since it depends for its essential characteristic upon the elements we want to fill it with. The mathematical importance
of this conclusion is closely rivaled by its philosophical
significance.
Another trend of thought opened by analytic geometry is
the four-dimensional point-space. One number fixes the position of a point on a straight line, two numbers determine a
point in the plane, and three numbers determine a point in
our three-dimensional space. By analogy four numbers may
then be interpreted as a point in four-dimensional space. Of
course, we have no intuition of such a space. But that has
not prevented the mathematician from accumulating, by
MATHEMATICS AND PHILOSOPHY
19
means of his analytical machinery, a vast number of theorems dealing with curves and surfaces in four-dimensional
space. More than that, he sees no good reason why he should
stop at four dimensions. Any n numbers may be interpreted
as a point in an n-dimensional space, and the use of equations with n variable enables the mathematician to study
n-dimensional space along the same lines along which he
studies the three-dimensional space which is so dear to our
intuition and in which we profess to feel so perfectly safe.
Later on we may say something about the "utility" of such
studies. For the present we would only insist on the intellectual side of this creation. The world of perception, the
world of intuition furnishes the suggestion of a three-dimensional geometry. But under this impulse the mathematician
spins a new thing which is patterned after the old one, but
which has nothing to do with experience,—just a product of
pure intellect. The bearing that such a performance has upon
the theory of knowledge and the sources of knowledge is
obvious.
As a fitting conclusion to the discussion of the unity between number and space we may consider the question:
What is it that makes such a unity possible? If it be granted
that there is a complete equivalence between analysis and
geometry, as Wilczynski so staunchly maintains, what is the
profound common residue that accounts for this relation?
A discussion of this question may have important consequences. If we could find out precisely what characteristics
of geometry enable us to identify it so closely with number,
we may, in the process, discover the conditions which any
science must satisfy in order that it may be identifiable with
mathematical analysis. These sciences, if there be such,
would immediately secure the powerful succor of this wonderful mathematical discipline, and their progress would
proceed by leaps and bounds. On the theoretical side such
an investigation would further enlighten us to the nature of
knowledge itself. But even if these expectations should turn
out to be too optimistic, the question limited exclusively to
analysis and geometry is important enough.
D • Postulational Mathematics
Mathematics is reputed to
"prove" the propositions it advances. Now what does a mathematical proof accomplish? It shows that a mathematical
proposition that has been proved is a logical consequence of
one or more mathematical propositions which were already
admitted to be true before. These latter propositions, in their
turn, are logical consequences of other propositions admitted
20
MATHEMATICS IN FUN AND IN EARNEST
to be true, and so on. But this chain cannot be receding forever. We finally arrive at a proposition which we admit as
being in no need of a proof, as being "self-evident", that is to
say we arrive at an axiom or axioms of the mathematical
science considered. Euclid started his geometry by laying
down the self-evident propositions which he classified as
axioms and postulates, and then derived, by the use of logic,
all the propositions of his famous Elements. The development
of mathematics during the nineteenth century taught us a
great deal about the role of postulates and axioms in the
body of a mathematical science.
In the first place a little reflection makes it clear that the
"self-evidence" of the postulates—this term will be used to
include both axioms and postulates—is a luxury we can
readily dispense with. Indeed, any logical deduction made
from the postulates will be logically valid, as long as we
admit the postulates to be valid, regardless of whether this
validity of the postulates is based on "self-evidence" or is
just a convention. This is so obvious that it would seem useless to insist upon it. But it remained hidden from the mental
eyes of mathematicians and philosophers alike, until the latter part of the nineteenth century.
TTie possibility of the conventional character of the postulates of mathematics is only one side of the medal. Let us
now focus our attention upon a definite branch of mathematics, say, geometry. We define a polygon, for instance, as
being made up of triangles, and a triangle as being made up
of straight lines, and so on. But here again we must come
to a point where we cannot reduce our terms to other terms
more simple or more familiar unless we agree to turn a
circle, as the ordinary dictionaries actually do. We may, for
instance, define a straight line as the shortest distance between two points. But this simply assumes that the notion of
distance is more familiar to us than the notion of a straight
line. We are thus confronted with the same situation as to
the basic terms of geometry as we were before when we
tried to trace the validity of our propositions back to its
origin. We must admit some terms of geometry to be "selfexplanatory", to be in no need of further elucidation beyond
an appeal to our common knowledge, to our intuition. A
further careful study of the postulates and propositions of,
say, plane geometry, reveals the astonishing fact that these
propositions do not involve any of the physical properties of
points and lines beyond the relations of these elements to one
another which are stated explicitly in the postulates of the
science, as for instance that two points determine a line.
MATHEMATICS AND PHILOSOPHY
21
The terms "point", "line", thus become mere words to designate things which satisfy the postulates of geometry, but
devoid of any other meaning. 3
Now if we combine the arbitrariness of the fundamental
postulates of geometry with the lack of meaning of the terms
involved in these postulates, we obtain a strange picture of
geometry, or of any other mathematical science, for that
matter. It is a perfectly coherent logical structure about
things the meaning of which we ignore, beyond a certain
number of formal relations, and of propositions which inform us about nothing more than that one statement is true,
if another statement be granted to be true. "A is true, if B
is true." What an enchanted world for the mathematician!
What an enormous freedom for intellectual endeavor, without
any restraints or impediments! If a mathematician takes a
notion to create a mathematical science, all he has to do is
to set up a group of postulates to suit his own taste, postulates which he by his own fiat decrees to be true, and involving things nobody, including the mathematician himself,
knows much about, and he is ready to apply formal logic
and spin his tale as far and as fast as he will. If any human
being ever was entitled to lose his head in a fit of megalomania, it was the mathematician at the opening of the present century. And he did lose his head. All mathematics—
just nothing, but a child of the mathematician's brain, a
structure in which not only the plan and design, but even
the foundation, even the very material it is built of—nothing
but a product of his imagination. The mathematician felt
himself to be the great master of creation, and so secure in
his greatness that he could afford to poke fun at himself and
at his own expense, as in the famous phrase: "A mathematician never knows what he is talking about or whether
what he says is true." 4
The mathematician was fortified in his conceit by the signal success of his new conceptions on the structure of mathematics. It has given him an understanding of the different
branches of his science that he could not have reached
otherwise. It has shown him a new way of erecting new disciplines with a firmness and security that he had not experienced before. Furthermore he found out that he has a
model to offer for others to imitate. C. J. Keyser (18621947) has shown, in the Yale Law Journal for February
1929, in an article entitled: "On the Study of Legal Science",
how the postulational method may apply in this domain,
apparently so remote from the mathematical world. R. D.
Carmichael (1879) suggests that the physicists ought to aban-
22
MATHEMATICS IN FUN AND IN EARNEST
don attempts to base physics on "self-evident," that is to say,
anthropomorphic postulates and terms, if they are to succeed
in their efforts to make physics a mathematical science. The
Frenchman J. Rueff advocates the extension of the postulational method to the social sciences in his book, Des Sciences
physiques aux sciences morales (Paris, 1922).
The arbitrariness, the lack of specific meaning of the
terms employed in a mathematical science, has a great
advantage. The whole theory of geometry, for instance, developed from point and line as basic elements, may be applied to any other two terms, if these two terms fit the
fundamental postulates. The whole doctrine thus becomes a
mold, a form ready to receive contents as soon as a set of
terms may be found to fit the mold. The meaning of the
propositions will be different with the different sets of terms
used, but the structure of the doctrine will remain the same.
C. J. Keyser calls such a doctrine a "doctrinal function". The
projective geometry in three dimensions elaborated for the
terms point, line, plane is perfectly valid, if we interchange
the terms point and plane. The individual propositions are
different, but the mold as a whole remains intact. A Russian
mathematicianA. A. Glagolev in his doctoral dissertation
devised a scheme which makes the geometry of all the circles in the plane fit the mold of the point geometry of the
ordinary three-dimensional space. The potential applicability
of these ready molds, of these doctrinal functions cannot be
foretold, but their very existence is a philosophical achievement of no mean magnitude.
We may now return for a moment to the question of the
unity of number and space, that we discussed before. If
there is a complete equivalence between analysis and geometry, that can readily be explained by assuming that these
two doctrines are two different forms of the same doctrinal
function. To be sure, the fundamental postulates in analysis and in geometry are very different. But this is no serious
objection. The same mathematical doctrine may be developed from several different sets of postulates. If we start
with the postulates of one set, the postulates of the other set
become propositions in the body of the doctrine developed,
and vice versa. It is therefore possible that a set of postulates and a set of undefined terms may be found, and a doctrinal function developed from them, which would become
in turn analysis and geometry, if we replace the undefined
terms of this doctrinal function, in turn, by the specific
terms of analysis and of geometry.
MATHEMATICS AND PHILOSOPHY
23
E ' The Question of Consistency Postulational mathematics is one of the great conquests of the human spirit. However, the paradise of freedom and sovereignty which the
mathematician arrogated to himself in connection with this
wonderful creation has early enough proved to be a precarious place to dwell in. The postulates of a mathematical
science may be laid down arbitrarily. The rest of the doctrine
is developed by pure logic and the test of its validity is that
it must be free from contradictions. Such a result cannot
possibly be attained, if the postulates themselves involve a
contradiction. Hence the first and cardinal requirement a set
of postulates must satisfy is that it be consistent with itself,
that it be free from inner contradictions. It is therefore the
first duty of the mathematician to verify that his postulates
possess this indispensable property. But how is that to be
accomplished? There is the a posteriori way: the mathematician may begin to derive propositions from his postulates,
and if he encounters no contradiction, he may assume that
his postulates are consistent. The question then arises, how
many such propositions does he have to derive to be entitled
to the desired conclusion? ten? a hundred? a thousand? a
million? It is clear that such an a posteriori proof cannot be
satisfying. The necessary thing is a test applied directly to
the postulates themselves, an a priori proof. The discouraging thing about the situation is that the shrewdest minds
among mathematicians have not been able to devise a logical
criterion by which to test the consistency of a set of postulates. The tests actually used are not of a logical, but of a
physical nature. If the postulates are true propositions about
a concrete set of objects, the postulates are judged to be
consistent, on the assumption that no existing thing can have
two properties which are contradictory. Little stress is laid
on this point by writers on the subject. This method of procedure is very significant. It expressed the fundamental belief
that logical consistency is identical with natural consistency.
It makes the consistency of nature to be one of the foundations, one of the cornerstones of the mathematical edifice.
This is already a far cry from the dictatorial powers of the
mathematician, as we have considered them before. 5
F ' The Empirical Origin of the Axioms
The restriction
that the postulates of a mathematical science must be consistent, and that this consistency can be tested only in an
empirical way is a severe blow at the principle of arbitrariness of these postulates. Nor is this all. The most famous
book that exhibits the postulational method of developing a
24
MATHEMATICS IN FUN AND IN EARNEST
mathematical science is David Hilbert's Foundations of Geometry. In spite of the theoretical freedom of choice of the
postulates, Hilbert happened to choose the same postulates
as did Euclid over two thousand years before him. Hilbert's
work is more systematic in this respect, it reflects all the
acquisitions made in this domain of thought by the intervening centuries, but the postulates are essentially the same, and
therefore the resulting geometry is Euclidean geometry. The
number of different sets of postulates which have been
worked out as a basis of geometry is considerable. But all of
them are equivalent and arrive at the same geometry. Why
do not these mathematicians use the freedom that is theirs,
that they have won at the cost of an immense effort? The
answer is very simple and at the same time of basic importance in the understanding of the role of mathematics
in relation to epistemology and to the other sciences. The
postulates of Euclid gave us a geometry which works in the
world we live in (leaving out the questions raised by
the theory of relativity), a geometry which is practical, which
tells us something about this world, a geometry that fits the
other branches of human knowledge. Furthermore, the fact
that the geometry deduced from these postulates is applicable
to the physical world shows that those postulates themselves
have a physical basis, that they are empirical laws, refined
and abstracted laws, but laws derived from experience, just
as the very notions of point and line are abstracted from
the physical world. Let us take the time to give one illustration. Given a triangle, it is not possible to draw a line
which would cut its three sides between the vertices. The
older mathematicians made use of this property, although this
is not a consequence of the postulates of Euclid. They took
this property to be so obvious that it required no proof, and
the reader, no doubt, would be in agreement with the mathematicians on this point. Now this "self-evidence" is nothing
else but empirical. Modern mathematicians when they are
trying to be complete in the statements of their postulates,
include this property in their list.6
The manner in which the postulates of Euclid have been
derived from experience is a psychological problem which
has been discussed by mathematicians like Poincare and Enriques, to mention only two. The classification of the postulates as empirical laws makes of the whole body of geometry
a physical science, the most perfect physical science, if you
will, but a physical science nevertheless. David Hilbert in an
address delivered before a congress of naturalists said: "Indeed, geometry is just that part of physics which describes
MATHEMATICS AND PHILOSOPHY
25
the relations of position of solids to one another in the
world of real things."7 And this by the famous author of the
modern classic on the foundations of geometry!
Geometry is a more perfect science than the other physical
sciences because the number of postulates, or of fundamental
laws, if you prefer, is small, and on that account we succeeded in collecting all, or nearly all those which were necessary for the erection of the stately geometrical edifice.
G * The Worth of Deductive Reasoning Within the limits
of logical consistency the choice of the postulates of geometry is arbitrary and the mathematician is free to make any
choice that would suit his fancy. But he sacrificed his freedom, first subconsciously, and then deliberately and knowingly. Or rather he has freely chosen the Euclidean postulates,
upon the suggestion of the environment he lives in, and the
rest has been done by logical deduction.
The question may be raised, in passing, about the value
of this deduction. Strictly speaking, deductive reasoning cannot teach us anything that is not contained implicitly in the
premises. Hence all deductive reasoning is a tautology, a
roundabout way of saying that A is A. Does this apply to
geometry? Are all propositions of geometry quite obvious to
anyone who masters the first postulates of the science?
When Newton first came across a copy of Euclid's Elements he read in it a page or two here and there and ended
up by throwing the book under his bed with the contemptuous remark: "Now, all this is too obvious." In the course of
a lecture on mathematical analysis the writer's professor remarked about Henri Poincare: "This man handles analysis
with such dexterity that he really believes the subject is easy."
And the lecturer went on to reinforce his remark by telling
his listeners that Poincare, when a student at the famous
£cole Polytechnique of Paris, attended the course of lectures in analysis without a book and without a notebook.
Apparently analysis may be too easy, too. Now these may be
just legends. But whether the stories are true or not, Newtons
and Poincares are mighty few and far between, and even
for those privileged darlings of fortune the going gets rough
ultimately, after a certain level is reached.
We have to admit that while the deductive reasoning is
not capable of producing anything that is not in the premises,
such efforts make explicit what those premises contain and
imply; and the results thus elicited, as far as the student, the
knower, is concerned, are quite new and form a valuable
addition to his knowledge. The nature of the game is such
26
MATHEMATICS IN FUN AND IN EARNEST
that the postulates become overwhelmed by the mass of
consequences to which they themselves give rise.
The postulates of geometry make the existence of a triangle logically possible, but they do not compel its consideration. A triangle is a further product of human experience or
of human imagination and, as such, is an addition, in a way,
to what is contained in the postulates. The consideration of
the circle circumscribed about the triangle is a further step
in this creative direction, and the comparison of two triangles another invention of the human mind. From triangles
we pass to polygons, from the consideration of one circle to
a group of two circles, of three circles, of the infinite number of circles passing through two given points. This creative
capacity of the human mind that accounts for the fruitfulness of the passage from a triangle to a polygon, from one
circle to a group of circles is nothing else but induction. The
familiar process which mathematicians call "generalizing" is
induction, quite comparable to the induction which is the
foundation of the physical sciences.8
H • Imagination and Imitation
Geometry, like mathematics in general, is a combination of basic postulates, which
are physical laws, and the creative imagination of the human
mind, the two elements being joined together by deductive
logic. What is the source, what is the impulse of the creative
capacity of the human mind, in connection with mathematics? The answer is twofold: observation and imitation. The
world around us includes many objects having the approximate form of triangles, circles, and so on. On a higher
plane, many advanced theories of mathematics are due to
questions raised by the physical sciences. The invention of
the calculus, for instance, is largely due to such questions.
We have spoken before of the basic terms, the so-called undefined terms of a mathematical science, and we pointed
out that they need not have any meaning beyond satisfying
the basic postulates of the science. But when this is actually
the case, if there is no physical picture attached to these
terms, and the imaginative capacity of the mind is left to its
own resources exclusively, progress in the science is slow and
laborious. Take four-dimensional geometry. We admit that
there is no direct intuition attached to it, and its propositions
are but theorems concerning equations in four variables.
Nevertheless, we insist on using the language of geometry,
because this language is suggestive, it points toward avenues
of investigation and thus helps the imagination which would
be very much hampered without such aid.
MATHEMATICS AND PHILOSOPHY
27
This four-dimensional geometry may also serve as an
example of what we called "imitation." We create mathematical theories which are based on intuition. But once we
have such a theory, we may create another similar to it, but
for which we have no empirical model to follow. Thus we
have four-dimensional geometry, non-Euclidean geometry,
etc. On a smaller scale this may be observed in the everyday work of the mathematician. Suppose the mathematician
comes across the problem of finding the path of a point
which moves so that the ratio of its distances from two
fixed points is constant, and he finds that this path is a circle.
He will immediately ask himself, what that path would be,
if instead of the ratio, the product of the distances were
constant, or the sum, or the difference. He is thus led to the
study of curves which may have little in common with the
circle.
It may be interesting to note that these two forms of
activity of the human mind, namely the following of the
world outside and the imagining of new things which have
no counterpart in the external world, far from being distinct
and separate, keep on crossing each other's path, intertwining
to the extent of making it impossible to tell them apart. The
history of mathematics is full of cases where problems taken
from the physical world have given rise to mathematical
theories, and conversely, the creation of the imagination of
the mathematician later found its application in physical
problems. The conic sections were little more than a mathematical pastime with the Greeks and are common structural
forms in our time. The imaginary numbers, as their very
name indicates, were created by the mathematician almost
in spite of himself, with little faith even in their legitimacy.
For the last half century or so these fancy numbers have become an indispensable part of the mathematical theory of
electricity, and a valuable tool in electrical engineering.
The great French mathematician, J. L. Lagrange (17361813), proposed the problem of determining the surface of
least area which would pass through a given curve. In 1873
Joseph Plateau (1801-1883), the blind physicist of the University of Ghent, described an experimental way of realizing
such a surface by means of soap bubbles made of glycerin
water. Such bubbles tend to become as thick as possible at
every point of their surface, and the surface thus becomes
as small as possible, that is a minimal surface. But a mathematical solution of the problem was not forthcoming. As
late as the first quarter of the present century competent
mathematicians were willing to venture the opinion that
28
MATHEMATICS IN FUN AND IN EARNEST
mathematics may not have developed far enough to cope
with this problem. In 1931 a complete solution of the problem was published by a young American mathematician,
Jesse Douglas (b. 1897).
/ • Conclusion It will be fitting to conclude this discussion with the following quotation from the address of David
Hilbert referred to before. Says Hilbert: "For the mathematician there is no ignorabimus, neither is there one for
the natural sciences, in my opinion. The philosopher, Auguste
Comte (1798-1857), said one day—in order to point out a
problem that is certainly insoluble—that science will never
succeed in piercing the secret of the chemical composition
of the celestial bodies. A few years later this problem was
solved by the spectral analysis of Kirchhoff and Bunsen, and
one may now say that the far away stars are important
physical and chemical laboratories of a kind that have not
their like on earth. In my opinion, if Comte has not succeeded in pointing out an insoluble problem it is because
there is no such. Instead of falling into a senseless agnosticism we ought to adopt the following slogan: 'We must
know, we will know.'"
2
•
Geometry and Experience
A ' Origins of Geometrical Knowledge
Students who
gather for their first lesson in geometry already know a good
deal about the subject. They are familiar with certain shapes
that textbooks on geometry call parallelepipeds, spheres, circles, cylinders, which the students would call boxes, balls,
wheels, pipes. Notions such as point, line, distance, direction,
and right angle are quite familiar and clear to them, in spite
of all the difficulties learned mathematicians profess to encounter when they try to clarify or define these concepts.
The question arises: how was this store of knowledge gathered, how was this information acquired? The empiricists
maintain that geometrical knowledge is the result of the experience of the individual in the world surrounding him.9
However, the universal acceptance of the basic properties of
space lead the apriorists to the conclusion that these spatial
relations are innate, that they constitute a fundamental characteristic or limitation of the mind which cannot function
without it or outside of it. The invention of non-Euclidean
geometry by Lobacevskii has done considerable damage to
MATHEMATICS AND PHILOSOPHY
29
the solidity of the apriorist armor but has not eliminated
the debate between the two schools of thought.
During the present century the eminent French sociologist
Emile Durkheim (1858-1917) advanced an intermediate
thesis. The source of our geometric knowledge is experience.
However, at a very early stage of civilization this individual
experience is pooled and codified by the group, owing to social necessity and in order to serve social purposes. Our
basic geometric knowledge is thus a social institution. It is
this social function of geometry that accounts for the fact of
its universal acceptance, for the inability of the individual
to act contrary to it, for the mind to reject it.
It is universally agreed that the actual experience of living
is the basic factor in the process of accumulating information
of the kind that we call spatial or geometrical. This in turn
amounts to saying that we come into possession of this information through our senses. Such being the case, the question naturally comes to mind, which of our senses is it that
performs this function?
The sense of hearing helps us acquire the notion of direction. To a lesser degree this is also true of the sense of
smell. The sense of taste need hardly be mentioned in this
connection. The sense of sight and the sense of touch remain. It does not take much effort to see that these two
senses play the dominant part in the shaping of our geometrical knowledge.
B • The Sense of Touch The sense of touch, considered
in its broader aspect of including also our muscular sense,
informs us of the shape of things. It is also our first source
of information about distance. By touch we learn to distinguish between round things and things that have edges,
things that are flat and things that are not flat. It is the
sense of touch that conveys to us the first notions of size.
This object we can grasp with our hand, and this other
cannot be so grasped; it is too big; this object we can surround with our arms, this other we cannot; it is too big.
These examples imply measuring, and the measuring stick
is the size of our hand, the length of our arm, and, more
generally, the size of our body. The whole environment
that we have created for ourselves in our daily life is made
to measure for the size of our body. That the clothes we
wear are adapted to the size of our body and our limbs
goes without saying. But so is the chair we sit on, the bed
we sleep in, the rooms and the houses we five in, the steps
30
MATHEMATICS IN FUN AND IN EARNEST
we climb, the size of the pencil we use, and so on, without
end. We take it so much for granted that things should fit
our size that we are startled when they fail to conform to
the adult standard, as, for example, in the children's room
of a public library where the chairs are tiny and the tables
so very low. The legendary robber Procrustes, of ancient
Greece, had his own ideas about matching the sleeper and
the size of the bed. He made his victims occupy an iron bed.
If the occupant was too short, he was subjected to stretching until he reached the proper length. If, on the contrary,
the helpless victim was too tall he was trimmed down to the
right size, at one end or the other. Hebrew writers placed
this famous bed in Sodom, and it was one of the iniquities
that caused Sodom's destruction, by a "bombardment from
the air."
In many cases the fact that things are made on the "human scale" may be less immediate but is no less real. The
clock on the wall has two hands, whereas, strictly speaking,
the hour hand alone should be sufficient. Owing to the limitations of our eyesight, we cannot evaluate with sufficient
accuracy fractional parts of an hour by the use of the hour
hand alone, unless the face of the clock was made many
times larger than is customary. But then the clock would
become an unwieldy object, out of proportion to the other
objects around us made to the "human scale."
The comparison of the size of objects surrounding us with
the size of our body is not just a kind of automatic reflex
but is a deliberate operation as well. When, in the course
of our cultural development, the need arose for greater precision in describing sizes and for agreement upon some
units of length, we turned to our body to provide the models. The length of the arms and of the fingers, the width
of the hand, the length of the body and of the legs all
served that purpose at one time or another, at one place or
another. The yard is, according to tradition, the length of
the arm of King Henry I. The origin of the "foot" measure
requires no explanation, and we still "step off" lengths.
C * The Sense of Vision The sense of vision is the other
great source of geometrical information. To a considerable
extent this information overlaps the data furnished by the
sense of touch. Sight informs us of the difference in sizes
of objects around us. Sight supplements and extends the notion of distance that we gain through touch. Sight tells us of
the shape of things, and on a much larger scale than touch
MATHEMATICS AND PHILOSOPHY
31
does. But sight asserts its supremacy as a source of geometrical knowledge when it comes to the notion of direction.
Moreover, sight tells us "at a glance" which object is closer,
which is farther, which is in front and which is behind,
which is above and which is below. Sight is supreme in telling us when objects are in the same direction from us, when
they are in a straight line. When we want to align trees
along our streets, we have recourse to sight. The fact that
light travels in a straight line is one of the main reasons for
the dominant position the straight line occupies in our geometrical constructs. Some learned persons will smile indulgently at the statement that a ray of light is rectilinear. The
writer will, nevertheless, stick to his assertion as far as our
terrestrial affairs are concerned, whatever may be true of
light on the vaster scale of the interstellar or intergalaxian
universe.
D ' Metrical Geometry and Projective Geometry
Up to
this point the geometrical knowledge that has been mentioned is the kind familiar to "the man in the street." Let us
now turn to the systematic study of the subject, to the
science of geometry. Are both empirical sources of geometrical knowledge reflected in systematic geometry? Is it possible to classify geometrical theorems on that basis?
If we examine Euclid, we see that he leaned heavily towards tactile geometry, or the geometry of size. His main
preoccupation was to establish the equality of segments and
angles, to prove the congruence of triangles. The method of
proving triangles to be congruent consists in picking up one
triangle and placing it on the top of the other, which implies that the moving triangle does not change while it is in
motion. The possibility of rigid motion was much insisted
upon by Henri Poincare( 1854-1912) and is now considered
by mathematicians to be the characteristic property of the
geometry of size, or, to use the professional term, of metric
geometry. Euclid's is thus metrical geometry exclusively, or
nearly so. This is not at all surprising, since metrical geometry is the geometry of action, the geometry that builds our
dwellings and makes our household utensils. The very origin
of Euclid's geometry is supposed to be connected with the
parcelling out of plots of land in Egypt after the recession
of the flood waters of the Nile.
Euclid did not know that his was metrical geometry. To
him it was just geometry, for he knew of no other kind.
Neither did his successors, in spite of the fact that they
32
MATHEMATICS IN FUN AND IN EARNEST
added to Euclid's Elements a considerable number of geometric propositions which in their nature are visual and
not metric. There are numerous such propositions, some of
them of fundamental importance, in the collection of Pappus, a Greek author of the third century of our present
era. A systematic study of visual geometry had to wait for
a millennium and a half before it found its apostle and
high priest in the person of the French army officer Jean
Victor Poncelet (1788-1867), the father of projective geometry. 11
Consider any geometrical figure, say a plane figure (triangle) F (Fig. 1) for the sake of simplicity, and let S be
a point (representing the eye) not in the plane of figure F.
Imagine the lines joining every point of figure F to the
point S. Now, if we place a screen between S and figure
F, every one of these lines will mark a point on the screen
and thus we obtain a new figure F" in the new plane, the
image of figure F.
If we compare the two figures F and F ' we notice some
very interesting things. The figure F ' in general will be different from F. It has suffered many distortions. If A, B,
are two points in F, and A', B' are their images in F', the
Figure 1
distance A H ' is not equal to the distance AB, as a rule,
and may be either smaller or greater than AB, and this
alone deprives the figure F' of any value in the study of the
figure F from a metrical point of view. There are, moreover, many other distortions of various kinds. But some
characteristics of F always reappear in F'. Of these the most
important is that a straight line p of F has for its image in
F ' a straight line p', and consequently any three points A,
MATHEMATICS AND PHILOSOPHY
33
B, C of F that lie on a straight line in F will have for their
images in F' three points A', B', C that also lie in a straight
line. If two lines p and q are taken in F, their images in
F' are two straight lines p' and q', but the angle p'q'
is not equal to the angle pq, as a rule, and may be either
smaller or larger than pq. In particular, the images of two
parallel lines are not necessarily parallel, and the images of
two perpendicular lines are not necessarily perpendicular.
If we call figure F' the projection of figure F from the
point S, we may say that projection preserves incidence
and collinearity. The systematic study of projective geometry, or visual geometry, is the study of those properties of
figures that remain unaltered by projection, just as it may
be said of metrical geometry that it is the study of those
properties of figures that remain unaltered in rigid motion.
From the point of view of the theory of knowledge it is
of great significance that the distinction between tactile geometry and visual geometry was not noticed by either philosophers or psychologists. Only after the patient labors of
mathematicians created the doctrine of projective geometry
did the distinction come to light. The credit for having
pointed out this distinction goes to Federigo Enriques (18711946), late Professor of Projective Geometry at the University of Rome.
In the study of the sources of our geometrical knowledge
too little attention is accorded to our own mobility, to our
ability to change places. Even the range of our knowledge
due to touch is considerably increased by our ability to move
our arms. In connection with our visual information our
mobility is of paramount importance. To mention only one
point, the shape of an object depends upon the point of
view, from which it is observed. It is our ability to change
places that makes it possible for us to eliminate the fortuitous features from our observations.
E • Conflicting Testimony of the Senses As has been mentioned before, our tactile and visual information do not
cover the same ground, but they overlap to a considerable
extent and thus complement each other. But do they always
agree? If a person drives his car over a stretch of straight
road, he observes that the road is of the same width all
along. He knows it to be so by comparison with the size of
his car and by comparison of the size of his car with his
own size; in other words, it is a tactile fact. Now, if he
turns around and looks at the road just traversed, he sees
34
MATHEMATICS IN FUN AND IN EARNEST
"with his own eyes" that the road is getting narrower as it
extends back into the distance and seems to vanish into a
point. These two items of information on the same subject
contradict each other. Which of them is true and which is
false? Which of them do we accept and which do we reject?
Above all, how do we go about telling which to accept and
which to reject?
When one puts a perfectly good spoon into a glass of
water, he sees that the spoon is unmistakably broken, or at
least bent at a considerable angle. He takes the spoon out,
and it is as good as it was before he put it in. He runs his
finger along the spoon while it is in the glass and feels
that it is straight as ever. But when he looks at it, there is
no doubt that the spoon is bent; contradictory testimony of
two different senses. Again the question arises, which of the
two pieces of information do we accept, and on what
ground do we make our choice?
A long time ago I read of a lake where the water was so
clear that on a bright moonlit night it was possible to see
the fish asleep on the bottom of the lake. Devotees of fishing would take advantage of this situation and go out in a
boat, as quietly as possible, to the middle of the lake and then
try to catch the fish by striking them with a harpoon. But
simply to aim the harpoon at the spot where a fish was
seen would spell disastrous failure. Successful practitioners
of the sport would know the spot at which to aim, although
the fish was seen to be elsewhere.
The moral of this fish story is of great importance. In
the case of the road and in the case of the spoon we all repudiate the testimony of our eyes and accept the verdict of
the sense of touch. We do so whenever the tactile and the
visual testimonies are in disagreement. But why?
The answer to this puzzling question may be found in
the activity of man. Moreover, his activities are purposeful
and must be co-ordinated so as to achieve success. Now,
man's organs of activity, his hands, are also the main organs
of touch. Man has thus developed a close coordination between his touch and his actions. At short range, he has implicit faith that his actions will be fruitful if he relies on the
data furnished by touch. Visual data concern objects at a
distance and serve well as a first approximation. They are
good in most cases but are always subject to control and
check. If light sees fit to indulge in such vagaries as reflection, refraction, and mirages, so much the worse for
light. The fish story told above points to just that moral.
MATHEMATICS AND PHILOSOPHY
35
Sight leads us to the fish. But if we want to act on it successfully, we must subject this information to the necessary
correction as learned by touch. Otherwise we shall have no
fish to fry.
3
* The Might and Plight of Reasoning
Introduction
Every inhabitant of this vast land of ours is
aware of the fact that "in the city of Boston, the city of
beans and of cod, the Lowells speak only to the Cabots, and
the Cabots speak only to God." Mathematicians are just
as exclusive a clique; they, too, speak only to their own
kind. But the intellectual heirs of Pythagoras have gone
those proud and haughty Bostonian families one better:
they promoted God into membership in their own clan. A
prominent British mathematician and astronomer figured out
mathematically that God is a pure mathematician. And long
before that, Plato decided that God (in His spare time)
"geometrizes."
However, mathematicians are not very happy about the
esoteric character of their science. In fact, they deplore it,
for it causes them a great deal of embarrassment. When he
has the opportunity of addressing a non-professional audience, the mathematician must leave behind, however regretfully, the field with which he is most familiar and move
'way out, on the fringes of his science, in order to find a
terrain on which he can meet and commune with people
who do not belong to his own fraternity. The adventure is
alluring, but is also fraught with danger. If he does not go
out far enough, he will bore his audience, and if he goes
out too far he may be caught trespassing on somebody
else's private territory and bore the experts, to say nothing
about the danger of his "sticking his neck out." It is no
simple task both to "satisfy the wolf and to save the sheep,"
to borrow a metaphor from the Russian peasants.
A • Reasoning and Psychology Reasoning is by no means
the exclusive prerogative of civilized man. Primitive man
reasons too, and so do animals, for that matter. The question is only of degree. It is a standard anthropological
method to try to understand man in his present state by studying him in his earlier stages of development. But this procedure is not suitable, if we want to analyze man's thinking
processes. The features of human reasoning can best be dis-
36
MATHEMATICS IN FUN AND IN EARNEST
cerned on samples where this reasoning has been checked and
rechecked by successive generations of thinkers, as is the case
with mathematical propositions.12 A textbook on plane geometry includes samples of reasoning which are as good
as any the human race is capable of producing.
Mathematicians not only use reasoning, but they also like
to reflect upon this subtle art. They have contributed a great
deal to the discussion of the logical aspect of reasoning, beginning, say, with Leibniz, to limit ourselves to modern
times. During the present century mathematicians appropriated logic altogether and turned it into a branch of mathematics under the name of "Symbolic Logic."
Besides the logical aspect of reasoning there is also the
psychological aspect. Cassius Jackson Keyser (1862-1947)
said: 13 "Select a well wrought demonstration and examine i t
What can you say of it? You can say this: A normal human
mind is such that if you begin with such-and-such principles
or premises and with such-and-such ideas and if you combine them in such-and-such order, it will find that it passed
from darkness to light—from doubt to conviction. Obviously
such a proposition is not mathematical; it is psychological—
it states a fact respecting the normal human mind." The
same idea was expressed even more pointedly by Henri
Lebesgue (1875-1941) who said: "Les raisons de se declarer
satisfait par un raisonnement sont de nature psychologique,
en mathematiques comme ailleurs." (The reasons for declaring oneself satisfied with a reasoning are of a psychological nature, in mathematics as in anything else.) It seems the
psychologists have not come to grips with this problem. But
they have approached the question of thinking from another
side.
B • The Role of the Body in the Reasoning Process
Everyone knows that if you want to think you have to use
your head. Does any other part of the body participate in
this process? When a little boy, before I reached my 'teens,
I made an astounding discovery. One day, after school, before I got ready to do my homework, I engaged in a hard
running game with some boys of the neighborhood. When I
finally yielded to the call of duty I was amazed that I could
not find an opening sentence for my composition, and that
reading my geography lesson instead was just as fruitless. I
finally decided to work my arithmetic problems. But I could
not solve the problems. I was sure the world was coming to
an end. It did not, however, and I drew the conclusion that
MATHEMATICS AND PHILOSOPHY
37
there must be some connection between my mental efficacy
and the physical state of my body. I know from repeated
personal experience, just as everyone else does, that one
may be too tired physically to be able to read a book, or
even a newspaper.
Ask the average man to explain to you the meaning of
the adjective "solid". He will tell you that it means something strong, something substantial, and while doing so he
will more likely than not clench his fist as tightly as he can.
Modern psychology has studied this subject in a methodical way. If you try to imagine a flying bird or a moving
automobile, there is a tension in the muscles of your eyes
and a tendency for you to roll your eyes in the direction of
the imagined motion. When you imagine that you are bending your right arm, or that you are lifting a weight, your
muscles become tense in the same way as though you were
actually trying to do those things. If you imagine that you
are counting one, two, three, etc., the tension in your speech
musculature is the same as it is when you actually count
aloud. On the other hand, one who is deprived, say, of the
left arm is not able to imagine that he is lifting that arm.
It is important to emphasize that these are not mere assertions or "hunches" on the part of the psychologist. He
has measured those tensions with instruments and has graphical records of his findings like those, say, which the weatherman has of the variations of the temperature during the
day. His experimental evidence is just as solid as the evidence on which the experimental physicist bases his findings.
The experimental psychologist is thus led to the inescapable
conclusion that mental acts are performed not in the brain,
or at least not in the brain alone, but that you perform those
mental acts with your muscles, in a rudimentary way, to be
sure, but your muscles come into play nevertheless. Their
role is somewhat like the role played in theoretical mechanics by "virtual velocities" and "virtual work." Thus, in a
paradoxical way, it may be said that we think with our
muscles. Some psychologists go even further. They maintain
that every part of our body participates in the process of
thinking.
It is tempting to mention here an interesting analogy
from the field of esthetics.
Music is a form of noise, "the most expensive of all disagreeable noises", as a very distinguished music hater once
put it. As sound, music is directed at your ears. You surely
will not be surprised if you are told that you usually listen to
38
MATHEMATICS IN FUN AND IN EARNEST
music not only with your ears but with your whole body.
Few people can listen to music, particularly if the music is
more or less familiar, without moving their body, or, more
specifically, some part of their body, say, the head, or an
arm, or a foot. When listening over the radio to a symphonic concert by a first rate orchestra, one is tempted to
direct that body of performers, although knowing full well
that at the other end of the line there is a competent conductor on the job. Strange as this behavior may be, one has
a very good reason for engaging in the competition. One
cannot derive all the enjoyment out of music unless one participates, so to speak, in its performance. This participation
finds its expression in the more or less pronounced motions
of one's body. By doing so you "feel yourself into the
music". Students of the psychology of esthetics describe this
attitude of the listener by the word "empathy." They insist
that analogous things may be said about other forms of art,
but we shall not go into that. Suffice it to point out here
that according to this theory it may be said, paradoxically
again, that "you appreciate beauty with your muscles."
C ' A Definition of Reasoning Having observed that thinking is helped or at least accompanied by muscular activity,
we may raise the question: what is reasoning? What is it
that we do when we reason? You need not be told that this
is no mean question to answer. Perhaps some light may be
derived from watching the reasoner at work.
Two men have an eight gallon keg of wine. They want
to divide the wine equally between them. They have at their
disposal two empty kegs of five and three gallons. How
could they accomplish the division? One who is not addicted
to reasoning may try it by the direct experimental method,
i.e., by pouring the liquid from one container into the other.
With sufficient patience and some good luck he may succeed.
One who is prone to reason will attack this well known
riddle in a different way. He will still do the pouring, but
only mentally. He may fill, say, the three gallon keg and
make a note of the fact that he has five gallons left in the
big keg. This is registering the result of a physical operation
performed only mentally, and this implies that the reasoner
is in possession of the information, acquired previously, that
if you take three gallons from eight gallons there remain
five gallons. Proceeding in this manner the reasoner will succeed in his task presumably much faster than the actual ex-
MATHEMATICS AND PHILOSOPHY
39
perimenter, especially if he is able to keep a record of the
various pourings, either in his memory, or by using some
mnemonical device, like, say, writing.14
The above description of the method of solving the riddle
suggests that the reasoning involved consists of a series of
physical operations performed in the imagination only, and
that the performance of the operations mentally is made possible by the reasoner's knowledge, from previous experience, of the outcome of each individual operation.
Let us try the same scheme on another example. A block
of wood in the form of a cube 3 x 3 x 3 inches is painted,
say, blue. If the block were sawed up into one inch cubes,
how many faces of each small cube would be blue?
Here again the question may be answered experimentally
by actually sawing up the big block and counting the number
of painted faces of the individual small blocks. But the reasoner may arrive at the outcome of the sawing without
having recourse to the actual operation, relying for his answer upon his knowledge, that is to say, his previous experience with the cube. The reasoner will say that a small cube
occupying a corner of the original block was a part of three
of the faces of the big cube and will thus have three of its
faces painted, and there are eight such little cubes. A small
cube that was a part of an edge of the block but not placed
at a vertex was part of two of the faces of the block and
will therefore have two painted faces. There will be twelve
such cubes. The six little cubes that were at the center of
the faces of the block will have one painted face, and the
little cube that occupied the center of the block will have
no paint on it.
This example confirms our observation that to reason is
to perform experimental work mentally, the outcome of
each step in the chain of experiments being known to the
reasoner from previous experience.
The definition of reasoning was given by the late professor of philosophy of the University of Pavia and editor of
the renowned periodical Scientia, Eugenio Rignano (18701930), in his book: La psicologia del ragionamento. This
work has been translated into French, German, and English,
but it does not seem to have received the attention it deserves.
Let us consider the problem: If A walks to the city and
rides back, he will require five and one. quarter hours; but
if he walks both ways he will require seven hours. How many
hours will he require to ride both ways?15
40
MATHEMATICS IN FUN AND IN EARNEST
The problem may be solved in various ways. It will illustrate our point best to argue the case as follows: If A
should make the trip to the city twice, walking one way and
riding back, he would require for that 514 x 2 = 1 0 V i hours.
Now two such trips are equivalent to one round trip on foot
and a round trip riding both ways. But the former, we
know, requires 7 hours, hence the latter trip will take lOVi —
7=3Vi hours. This reasoning confirms Rignano's definition so
well that comments are unnecessary.
Rignano, himself, considers the following example: A
pendulum clock is keeping good time in a given room. How
will the clock be affected if it is transported into a room
where the temperature is markedly lower than in the first
room? The question may be answered experimentally by observing the clock in the new location. But one may reason
out the answer, if he is in possession of some experimental
facts, namely, the effect of temperature upon the length of
a metal bar, and the relation of the length of a pendulum
to the length of its swinging period. The reasoner so
equipped will be able to say that the pendulum will become
shorter in the cold room, and the shorter pendulum will
swing faster, hence the clock will be fast.
However, as I mentioned before, the best place to check
the validity of this definition of reasoning is in mathematics,
and a textbook on plane geometry would do as well as anything.
Consider the proof, by superposition, that a triangle is
congruent to a second triangle if two sides and the angle included between them of one triangle are respectively equal
to the corresponding elements of the other. The steps in the
proof are nothing else but physical operations performed
mentally, "in the imagination." We put one vertex upon the
other, and we know, by previous experience, that without
changing the position of that point we still can revolve one
triangle about that point. We revolve that triangle so as to
make one side of it fall on the corresponding side of the
other triangle, and so on. All these operations we perform
mentally, and we are able to perform these successive steps
because we know the outcome of each step from previous
experience.
D • Applications of the Definition
The examples to which
our analysis was applied were deliberately chosen for their
simplicity, in order to bring out as clearly as possible the
salient features of what reasoning is. However, the same
MATHEMATICS AND PHILOSOPHY
41
features will be found if we examine reasoning on
any level, no matter how abstract. The subject matter of the
reasoning will in such cases be not the physical facts, but
abstractions, symbols representing such facts and groups of
facts. The methods of operation remain the same.
Let us now apply our definition of reasoning to some outstanding intellectual problems.
One of the questions that has preoccupied philosophers
through the ages is that of the rationality of the world we
live in. How is it that the results of reasoning, the products
of our inner intellectual effort are applicable to the external
world and find their verification in it? Is it that reason pervades that world? If this be accepted as the answer, it is still
possible that only a part of nature is rational and thus accessible to our mind, while the rest of the universe is irrational
and thus completely closed to our intellectual perspicacity.
The world would thus be divided into two distinct and
mutually exclusive parts: The knowable and the unknowable.
This whole question with its mysterious profundity vanishes, if our description of reasoning is correct. If thinking
about the external world is to perform a series of physical
operations mentally, the outcome of each individual operation being already known from actual experience, then reasoning is simply returning to nature what we have learned
about it directly. It is therefore not surprising that the results
of correct thinking are in conformity with what happens
around us. If the predictions of the astronomer concerning, say, an eclipse of the moon, come true, this is simply
because each step in the reasoning which predicts the phenomenon is a verified fact of the physical world. We know
the external world first and reason about it afterwards, and
not the other way about.
Is it possible to learn to reason? Or better, is it possible
to improve one's reasoning abilities? This is a pedagogical
problem of basic iirroortance. We all know that the native
endowment regarding people's ability to reason varies, and
the gamut of variation is enormous. It is the experience of
any teacher that some of his pupils take to reasoning as
naturally as a duck takes to water, while it is almost the
exact opposite with others. Some teachers of voice maintain
that anybody can learn to sing. Those singing enthusiasts will
agree, no doubt, that it would take many years of training
and a lot of sustained effort for some people to attain the
skill in singing that others possess by accident of birth. But
nobody doubts that the singing proficiency of a person can
42
MATHEMATICS IN FUN AND IN EARNEST
be improved by training. Is the same true about reasoning
abilities?
Some decades ago a theory was current in pedagogical
circles that training is not transferable from one domain to
another. The learning skill acquired in one branch of knowledge does not help in mastering another. In particular, the
practice of reasoning in one field carries with it no advantage when it comes to reason in another field. The proponents of this doctrine offered a lot of experimental data in
support of their contention. In spite of that they could not win
over the skeptics. In more recent years the original doctrine
was modified. The reasoning ability acquired in one field is
useful in another, but this transfer is not automatic. Students must be trained in acquiring skill in transfer, and
"teaching for transfer" became a sound educational practice.
E ' Pitfalls and Merits of Reasoning The original nontransfer theory is obviously in contradiction with what we
have been saying about the nature of reasoning. If reasoning
consists in mentally performing physical acts in some purposeful succession, the ability to do so should clearly be
subject to improvement when practiced on any subject matter, and this improvement should have noticeable effects
when reasoning is applied to a different field. But this transfer is indeed not automatic. The reasoner, in order to be
successful, needs to know the result of each of his mental
operations beforehand, from past experience. In other
words; effective reasoning requires on the part of the reasoner a familiarity with the new situation, the knowledge of
the facts in the case. It is therefore impossible to transfer
reasoning ability acquired in one domain to another, if the
subject matter of the new domain is not known to the reasoner. The point may be illustrated by the following example, which makes up in effectiveness what it lacks in
depth. A charming young lady boasted before her chum
that she had already had six marriage proposals. "And I am
only eighteen," she added demurely. "That ain't so many,"
piped up her little brother Johnny, age eight, "that's just
one proposal in three years."
On the authority acquired through many years of teaching
the subject the writer can assure you that the little boy's
arithmetic is absolutely impeccable. It is perhaps less certain that Johnny is quite familiar with all the facts regarding
courting, proposals, marriages, etc., involved in the situation.16
MATHEMATICS AND PHILOSOPHY
43
Lack of familiarity with the domain to which the reasoning is applied is not the only trap into which the reasoner
may fall. The chain of mental experiments may be so long
that he may not be able to keep track of all the links and
thus arrive at an erroneous conclusion. The reasoner must
at all times be aware of this danger and check his results
whenever there is an opportunity. The fellow whom we
watched as he sawed that wooden block in his mind would
do well, before he quits his job, to ascertain that the various
little blocks that he produced mentally when taken together
will account for the twenty-seven small cubes he expects to
have.
He would also do well to check it in another way. He
may count the number of painted faces on all the small
cubes and see whether they add up to the 3 x 3 = 9 , 9 x 6 =
54 little squares on the faces of the original block.
Another danger that lurks in the path of the reasoner
is that he may overlook the fact that a new link that he
brought in has a bearing upon the results he already admitted and that those results thus require some modification or
adjustment. Here is an illustration. A man whose back was
much stronger than his head dug a hole in the ground.
When filling the hole up again he was very much perturbed
that the hole could not hold all the dirt he took out of it.
After much head scratching he finally discovered the reason
for his predicament. "I must not have dug that hole deep
enough."
Mistakes in reasoning would not be so disturbing if we
could console ourselves that only inexperienced or poorly
endowed individuals are the victims of those pitfalls. But
unfortunately this is not the case. Sometimes we blame the
social sciences when their arguments and predictions are off
the mark. But we like to think that the so-called exact
sciences are free of errors, and above all that mathematics
is always correct. You may be surprised to learn that the
mathematical literature includes a great many mistakes.
Some of them are rectified rather promptly, others remain
a long time unnoticed, and presumably a good many may
remain undetected. Will all the mistakes in the mathematical
literature ever be eliminated? This is a question that cannot
be answered by "yes" or "no". What is more aggravating
is that among the papers in which errors have been found
are some that came from the pens of the bearers of the
most illustrious names that adorn the annals of mathematics.17
44
MATHEMATICS IN FUN AND IN EARNEST
In spite of these weaknesses, and many others, we are
not likely to give up our mental experimentation for direct
action, for various reasons.
It is much easier and more convenient to perform operations mentally. The mental process requires no equipment,
no apparatus, no installations of any kind. Furthermore, it
saves a great deal of time, to say nothing about expense.
What is even more important, the "thought experiment"
has a much greater degree of generality than any physical
experiment can have. If we superpose two material triangles
we are tied by and to the two triangles at hand. When performing the same operation mentally, we can also mentally
vary the two triangles and notice that the result of the operation is quite independent of the two triangles used. Thus
we perform not one experiment at a time, but a great many
experiments in the same time, and are able to embody the
results of all of them in one statement, in one proof.
The mental, instead of the material performance of the
experiments has also the merit of showing clearly the interdependence of the component parts of the "experiment"
or "reasoning", a thing that would escape notice in the material execution of the experiment. After the cube we considered before is sawed up it is easy to establish that there
are small cubes having paint on two faces, and others that
have paint on three faces. But why are there twelve of the
first kind and only eight of the second7 The wielder of the
saw has no answer for that, but the "reasoner" had no
trouble accounting for it. In fact, he could not have done
his mental sawing had he not known this beforehand.
F ' More Checks on the Definition
The mathematicians
of the nineteenth century made a great contribution to human understanding when they invented postulational, or axiomatic mathematics. New branches of geometry that came
into being during the early part of that century, non-Euclidean geometry among them, led mathematicians to the surprising conclusion that a mathematical science can be built
by choosing arbitrarily a set of objects or entities and by
promulgating, just as arbitrarily, a set of rules or axioms
which those entities are to obey. Those "undefined terms" 18
and "unproved propositions" are the clay out of which the
proposed science can be fashioned. Plane geometry has been
built according to this model. The undefined terms are
"point" and "line", and the "unproved propositions" are,
roughly speaking, the axioms and postulates of Euclid.13
MATHEMATICS AND PHILOSOPHY
45
In principle, the choice of the undefined terms and of the
unproved propositions for the building of the science of
plane geometry is practically unrestricted. These terms and
propositions need not be anything else or anything more
than the creation of the human mind, a lucky product of
the excited imagination, without any relation to or connection with the external world.
"The breath of life" that is supposed to pulsate in this
postulational clay is, of course, reasoning. Postulational
mathematics begins by providing the material to which reasoning is to be applied. It thus conforms to the requirement
stated before that we must have subject matter first and
reason upon it afterwards. It may further be said that the
postulates furnished provide enough "results of experiments
known to the reasoner from previous experience" and thus
satisfy this other requirement that would make it possible
for reasoning to function in the way we said reasoning does.
Thus, postulational mathematics seems to confirm, or at
least does not conflict with our theory of reasoning.
The enthusiasts of postulational mathematics were outdone by a school of thought that has become known as logicalism. The exponents of this school blame the postulationists
for granting too much. The logicalists maintain that in
order to erect the whole edifice of mathematics nothing
more need be assumed than the power to reason correctly,
according to well defined rules. Weighty volumes have been
written by foremost thinkers in support of logicalism.
As far as our present discussion is concerned logicalism
flatly contradicts what we have attempted to present as the
nature of reasoning. Contrary to what we have said, logicalism begins with reasoning and undertakes to produce the
subject matter of mathematics as a result of it. Even if one
would attempt to find consolation in the fact that the rules
of logic which the logicalists begin by setting up may be
taken to be a sort of preliminary subject matter, the discrepancy still remains wide enough to "give us pause," as Hamlet said.
But if we seem to disagree with logicalism, we may find
comfort in the fact that we are in this respect fellow travelers of another, more recent school of mathematical
thought known as intuitionism. The institutionists take sharp
issue with logicalism. They categorically deny that mathematics is a corollary to logical reasoning. In fact, they reverse the entire situation and maintain that logical thinking
is a by-product of mathematics, a technique developed in
46
MATHEMATICS IN FUN AND IN EARNEST
and for the study of mathematics, or any science, for that
matter. Thus the attitude of the intuitionists towards reasoning comes so close to coinciding with the idea we have
been discussing that one hardly could hope for a more striking confirmation.
You listen to an argument and then you declare: "I do
not understand." What do you mean by that statement? In
the light of what was said about the nature of reasoning you
would mean that you do not visualize the chain of operations involved in the argument developed. And when you
grasp the connection, you declare: "I see," with a sigh of
relief, or a feeling of triumph, as the case may be.
If this interpretation is correct, it may help us to understand what Keyser and Lebesgue meant when they made
the startling and disconcerting statement that the reasons for
which we accept a logical argument are of a psychological
nature. That "nature," according to our way of looking at
reasoning, is the need to see the chain of mental experiments involved and to be sure of the outcome of each of
them individually. If we can follow these steps, the argument
is acceptable, and not otherwise.
G ' Reasoning, Memory, and Invention In what preceded
I have tried to present in a plausible way what reasoning
is, a definition which may have been repeated too many
times already. We shall now mention briefly some other
facets of the subject.
The reasoner has to have recourse to facts of experience
which he learned previously. Obviously, he cannot make use
of the necessary facts, unless he remembers them. Abel
Rey lays great stress on the important role memory plays
in reasoning and in the acquisition of knowledge generally.
It is our memory which enables us to "fuse the past with the
present in order to foresee the future." 20
Think of any proof in plane geometry, say, the proof of
the Pythagorean theorem. Before you begin your argument
you say that you draw such a line, or you join two such
points, etc. How do you know which line to draw, or which
points to join, or to do one rather than the other? This
choice is invention. Even if you would argue that many false
moves were made first, and that the lines drawn in the textbooks of plane geometry in connection with this theorem
are the result of trial and error, all possibilities could not
have been tried, for they are too numerous, and the element of invention still remains.
MATHEMATICS AND PHILOSOPHY
47
Let us consider another example. A cylindrical tube sealed
at both ends stands on one end, on a table. The upper part
of the tube is opaque, while the lower part is transparent
and is filled with a liquid. How far into the opaque part of
the tube does the liquid extend? All it takes to answer the
question is to turn the tube upside down. We know, from
experience, that the column of liquid will reach the same level
regardless of the end the tube may stand on. If the liquid
reaches above the middle line of the tube in one case, it
will be just as much above that line in the other. "If equals
are taken from equals, the results are equal." Does not that
sound familiar? All this is good reasoning and conforms
perfectly to the pattern we have been considering. But what
made you think of turning the tube upside down? This is
invention. It is an indispensable part of reasoning, but it is
not reasoning, or at least it is a different aspect of reasoning. In the case considered another method, another invention could have served the purpose just as well. A plumber,
under the circumstances, would have taken a different
course. He would leave the tube in its original position and
would drill a row of holes in it, beginning from the top. He
would stop when the liquid would begin to flow out of the
tube and would conclude that the level of the liquid was
between the last two holes drilled. The plumber is inventive,
too. But whichever method you might choose, you have to
invent it. Abel Rey maintains that there exist positive if
obscure ties between intuition, invention, and the subconscious. Any invention, from the humblest and up to one
which ^upsets the whole economy of human thought, is
just an analogy which escaped notice up to that moment.
A French writer of the seventeenth century said: "Le
choixdesidees est invention." (The choice of ideas is invention.) How one goes about the business of choosing one's
ideas, about the business of invention, is a question that
preoccupied and baffled many of the greatest mathematicians. A most significant contribution along this line is due to
Jacques Hadamard of the College of France. 21
4
•
Plane Geometry and Plain Logic
A • The Impact of Non-Euclidean Geometry and of Projective Geometry
The foundations on which Euclid reared
his marvelous Elements endured for more than 2000 years.
The first effective thunderbolt that struck these foundations
48
MATHEMATICS IN FUN AND IN EARNEST
originated in a remote Russian town, Kasan, located on the
lower reaches of the river Volga, and was hurled at them
in 1826 by an obscure professor of mathematics, Nikolai
Ivanovich Lobacevskii (1793-1856). Lobacevskii's object
was to prove that the parallel postulate of Euclid was not an
obvious truth. In that he was eminently successful.22 As a
by-product, he wrought a change in our conception of the
world we live in that has been compared, and rightly so, to
the epoch-making achievement of the other Slav, Mikolai
Kopernik, better known as Copernicus (1473-1543).
By a queer whim of history, about the time when Lobachevskii meditated on his new geometry in Kasan, in another Russian town, Saratov, further down the Volga from
Kasan, a young French officer of Napoleon's Grand Army
of 1812, J. V. Poncelet, was whiling away his long solitary
prison days in another kind of geometrical speculation. His
lonely efforts were destined to become the foundation of a
new branch of geometry and to form the contents of his
famous Traite des proprietes projectives des figures, which
was published in 1822.
One of the strange ideas contributed by projective geometry is the principle that any of its propositions, in plane
geometry, remains valid if we replace in it the points by
straight lines and the straight lines by points. As a consequence of this "principle of duality," each theorem that is
proved provides another theorem as well, which does not
require a new, direct proof. The number of theorems is
thus automatically doubled. 23
The principle of duality was a great surprise. It was important enough to give rise to an acrimonious dispute over
its paternity between two geometrical luminaries, J. D. Gergonne (1771-1859) and "the father of projective geometry,"
Poncelet himself. 24 Nowadays novices to the mysteries of
projective geometry are confronted with this principle right
at the start, as with a proposition which is practically selfevident. Those of the neophytes who shamefacedly confess
that they do not grasp this idea quite clearly are assured
by their elders that further progress in their studies will
bring more light, and faith will be sure to follow. It does,
usually.
The importance of the "windfall" that the principle of
duality contributed to geometry is quite obvious. But the
philosophic by-product which that principle entailed is even
more far-reaching. We have a theorem dealing with certain
entities—namely, points and lines. If this theorem remains
MATHEMATICS AND PHILOSOPHY
49
valid when these entities are replaced by some others (in
this case by lines and points, respectively), then our original
theorem is not specifically a statement about points and
lines. If we press this trend of ideas further, we are in the
end confronted with the devastating question: What is it
that we are talking about when we make our statements in
geometry?25
B ' The Formalist Approach to Geometry
Non-Euclidean
geometry and the principle of duality called into question the
foundations of geometry and of mathematics in general.
This was a much-discussed topic during the 19th century,
both by philosophers and by philosophically minded mathematicians. Toward the turn of the century, the interest in
these matters was greatly stimulated, among professionals
and laymen alike, by the philosophical writings and lectures
of Henri Poincare (1854-1912) because of his towering
scientific eminence, and perhaps even more because of his
literary talent. These essays are excellent reading even today. The best English edition of most of Poincare's contributions along these lines was prepared by G. B. Halstead. 26
One outcome of the 19th century discussions was a deeper insight into and a more systematic use of the axiomatic,
or formalist approach to mathematics in general, and to
geometry in particular. In the case of plane geometry, the
method consists in starting out with two kinds of objects
named "points" and "lines" about which we profess to know
exactly nothing. These objects have for us absolutely no
other connotations than those bestowed on them by the
propositions we explicitly formulate about them, and by
which we are to be governed. These propositions are selected arbitrarily and declared to be true. When sufficient
"axioms" have been accumulated, we are set up in business
and are ready to start on the erection of the superstructure,
with the help of the powerful lever of pure logic. The most
highly regarded work along this line was done by David
Hilbert (1862-1943). Thus, in this conception, plane geometry is just one grandiose creation of the human mind, one
in which the senses and the sensory world have no part
whatever.27
Imposing, even inspiring, as the edifice of the formalist
may be, the obscurities of its starting point seemed to some
to smack of sheer mysticism, and its proud aloofness from
the world around us appeared to others to border on the
absurd. But the actual heel of Achilles of this purely intel-
50
MATHEMATICS IN FUN AND IN EARNEST
lectual doctrine is that it suffers from an inherent intellectual
weakness.
The arbitrary choice of the fundamental axioms is subject
to an obvious limitation: the axioms have to be logically
consistent with each other. Hilbert labored for many years
trying to produce a proof that the axioms of his Grundlagen
der Geometrie satisfied this requirement. But all his persistent
zeal and his enormous intellectual resources proved unequal to the task, though he could find some personal consolation in the proposition, proved by K. Goedel in 1931,
that the "Grundlagen" could not yield a proof of its own
consistency.28 Georges Bouligand formulates the argument
as follows: 29 "To find within a body of doctrine G a proof
that G is consistent is impossible, for to accept the validity
of such a proof is to concede to a part of G a special privilege: an abusive procedure, if the coherence of G as a
whole is in doubt." Simple and obvious, David Hilbert to
the contrary notwithstanding.
The shortcomings of formalism have brought out the
limitations of the axiomatic method but have not impaired
its value. Originated more than 2000 years ago in geometry,
this method continues to lure other sciences by its undeniable advantages. Among the more recent conquests of, or
converts to, the axiomatic method are the biological
sciences.30
C * Role of the Knower The geometrical advances that
were realized in the first third of the 19th century called
into question the validity of the postulates of geometry as
well as the nature of the entities it deals with. It was inevitable that sooner or later the instrument that geometry
uses to manipulate these materials—namely, logic—should
in turn be subject to scrutiny. What are the inviolate laws
of logic? How and where have they acquired their infallibility? On what is based their tyrannical power over the
mind of man?
And while we are in the questioning mood, would it not
be appropriate to cast an inquiring eye on the manipulator
of this powerful tool—the geometer himself? Does not the
mental and physical make-up of the investigator have a
bearing on the results obtained in the investigation? May not
the Knower's knowledge depend on the Knower himself?
Or, to put it broadly, is not the conception we make for
ourselves of the world we live in influenced by the kinds of
creatures we are ourselves?
MATHEMATICS AND PHILOSOPHY
51
Let us deal with the latter problem first. The questions of
the dependence of our knowledge on our own physical and
mental constitution are of rather recent origin. In the mental domain they were first considered by Kant. An adequate
discussion of the entire problem would require knowledge
of our nervous system that at present is not available. But
once we raise these questions, the nature of the answer is
beyond doubt.
When we look at an object, or at a landscape, and are
not quite certain what we see, we turn our heads, or we
move closer, or we walk around the object. Our knowledge
thus depends on our ability to move—that is, on our physical structure. How utterly different this world of ours
would be to us if we were immobile!
We explore our surroundings with our five senses (or is
it six?). But what is so fixed and immutable about this number? Could we not have a larger number of them? The
question is not quite as preposterous as it might seem at
first. We have a sense for light. Why could we not have a
sense for electricity? As matters stand now, the only way
we can feel that mysterious stuff directly is to be shocked
by it, sometimes to death, sometimes into health. If we want
to make electricity accessible to our senses, in a less violent
form, we resort to the expedient of transforming it into
light. But we could conceivably have nerve ends that would
convey to us the sensation of such electromagnetic waves
directly. We know that our nerve ends that convey the sensations of high temperatures are different from those that
convey sensations of low temperatures. That such an extension of our perceptivity within the domain of electromagnetic waves is not a physiological impossibility is attested
by the fact that the visual spectrum of some animals
reaches beyond the limits of the visual spectrum of man.
For instance, insects, as a class, respond to electromagnetic
radiations from both the ultraviolet and the infrared. 31
A substantial extension of the range of electromagnetic
waves directly perceptible to us through our senses would
obviously materially affect our conception of the world
around us. Radio astronomy would not have had to lag
several thousand years behind optical astronomy. 32 At any
rate, it certainly would have saved us all the time and all
the effort that we had to spend, and still do, to study this
form of energy by our indirect methods.
If we may find it difficult to talk about additional senses,
it is easy to imagine that we might have been deprived of
52
MATHEMATICS IN FUN AND IN EARNEST
some of those we have. We all know unfortunates who
are handicapped that way. Certainly for writers of science
fiction such a conjecture is no triek at all—witness the
story of H. G. Wells, The Country of the Blind. Those who
write just science, pure and simple, should not have much
trouble either. They know full well that we are blind, at
least relatively, compared with other creatures.
The well-known limitations of our auditory perceptions
offer occasion for analogous remarks.
D • Axiomatic Method Federigo Enriques in his Problemi della Scienza of 190633 points out that the foundations of knowledge are more clearly discernible in knowledge
that has already evolved than when it is still at its crude
beginnings. This idea was taken up by Ferdinand Gonseth,
author of Les fondements de mathematiques, Les mathematiques et la realite, Qu'est-ce que la logique, and so forth.
The Cumulative Index of Books in English does not list any
books by Gonseth. In order to find an answer to the question bearing on the nature of logic, Gonseth first subjected
to a psychological examination the axiomatic method as
applied to plane geometry.
A city is described by its plan in a schematic way. This
schema usually furnishes information about the location of
the streets, the public buildings, the transportation lines,
and so forth, but has nothing to say about private residences
or the location of the taxicab stations. The plan is thus only
a simplified or summary description of the city.
The plan, or schema, is obviously incomplete, and additions may be made to it if necessary. The plan of the city
may always be enriched, say, by marks indicating the location of the service stations of any enterprising oil company.
Moreover, the things that are indicated on the plan are
represented by conventional marks or symbols.
Thus the schema is a summary, symbolic and unfinished.
The city that the schema represents may be said to be the
exterior meaning or the exterior significance of the schema.
However, we may consider this schema by itself and for
itself, without reference to the thing that it is supposed to
represent. As such, the schema has its own reality and may
be an object of study for its own sake. We may, for instance, examine the network of lines indicating the one-way
streets or the pattern formed by the points marking the locations of the post offices and relate it to the similar pattern
formed by the telegraph offices. We may even solve some
MATHEMATICS AND PHILOSOPHY
53
geometrical problems that those figures might suggest. Of
course, by these intrinsic considerations regarding the
schema, we are diverting our attention from the plan's
original purpose. On the other hand, such studies may well
be undertaken in order to serve that very purpose with
greater efficiency. The profound analogy between this example and geometry is so transparent and so striking that it
can hardly be overlooked.
In order to accept the edge of a ruler as a realization
of the abstract concept of a straight line, we must, in the
first place, reconcile ourselves to the approximate character
of this realization. But this is not enough. We must also be
willing to forget that the correspondence between the concept and its realization holds only macroscopically and that
it vanishes completely when the edge of the ruler is put
under a microscope. In other words, the realization of the
concept is only summary. What we said of the straight line
can obviously be repeated about any other concept used in
geometry. In the light of our example with the plan of a
city, we say that rational geometry is a schema of ideas
whose exterior significance is to be sought in a certain natural structure of the physical world. We are thus quite far
from the much quoted quip of Bertrand Russell: "In mathematics we don't know what we are talking about, or
whether what we say is true." 34
Pursuing our analogy between rational geometry and the
plan of a city, we may say that to set up our geometrical
schema means to conceive, in a summary and schematic
fashion, a set of simplified notions and a number of relationships among them. To reason intrinsically on this schema
means to render explicit the consequences implied in those
relationships. In other words, to develop the reality of
schema is to set up a system of statements having the
value of axioms, and the business of the geometer is to reason intrinsically on this schema.
The process of constructing an abstract schema in correspondence with a given exterior significance may be called
"abstraction by axiomatization."
Let us observe that the schema is the abstract of its exterior significance and that the latter is the concrete of the
schema. Abstract and concrete are thus relative to one
another. Their mutual correspondence, as well as their opposition, constitutes a part of their meaning.
E • Meaning of Intuition
There is, however, an impor-
54
MATHEMATICS IN FUN AND IN EARNEST
tant difference between the schematization of a city and
that of geometry. We have no hesitation how to perform
the first task. But it is not quite as clear how to go about
picking for geometry "a system of statements having the
value of axioms." When we considered the axiomatization
of the straight line, we assumed that the notion of a straight
line is familiar to us. We know the thing "by intuition."
Euclid's axioms have been accepted through the ages, as
given "by intuition." Let us try to examine what this notion
by intuition means.
Our accumulated knowledge is perpetuated and transmitted from generation to generation largely through books.
More than 3000 years ago, a wise man voiced a complaint
that "of making many books there is no end." 35 No part of
this blame may be attributed to children, for children do
not write books. But children, even infants, acquire a considerable amount of knowledge about the surroundings they
live in and make a vast number of adaptations to it.
The adult, however, by the time he is ready to write a
book, is prone to forget about the things he learned in his
early life without the benefit of books. Relying on his personal recollections, any adult would, for instance, staunchly
maintain that he had always been able to walk, if his observation made on his very youngest contemporaries did not
shatter this quaint illusion.
Suppose you take two flat sticks, say two rulers, of equal
length, hold them in your two hands so that they cover each
other, and then slide them part way one on the other. It
would not occur to you to resort to measurements in order
to settle the question: Which of the two uncovered parts is
longer? You know for sure that those two parts are equal,
and you have always known that to be true. Now, have
you? No, there was a time when you did not know it. That
was the time when, after having counted up five of your
wooden toy blocks, you entertained high hopes that there
might turn out to be six of them, if you arranged those
same blocks in a different order.36
Knowledge of this sort is "intuitive" geometry. We accumulate a considerable body of this kind of information at
an early age. By the time we are confronted with our first
textbook on geometry, we are pleasantly surprised to find
how much of the stuff we already know.37 And by the
time we feel called on to write on or about geometry ourselves, we pass those things on for "common sense" and
as "self-evident, intuitive truths" (Euclid), or for "knowledge
MATHEMATICS AND PHILOSOPHY
55
a priori" (Kant), while they are no more and no less than
empirical information that had been acquired very early in
the hard and exacting school of living and acting in a certain environment.
F ' Foundations of Logic One of the basic entities that
preoccupy the logician is the concept of "object." One commonly conceives of an object as a quantity of matter
packed into a portion of space and having well defined
boundaries. This is a very useful idea, well adapted to our
human needs. But when they are subjected to the closer
scrutiny of the physicist, these qualities of common objects,
as well as others that we associate with such objects, retain
a validity that is only approximate and provisional. We thus
arrive at the conclusion that the conception of a "given
object" is the outcome of an effort to abstraction bearing
on the shape, motion, and other preceptive qualities of common objects. In other words, the idea of an "object" derived
from our everyday experience is only summarily correct,
just as is the idea of a straight line that is suggested by a
stretched string.
Following up this analogy, we may attempt to axiomatize
the concept of "any object" in order to facilitate its study
from the point of view of logic.
The process of systematic axiomatization will require a
further schematization of the idea of "object" itself, as well
as of the conjoint idea of the presence of the object somewhere, or of its complete absence.
In the first place, the idea of "object" will have to be
divested of the notion of the object being present at a
definite spot or place. The idea of object will just remind
us that the object is, or is not. This idea of presence or
absence, when pushed farther, results in the idea of pure
existence or non-existence.
Furthermore, the strictly practical notion of the permanence of the object with regard to its own properties leads
to the abstract notion of its pure identity with itself. This
notion, combined with the notion of pure existence or
non-existence, results in the abstract of existential identity.
To render this idea of existential identity more concrete,
let us consider two different marks, say A and B. Suppose
it so happens that neither could be written down without
the other likewise being written, and that neither could be
crossed out or erased without the same happening to the
other. Moreover, the only matter of concern is to ascertain
56
MATHEMATICS IN FUN AND IN EARNEST
their presence or their absence. Under such circumstances,
no confusion could arise if either one of the two marks A,
B, is taken for the other. This practical equivalence of A
and B is a good realization of their purely existential equivalence.
Going a step further, the same thing may be restated if,
instead of two different marks, the same mark were drawn
twice—that is, two distinguishable realizations of the same
mark, say A, but again provided that only the question of
presence or absence is involved. The two realizations may
then be considered as identical. This practical identity realizes the abstract idea of existential identity.
Let us take the idea any object as our undefined term. We
are now ready to formulate the axioms governing that term.
It should be recalled that the axioms aim to point out, or
reproduce, in connection with the abstraction considered,
the salient properties of the concrete representations of that
abstraction, or better, the properties that are inherent in
the intuitive image we have of those concrete representations.
The realization of the abstract notion of existential identity by the practical equivalence of two copies of the same
symbol quite naturally suggests the following.
Axiom 1. Any object is identical with itself.
To this axiom we shall add two more, based on our practical experience and on our everyday knowledge of the rules of
presence and absence of material objects:
Axiom 2. Any object is, or is not.
Axiom 3. No object can be and not be at the same time.
The three axioms represent, respectively, the principle of
identity, the principle of the excluded middle, and the principle of contradiction.
The upshot of our efforts at abstraction by axiomatization
is thus the idea of "any object" governed by the laws of
being, or not being, and the law of existential identity with
itself, but which is otherwise undetermined. This eminently
abstract idea of an object might be called the "abstract object" or the "logical object."
The concrete object realizes the logical object in the same
way that the stretched string realizes the geometrical
straight line. The idea of pure existence of the logical object is realized by a natural or concrete object in the same
fashion and to the same degree as the ideal rectitude of a
straight line is realized by the crude rectitude of the string.
But no concrete object realizes the abstract object any more
MATHEMATICS AND PHILOSOPHY
57
closely than it can realize the geometrical idea of a point.
These observations simply emphasize the fact that in the
present case, as in any other abstraction by axiomatization,
the relationship between the concrete and the abstract is
adequate only in a schematic way. The degree to which it
is necessary to simplify the common, intuitive notion in order to achieve this correspondence is clearly shown by our
effort to establish the principle of identity.
G ' Symbolic Representation, or Miniature Realization Let
A be the symbol of an object whose existence or non-existence has not been specified, and let A and A be the symbols
for A's existence and non-existence, respectively. The three
marks A, A, A are three concrete objects. With reference
to them, our three fundamental axioms take on the following
forms:
Axiom 1. The letter A has everywhere the same significance.
Axiom 2. Each determination of A is expressed by its
being either underlined or overlined.
Axiom 3. The latter two cases are mutually exclusive.
It is essential to observe that in those three statements all
the words used have their ordinary meanings. There is no
mention of purely existential identity or of pure existence.
The reason for it is that in formulating these statements we
have entrusted ourselves to our intuitive and practically certain knowledge concerning three signs drawn on paper.
To put it in other words, these three symbols and the
formal rules that they have to obey are a miniature realization of the abstract schema that we have devised. This
abstract schema is the link that establishes a correspondence
between the three symbols A, A, A the concrete number
2, so to speak, and the concrete number 1—namely, the
common objects with which we started in the first place.
However, we are prone to forget about the existence of
this connecting link and see only the two concretes facing
each other; the original concrete upon which we were reluctant to operate and the new concrete, much reduced and
more readily handled.
The undetermined object symbolized by the "abstract
form" A may be filled, or it may be empty. The concrete
realization, or model of this form, may be perceived in any
object that may, at will, be brought into the field of attention,
or may be far from it. The form A on the other hand,
symbolizes an existing object.
58
MATHEMATICS IN FUN AND IN EARNEST
Now consider two objects that have no apparent tie and
that may be treated independently of one another. True, a
sufficiently close examination of any two objects is likely to
result in the discovery of some kind of a relationship between them. But we shall gloss this over. If our two objects
are, for instance, two books, their independence, for our
purpose, may manifest itself in the fact that the two books
may be in, or out of, the library independently of one
another. This schematic form of independence is an abstract
concept that may be adequately represented by two "forms
of objects" admitting, without preference or distinction, the
four eventualities shown in Table 1.
No.
Table 1. Four eventualities.
In Words
1
A is and B is
A and B
2
A is, but B is not
A and ~B
3
B is, but A is not
A and B
4
Neither A is nor B is
A and "B
In Symbols
The enumeration of these eventualities may perhaps incline the reader to think that any two objects are always
independent. But that is not the case. The two independent
objects that are existentially equivalent, considered in a
previous paragraph, are just objects for which the eventualities 2) and 3) do not exist. We could also conceive of
two objects for which the eventualities 1) and 4) are excluded, by definition. This is the case of mutual exclusion.
The case when only eventualities 1 and 4 are valid may
be rendered concrete by two persons who always enter and
leave a certain room at the same time; and the case when
only eventualities 2 and 3 are valid is made concrete if one
of those two persons leaves the room whenever the other
enters, and vice versa.
These two examples show, by the way, that the relationships of equivalence and exclusion that we imagined between
two forms are schematizing some close relationships that
may exist between material objects.
H ' Rational Theory of Objective Existence The concept
of the set of the eventualities 1, 2, 3, and 4 considered as
MATHEMATICS AND PHILOSOPHY
59
freely admissable may be referred to as the "abstract form
relatives to two abstract objects." If one, or two of these
eventualities is left out, a less extended form, or a "subform," is obtained. Thus a pure and simple eventuality is
the least extended form. We may also say that a subform
"enters" into a more extended form, or that the latter "contains" the former, if all the eventualities of the former are
also eventualities of the latter. Every subform enters into the
complete form. Two subforms overlap, or are mutually exclusive, according to whether they do or do not possess a
common eventuality.
Using this terminology, we may state the following new
axiom: Two determined objects which enter into the form
of equivalence cannot enter into the form of mutual exclusion. The realization (involving two persons) that we have
considered makes it clear that this statement (axiom) formulates an empirical law of the world of material objects,
a very primitive law, and therefore one of practically unfailing validity.
Thus, starting with the most common and the most elementary properties of material objects, and applying the
axiomatic method as exemplified in plane geometry, we arrived at the concepts and the rules of pure existence. These
ideas are basic in Gonseth's Rational Theory of Objective
Existence; Gonseth's theory axiomatizes one of the first
chapters of physics, if not the very first—namely, the one
dealing with the existence, the presence, and the absence
of objects of any kind. In other words, the physics of any
object and the rational theory of pure existence are two
phases of the same undertaking, the former being the external significance of the latter.
For the subject at hand, the importance of the rational
theory of existence is that all the laws of elementary logic
may be expressed in the form of rules of existence, as was
quite apparent when we sketched those ideas, and may be
substantiated by further analysis. But we shall not pursue this
argument. The detailed treatment may be found in Gonseth's
own writings.
I ' Logic The conclusions to be drawn from the preceding discussion are as illuminating as they are far-reaching.
The rules of pure existence being the schematized properties of common objects, the same holds for the equivalent
60
MATHEMATICS IN FUN AND IN EARNEST
laws of logic. Hence, the common sense of logic and its
intuitive rules are seen to be the outcome of a schematization that is based upon our experience in the world of
common objects.
Furthermore, since the abstract laws that logic formulates have their origin and their realization in the domain of
concrete objects, those laws take on the significance of very
primitive natural laws and are therefore practically infallible. That is what accounts for their usefulness, on the one
hand, and for their irresistible power over us, on the other
hand.
Our distinguished contemporary Maurice Frcchet (b.
1878), professor of mathematics at the Sorbonne, put the
question of the origin of logic in a nutshell: "The rules of
logic start with an approximation of the real, and that reality is rediscoverable even in the remotest conclusions drawn
from those laws. Is it just by lucky accident, independently
of all experience, that those laws impose themselves upon
our mind? Or is not our acceptance of those laws from our
predecessors due to the fact, taught us by our daily experience, that if we apply those laws correctly, we are never
mistaken? We are thus not far from concluding that logic
itself is a product of our experience, that logic is the result
of an inductive synthesis. It is therefore quite legitimate, and
even very useful, to submit logic to a process of axiomatization. This axiomatization, like that of any other science,
must be considered as being only an essentially revisable
schematization of the practical rules of reasoning. But we
are certain that we shall always be able to utilize our
logic, without change, in the major part of our scientific
research." In brief, the empirical origin of logic is obvious
a priori, despite the firm conviction of all those thinkers
through the ages for whom the laws of logic were inherent
laws of the mind. But then, the principle of quality turns
out to be obvious, and so does Goedel's theorem. It would
seem that nothing is more effectively hidden in the farthest
recesses of obscurity than the obvious.
Since our laws of logic are derived from our practical
experience, our reasoning can be valid only as long as we
apply it to our environment as it is here and now, so to
speak. This may cause trouble in some unsuspected and unsuspecting quarters. Take the great phalanx of enthusiastic
space travelers, young and otherwise, that has sprung up in
MATHEMATICS AND PHILOSOPHY
61
the wake of the rockets of recent invention. These travelers
are beset by a great many worries and difficulties. But the
fun-seeking excursionists to neighboring planets, as well as
the intrepid conquistadores of new galaxies, may discover,
to their amazement and chagrin, that the logic which they
had found so reliable as long as they stayed home, goes
"haywire" when they get abroad.
But remaining peacefully at home offers no guaranty of
the permanent validity of our logic. Should our environment
change, we would have to change our logic accordingly.
This may sound fantastic, but it is not outside the realm of
the possible. In fact, in a way, we are already in the
midst of such a change right now, and have been for about
half a century. Quantum theory, the new atomic theory, and
the theory of relativity have confronted us with phenomena
that operate on a scale either too vast or too minute compared with those on which our senses received their education
and training. No wonder that we run into "inconsistencies"
and "contradictions." The physicists have had to re-examine
many of our notions that were well established according to our "common sense."38 The logicians, for their part,
try to meet the newly arisen problems by introducing
multivalued logic.39
Our discussion of the origin of the laws of logic has
brought into the open the limitations of those laws and
should thus contribute to a better understanding of that wonderful instrument our human race is so proud of—the power
of reasoning. We have convinced ourselves once more that
our source of knowledge lies in closer contact with our environment. Is it not this idea that the Greek mythology
wanted to express by imagining the demigod Anteus whose
power endured as long as he maintained contact with the
earth?
The same idea may be found in the play Chantecler of
the French poet Edmond Rostand (1868-1918). This is the
way the mighty Chantecler explains to his friend the pheasant hen where he derives his power to call out the sun
from below the horizon: "I never start to sing until my
eight claws, after clearing a space of weeds and stones, have
found the soft, dark turf underneath. Then placed in direct
contact with the good earth, I sing." We, too, have to be
"in contact with the earth" if we want the light of knowledge to shine on us.
62
MATHEMATICS IN FUN AND IN EARNEST
FOOTNOTES
1
See Chapter I, Section 2c; Chapter III, Section 2.
Bulletin American Mathematical Society, 1912-1913, p. 332.
3
See Chapter I, Section 3f, section 4b; Chapter V, Section li.
I
See Chapter I, Section 4c.
5
For K. Goedel's contribution to the question of consistency
of the postulates see Chapter I, Section 4b.
9
Cf. Chapter I, Section 2a.
7
Enseignement Mathematique,
1931, p. 29.
8
See Chapter IV, Section 2d.
9
Chapter I, Section If.
10
See Chapter I, Section 4a.
II
Cf. Chapter I, Section 4a; Chapter III, Section 2b.
12
See Chapter I, Section 4d.
13
Mathematical
Philosophy, New York, 1922, pp. 412-413.
14
See Chapter VII, Section lc.
15
Education Times, Reprints, V. 65, 1896, Question 12854.
18
Cf. Chapter III, Section 3i.
17
See Chapter V, Section le.
18
Cf. Chapter III, Section 3f; Chapter V, Section lj.
19
See Chapter I, Section Id, Section 4b; Chapter V, Section li.
20
Encyclopedic Frangaise, Vol. I, p. 20-26, 1937.
21
An Essay on the Psychology of Invention in the Mathematical
Field, Princeton University Press, 1945.
22
Cf. Chapter V, Section li.
23
Cf. Chapter I, Section 2d; Chapter III, Section 2.
21
Cf. Chapter V, Section lc.
20
See Chapter I, Section Id, 3f; Chapter V, section li.
28
Foundations of Science, H. Poincare, Edited by G. B. Halstead, Science Press, Lancaster, Pa., 1913, 1946.
27
Cf. Chapter I, Section Id; Chapter V, Section li.
28
Chapter I, Section le.
29
Le decline des absolus mathematico-logiques,
G. Bouligand
and J. Desgranges, Sedes, Paris, 1949, p. 16.
30
The Axiomatic Method in Biology, 1. H. Woodger, Cambridge
University Press, Cambridge, 1937.
31
"Vision" in Insect Physiology, V. G. Dethier, Edited by K. D.
Roder, Wiley, New York, 1953, p. 488.
32
Radio Astronomy,
B. Lovell and J. A. Clegg, Wiley, New
York, 1952. Scientific Monthly,
W. L. Roberts, 79, 170, 1954.
Astrophysical Journal, articles by Minkowski and others, 1954.
33
Problems of Science, F. Enriques, Open Court, Chicago, 1914.
34
Cf. Chapter I, Section 4d.
35
Ecclesiastes: 12, 12.
30
Pedagogical Seminary, 27, 75, "Number, time and space in
the first five years of a child's life." S. R. A. Court, 1920.
37
Cf. Chapter I, Section 2a.
38
Scientific Monthly, 19, '2, P. W. Bridgeman, 1954.
39
See Chapter V, Section 1/.
2
II
SOME SOCIOLOGIC ASPECTS OF MATHEMATICS
1
•
Mathematics and Civilization
A • The Early Beginnings of Counting and Reckoning
In
a country where compulsory school attendance has been the
practice for several generations the vast majority of children learn to count, that is, to recite the series of words
one, two, three, four, and so on, very early, long before
they reach school age. One of the consequences of this situation is that when they grow up they cannot remember
when they could not perform this very useful trick.
The same situation prevails with regard to the human
race as a whole. We know quite definitely that there was a
time when the notion of number was totally alien to mankind. Who was the genius who first asked the momentous
question: "How many?" We will never know. At a certain
stage of social development the need arises to determine how
many objects constitute a given collection. The answer to
the question becomes a social necessity. Contributions toward finding that answer are made by individuals confronted with the same need, and the notion of number
slowly emerges.
How slow and painful a process of creation this was may
be judged from the fact that there are human tribes whose
languages have no words for numbers greater than four, and
even no greater than two. Beyond that any group consists of
"many" objects.
A bright light is shed upon this subject by a story told
the writer by a colleague, from the latter's personal experience, about a flock of crows.
The birds were infesting a cornfield. One morning, when
two men armed with shotguns approached the field, the
crows took refuge in a grove of trees at one end of the
field. They remained there as long as the two suspected enemies occupied a shed at the other end of the field. When
one of the men emerged from the hiding place and left the
scene, the birds were not impressed: they remained where
they were. But when the other man left the shed and vanished in the distance, the hungry birds resumed their feasting.
Next day three men entered the shed and the flock
63
64
MATHEMATICS IN FUN AND IN EARNEST
p e r c h e d in t h e trees. T h e cautious b i r d s w e r e not f o o l e d
w h e n t w o of the m e n c a m e out of t h e shed a n d walked
away. T h e y w a i t e d until t h e t h i r d o n e did likewise. T h e following d a y t h r e e of the f o u r m e n in t h e shed c a m e o u t of
t h e h i d i n g place. A s soon as t h e y got o u t of sight, the crows
d e s c e n d e d u p o n the field in f o r c e . T h e wise crows c o u l d
a p p a r e n t l y tell the d i f f e r e n c e b e t w e e n o n e a n d two, also b e t w e e n t w o a n d three, a n d acted accordingly. But the differe n c e b e t w e e n t h r e e a n d f o u r passed the limits of their
a r i t h m e t i c a l w i s d o m , a n d t h e flock p a i d a high tribute f o r
their i g n o r a n c e .
O u r n u m b e r s a r e a p p l i e d t o a n y k i n d of object in the
s a m e w a y , w i t h o u t d i s c r i m i n a t i o n . T h e y h a v e a kind of " i m p e r s o n a l i t y , " w h i c h w a s n o t t h e case with primitive m a n .
W i t h h i m the n u m b e r a p p l i e d t o a g r o u p is modified in acc o r d a n c e w i t h t h e n a t u r e of t h e g r o u p . T h e n u m b e r characterizes the g r o u p in t h e s a m e w a y as a n adjective applied to
a n o u n modifies t h e object t o w h i c h it is applied. T h e E n g lish l a n g u a g e h a s p r e s e r v e d s o m e traces of that attitude. A
g r o u p of cattle is a herd, while a g r o u p of birds is a flock;
a g r o u p of wolves is a pack, while a g r o u p of fish f o r m
a school. It w o u l d be s h o c k i n g i n d e e d t o speak of a school
of cows. O t h e r l a n g u a g e s offer m u c h m o r e striking p r o o f s
of such a n a t t i t u d e t o w a r d s n u m b e r s in their relation tow a r d s t h e objects t h e y a r e applied to. T h u s in English w e
u s e the singular g r a m m a t i c a l f o r m w h e n one object is involved, a n d w e use the p l u r a l g r a m m a t i c a l f o r m f o r a n y
n u m b e r of objects larger t h a n one. S o m e of the languages of
t h e W e s t e r n w o r l d , in t h e i r earlier stages of d e v e l o p m e n t ,
h a d a special g r a m m a t i c a l f o r m , a d u a l f o r m , w h e n two
objects w e r e s p o k e n of. S o m e l a n g u a g e s even h a d separate
g r a m m a t i c a l f o r m s w h e n r e f e r e n c e w a s m a d e to t h r e e objects, a n d still a n o t h e r f o r f o u r objects. A n instructive exa m p l e of t h e w a y t h e f o r m of t h e s a m e n u m b e r m a y be
m o d i f i e d to fit the g r o u p to w h i c h it is applied is f u r n i s h e d
b y t h e Polish l a n g u a g e in its use of the n u m b e r two. I n
that l a n g u a g e a different f o r m of " t w o " is used w h e n a p plied t o t w o m e n , t o t w o w o m e n , t o a m a n a n d a w o m a n ,
a n d t o i n a n i m a t e objects o r animals. T h e s e f o r m s are, respectively: dwaj, dwie, dwoje,
dwa.
T h e process of a c c u m u l a t i n g e n o u g h w o r d s to answer t h e
q u e s t i o n : h o w m a n y ? t o satisfy the g r o w i n g needs was slow
a n d laborious. M a n derived a great deal of help f r o m t h e
n a t u r a l set of c o u n t e r s h e always carried with h i m — h i s
fingers.1 W e still use the w o r d " d i g i t " b o t h f o r fingers a n d
SOME SOCIOLOGIC ASPECTS OF MATHEMATICS
65
to designate a n u m b e r less t h a n ten. Of t h e m a n y e x a m p l e s
that could be cited to illustrate t h e u s e of fingers as c o u n t e r s
let us q u o t e a r e p o r t of F a t h e r Gilij w h o describes t h e arithmetic of t h e I n d i a n tribe of t h e T a m a n a c a s , o n the O r i n o c o
River.
T h e T a m a n a c a s h a v e w o r d s f o r t h e first f o u r n u m b e r s .
W h e n t h e y c o m e to five t h e y express it b y a p h r a s e w h i c h
literally m e a n s " a w h o l e h a n d ; " t h e p h r a s e f o r t h e n u m b e r
six m e a n s literally " o n e o n t h e o t h e r h a n d , " a n d similarly
f o r seven, eight a n d nine. W h e n t h e y c o m e to ten t h e y u s e
the p h r a s e " b o t h h a n d s . " T o say eleven they stretch out b o t h
hands, a n d a d d i n g a f o o t , they say " o n e o n the f o o t , " a n d
so on, u p to 15, w h i c h is " a w h o l e f o o t . " T h e n u m b e r 16
is " o n e o n t h e o t h e r f o o t . " F o r t w e n t y they say " o n e
I n d i a n , " a n d 21 is expressed b y saying " o n e o n the h a n d s of
the o t h e r I n d i a n " ; f o r t y is " t w o I n d i a n s , " sixty " t h r e e
I n d i a n s , " a n d so on.
In this c o n n e c t i o n it m a y be of interest to p o i n t out t h a t
the R u s s i a n w o r d f o r five ( " p i a t " ) is a slight modification of
the w o r d f o r first ( " p i a s t " ) . T h e s a m e is t r u e f o r o t h e r Slavic
languages.
W h e n the q u e s t i o n : h o w m a n y ? h a s o n c e been raised,
m e r e c o u n t i n g b e c o m e s insufficient. F u r t h e r steps in civilization b r i n g a b o u t the need of c o m p u t a t i o n . T h e strongest
single f a c t o r that stimulated the d e v e l o p m e n t of m e t h o d s of
c o m p u t a t i o n was trade. A c c o r d i n g t o the m y t h o l o g y of t h e
ancient Egyptians, a r i t h m e t i c w a s invented b y their G o d of
c o m m e r c e . A s with c o u n t i n g , t h e beginnings of r e c k o n i n g
were slow a n d laborious, a w k w a r d a n d p a i n f u l . A t r a d e r in
tropical S o u t h A f r i c a d u r i n g t h e last c e n t u r y has this t o
say a b o u t the m e m b e r s of the D a m m a r a tribe. " W h e n b a r tering is going on, each sheep m u s t b e p a i d f o r separately.
T h u s , suppose t w o sticks of t o b a c c o t o be the rate of exc h a n g e f o r o n e sheep; it w o u l d sorely puzzle a D a m m a r a
to take t w o sheep a n d give h i m f o u r sticks. I h a v e d o n e so,
and seen a m a n put t w o of t h e sticks a p a r t a n d take a sight
over t h e m at o n e of the sheep h e was a b o u t to sell. H a v i n g
satisfied himself that that o n e w a s h o n e s t l y p a i d f o r , a n d
finding t o his surprise that exactly t w o sticks r e m a i n e d in
his h a n d t o settle the a c c o u n t f o r t h e o t h e r sheep, he w o u l d
be afflicted with d o u b t ; t h e t r a n s a c t i o n seemed to c o m e out
t o o " p a t " to be correct, a n d he w o u l d r e f e r b a c k to t h e
first couple of sticks; a n d t h e n his m i n d got hazy a n d c o n f u s e d , a n d h e w a n d e r e d f r o m o n e s h e e p to t h e o t h e r , a n d
he b r o k e off the transaction, until t w o sticks w e r e p u t in his
66
MATHEMATICS IN FUN AND IN EARNEST
h a n d , a n d o n e sheep d r i v e n away, a n d t h e n t w o o t h e r
sticks given h i m a n d the second sheep driven a w a y . " It
w o u l d s e e m t h a t at least to this representative of h u m a n i t y
it was n o t o b v i o u s t h a t t w o times t w o m a k e s f o u r .
T h e s t o r y illustrates t h e b l u n d e r i n g beginnings of the art
of r e c k o n i n g . T o relate the evolution of this art f r o m its
h u m b l e b e g i n n i n g s t o t h e heights of p o w e r a n d p e r f e c t i o n
it h a s a c h i e v e d in m o d e r n times, a n d h o w this art has followed a n d served t h e ever g r o w i n g n e e d s of m a n k i n d is to
tell o n e of the m o s t exciting sagas in the history of civilization. O n l y a m e r e outline can be a t t e m p t e d here.
V a r i o u s h u m a n activities, a n d in p a r t i c u l a r c o m m e r c e , req u i r e t h e k e e p i n g of s o m e n u m e r i c a l records. S o m e kind of
m a r k s h a d t o be invented f o r the p u r p o s e . T h e devices
used t h r o u g h t h e ages w e r e k n o t s tied in a r o p e a n d notches
c u t in sticks. It m a y surprise s o m e r e a d e r s that such sticks,
called tallies, w e r e used as a m e t h o d of b o o k k e e p i n g by the
B a n k of E n g l a n d well i n t o the n i n e t e e n t h century.
T h e first w r i t t e n s y m b o l s f o r n u m b e r s were, naturally,
sticks: One stick, two sticks, three sticks, a n d so on, to represent " o n e , " " t w o , " " t h r e e " etc. T h i s w o r k e d fairly well
as long as t h e n u m b e r s to be r e p r e s e n t e d were small. F o r
larger n u m b e r s t h e sticks o c c u p y too m u c h space, it becomes
difficult to c o u n t t h e m , a n d it takes t o o m u c h time. T h e
sticks h a d to be c o n d e n s e d i n t o groups, thus representing
larger units, a n d these n e w units in t u r n h a d to be condensed
i n t o larger u n i t s a n d t h u s a h i e r a r c h y of units h a d to be
formed.
T h i s n e e d f o r c o n d e n s a t i o n of n u m e r i c a l symbols is readily b r o u g h t h o m e to us by a f a m i l i a r example. In theory the
t r e a s u r y of the U n i t e d States should mint only o n e kind of
coin, n a m e l y a p e n n y , f o r every s u m of m o n e y can be realized with pennies. In practice, h o w e v e r , this would be a
m o s t a w k w a r d p r o c e d u r e , even w h e n only small s u m s are
involved. T o help m a t t e r s the t r e a s u r y m i n t s also nickels
a n d several k i n d s of silver coins. M o r e o v e r , the treasury
considers t h a t o n e is justified in r e f u s i n g to accept m o r e
t h a n t w e n t y five p e n n i e s in a n y single p a y m e n t . F o r larger
a m o u n t s the t r e a s u r y c o n d e n s e s o n e h u n d r e d pennies into
a single p a p e r dollar bill, a n d then c o n t i n u e s the process by
issuing bills of several h i g h e r d e n o m i n a t i o n s .
T h e G r e e k s a n d t h e H e b r e w s used the letters of their
a l p h a b e t s as n u m e r a l s . T h e B a b y l o n i a n s h a d special n u m e r i cal symbols. T h e R o m a n n u m e r a l s are still in use occasionally, as f o r i n s t a n c e o n o u r clocks. All these symbols o r
SOME SOCIOLOGIC ASPECTS OF MATHEMATICS
67
m a r k s f o r n u m b e r s h a d o n e f e a t u r e in c o m m o n — t h e y did
not lend themselves to a r i t h m e t i c a l c o m p u t a t i o n s . T h e art
of r e c k o n i n g h a d to be c a r r i e d out with t h e help of different
devices, the chief a m o n g t h e m b e i n g t h e c o u n t i n g f r a m e , o r
the abacus. This i n s t r u m e n t most o f t e n consisted of a rectangular f r a m e with b a r s parallel to o n e side. T h e o p e r a t i o n s
were p e r f o r m e d o n the b e a d s o r c o u n t e r s s t r u n g o n these
bars. This i n s t r u m e n t was w i d e s p r e a d b o t h in Asia a n d in
E u r o p e . W h e n the E u r o p e a n s arrived in A m e r i c a they f o u n d
that a f o r m of a b a c u s was in use b o t h in M e x i c o a n d in
Peru.2
T h e m e t h o d of writing n u m b e r s a n d c o m p u t i n g with
t h e m that we use n o w h a d its origin in India. T h e most original f e a t u r e of that system, n a m e l y the zero, the s y m b o l f o r
nothing, was k n o w n in B a b y l o n a n d b e c a m e c o m m o n in
India d u r i n g the early c e n t u r i e s of t h e C h r i s t i a n E r a . T h i s
system of c o m p u t a t i o n was b r o u g h t to E u r o p e by the A r a b i c
and Jewish m e r c h a n t s d u r i n g t h e t w e l f t h c e n t u r y . T h e first
printing presses set u p in E u r o p e , in t h e m i d d l e of the fifteenth c e n t u r y , rolled off a c o n s i d e r a b l e n u m b e r of c o m mercial arithmetics. T w o centuries later the a b a c u s in
Western E u r o p e was little m o r e t h a n a relic of the past. It is
still widely a n d efficiently used in t h e O r i e n t .
T h e very heavy d e m a n d s that m o d e r n life in its v a r i o u s
phases m a k e s u p o n c o m p u t a t i o n seem to be t u r n i n g the tide
against p a p e r a n d pencil r e c k o n i n g . W e are a b o u t to e n t h r o n e the a b a c u s back again, in a m u c h i m p r o v e d f o r m ,
to be sure, but nevertheless in t h e f o r m of an i n s t r u m e n t .
In fact, we are using a c o n s i d e r a b l e n u m b e r of t h e m , like
the slide rule, the cash register, the v a r i o u s electrically o p erated c o m p u t e r s , to say n o t h i n g of the c o m p u t i n g m a chines which o p e r a t e on a m u c h h i g h e r level, like those
which give the solutions of differential e q u a t i o n s . Such is the
devious a n d puzzling r o a d of h u m a n progress.
B • Measuring.
Beginnings
of Geometry
and
Chronology
" H o w m a n y ? " T h i s question is t h e origin of a r i t h m e t i c a n d
is responsible f o r m u c h of its progress. But this question c a n not claim all the credit. It m u s t s h a r e the credit with a n other, a later arrival on the scene of civilization, but w h i c h
is even m o r e f a r reaching. T h i s question is: " h o w m u c h ? "
H o w m u c h does this rock weigh? H o w m u c h time h a s
passed between t w o given events? H o w long is the r o a d
f r o m town A to town B? etc. T h e answers to these questions
are n u m b e r s , like the a n s w e r to the q u e s t i o n : " h o w m a n y ? "
68
MATHEMATICS IN FUN AND IN EARNEST
T h e r e is, h o w e v e r , a vast difference b e t w e e n the n u m b e r s
w h i c h a n s w e r t h e t w o k i n d s of questions.
T h e a n s w e r t o t h e q u e s t i o n : " h o w m a n y ? " is obtained by
c o u n t i n g discreet objects, like sheep, trees, stars, warriors,
etc. E a c h of t h e objects c o u n t e d is entirely separate f r o m
the o t h e r s . T h e s e objects c a n be " s t o o d u p a n d be c o u n t e d . "
S o m e t h i n g vastly different is involved in the q u e s t i o n : H o w
m u c h does this r o c k w e i g h ? T h e a n s w e r c a n only be given
b y c o m p a r i n g t h e weight of t h e given r o c k to the weight of
a n o t h e r r o c k , o r t o the w e i g h t of s o m e o t h e r object taken
f o r the unit of weight, say a p o u n d or a ton. Obviously this
is a m u c h m o r e involved process a n d implies a m u c h m o r e
a d v a n c e d social a n d intellectual level t h a n the answer to the
question: how many?
T h e q u e s t i o n : " h o w m a n y ? " is always a n s w e r e d b y a n
integer. N o t so the q u e s t i o n : " h o w m u c h ? " G i v e n seventeen
trees, is it possible t o p l a n t t h e m in five rows so that each r o w
h a s t h e s a m e n u m b e r of trees? T h e a n s w e r is: " N o , " a n d
this is t h e e n d of the story. B u t given seventeen p o u n d s of salt
in a c o n t a i n e r , it is possible t o distribute this salt into five
c o n t a i n e r s so t h a t each of t h e m will hold the s a m e a m o u n t
of salt. B u t t h e q u e s t i o n : " H o w m a n y p o u n d s of salt does
e a c h c o n t a i n e r h o l d ? " c a n n o t be a n s w e r e d by an integer.
T h u s , t h e q u e s t i o n : " h o w m u c h ? " is responsible f o r the inv e n t i o n of f r a c t i o n s . It is also responsible f o r t h e introd u c t i o n of i r r a t i o n a l n u m b e r s . But a b o u t that w e m a y say
s o m e t h i n g later o n .
T h e q u e s t i o n : " h o w m u c h ? " that is, the introduction of
m e a s u r e m e n t s , h a s involved us in a n o t h e r kind of difficulty
w h i c h did n o t b o t h e r us in c o n n e c t i o n with the q u e s t i o n :
" h o w m a n y ? " W e c a n ascertain that the g r o u p at the
p i c n i c consisted of f o r t y boys. But w h e n we say that this
table is f o r t y i n c h e s long, w e c a n only m e a n that it is closer
t o f o r t y i n c h e s t h a n it is either t o thirty-nine or f o r t y - o n e
inches. W e m a y , of course, use m o r e precise instruments of
m e a s u r e m e n t w h i c h m a y n a r r o w d o w n the d o u b t f u l area, b u t
it will not r e m o v e it. Results of m e a s u r e m e n t s are necessarily
o n l y a p p r o x i m a t i o n s . T h e degree of a p p r o x i m a t i o n to which
w e c a r r y o u t these m e a s u r e m e n t s d e p e n d s u p o n the use w e
are to m a k e of these m e a s u r e d things.
T h e h e r d s m a n is m u c h c o n c e r n e d with the q u e s t i o n : " h o w
m a n y ? " T h e s h e p h e r d , in addition, is also interested in the
q u e s t i o n : " h o w m u c h ? " a f t e r he is t h r o u g h sheering his
flock. W h e n a h u m a n tribe t u r n s to agriculture, the question:
" h o w m u c h ? " imposes itself with increased insistence. Agri-
SOME SOCIOLOGIC ASPECTS OF MATHEMATICS
69
culture requires s o m e m e t h o d s of m e a s u r i n g land, of m e a s uring the size of t h e c r o p , t h a t is m e a s u r i n g areas a n d v o l u m e s
as o u r school b o o k s call it. F u r t h e r m o r e , the agricultural
stage of society implies a l r e a d y a c o n s i d e r a b l e d e g r e e of
social organization, a n d the tax collector a p p e a r s o n t h e
scene. This official is vitally interested in t h e size of the c r o p .
H e also h a s to h a v e s o m e n u m e r i c a l r e c o r d s of the a m o u n t
of taxes collected a n d of the a m o u n t of taxes due. N o w y o u
m a y not like the tax collector. F e w p e o p l e waste t o o m u c h
love on this maligned official. It is nevertheless quite obvious that n o o r g a n i z e d society is possible w i t h o u t the
collection of taxes, that is w i t h o u t c o n t r i b u t i o n s f r o m t h e
individual m e m b e r s of t h a t society t o w a r d s the necessary
enterprises that are of benefit to t h e m e m b e r s of the e n t i r e
c o m m u n i t y . A n d such collections c a n n o t be m a d e in a n y
orderly f a s h i o n , unless a n s w e r s c a n b e given to t h e t w o
questions: " h o w m u c h ? " a n d " h o w m a n y ? "
H o w f a r back in the history of m a n k i n d the question " h o w
m u c h ? " was first asked w e c a n o n l y guess, a n d t h a t very
roughly. T h e s e surmises are h e l p e d b y the study of the cult u r e of s o m e of the primitive tribes still i n h a b i t i n g this e a r t h ,
or did so in recent past.
H o w e v e r , c o n j e c t u r e s a r e r e p l a c e d b y d o c u m e n t a r y evidence w h e n we t u r n to t h e p e r i o d of h u m a n history w h i c h
starts about six o r seven t h o u s a n d y e a r s ago in M e s o p o t a m i a ,
Egypt, India. This is the p e r i o d of t h e B r o n z e Age, the beginning of u r b a n civilization. T h e B a b y l o n i a n tablets, the
Egyptian papyri, a n d o t h e r d o c u m e n t s tell us a n e w a n d
w o n d r o u s story of n e w f o r m s of social a n d g o v e r n m e n t a l
organizations, of r e m a r k a b l e c o n q u e s t s in the d o m a i n of
arts and crafts, of great e x p a n s i o n of t r a d e a n d c o m m e r c e .
T h e same d o c u m e n t s tell us of astonishing a c h i e v e m e n t s in
the field of m a t h e m a t i c s a n d a s t r o n o m y . T h e historian of
civilization m a k e s it clear that this n e w k n o w l e d g e w a s
called f o r t h — a n d c o n t r i b u t e d to—-by the artisan, t h e builder,
the m e r c h a n t , the surveyor, the w a r r i o r . 3
T h e cultivation of the land f a c e d t h e h u m a n r a c e w i t h
problems of geometry. E g y p t with its p e c u l i a r d e p e n d e n c e
u p o n the flood w a t e r s of the river N i l e was c o n f r o n t e d with
extra difficulties of a geometrical n a t u r e . T h a t is the r e a s o n
w h y geometry f o u n d such a fertile soil in the valley of
the Nile.
M u c h g e o m e t r y h a d t o b e discovered in o r d e r to c o n struct h u m a n habitations. W h e n civilization progresses beyond the cave dwelling stage, shelter b e c o m e s a p r o b l e m
70
MATHEMATICS IN FUN AND IN EARNEST
of the first m a g n i t u d e . T h e c o n s t r u c t i o n of dwellings involves in t h e first place k n o w l e d g e of t h e vertical direction,
as given b y t h e p l u m b line. It was observed very early that
t h e p l u m b line o r a pole having t h e s a m e direction as the
p l u m b line m a k e s e q u a l angles with all the lines passing
t h r o u g h its f o o t a n d d r a w n o n level g r o u n d . W e have thus
w h a t w e call a right angle, as well as the f a m o u s t h e o r e m
of o u r t e x t b o o k s t h a t all right angles are equal.
H o w e v e r i m p o r t a n t t h e answers t o the q u e s t i o n : " H o w
m u c h ? " m a y h a v e been in the c o n n e c t i o n s w e just considered, t h e m o s t i m p o r t a n t a n s w e r t o this question is t h e
o n e c o n n e c t e d with t h e m e a s u r i n g of time. W i t h the most
r u d i m e n t a r y a t t e m p t s at agricultural activity comes the realization t h a t success is d e p e n d e n t u p o n the seasons; this dep e n d e n c e is even e x a g g e r a t e d . W e still w o r r y about the
phases of the m o o n w h e n we want to plant o u r potatoes.
V a r i o u s tribes o n the s u r f a c e of t h e globe noticed that
the shortest s h a d o w cast b y a vertical pole during the day
always has t h e s a m e direction. This is the n o r t h a n d south
direction. T h e sun at that time occupies the highest point in
the sky. It is essential to h a v e a w a y of m a r k i n g this direction. H e r e is h o w it can be d o n e .
A circle is d r a w n o n the g r o u n d having f o r center the
f o o t of t h e pole used in the observation. T h e t w o positions of the s h a d o w a r e m a r k e d , the tips of which just fall
o n the c i r c u m f e r e n c e . T h e n o r t h - s o u t h line sought is the line
m i d - w a y b e t w e e n the t w o lines m a r k e d , a n d that north-south
line was f o u n d b y m a n y h u m a n tribes by bisecting this angle, a n d was d o n e b y the m e t h o d s still in use in o u r textbooks.
M e a s u r e m e n t s c o n n e c t e d with the sun, the m o o n , and the
stars in general c a n n o t be m a d e directly. Some r o u n d - a b o u t
m e t h o d m u s t b e used. N e i t h e r could the size of the earth be
d e t e r m i n e d directly. O n the e l e m e n t a r y level such artifices
are based o n g e o m e t r y a n d t r i g o n o m e t r y . T w o centuries
B. c., E r a t o s t h e n e s , the librarian of the f a m o u s A l e x a n d r i a n
L i b r a r y , s u c c e e d e d b y the use of such m e t h o d s in determ i n i n g the length of t h e d i a m e t e r of the earth with a surprising d e g r e e of a c c u r a c y . H e t h u s m a d e his c o n t e m p o r a r i e s
realize t h a t the w o r l d they k n e w was only a very small part
of the s u r f a c e of the earth.
T h e s e s k e t c h y indications give a n idea of the role m a t h e m a t i c s p l a y e d in t h e d e v e l o p m e n t of m a n k i n d f r o m the
earliest times u p until the g r e a t civilizations of antiquity.
SOME SOCIOLOGIC ASPECTS OF MATHEMATICS
71
C * The Renaissance
Period. The Great Voyages.
The Invention
of Analytic
Geometry
and of the Calculus
The
Renaissance was the age of t h e revival of secular learning
in E u r o p e . It w a s also the age of the great voyages a n d of
the discovery of A m e r i c a , t h e age of g u n - p o w d e r a n d of
m e c h a n i c a l clocks.
T h e n e w interest in s e a f a r i n g h a d raised m a n y pressing
p r o b l e m s t h a t h a d to be solved. T h e most obvious o n e w a s
the need f o r a w a y of d e t e r m i n i n g the position of a ship
o n the high seas, that is the n e e d of d e t e r m i n i n g the longit u d e a n d the latitude of the ship at a n y time. T h i s involved
a great deal of laborious c o m p u t a t i o n . T h e i n v e n t i o n of
logarithms r e d u c e d this l a b o r to a f r a c t i o n of the w o r k it
used to require. T h i s a c c o u n t s f o r the great success t h a t t h e
invention of l o g a r i t h m s e n j o y e d , as soon as it b e c a m e available.
T h e process of finding t h e l o n g i t u d e r e q u i r e d a n a c c u r a t e
clock which could be relied u p o n . W e m e n t i o n e d b e f o r e the
i m p o r t a n t role t h e need of d e t e r m i n i n g the seasons p l a y e d
in the history of civilization, a n d the m a t h e m a t i c a l p r o b l e m s
that h a d to be solved in this c o n n e c t i o n . T h e navigation of
the R e n a i s s a n c e r e q u i r e d the m e a s u r i n g of t i m e with great
precision. It w a s a question n o t of seasons a n d days b u t of
minutes a n d seconds. T h e i n s t r u m e n t that m a d e such acc u r a c y possible w a s the m e c h a n i c a l clock m o v e d by a p e n d u l u m , then by springs. T h i s m o v i n g m e c h a n i s m raised m a n y
p r o b l e m s of a m a t h e m a t i c a l n a t u r e t h a t the available m a t h ematical resources were insufficient t o c o p e with. N e w
mathematical methods were needed.
New. m a t h e m a t i c a l p r o b l e m s w e r e also raised b y the c a n non. It m a y be observed, in passing, that a c a n n o n w a s
just as m u c h a necessary piece of e q u i p m e n t of a ship
starting out on a long voyage t o w a r d s u n e x p l o r e d shores as
was a m a p , or a clock.
A g u n n e r f r e q u e n t l y n e e d s to d e t e r m i n e t h e distance t o
certain inaccessible objects. T h e i n f o r m a t i o n has t h u s to be
obtained by indirect m e a s u r e m e n t s . T h i s is a p r o b l e m that
was m e t with m u c h earlier in the history of civilization
and was solved in v a r i o u s ways. T h e c a n n o n has stimulated
f u r t h e r d e v e l o p m e n t in this c o n n e c t i o n , t h u s c o n t r i b u t i n g t o
the progress of t r i g o n o m e t r y .
But artillery presented p r o b l e m s of a n e w type. T h e cannon ball was an object w h i c h m o v e d with a speed that w a s
u n p r e c e d e n t e d in the experience of m a n . M o t i o n t o o k on a
new significance a n d called f o r m a t h e m a t i c a l t r e a t m e n t a n d
72
MATHEMATICS IN FUN AND IN EARNEST
study. It r e q u i r e d the study of t h e p a t h t h a t the projectile
describes in the air, the distance it travels, the height it
r e a c h e s at a n y given distance f r o m the starting point, a n d so
on. I n s h o r t it r e q u i r e d w h a t w e n o w call a graph. 4
T h e c o m p u t a t i o n of t h e longitude of a ship at sea is
based on a s t r o n o m i c a l observations a n d c o m p u t a t i o n s m a d e
in a d v a n c e a n d p u b l i s h e d f o r t h a t p u r p o s e . T h e greater t h e
a c c u r a c y of these data, t h e m o r e correctly can the position
of t h e ship b e d e t e r m i n e d . T h u s navigation m a d e necessary
a m o r e a c c u r a t e k n o w l e d g e of the m o t i o n of h e a v y bodies.
T h e m a t h e m a t i c s that t h e R e n a i s s a n c e inherited f r o m preceding p e r i o d s w a s i n a d e q u a t e f o r the study of motion. T h e
new m a t h e m a t i c a l tools t h a t w e r e invented f o r the p u r p o s e
of a n s w e r i n g the n e w questions raised w e r e : ( 1 ) Analytic
G e o m e t r y , i n v e n t e d b y R e n e D e s c a r t e s ( 1 6 3 7 ) 5 a n d ( 2 ) the
Infinitesimal Calculus, the c o n t r i b u t i o n of N e w t o n and Leibniz to t h e l e a r n i n g a n d technical proficiency of m a n . 6
T h e p a t h of a c a n n o n - b a l l , or, f o r that m a t t e r , the m o tion of a n y o b j e c t is m o s t readily studied by a graphical
p r e s e n t a t i o n of t h a t m o t i o n . N o w a d a y s g r a p h s are very c o m m o n . W e see t h e m even in the n e w s p a p e r s w h e n things like
the fluctuation of t h e price, say, of w h e a t is discussed. B u t
it t o o k n o t h i n g less t h a n t h e invention of Analytic G e o m e t r y
to p u t this s i m p l e a n d p o w e r f u l device at the service of
man.
If a b o d y travels a l o n g a c u r v e d p a t h , it does so u n d e r
the action of a f o r c e exerted u p o n it. If the f o r c e suddenly
stops, the m o v i n g object c o n t i n u e s nevertheless to move, not
along the c u r v e , h o w e v e r , b u t along the t a n g e n t to that curve
at the p o i n t w h e r e t h e object was w h e n the action of the
f o r c e ceased. T h u s , in the study of m o t i o n , it is i m p o r t a n t
to be able to d e t e r m i n e the t a n g e n t to t h e p a t h at any point
of that curve. T h e r e s o u r c e s that m a t h e m a t i c s h a d to offer
u p to the m i d d l e of t h e seventeenth c e n t u r y w e r e insufficient
to solve t h a t a p p a r e n t l y simple p r o b l e m . T h e differential
calculus p r o v i d e d t h e answer. 7
T h e calculus provides t h e tools necessary t o cope with
t h e questions involving the velocity of m o v i n g bodies a n d
their a c c e l e r a t i o n , o r pick-up. T h e ancients h a d only very
h a z y notions a b o u t these concepts. T h e u n a i d e d imagination
seems to find it very difficult t o h a n d l e t h e m successfully.
T h e m e t h o d s f u r n i s h e d by the calculus take all the sting
a n d all t h e bitterness out of t h e m . W h e n velocity and acceleration a r e presented t o students of m e c h a n i c s w h o d o
not h a v e the calculus at their disposal, these notions are still
SOME SOCIOLOGIC ASPECTS OF MATHEMATICS
explained in t e r m s of t h e calculus, in a r o u n d - a b o u t ,
guised f a s h i o n .
73
dis-
D • Mathematics
for the Modern
Age
O u r o w n age is
c o n f r o n t e d with technological p r o b l e m s of great difficulty.
T h e m a t h e m a t i c a l tools they call f o r w e r e not in existence at
the time of N e w t o n , t w o centuries ago. T h e a i r p l a n e alone
is sufficient to m a k e o n e t h i n k w h a t a variety of questions of
an u n p r e c e d e n t e d kind h a d to be a n s w e r e d , w h a t c o m plicated p r o b l e m s h a d to b e solved to e n a b l e t h e flier to
accomplish all the w o n d e r s of w h i c h w e a r e t h e surprised
and a d m i r i n g witnesses. T h e difficulties of c o n s t r u c t i n g the
airplane wings necessitated the c o n c e n t r a t i o n of m a t h e m a t i cal talent, a n d m a t h e m a t i c a l i n f o r m a t i o n t h a t has h a r d l y
any parallel in history.
As h a s been pointed out, s e a - f a r i n g called f o r the solution of m a n y p r o b l e m s . H o w e v e r , a ship sailing the high
seas has o n e i m p o r t a n t f e a t u r e in c o m m o n with a vehicle
traveling on l a n d : b o t h m o v e on a s u r f a c e . F r o m a geometric point of view the p r o b l e m s related to their m o t i o n a r e
two-dimensional. A n a i r p l a n e that r o a m s in the air a b o v e
is engaged in t h r e e - d i m e n s i o n a l navigation. T h e geometrical aspect of flight belongs to t h e d o m a i n of Solid G e o m e t r y ,
and the p r o b l e m s c o n n e c t e d with it a r e t h u s m u c h m o r e d i f ficult, o t h e r things being equal.
M a t h e m a t i c s plays an e n o r m o u s role in t h e field of social
problems, t h r o u g h the use of statistics. I h a v e a l r e a d y p o i n t e d
out the value of m a t h e m a t i c s in c o n n e c t i o n with the collecting of taxes, at earlier stages of civilization. T h e f u n c tions of a m o d e r n g o v e r n m e n t are vastly m o r e complex,
m o r e varied, a n d applied on a n e n o r m o u s scale. T h e variety
and scope of p r o b l e m s m o d e r n g o v e r n m e n t is interested in
can be gleaned f r o m the questions t h e citizen is asked w h e n
he receives the census b l a n k , every ten years. T o s t u d y t h e
wealth of i n f o r m a t i o n t h a t is t h u s g a t h e r e d o n millions of
blanks is the f u n c t i o n a n d the task of the census b u r e a u . T h e
inferences that c a n be d r a w n f r o m these d a t a are as involved as they are f a r - r e a c h i n g in their applications. Such a
statistical study requires a wide r a n g e of m a t h e m a t i c a l e q u i p m e n t , f r o m the most e l e m e n t a r y a r i t h m e t i c to t h e most abstruse b r a n c h e s of m a t h e m a t i c a l analysis. If one thinks of
the new f u n c t i o n s of social w e l f a r e that the g o v e r n m e n t has
taken on, like social security or old age pensions, as well as
of those that are in the offing, like health insurance, and
the millions of individuals that these services cover, o n e is
readily led to the realization t h a t t h e intelligent dealing with
74
MATHEMATICS IN FUN AND IN EARNEST
these services sets b e f o r e t h e g o v e r n m e n t n e w statistical
p r o b l e m s of vast m a g n i t u d e . 8
T h e g o v e r n m e n t is n o t t h e o n l y social agency t o use statistics. F a r f r o m it. I n s u r a n c e c o m p a n i e s h a v e been using
statistics f o r a long time. B a n k s a n d o t h e r organizations
w h i c h s t u d y t h e t r e n d s of business arrive at their predictions
by statistical analysis. T h e study of t h e w e a t h e r raises m a n y
very difficult statistical p r o b l e m s . Statistics are used t o determ i n e t h e efficiency of t h e m e t h o d s of instruction in o u r
public schools. T h i s list c o u l d be m a d e m u c h longer and bec o m e b o r i n g b y its m o n o t o n y . A s it is it will suffice to convey
t h e i d e a of t h e all-pervading role this b r a n c h of applied
m a t h e m a t i c s plays in o u r m o d e r n life.
E • Conclusion
W e h a v e alluded several times t o t h e f a c t
that d u r i n g t h e c o u r s e of the centuries m a t h e m a t i c s was
called u p o n to p r o v i d e solutions f o r p r o b l e m s that have
arisen in v a r i o u s h u m a n pursuits, f o r w h i c h n o solution was
k n o w n at the time. This, h o w e v e r , is not always the w a y
things o c c u r . I n m a n y cases the reverse is true. W h e n the
need arises a n d the question is asked, m a t h e m a t i c s reaches
out into its vast store of k n o w l e d g e a c c u m u l a t e d t h r o u g h
t h e centuries a n d p r o d u c e s the answer. T h e a s t r o n o m e r
K e p l e r h a d b e f o r e h i m a vast n u m b e r of observations conc e r n i n g the m o t i o n of the planets. T h e s e figures were m e a n ingless until he noticed t h a t they w o u l d h a n g nicely together
if t h e planets f o l l o w e d a p a t h of the f o r m which the Alexa n d r i a n G r e e k A p o l l o n i u s called a n ellipse. A n o t h e r playt h i n g of the s a m e Apollonius, t h e h y p e r b o l a , c a m e in very
h a n d y to locate e n e m y guns d u r i n g W o r l d W a r I, w h e n the
flash of t h e g u n c o u l d be observed twice.
T h i s readiness of m a t h e m a t i c s goes m u c h f u r t h e r . V a r i o u s
b r a n c h e s of science, w h e n they pass and o u t g r o w t h e purely
descriptive stage a n d are r e a d y t o e n t e r the following, the
q u a n t i t a t i v e stage, discover that the m a t h e m a t i c a l problems
w h i c h these n e w studies p r e s e n t h a v e already b e e n solved
a n d a r e r e a d y f o r use. T h u s Biology has in the last decades
raised m a n y questions, answers f o r w h i c h were available in
the s t o r e r o o m of m a t h e m a t i c s . A t present the scope of m a t h e m a t i c s used in " M a t h e m a t i c a l Biology" exceeds by f a r the
m a t h e m a t i c a l e d u c a t i o n w h i c h o u r engineering schools equip
their g r a d u a t e s with. A similar tale c a n be told of psychology,
e c o n o m i c s , a n d o t h e r sciences.
W e h a v e tried to p o i n t out t h e close relation of the m a t h e m a t i c s of a n y p e r i o d of civilization to t h e social and eco-
SOME SOCIOLOGIC ASPECTS OF MATHEMATICS
75
n o m i c needs of t h a t period. M a t h e m a t i c s is a tool in t h e
w o r k - a - d a y life of m a n k i n d . It is closely c o n n e c t e d with
the well-being of the race a n d h a s played a n i m p o r t a n t role
in the slow and p a i n f u l m a r c h of m a n k i n d f r o m savagery t o
civilization. M a t h e m a t i c s is p r o u d of t h e m a t e r i a l h e l p it h a s
rendered the h u m a n race, f o r the satisfaction of these needs
is the first a n d indispensable step that m u s t be t a k e n b e f o r e
higher a n d nobler pursuits c a n be cultivated.
T h e Russian f a b l e writer Ivan K r y l o v o b s e r v e d : " W h o
cares to sing on a h u n g r y s t o m a c h ? " T h e H e b r e w sages of
yore put it m o r e c o n c r e t e l y : " W i t h o u t b r e a d t h e r e is n o
learning," a n d t h e y are not slow to f o l l o w it u p with the
converse p r o p o s i t i o n : " W i t h o u t learning t h e r e is n o b r e a d . "
T h r e e millennia or so later b o t h p r o p o s i t i o n s f o u n d an eloquent d e f e n d e r in o n e of the greatest m i n d s of all time.
H e n r i P o i n c a r e in his philosophical writings r e f e r s to a disp u t e between those of his c o n t e m p o r a r i e s w h o t h o u g h t that
w e should study m a t h e m a t i c s in o r d e r to build m a c h i n e s ,
and their o p p o n e n t s w h o t h o u g h t that w e should build m a chines in o r d e r to h a v e leisure to study m a t h e m a t i c s . P o i n care opines that he is in c o m p l e t e and p e r f e c t a g r e e m e n t with
both c a m p s and endorses b o t h propositions.
2
•
Mathematics and Genius
A ' The "Heroic"
and the "Objective"
Interpretations
of
History
O n c e u p o n a time, m a n y , m a n y years ago, so the
story goes, a b e a u t i f u l stallion was b r o u g h t to the royal
court and presented to the king. T h e stallion was very wild.
T h e king was w a r n e d that n o m a n h a d ever m a n a g e d to
m o u n t the fiery beast. T h e heir a p p a r e n t w h o h a p p e n e d to
witness the presentation c e r e m o n y of this u n u s u a l gift,
j u m p e d u p o n the back of the spirited horse, a n d b e f o r e
anybody h a d time t o realize w h a t was h a p p e n i n g , t h e y o u n g
prince was a l r e a d y way out of sight. T h e king's anxiety f o r
the safety of his beloved son was very great. A f t e r a certain
lapse of time, the y o u n g m a n r e a p p e a r e d , s a f e a n d sane, o n
the back of the s u b d u e d , t a m e animal. T h e p r o u d a n d loving f a t h e r was so elated that he exclaimed in e x a l t a t i o n : " M y
son, find f o r yourself a n o t h e r k i n g d o m . M i n e is t o o small
f o r y o u . " T h e s e accidental w o r d s of the king took deep root
in the sensitive soul of the y o u n g prince. H i s t o r y k n o w s this
young m a n u n d e r the n a m e of A l e x a n d e r the G r e a t ( 3 5 6 323 B. c . ) , the f a m o u s c o n q u e r o r of the ancient world.
I read this story in m y school-text o n ancient history, a
76
MATHEMATICS IN FUN AND IN EARNEST
fine b o o k , f u l l of n a m e s a n d dates. E v e r y historical event
h a d its precise m o m e n t of o c c u r r e n c e r e c o r d e d . Y o u were
told exactly b y w h a t king, o r general, o r by w h a t great
l e a d e r a n y given event w a s b r o u g h t a b o u t . F o r the sake of
brevity let us r e f e r t o this w a y of conceiving historical events
as t h e " h e r o i c v i e w " of history.
T h i s h e r o i c i n t e r p r e t a t i o n of history is very attractive, because of its simplicity a n d its definiteness. All the whys a n d
w h e r e f o r e s are readily a n s w e r e d b y t h e n a m e s of the great
m e n w h o m a d e t h e history of t h e nation, o r of the race.
H o w e v e r , this h e r o i c view h a s a n obvious w e a k n e s s : it m a k e s
h i s t o r y whimsical, capricious, a n d accidental, to the point of
triviality. S u p p o s e that o u r stallion of a m o m e n t ago, in its
f r a n t i c effort to rid itself of its u n s u c c e s s f u l a n d u n l u c k y
t a m e r s , h a d b r o k e n a leg, o r two. K i n g Philip would h a v e
been d e p r i v e d of t h e occasion to utter those f a t e f u l w o r d s
of his, a n d his son A l e x a n d e r w o u l d h a v e lived out his
life as a n o b s c u r e a n d i n c o n s e q u e n t i a l ruler of the little
k i n g d o m of M a c e d o n i a .
A c c o r d i n g t o a m u c h r e p e a t e d saying, of u n d e t e r m i n e d
origin, " G o d m a d e G e o r g e W a s h i n g t o n childless, so he could
b e c o m e t h e f a t h e r of his c o u n t r y . " T h u s , if it were not f o r
s o m e physiological peculiarity o r deficiency of M a r t h a W a s h i n g t o n ( o r w a s it of G e o r g e h i m s e l f ? ) this c o u n t r y w o u l d
h a v e r e m a i n e d a British colony, even u n t o this very day
and generation.
D u r i n g the n i n e t e e n t h c e n t u r y v a r i o u s writers, like t h e
E n g l i s h m a n H e n r y T h o m a s Buckle ( 1 8 2 1 - 1 8 6 2 ) , the F r e n c h m a n H i p p o l y t e T a i n e ( 1 8 2 8 - 1 8 9 3 ) , best k n o w n in t h e
English s p e a k i n g w o r l d f o r his history of English literature,
the G e r m a n K a r l M a r x ( 1 8 1 8 - 1 8 8 3 ) h a v e a d v a n c e d the view
that h u m a n history is not m a d e b y individuals, but is domin a t e d by objective f a c t o r s , like climate, geographic environm e n t , n a t u r a l resources, e c o n o m i c a n d social conditions, etc.
T h i s objective i n t e r p r e t a t i o n of history h a s since gained a
great deal of g r o u n d . A f o r c e f u l presentation of this conc e p t i o n m a y be f o u n d in the presidential address delivered
b e f o r e the A m e r i c a n Historical Association by E d w a r d P.
C h e y n e y ( 1 8 6 1 - 1 9 4 7 ) , u n d e r the title " L a w in H i s t o r y " in
w h i c h the f o l l o w i n g t w o passages o c c u r : "History, the great
c o u r s e of h u m a n affairs, has not been the result of v o l u n t a r y
action o n the p a r t of individuals o r groups of individuals,
m u c h less of c h a n c e , b u t has been subject to L a w . " " M e n
h a v e o n the w h o l e played the p a r t assigned to t h e m : they h a v e
not written the play. P o w e r f u l rulers and gifted leaders
h a v e s e e m e d to c h o o s e their policies and carry t h e m out,
SOME SOCIOLOGIC ASPECTS OF MATHEMATICS
77
but their choice a n d success with w h i c h t h e y h a v e b e e n able
t o impose their will u p o n their t i m e h a v e alike d e p e n d e d o n
condition over w h i c h t h e y h a v e h a d n o c o n t r o l " .
T h e heroic a n d t h e " o b j e c t i v e " i n t e r p r e t a t i o n of history
are obviously poles a p a r t . W h i c h of t h e m is right? G e n e r a l
h u m a n history is so m a n y - s i d e d , so c o m p l e x , t h a t it is easy
e n o u g h to e m p h a s i z e o n e e l e m e n t o r a n o t h e r of its vast c o n tents and arrive at conclusions w h i c h are c o n t r a d i c t o r y , a n d
still have each a good deal of t r u t h in t h e m . W e m a y try to
simplify the p r o b l e m , as w e o f t e n d o in m a t h e m a t i c s , r e d u c e
the n u m b e r of variables, a n d e x a m i n e a f e w of t h e m at a
time.
O u r objective m a y p e r h a p s be achieved m o r e readily if
we e x a m i n e the history of a restricted, p a r t i c u l a r d o m a i n ,
say, that of m a t h e m a t i c s .
B ' Are Inventions
Inevitable?
W e a r e a c c u s t o m e d to p r o n o u n c e with respect a n d a d m i r a t i o n , not t o say with reverence and awe, n a m e s like Euclid, A r c h i m e d e s , Descartes,
N e w t o n , Leibniz, L a g r a n g e , G a u s s , P o n c e l e t , Klein, P o i n c a r e ,
a n d m a n y others. W e k n o w the b o o k s those m e n h a v e
written, the t h e o r e m s w h i c h b e a r their n a m e s . I n o u r o w n
time we k n o w b y n a m e m e n w h o live in o u r m i d s t a n d
s o m e of w h o m w e k n o w personally, m e n w h o lend luster
a n d glory to o u r g e n e r a t i o n , m e n w h o give us c o u r a g e a n d inspiration. T h r o u g h the study, direct a n d indirect, of the
w o r k s of these e m i n e n t scholars w e k n o w w h a t t h e y h a v e
contributed to the g r o w t h and a d v a n c e m e n t of m a t h e m a t i c a l
science. T h e r e h a r d l y c a n be a m o r e f o r c e f u l c o n f i r m a t i o n
of the i m p o r t a n c e of t h e individual in history, of t h e h e r o i c
interpretation of history, if y o u will. Nevertheless, t h e r e is
a n o t h e r side to this m e d a l .
O n D e c e m b e r 21, 1797, in Paris, t h e great m a t h e m a t i cians L a p l a c e a n d L a g r a n g e w e r e b o t h p r e s e n t at a brilliant
social gathering w h i c h i n c l u d e d a great m a n y celebrities.
A m o n g the guests w a s also a victorious y o u n g general w h o s e
star was ascending rapidly, a n d w h o h a p p e n e d t o b e a f o r m e r student of L a p l a c e . I n the c o u r s e of the evening t h e
general, while talking t o t h e t w o w o r l d f a m e d scholars,
entertained t h e m with s o m e u n u s u a l a n d c u r i o u s solutions
of well k n o w n p r o b l e m s of e l e m e n t a r y g e o m e t r y , b u t solutions with w h i c h n e i t h e r of his t w o e m i n e n t listeners w e r e
familiar. Laplace, a bit peeved, finally said to his erstwhile
pupil, " G e n e r a l , w e e x p e c t everything of y o u , except lessons
in g e o m e t r y . " T h e n a m e of the y o u n g general was N a p o l e o n
Bonaparte. Napoleon h a d learned about those strange con-
78
MATHEMATICS IN FUN AND IN EARNEST
structions d u r i n g his f a m o u s c a m p a i g n s in Italy, w h e n c e h e
h a d just r e t u r n e d . W h i l e t h e r e , h e m e t L o r e n z o M a s c h e r o n i ,
a p r o f e s s o r at the U n i v e r s i t y of Pavia, w h o that very year,
1797, p u b l i s h e d a b o o k Geometria
del Compasso
in w h i c h
the a u t h o r s h o w e d t h a t all the c o n s t r u c t i o n s that can be
carried out with r u l e r a n d c o m p a s s c a n also be carried out
with c o m p a s s alone, a very astonishing result, indeed. H a d
M a s c h e r o n i died in i n f a n c y , w o u l d science have been deprived f o r e v e r of those M a s c h e r o n i a n constructions? O n e
m a y t h i n k t h e question p r e p o s t e r o u s , f o r such a hypothetical q u e r y a d i m t s of n o a n s w e r , o n e w a y o r the other. C u r iously e n o u g h , in the p r e s e n t case the question can be
a n s w e r e d , in a very definite w a y . A c e n t u r y a n d a q u a r t e r
b e f o r e the p u b l i c a t i o n of M a s c h e r o n i ' s b o o k a D a n i s h m a t h e m a t i c i a n G e o r g M o h r published in A m s t e r d a m a b o o k in
two languages, o n e in D a n i s h a n d the o t h e r in D u t c h ,
s i m u l t a n e o u s l y , in w h i c h h e gives M a s c h e r o n i ' s m a i n result,
as well as t h e solutions of a good m a n y of the p r o b l e m s
solved later b y the Italian scholar. M o h r ' s book passed entirely u n n o t i c e d by his c o n t e m p o r a r i e s . It c a m e to light in
the p r e s e n t c e n t u r y b y accident. In the p r e f a c e to his book
M a s c h e r o n i states explicitly t h a t h e k n o w s of n o previous
w o r k along t h e s a m e lines as his b o o k , a n d t h e r e is not t h e
slightest r e a s o n to d o u b t his w o r d .
T h e story e m p h a s i z e s the f a c t that so m a n y m a t h e m a t i c a l
discoveries, great a n d small, h a v e been m a d e independently
b y m o r e t h a n o n e scholar. T h i s multiplicity of claims to the
discovery of o n e a n d the s a m e thing is p r o b a b l y the most
o u t s t a n d i n g f a c t in t h e history of m a t h e m a t i c s .
T h e dispute as to w h e t h e r N e w t o n o r Leibniz invented
t h e calculus is well k n o w n . 1 0 T h e F r e n c h claim, with a
good deal of justice, t h a t F e r m a t anticipated both of t h e m . It
is only F e r m a t ' s s t r a n g e a n d persistent aversion to the pen
t h a t deprived h i m of the credit as i n v e n t o r of that p o w e r f u l
m a t h e m a t i c a l tool.
A similar story m a y be related a b o u t the e p o c h - m a k i n g
discovery of analytic g e o m e t r y . T h e r e is as m u c h reason to
r e f e r to this discovery as " F e r m a t i a n " as there is to call it
" C a r t e s i a n . " C a r l B. B o y e r in the p r e f a c e to his History of
Analytic
Geometry11
s a y s : " H a d D e s c a r t e s not lived, m a t h e m a t i c a l history p r o b a b l y w o u l d h a v e been m u c h the same,
b y virtue of F e r m a t ' s s i m u l t a n e o u s discovery (of analytic
geometry)."
T h e g e o m e t r i c i n t e r p r e t a t i o n of c o m p l e x n u m b e r s was disc o v e r e d i n d e p e n d e n t l y a n d almost simultaneously by f o u r
different m e n , at t h e b e g i n n i n g of the n i n e t e e n t h century. A n
SOME SOCIOLOGIC ASPECTS OF MATHEMATICS
79
i n s t r u m e n t f o r d r a w i n g a straight line w i t h o u t t h e use of a
ruler, k n o w n as the "cell of P e a u c e l l i e r " ( 1 8 3 2 - 1 9 1 3 ) , was
also invented b y a y o u n g s t u d e n t L i p k i n of St. P e t e r s b u r g
(Leningrad).12
E v e n w h o l e theories h a v e g r o w n u p , t h e p a t e r n i t y of
which n o b o d y c a n claim with justice. A good a n d simple
e x a m p l e of this k i n d is offered b y t h e t h e o r y of inversion.
This theory c a m e i n t o being early in the n i n e t e e n t h c e n t u r y ,
a n d f r o m so m a n y different q u a r t e r s t h a t it is impossible to
associate a n y p a r t i c u l a r n a m e with it. T h e only t h i n g t h a t
can be said a b o u t it is that, like T o p s y , it "just g r o w e d . "
T h e multiplicity of claims to the s a m e discovery is so
c o m m o n that not only h a v e w e s t o p p e d t o b e surprised b y
it, b u t w e h a v e g r o w n a c c u s t o m e d to expect it. Better to be
able to protect the priority right of c o n t r i b u t o r s , m o s t of t h e
editors of m a t h e m a t i c a l j o u r n a l s a d d to e a c h article they
publish, the date w h e n t h a t p a p e r was received in the editorial office.
W h a t was said a b o u t m a t h e m a t i c s m a y be r e p e a t e d with
equal f o r c e a b o u t a s t r o n o m y , physics, c h e m i s t r y , m e c h a n i c s ,
in f a c t about a n y science, p u r e or applied. T w o industrious
sociologists c o m p i l e d a list of inventions, each of w h i c h h a s
m o r e than o n e c l a i m a n t to its paternity. T h e list c o n t a i n s 148
entries a n d is f a r f r o m being exhaustive. A r m e d with their
incredible, but c o r r e c t list, the t w o a u t h o r s fire, point b l a n k ,
a n a m a z i n g question at their readers, n a m e l y : " A r e inventions inevitable?" 1 3
C ' Genius and Environment
W e are p r o n e t o t h i n k that
the essence of genius is f r e e d o m . D o e s not genius invent o r
create what he will? O n closer e x a m i n a t i o n , h o w e v e r , it is
seen that this c o n c e p t i o n of genius is a n exaggeration. W h a t
a genius m a y a c c o m p l i s h d e p e n d s u p o n c i r c u m s t a n c e s w h i c h
can be controlled b y n o individual. T h e invention of the
creative individual is necessarily a n extension of t h e k n o w l edge of his time, or is s o m e t h i n g that satisfies the needs of
his c o n t e m p o r a r i e s . T h e s e characteristics h a v e to be incorporated in the invention, if the genius is t o be recognized
as such. If a self-taught scholar f r o m s o m e w h e r e in the
hinterland w o u l d send to the editor of a j o u r n a l o r to the
A c a d e m y of Science a m a n u s c r i p t w h i c h in s u b s t a n c e w o u l d
a m o u n t to the discovery, say, of n o n - E u c l i d e a n g e o m e t r y , o r
of the sextant, not m u c h fuss would be m a d e a b o u t the a u thor, even if his honesty w o u l d not be called into question.
A n d such things h a p p e n , on various levels of a c h i e v e m e n t .
A b o u t the m i d d l e of t h e n i n e t e e n t h c e n t u r y t h e A c a d e m y of
80
MATHEMATICS IN FUN AND IN EARNEST
St. P e t e r s b u r g w a s o f f e r e d b y a t e a c h e r in s o m e r u r a l elem e n t a r y s c h o o l a c r u d e exposition of t h e basic ideas of t h e
calculus.
O n t h e o t h e r h a n d , w h a t genius c a n accomplish d e p e n d s
u p o n w h a t o t h e r s h a v e d o n e b e f o r e . N e w t o n realized that
if h e h a d seen f a r t h e r t h a n others, it is because h e w a s
" s t a n d i n g o n t h e s h o u l d e r s of giants." " P e r h a p s n o w h e r e
does o n e find a b e t t e r e x a m p l e of the value of historical
k n o w l e d g e f o r m a t h e m a t i c i a n s t h a n in the case of F e r m a t ,
f o r it is s a f e t o say t h a t , h a d h e not been intimately acq u a i n t e d with the g e o m e t r y of A p o l l o n i u s and Viete, h e
w o u l d n o t h a v e i n v e n t e d analytic geometry." 1 4 O n the other
h a n d , as g r e a t a genius as A r c h i m e d e s could not invent analytic g e o m e t r y , f o r the algebraic k n o w l e d g e necessary f o r
such a n a c h i e v e m e n t was n o t available in his time.
T h e relation b e t w e e n t h e genius a n d the culture he is
b o r n into is expressed b y A . L . K r o e b e r in the following
w a y : 1 5 " K n o w i n g t h e civilization of a l a n d a n d of a n age,
w e c a n t h e n substantially affirm that its distinctive discoveries, in this o r t h a t field of activity, w e r e not directly contingent u p o n t h e personality of the actual inventors that
graced that period, but would have been made without them;
a n d t h a t , conversely, h a d t h e great illuminating minds of
o t h e r centuries b e e n b o r n in t h e civilization r e f e r r e d t o instead of their o w n , its first a c h i e v e m e n t s would have fallen
to their lot. E r i c s s o n o r G a l v a n i , eight t h o u s a n d years ago,
w o u l d h a v e polished or b o r e d the first stone; a n d in turn, the
h a n d a n d m i n d w h o s e o p e r a t i o n s set in inception the neolithic age of h u m a n culture, if held in its i n f a n c y in u n c h a n g e d catalepsy f r o m t h a t t i m e until today, would n o w
be devising wireless t e l e p h o n e s a n d nitrogen extracts," or
(let us a d d ) n u c l e a r w e a p o n s a n d interstellar ships, a generration o r t w o later.
T h e d e p e n d e n c e of t h e individual, w h a t e v e r his n a t u r a l
e n d o w m e n t s , u p o n the t i m e a n d civilization he h a p p e n s to
live in, b e c o m e s quite obvious, o n c e attention is called to
this p h e n o m e n o n . W e are n o t a bit surprised to see that
t h e F r e n c h children are so very partial to the F r e n c h language, a n d t h a t the C h i n e s e children, not to be o u t d o n e by
t h e F r e n c h , speak as u n a n i m o u s l y the Chinese tongue. T h e
s a m e m a y be said, in a b r o a d e r sense, a b o u t arts and crafts,
music, or a n y o t h e r c o m p o n e n t element of culture. On a
larger scale, a n a l o g o u s r e m a r k s m a y be m a d e about those
p a r t s of c u l t u r e w h i c h h a v e b e c o m e c o m m o n to a considerable p a r t of m a n k i n d , like the sciences, a n d m a t h e m a t i c s in
particular.
SOME SOCIOLOGIC ASPECTS OF MATHEMATICS
81
T h e s e o b s e r v a t i o n s m a y help us to c o m p r e h e n d the reasons f o r the multiplicity of claims f o r the s a m e discovery.
T h e a n t h r o p o l o g i s t Leslie A . W h i t e p u t s it this w a y : 1 6 " I n t h e
b o d y of m a t h e m a t i c a l c u l t u r e t h e r e is action a n d r e a c t i o n
a m o n g t h e various elements. C o n c e p t reacts u p o n c o n c e p t :
ideas mix, f u s e , f o r m n e w syntheses. W h e n this process of
interaction a n d d e v e l o p m e n t r e a c h e s a c e r t a i n point, n e w
syntheses a r e f o r m e d of themselves. T h e s e are, to be sure,
real events a n d h a v e their location in t i m e a n d space. T h e
places are, of course, the brains of m e n . Since the cultural
process h a s been going o n r a t h e r u n i f o r m l y o v e r a wide a r e a
of p o p u l a t i o n , the new synthesis t a k e s place in a n u m b e r
of brains at o n c e . "
T o b i a s D a n t z i g ( 1 8 8 4 - 1 9 5 6 ) in his a d m i r a b l e b o o k , Number—The
Language
of Science,17
says the same thing, with
a different e m p h a s i s : " I t seems t h a t the a c c u m u l a t e d experience of the r a c e at times r e a c h e s a stage w h e n a n outlet
is imperative a n d it is m e r e l y a m a t t e r of c h a n c e w h e t h e r
it will fall t o the lot of a single m a n , t w o m e n , o r a t h r o n g
of m e n to g a t h e r t h e rich harvest.
D ' Genius and the "Instinct
of Workmanship"
Granting that objective c o n d i t i o n s d e t e r m i n e the kind of discoveries that can be m a d e at a n y given period of history o n t h e
one h a n d , a n d that on the o t h e r h a n d such inventions are
"inevitable," such f o r w a r d steps d o not t a k e place a u t o m a t i cally. E a c h p a r t i c u l a r a d v a n c e requires an effort, a n d o f t e n
a very s t r e n u o u s one, on the p a r t of the gifted individual,
the " g e n i u s " w h o brings it a b o u t . W h a t impulse does the individual r e s p o n d to, w h e n he m a k e s the requisite effort?
M a t h e m a t i c s , like a n y o t h e r science, in its early stages
developed empirically f o r practical, utilitarian purposes. T h e
d e m a n d f o r its services n e v e r cease t h r o u g h the ages, alt h o u g h the extent a n d the p r e s s u r e m a y v a r y widely f r o m
one period to a n o t h e r , a n d m a t h e m a t i c a l inventiveness m a y
vary accordingly. This is quite clear, f o r instance, in the
case of the rapid strides m a d e b y m a t h e m a t i c s d u r i n g the
brilliant seventeenth century. 1 8 T h e m a t h e m a t i c i a n , like any
other scientist, is not u n m i n d f u l of the needs of his t i m e
and is not indifferent to the acclaim that w o u l d be his if
he supplied the a n s w e r t o a pressing question of his day.
T h e r e is, h o w e v e r , a n o t h e r phase of the situation to be
considered. A f t e r a sufficient a m o u n t of
mathematical
knowledge has been a c c u m u l a t e d , the cultivation of this
d o m a i n of learning m a y b e c o m e a n interest in itself. T h o s e
versed in its secrets a n d a d e p t in m a n i p u l a t i n g t h e m m a y
82
MATHEMATICS IN FUN AND IN EARNEST
find it attractive t o strive f o r n e w results just to satisfy w h a t
T h o r s t e i n V e b l e n ( 1 8 5 7 - 1 9 2 9 ) called the "instinct of w o r k m a n s h i p . " 1 9 T h e only e x t r a n e o u s element in the case m a y be
the wish to gain the a p p r o v a l of the restricted audience of
likeminded p e o p l e o r p e r h a p s to c o n f o u n d s o m e rivals. O u t side of t h a t t h e r e w a r d that m a y a c c r u e to the m a t h e m a t i cian f o r his efforts is t o live t h r o u g h the pains of creation
a n d to experience the exhilarating joy of discovery. His is
a l a b o r of love. H e considers himself a m p l y repaid if he
feels that he a d d e d , be it ever so little, to the luster of the
brightest jewel in the intellectual c r o w n of m a n k i n d — T h e
Science of M a t h e m a t i c s .
E ' Mathematics—the
Patrimony
of the Race
O u r discussion h a s t h u s led us to ascribe less i m p o r t a n c e to the
role of the individual in the d e v e l o p m e n t of m a t h e m a t i c s
a n d to give m o r e credit f o r the creation of this magnificent
edifice to the h u m a n r a c e as a whole. T o be sure, it is alw a y s t h r o u g h t h e gifted individuals t h a t the progress takes
place. But n o individual is indespensable in this task of f u r t h e r i n g m a t h e m a t i c a l k n o w l e d g e . T h e h u m a n race p r o d u c e s
e n o u g h ability of a high degree to m a k e the progress ind e p e n d e n t of a n y individual. Albert Einstein said in a press
interview: " I n d i v i d u a l w o r s h i p , as I look at it, is always
s o m e t h i n g unjustified. T o be sure, n a t u r e does distribute
h e r gifts in rich variety a m o n g h e r children. But of those
richly gifted ones t h e r e are, t h a n k G o d , m a n y , and I a m
firmly c o n v i n c e d that m o s t of t h e m lead a quiet unobtrusive
existence."
M a t h e m a t i c s is the p a t r i m o n y of the h u m a n race. It is
the result of slow a n d patient labor of countless generations
o v e r a p e r i o d of a great m a n y centuries. V a r i o u s practical
callings h a v e c o n t r i b u t e d t o w a r d s this a c c u m u l a t i o n of m a t h ematical k n o w l e d g e and h a v e f u r t h e r e d its development in
t h e early and difficult stages. M o d e r n technology provides
such stimulation at an ever accelerating pace. T h e effort
w h i c h has been e x p e n d e d in erecting the stately and imposing s t r u c t u r e w h i c h w e call m a t h e m a t i c s is e n o r m o u s . But
m a t h e m a t i c s h a s repaid the race f o r this effort. T h e practical value of m a t h e m a t i c s c a n n o t be overemphasized. T o
those privileged to a p p r e c i a t e the intellectual greatness of
m a t h e m a t i c s , t h e c o n t e m p l a t i o n of this g r a n d e u r is an endless s o u r c e of p u r e joy. T h e esthetic appeal of mathem a t i c s has f o u n d its enthusiastic and eloquent exponents. It
w o u l d be p r o p e r to m e n t i o n here a n o t h e r phase of the
m e r i t a n d value of m a t h e m a t i c s to m a n k i n d .
SOME SOCIOLOGIC ASPECTS OF MATHEMATICS
83
T h e superiority of the h u m a n r a c e o v e r all the c r e a t u r e s
inhabiting the e a r t h , the r e a s o n t h a t m a n k i n d is the m a s t e r
of this globe is d u e p r i m a r i l y to the f a c t t h a t the e x p e r i e n c e
of each g e n e r a t i o n does n o t die with t h a t generation, but is
transmitted to the next. T h i s t r a n s m i s s i o n of a c c u m u l a t e d
experience f r o m g e n e r a t i o n to g e n e r a t i o n is the real p o w e r
of the race, its greatest asset, its m o s t p o w e r f u l w e a p o n in
the conquest of n a t u r e , its surest tool in the a c c u m u l a t i o n of
intellectual treasures. N o w h e r e is this m o r e m a n i f e s t t h a n in
m a t h e m a t i c s . T h e c u m u l a t i v e c h a r a c t e r of m a t h e m a t i c s is
really astonishing. T h e r e is little in m a t h e m a t i c s t h a t ever
becomes invalid, a n d n o t h i n g ever gets old. W e m a y h a v e all
sorts of n o n - E u c l i d e a n geometries, n o n - A r c h i m e d i a n geometries, n-dimensional geometries, b u t all this m a k e s the venerable elements of Euclid neither invalid n o r obsolete. T h e y
r e m a i n , g r a c e f u l a n d solid, a n object of studies as m u c h as
ever, all in their own right. T h i s c u m u l a t i v e process, this
constant e n l a r g e m e n t a n d perfectibility of m a t h e m a t i c s , is
the most precious of its c h a r a c t e r s , f o r it has given t o m a n kind the idea of progress, with a clearness a n d distinctness
that n o t h i n g else can equal, let alone surpass.
C. J. K e y s e r in his b o o k Humanism
and Science20 goes a
step f a r t h e r a n d points out that the idea of progress suggested by science, a n d particularly by m a t h e m a t i c s , has reflected u p o n the race itself. It has given m a n k i n d the idea
that h u m a n n a t u r e in its t u r n m a y be p e r f e c t e d , t h a t with
the growth of k n o w l e d g e a n d i m p r o v e d living conditions
the h u m a n r a c e will k e e p on rising to greater a n d greater
heights on t h e r o a d t o w a r d civilization. M a t h e m a t i c s has
given the h u m a n r a c e not o n l y the technical tools t o b e n d
n a t u r e to its uses, not only a great a n d u n e q u a l l e d storehouse of intellectual b e a u t y a n d e n j o y m e n t , but it also h a s
given m a n k i n d a f a i t h in itself a n d its destinies, h o p e a n d
courage to c a r r y o n this u n c e a s i n g struggle f o r a better,
m o r e noble, and m o r e b e a u t i f u l life.
FOOTNOTES
1
Cf. Chapter VII, Section lb.
Handbook of South American Indians, James H. Steward,
editor, Vol. 5 (Smithsonian Institution, Washington, D. C., 1929)
p. 614.
3
For instance, What Happened in History, V. Gordon Childe,
(Pelican Books, A 108, London and Baltimore).
1
Cf. Chapter I, Section lb.
2
84
MATHEMATICS IN FUN AND IN EARNEST
5
Ibid.
See Chapter III, Section 3c.
7
Cf. Chapter III, Section 3h; Chapter VI, Section 2e.
8
See Chapter V, Section 2b.
9
"Law in History", American Historical Review, Edward P.
Cheyney, Vol. 29 (1923-1924), pp. 231-248.
10
Cf. Chapter V, Section lc.
11
History of Analytic Geometry, Carl B. Boyer (Scripta Mathematica Studies, New York, 1956).
12
Outline of the History of Mathematics,
R. C. Archibald,
(Mathematical Association of America, 1949), p. 99, note 280.
13
"Are Inventions Inevitable?" William F. Ogburn and Dorothy
Thomas, Political Science Quarterly, Vol. 37 (1922), p. 83.
14
History of Analytic Geometry, op. cit.
16
"The Superorganic", A. L. Kroeber, The American
Anthropologist, Vol. 19 (1917, p. 201.
19
"The Locus of Mathematical Reality. An Authropological
Footnote", Leslie A. White, Philosophy of Science, Vol. 14, No.
4 (October, 1947), p. 298.
17
Number—The
Language of Science, Tobias Dantzig (First
edition, New York, 1930), pp. 195-196.
18
Cf. Chapter II, Section lc.
19
The Instinct of Workmanship, Thorstein Veblen (New York,
1914).
20
Humanism
and Science, C. J. Keyser (olumbia University
Press, New York, 1931).
8
Ill
1
T H E LURE OF T H E INFINITE
•
T h e Vagaries of The Infinite
A ' No Largest Number
H a v e you ever h a d t h e o p p o r tunity of w a t c h i n g a bright youngster mastering the m e c h a nism of n a m i n g n u m b e r s ? It is a w o r t h while experience,
b o t h e n t e r t a i n i n g a n d instructive. A f t e r the child h a s learned
to n a m e the n u m b e r s , say, u p to twenty, h e readily notices
t h a t c o u n t i n g b e y o n d that, n a m e l y , twenty-one, twentyt w o , . . . twenty-nine, consists in repeating the n a m e s of the
first nine n u m b e r s he k n o w s so well already, with the w o r d
twenty preceding them.
W h e n you s u p p l y h i m , at the p r o p e r m o m e n t , with the
w o r d thirty, he will c o n t i n u e the s c h e m e to forty, and so
on, until in great t r i u m p h he comes to one h u n d r e d . N o w
r e p e a t i n g all the n a m e s he k n o w s already, in t h e same
o r d e r , preceded b y the w o r d o n e h u n d r e d he arrives at the
n u m b e r two hundred, then three h u n d r e d , . . . a thousand.
THE LURE OF THE INFINITE
85
One w o u l d n a t u r a l l y s y m p a t h i z e w i t h t h e y o u n g s t e r in his
feeling of a c h i e v e m e n t .
But he is not likely t o rest o n his laurels f o r very long.
T h e child will w i t h little o r n o help p u s h f o r w a r d , c a t c h i n g
o n m o r e a n d m o r e readily to the n a t u r e of the almost automatic m e c h a n i s m of a d v a n c i n g o n t h e r o a d t o w a r d s t h e
n a m e s of larger a n d larger n u m b e r s .
Of course, this will not h a p p e n all in o n e day, o r o n e
m o n t h . It m a y take a year, o r m o r e . In the m e a n t i m e o u r
youngster m a y learn, w i t h o u t m u c h effort, the symbols w e
use to represent n u m b e r s , a n d n a m e s of n u m b e r s like millions, billions, trillions. . . . T h i s m e c h a n i c a l way of e x t e n d i n g
the range of the n a m e s a n d symbols of n u m b e r s will finally
lead a bright y o u n g s t e r to raise the inevitable q u e s t i o n :
w h e r e does it stop? w h e r e is the e n d of it? A n d at a t e n d e r
age he will thus c o m e to the realization that t h e r e is n o
largest n u m b e r , that the series of integers is endless. It h a s
a beginning, b u t n o end. T h e little fellow has n o difficulty
in u n d e r s t a n d i n g that. T h e notion r e d u c e s itself to the simple
idea that w h a t e v e r the n u m b e r , you c a n add o n e m o r e unit
and you have a larger n u m b e r . A n u m b e r is like a b u s : n o body ever d o u b t s that " t h e r e is always r o o m f o r o n e m o r e . "
B ' A Part as Big as the Whole
It is n e a r l y b e y o n d belief
that a notion t h a t seems to be within the grasp of a child
should h a v e baffled the greatest m i n d s a m o n g b o t h m a t h e maticians a n d p h i l o s o p h e r s all t h r o u g h the ages. B u t this is
literally the case. F o r this quite i n n o c e n t looking " e n d l e s s "
series of n u m b e r s conceals b e h i n d its simple a p p e a r a n c e
m a n y a joker that c a n n o t readily be disposed of. Let us
consider some of t h e m .
Suppose we write d o w n in a r o w the n a t u r a l series of
n u m b e r s , a n d directly u n d e r n e a t h each n u m b e r we p u t d o w n
its double, I m e a n the s a m e n u m b e r multiplied b y t w o , like
this:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 . . .
2 4 6 8 10 12 14 16 18 20 2 2 2 4 26 28 . . .
N o m a t t e r h o w m a n y n u m b e r s w e m a y h a v e in t h e first
row, we will h a v e just as m a n y in the second. B u t this is
absurd, f o r we k n o w perfectly well that the even n u m b e r s
also a p p e a r in the first r o w as a part of the series of n a t u r a l
n u m b e r s . T h e n w h e r e is the catch? T h e e x a m p l e given is
one of the mildest possible. W e could m a k e m a t t e r s w o r s e
by writing in the second row the n u m b e r s of the first row
multiplied by three, or by seven, or by thirty-seven, w h i c h
86
MATHEMATICS IN FUN AND IN EARNEST
w o u l d m a k e the a b s u r d i t y m o r e p r o n o u n c e d . A still m o r e
striking e x a m p l e is o b t a i n e d if we write in the second line the
s q u a r e s of the n u m b e r s of the first line, so that the second
line will consist of the n u m b e r s
1, 4, 9, 16, 25, 36, 4 9 . . .
W e w o u l d t h u s be led to the conclusion that t h e r e are just
as m a n y p e r f e c t s q u a r e s as t h e r e are n u m b e r s in the natural
series of n u m b e r s . T h e " a b s u r d i t y " of the conclusion is the
m o r e e m b a r r a s s i n g in that between two consecutive perfect
s q u a r e s n 2 a n d ( n + 1 ) 2 t h e r e are 2n n u m b e r s which are not
p e r f e c t s q u a r e s . T h u s b e t w e e n the s q u a r e of 5 0 0 , 0 0 0 a n d
the s q u a r e of 5 0 0 , 0 0 1 there are a million n u m b e r s which
a r e not p e r f e c t squares. If instead of squares we take cubes,
f o u r t h p o w e r s , a n d so on, m a t t e r s are going f r o m b a d t o
worse all the time.
Similar difficulties, w h e n the infinite is involved, are m e t
with in geometrical considerations. Let us d r a w a fairly
large s e g m e n t A B (Fig. 2 ) of a straight line a n d a n o t h e r
segment C D , c o n s i d e r a b l y shorter. N o w join the points A
a n d C, B a n d D , a n d let the lines A C , B D meet in the
point M . If we t a k e a n y point, say E , on the segment A B
a n d join it to M , the line M E will meet the segment C D in
a point, say, F . If w e reverse the o r d e r of operations and
start with a p o i n t , J, on the segment C D , the line M J will
m a r k off a point I on A B . W e can t h u s m a t c h every point
of C D with a point of A B . It is n a t u r a l to conclude f r o m
these c o n s t r u c t i o n s that there are as m a n y points on the
s e g m e n t A B as there are o n the segment C D . But this is a
puzzling, indeed an a b s u r d conclusion, for we have deliberately t a k e n the s e g m e n t A B considerably longer than C D .
H o w is it that the n u m b e r of points they contain are equal?
W e could go on piling u p such difficulties, a c c u m u l a t i n g
such e m b a r r a s s i n g conclusions. But this m a y b e c o m e somew h a t m o n o t o n o u s , a n d p e r h a p s a little u n c o m f o r t a b l e . So instead of a d d i n g n e w troubles it m a y be better to try to get
out of the troubles we are in already. Is there a way out of
the difficulties we e n c o u n t e r e d ? W e m a y p e r h a p s be able to
c o p e with the p r o b l e m by taking heed of the way w e got
into that mess. It is all d u e to the youngster w h o m we
w a t c h e d a while b a c k , w h e n he so complacently accepted
the idea that the series of n a t u r a l n u m b e r s is endless. T h a t
w a s reckless on his part, reckless indeed. F o r collections of
objects we h a v e direct experience with are all finite. W e
h a v e ten fingers o n o u r h a n d s , we have a b o u t one h u n d r e d
t h o u s a n d hairs o n o u r h e a d , w e h a v e o n e h u n d r e d and sixty-
87
THE LURE OF THE INFINITE
five million co-citizens in o u r c o u n t r y , a n d w e h a v e n e a r l y
a three h u n d r e d billion dollar n a t i o n a l d e b t largely d u e t o
military e x p e n d i t u r e s . B u t w h e t h e r s m a l l o r large, t h e s e col-
8
h
Figure 2
lections are finite.1 W h a t w e k n o w a b o u t collections of o b jects w e learned o n finite collections. All the r e a s o n i n g t o
which we are a c c u s t o m e d is applicable only to finite collections, a n d all of it goes topsy-turvy w h e n w e try to a p p l y it
to infinite collections.
W e m a y p e r h a p s be able to see h o w this h a p p e n s if w e
re-examine o n e of the examples w e considered b e f o r e . Let
us take the r o w of integers in the first line a n d the row of
t h e s a m e integers multiplied by t w o in the second line. If
we stop the first line at, say, 18, the lower r o w stops at 36.
T h e u p p e r row contains t h e first half of the lower r o w ,
but not the second half, f r o m 2 0 on. T h e s a m e will be t r u e
n o m a t t e r w h e r e we stop the first row. T h u s it is not t r u e
that the lower r o w is a p a r t of t h e u p p e r one. B u t s t o p p i n g
a n y w h e r e m a k e s o u r collection a finite one. A n d w e a r e not
supposed to stop. But if we a r e not to stop, if o u r collection
of n u m b e r s is to go o n w i t h o u t e n d , it does not m a t t e r
which n u m b e r of the lower r o w w e m a y t a k e into c o n s i d e r ation, sooner o r later it will a p p e a r also in the u p p e r row.
It does not m a t t e r in t h e least that the u p p e r row is always
b e h i n d the lower one. U l t i m a t e l y the u p p e r row will catch
u p with a n y assigned place of t h e l o w e r row. T h i s is d u e
to the f a c t t h a t the process h a s n o s t o p p i n g place, that it
h a s n o u l t i m a t e end. W e a r e t h u s led to say that the n u m ber of even integers is as large as the n u m b e r of all integers
88
MATHEMATICS IN FUN AND IN EARNEST
a n d at the s a m e t i m e m a i n t a i n t h a t the even integers are
only a p a r t of all t h e integers.
W e a g r e e d a while ago that we h a v e ten fingers on our
two h a n d s . N o b o d y in his senses will a r g u e that w e have
as m a n y fingers on o n e h a n d as w e h a v e on both hands.
T h a t is p a t e n t l y a b s u r d . B u t the collection of fingers we are
talking a b o u t is finite. T h e n u m b e r of even integers seems
to be half the total n u m b e r of integers, but they are as
n u m e r o u s as all the integers p u t together. T h i s is not absurd,
f o r a n infinite collection.
C ' Arithmetical
Operations
Performed
on the
Infinite
N o w t h a t w e are a little better a c q u a i n t e d with the n a t u r e
of infinite collections of objects and with the difference between finite collections a n d infinite collections, we m a y try
to t a k e a closer look at the b e h a v i o r of infinite collections.
T o simplify the language, let us agree to replace the phrase
"infinite collections of o b j e c t s " by the single word "infinity."
W h e n we say " i n f i n i t y " we will still m e a n "an infinite collection of o b j e c t s , " b u t w e will use the simpler, the shorter
designation.
Infinite plus infinity is obviously infinity. So is infinity
plus a finite collection. On the o t h e r h a n d the sum of two
finite collections is necessarily a finite collection. But how
a b o u t the d i f f e r e n c e b e t w e e n two infinite collections? H o w
m u c h is infinity m i n u s infinity? Y o u m a y say that it is infinity, a n d y o u m a y be right. If in the infinite collection of
integers we p r o p o s e to cross out all the even integers, this
a m o u n t s to s u b t r a c t i n g an infinite n u m b e r of m e m b e r s of
the total collection. T h e r e m a i n d e r of the collection will consist of all the o d d integers which are still infinite in n u m ber, w h i c h still f o r m an inexhaustible collection. A given
segment of a straight line is an infinite collection of points,
a n d so is a p a r t of that segment. If f r o m the whole segment
we t a k e a w a y a p a r t , w h a t r e m a i n s is in infinite collection
of points.
T h e above e x a m p l e s s h o w that if w e say that infinity min u s infinity is e q u a l to infinity, this statement m a y be true,
b u t is it always true? T h e a n s w e r to this question is: N o .
Infinity m i n u s infinity m a y be a finite n u m b e r , say, seven.
A simple e x a m p l e m a y p e r h a p s c o n v i n c e you of that. Suppose that in the n a t u r a l series of integers we propose to
strike out all the integers larger than seven. W e would thus
s u b t r a c t an infinite collection f r o m a n infinite collection.
A n d w h a t w o u l d be the difference? Seven, exactly seven.
If y o u w o u l d r a t h e r h a v e the difference equal to a n o t h e r
THE LURE OF THE INFINITE
89
n u m b e r , say 5 o r 13, y o u are at p e r f e c t liberty t o h a v e it
y o u r way.
If a s u m involves a n infinite n u m b e r of terms, each of
t h e m of finite size, is the s u m finite o r infinite? If y o u w e r e
offered the choice o n a bet, w h i c h side w o u l d y o u t a k e ?
T h e r e is a saying, a t t r i b u t e d t o N a p o l e o n , that if t w o m e n
engage in a bet, o n e of t h e m is a c r o o k , a n d the o t h e r m a n
a fool. Should o n e offer y o u a bet o n the a n s w e r to t h a t
question, he w o u l d definitely be the c r o o k in the deal, f o r h e
would win n o m a t t e r w h i c h side y o u chose.
T h a t t h e r e are cases in which the s u m of a n infinite n u m ber of finite t e r m s is infinite is clear e n o u g h . T a k e the n u m ber three a n d keep o n a d d i n g it to itself. Y o u will o b t a i n
the s u m s 3, 6, 9, 12, . . . a n d this r o w of n u m b e r s h a s n o
largest t e r m , that is to say that the s u m sought is infinite.
But there a r e cases in w h i c h the addition of an infinite n u m ber of t e r m s gives a result t h a t is finite. A little story m a y
bring out the point.
J o h n , age ten, b o u g h t a p o u n d of cherries f r o m t h e neighb o r h o o d grocer, p l a c e d himself c o m f o r t a b l y on the stairs of
the back p o r c h and h a d a feast. W h e n he was all t h r o u g h
and on the w a y to the garbage c a n w i t h the collected pits
he s u d d e n l y h a d a brilliant idea. H e r a n b a c k to the grocer
a n d told the m a n t h a t he felt cheated. H e b o u g h t cherries, but
he has n o use w h a t s o e v e r f o r pits. T h e grocer, f o r reasons
of his o w n , gave J o h n half a p o u n d of cherries in e x c h a n g e
f o r his pits. T h e boy disposed of the cherries in the s a m e
w a y as b e f o r e , a n d he c a m e b a c k to the grocer with the
pits, f o r w h i c h h e got a q u a r t e r of a p o u n d of cherries. If
J o h n keeps this g a m e u p indefinitely, h o w m a n y p o u n d s of
cherries will he get f r o m the grocer, cherries, m i n d y o u , n o t
pits? In o t h e r w o r d s , let us n o w consider w h a t is the ultim a t e value of the sum
y2 + 1 / 4 + 1 / 8 + W 6 + y 3 2 +
...
if we should keep on a d d i n g t e r m s indefinitely a c c o r d i n g to
the same rule o r scheme?
T h e a n s w e r m a y not seem quite clear, but it b e c o m e s obvious if we p u t the question in a different f o r m . Let A B be
a segment of a straight line o n e f o o t long, a n d let C be its
mid-point ( F i g . 3 ) . T o the s e g m e n t A C a d d half the segm e n t CB, thus f o r m i n g the segment A D . T o A D add half
of D B to f o r m the segment A E , a n d so o n . This geometric
p r o c e d u r e p e r f o r m s the addition of J o h n ' s c h e r r y p r o b l e m .
W e m a y leave open the question w h e t h e r we will reach the
point B by this m e t h o d . B u t w h a t is q u i t e clear is that o n
90
MATHEMATICS IN FUN AND IN EARNEST
t h e o n e h a n d w e get as close to B as y o u wish, if we keep
t h e p r o c e s s u p l o n g e n o u g h , a n d o n the o t h e r h a n d we will
n e v e r get b e y o n d B. P o o r J o h n , his c h e r r y racket will never
yield h i m m u c h m o r e t h a n o n e m i s e r a b l e p o u n d of cherry
m e a t , if t h a t m u c h .
c
o
e
Jigure S
O n e m a y b e inclined t o a r g u e t h a t these p r o b l e m s are
p u r e l y artificial, t h a t m a t h e m a t i c i a n s just invent t h e m in ord e r to h a v e s o m e t h i n g to w o r r y a b o u t , or s o m e t h i n g to talk
a b o u t to e a c h o t h e r , o r p e r h a p s to write b o o k s about. But
actually it is f a i r e r to say t h a t p r o b l e m s of this sort are
t h r u s t u p o n t h e m a n d the p o o r m a t h e m a t i c i a n s d o the best
they c a n with a t o u g h assignment. Surely, there is nothing
e x t r a o r d i n a r y a b o u t a f r a c t i o n like Vs, a n d it is nothing out
of the w a y to a t t e m p t to c o n v e r t it into a decimal fraction.
T h e result is 0 . 3 3 3 . . . w h e r e the n u m b e r of decimals continues indefinitely. N o w this d e c i m a l f r a c t i o n m a y be writt e n in t h e f o r m
3/10+3/100+3/1000+3/10000+
. . . .
a n d w e are t h u s led to consider a sum with an endless n u m b e r of terms. Obviously the ultimate value, or the limit of
this s u m , as m a t h e m a t i c i a n s like to call it a m o n g themselves,
h a s t o b e o n e t h i r d , the i n n o c e n t little f r a c t i o n we started
with.
D • No escape from the Infinite
Such are s o m e of the
vagaries of the infinite. Y o u m a y p e r h a p s feel s o m e w h a t dist u r b e d , o r just a bit puzzled. But you need not let that
w o r r y y o u : y o u are in very good c o m p a n y . Since ancient times
the m o s t p r o f o u n d t h i n k e r s h a v e struggled with those questions. In a n c i e n t G r e e c e t h e p r o b l e m s raised by the consideration of t h e infinite w e r e f o c u s e d in t h e f a m o u s a r g u m e n t s
of Z e n o . L i g h t - o f - f o o t Achilles, Z e n o argues, c a n never
catch u p with t h e proverbially slow m o v i n g turtle; or an
a r r o w , t h e fastest m o v i n g t h i n g k n o w n to antiquity, c a n n o t
m o v e at all a n d m u s t always r e m a i n in the same spot. 2
A simple w a y of getting out of the t r o u b l e is to avoid the
c o n s i d e r a t i o n of the infinite. Just give it u p as a b a d job.
THE LURE OF THE INFINITE
91
But this is m u c h m o r e readily said t h a n d o n e . W e h a v e
already considered the p r o b l e m of r e p r e s e n t i n g V3 as a decim a l f r a c t i o n as an e x a m p l e of the w a y the infinite h a s of
imposing itself. All of the calculus, that p o w e r f u l tool of the
m a t h e m a t i c i a n , the physicist, the engineer, is squarely based
on considerations of the infinite. T h e F r e n c h m a t h e m a t i c i a n
H e n r i P o i n c a r e ( 1 8 5 4 - 1 9 1 2 ) , o n e of the greatest m i n d s of all
ages, said explicitly: " T h e r e c a n be n o science b u t of t h e
infinite."
2
•
T h e Infinite in Geometry
A ' Parallelism
in Euclid's
Elements
Parallel lines a n d
parallel planes are all a r o u n d us practically all the time.
T h e opposite walls of the r o o m s w h e r e w e spend such a
large part of o u r days and nights are parallel, a n d the f o u r
corner lines of those r o o m s are parallel to each other. T w o opposite edges of the tops of o u r tables are usually parallel
lines, and so are the legs w h i c h s u p p o r t those tops. M a n y
of the streets we walk o n in o u r t o w n s are parallel, a n d
the two c u r b i n g s on the t w o opposite sides of a street between which we drive o u r cars are parallel lines. W e p l a n t
m a n y of o u r c r o p s in parallel rows, a n d so on a n d on.
T h e r e is h a r d l y a notion m o r e f a m i l i a r to us t h a n parallelism. Nevertheless, this seemingly innocent and h a r m l e s s
thing has been f o r the professional m a t h e m a t i c i a n the "enfant terrible"
of g e o m e t r y , since the t i m e of Euclid, a n d
most likely even b e f o r e that.
Euclid based his t h e o r y of parallelism on a definition, the
last of the thirty five w h i c h he lists at the o p e n i n g of his
Elements,
and o n a t h e o r e m ( p r o p . 29, Book I ) , o n e of the
clumsiest in the b o o k . T h o s e two propositions were the first
to which objections were raised. T h e attack o n t h e m c a m e
very early. It c o n t i n u e d t h r o u g h the ages in v a r i o u s f o r m s
until it was discovered that this p a r t i c u l a r t h e o r e m of E u clid's was dissimulating b e h i n d its u n a t t r a c t i v e exterior n o t h ing less than n o n - E u c l i d e a n G e o m e t r y , and that the definition
referred to was trying to d o d g e the question of the infinite in
geometry.
H e r e is Euclid's definition: "Parallel straight lines are such
a are in the s a m e plane, a n d w h i c h , being p r o d u c e d ever
so f a r both ways, d o not m e e t . " T h i s is quite simple, to be
sure, and plausible e n o u g h . But the statement seems to invite a very pertinent, if obvious, q u e s t i o n : H o w f a r is "ever
so f a r ? " ten feet? a h u n d r e d yards? a t h o u s a n d miles? a
million light-years? Euclid himself is c a r e f u l not to raise
92
MATHEMATICS IN FUN AND IN EARNEST
this question. P e r h a p s b e c a u s e he w a s a w a r e of the skeleton
in his c l o s e t — t h e infinite. B u t a t i m e c a m e w h e n the d o o r
of t h a t closet w a s t h r o w n wide o p e n . T h a t valiant deed was
a c c o m p l i s h e d by P r o j e c t i v e G e o m e t r y , d u r i n g the first q u a r ter of the 19th c e n t u r y , t o say n o t h i n g a b o u t its precursors.
B
' The Difference Between Metric and Projective Geom-
etry
P r o j e c t i v e G e o m e t r y 3 starts out with the same basic
m a t e r i a l s as does E u c l i d e a n g e o m e t r y , n a m e l y , points, lines,
planes, triangles, etc. But the t w o geometries emphasize two
different kinds of p r o p e r t i e s of the figures considered. F o r
instance, if Euclid c o m e s across a c o u p l e of triangles, he inquires w h e t h e r a side of o n e of t h e m h a p p e n s to be as long
as o n e of the sides of the o t h e r triangle, a n d if so, whether
the angles a d j a c e n t to those t w o equal sides in the two triangles are respectively e q u a l . If this, too, h a p p e n s to be the
case, Euclid d r a w s the c o n c l u s i o n that the r e m a i n i n g sides
of the t w o triangles a r e respectively equal.
P r o j e c t i v e G e o m e t r y also takes a n interest in the t w o triangles, b u t in a different way. In Projective G e o m e t r y we
w o u l d join a vertex, say, A (Fig. 4 ) of the first triangle A B C
to a vertex, say, A ' of the second triangle A ' B ' C ' , a second
vertex B of the first to a second vertex, say, B' of the second,
a n d finally d r a w the line C C . N o w if it should h a p p e n
that the t h r e e lines A A ' , BB', C C ' m e e t in the s a m e point,
h
say S, this f a c t w o u l d justify the following conclusion: if X
d e n o t e s t h e p o i n t of intersection of the two sides BC, B ' C '
THE LURE OF THE INFINITE
93
of the t w o triangles opposite t h e vertices A , A ' , a n d if Y , Z
are similarly the points of intersection of t h e pairs of sides
C A , C ' A ' a n d A B , A ' B ' , respectively, t h e t h r e e points X ,
Y, Z are alined, t h a t is, they lie on the s a m e straight line.
T h e point S a n d the line X Y Z are said to be t h e c e n t e r
a n d the axis of perspectivity of the t w o triangles A B C ,
A'B'C'.
It is clear f r o m this e x a m p l e t h a t E u c l i d is p r i m a r i l y interested in the size of the e l e m e n t s of his figure. P r o j e c t i v e
G e o m e t r y , o n the o t h e r h a n d , ignores the metrical aspect of
the figure a n d c o n c e n t r a t e s its attention o n the relative position of the e l e m e n t s of the g e o m e t r i c a l figure.
T h e difference of a p p r o a c h a n d interest of the t w o g e o m e tries a c c o u n t s f o r t h e f a c t t h a t Euclid got along w i t h o u t
considering the infinite, while Projective G e o m e t r y h a d t o
f a c e the m u s i c squarely. M o r e t h a n t h a t , as w e shall see
presently, Euclid could n o t b r i n g in the infinite w i t h o u t
h a r m i n g t h e c o h e r e n c e of his m o n u m e n t a l w o r k . It m a y
also be p o i n t e d out that the n a t u r e of his b e a s t — t h e infinite
Figure 5
— i s such that it is difficult to t a m e it. A f t e r h a v i n g been in
t h e harness of Projective G e o m e t r y f o r a c e n t u r y and a
half, the infinite still leaves s o m e r o o m f o r discussion a n d
clarification, even a m o n g authorities in this b r a n c h of m a t h ematics.
C • The Point at Infinity of a Line
Projective G e o m e t r y
in the p l a n e considers t w o f u n d a m e n t a l f o r m s : the range of
points ( A , B, C , . . . ) situated o n a straight line m , a n d the
pencil of r a y s a, b, c , . . . passing t h r o u g h the s a m e point S
(Fig. 5 ) . G i v e n the r a n g e m , we o b t a i n a pencil S, if w e
join the points of m t o a p o i n t S ( n o t o n m ) . T h u s t h r o u g h
every point of m t h e r e will pass a ray of S; conversely,
every ray of S passes t h r o u g h a p o i n t of m , every r a y , t h a t
94
MATHEMATICS IN FUN AND IN EARNEST
is, but o n e , n a m e l y the r a y t passing t h r o u g h S a n d parallel
t o m . T h i s is a t r o u b l e s o m e exception, w o r t h y of f u r t h e r
scrutiny.
T h e t w o lines m a n d t d o n o t h a v e a point in c o m m o n .
D o e s t h a t m e a n t h a t they h a v e n o t h i n g in c o m m o n ? A line,
in addition t o the m a n y points that it has, possesses also a n
additional quality or p r o p e r t y w h i c h we call " d i r e c t i o n . " T h e
t w o lines m a n d t h a v e this quality in c o m m o n : they h a v e
the s a m e direction. W e c o u l d t h e r e f o r e m a k e the statement
t h a t a r a y of the pencil S h a s either a p o i n t or the direction
in c o m m o n with the line m .
T h e f a m o u s postulate of Euclid c o n c e r n i n g parallel lines
m a y be stated as f o l l o w s : T h r o u g h a given point o n e a n d
only o n e line can be d r a w n h a v i n g a given direction. H e n c e
t h e usual s t a t e m e n t t h a t " a line is d e t e r m i n e d b y t w o of its
p o i n t s " m a y be s u p p l e m e n t e d to r e a d : " o r by one point and
t h e direction of the line." T h u s in the d e t e r m i n a t i o n of a line
t h e direction of t h e line plays the role of a point.
T h e s e r e m a r k s m a k e it clear that the difficulty w e enc o u n t e r e d in c o n n e c t i o n with the r a n g e of points a n d the
pencil of lines c a n readily be r e m o v e d by identifying "direct i o n " with a point. W e c a n eliminate f r o m o u r geometrical
l a n g u a g e the w o r d " d i r e c t i o n " and e n d o w the line, in addit i o n to all the " o r d i n a r y " points that it has, with a n e w
" e x t r a o r d i n a r y " point. W e will thus be able to m a k e t h e
s t a t e m e n t t h a t a line t h r o u g h S meets m in a point. In certain cases w e m a y h a v e to i n q u i r e w h e t h e r the c o m m o n
p o i n t is an o r d i n a r y o r a n " e x t r a o r d i n a r y " point, i.e., w h e t h e r
w e are dealing with a case of intersecting lines or of parallel
lines. B u t in general, w e will p a y n o attention to this distinction, not a n y m o r e t h a n w e pay, in algebra, to the
question as to w h e t h e r a is greater t h a n b w h e n w e write
a—b. O u r " e x t r a o r d i n a r y " p o i n t is usually called the " p o i n t
at infinity" of t h e line. T h i s n a m e is justified on the g r o u n d
that the p o i n t of intersection of a line t h r o u g h S with the
line m k e e p s on r e c e d i n g indefinitely f r o m any fixed point
o n m ( s a y the f o o t of the p e r p e n d i c u l a r f r o m S to m ) as
the line t h r o u g h S a p p r o a c h e s the limiting position of
parallelism with the line m . S o m e a u t h o r s r e f e r to this
p o i n t as the " i m p r o p e r " point of the line, while others
go t o t h e opposite e x t r e m e a n d call it the " i d e a l " point of
the line.
D * The Line at Infinity of a Plane and the Plane at Infinity
of Space
In space, given a plane /u, ( m u ) and a point S
( F i g . 6 ) , a n y line v of yx a n d the point S d e t e r m i n e a plane;
THE LURE OF THE INFINITE
95
conversely, every p l a n e passing t h r o u g h S cuts t h e p l a n e
(i along a line, every plane, that is, except o n e , n a m e l y t h e
plane A. ( l a m b d a ) t h r o u g h S w h i c h is parallel to jju. H e r e
again it is not correct to say t h a t since t h e planes fj, a n d \
have n o line in c o m m o n , they have n o t h i n g in c o m m o n . T h e
two planes h a v e the s a m e " d i r e c t i o n , " o r let us b e t t e r say
the s a m e " o r i e n t a t i o n , " to avoid o v e r w o r k i n g the s a m e
t e r m a n d to t a k e a d v a n t a g e of t h e a b u n d a n c e of w o r d s in
t h e English language.
Figure 6
1
T h r o u g h a p o i n t one a n d only one p l a n e c a n be d r a w n
parallel to a given p l a n e . T h i s p r o p o s i t i o n m a y be restated
by saying: " A plane is d e t e r m i n e d by a p o i n t and the orientation of the p l a n e . " O n the o t h e r h a n d , a plane is determ i n e d by a point a n d a line. H e n c e in the d e t e r m i n a t i o n of
a plane, the orientation of t h e p l a n e plays t h e s a m e role as
a line. W e can t h u s eliminate the e x c e p t i o n n o t e d , if w e
d r o p f r o m our geometric v o c a b u l a r y the w o r d " o r i e n t a t i o n "
a n d in its place e n d o w t h e plane with an " e x t r a o r d i n a r y "
line which we m a y call the "line at infinity," or t h e " i m p r o p e r " line, o r the " i d e a l " line of the plane. T h i s c o n v e n t i o n
enables us to say that a p l a n e t h r o u g h S always cuts the
plane /jl along a straight line. Occasionally we m a y again
have to inquire as to w h e t h e r this line is a line in the o r d i n a r y sense, o r the fictitious line, i.e., w h e t h e r w e are c o n sidering intersecting planes o r parallel planes. But in general
we have n o c o n c e r n a b o u t this distinction. If it w e r e o t h e r wise, the whole s c h e m e w o u l d serve n o useful p u r p o s e .
T o r e t u r n to p l a n e g e o m e t r y , the introduction of the
point at infinity f r e e d us f r o m a certain inconvenience. But
this new point raises t r o u b l e s o m e questions of its o w n . D o
96
MATHEMATICS IN FUN AND IN EARNEST
m7
Figure 7
t h e points at infinity f o r m a locus, t h a t is, s o m e k i n d of a
figure, a n d if so, w h a t is t h a t figure? T h e difficulty, h o w ever, is m o r e a p p a r e n t t h a n real. Since every line in t h e
p l a n e h a s o n e a n d only o n e p o i n t at infinity, the locus of
t h e s e points, if t h e r e be s u c h , m u s t be m e t by every line in
t h e p l a n e in o n e a n d only o n e p o i n t ; h e n c e that locus c a n
only be a straight line, the "line at infinity" of the plane.
T h i s is a very f o r t u n a t e c i r c u m s t a n c e , since it h a p p e n s t o
agree with t h e "line at infinity" w e a t t r i b u t e d to the p l a n e
w h e n c o n s i d e r i n g the p l a n e in space. T h i s c o n c o r d a n c e is
f u r t h e r s t r e n g t h e n e d b y t h e c o n s i d e r a t i o n of a line v a n d a
plane v ( N u ) parallel to e a c h other. T h r o u g h v (Fig. 7 ) a
p l a n e cr ( s i g m a ) m a y be d r a w n parallel t o v, a n d the point of
intersection of cr a n d v lies o n t h e line c o m m o n to cr a n d y,
i.e., the line m e e t s v o n t h e l i n e at infinity of v• Indeed, t h e
p o i n t at infinity of the line belongs to the line at infinity
of the p l a n e v, a n d the latter line is c o m m o n to the t w o
planes cr a n d v.
W e m a y ask, b y a n a l o g y with t h e case of the plane, w h a t
is t h e locus in space c o n t a i n i n g all the lines at infinity of all
t h e planes is space? T h e a n s w e r is based o n the consideration t h a t the locus m u s t b e a g e o m e t r i c entity with w h i c h
every plane in space h a s a line in c o m m o n a n d only one;
h e n c e that entity m u s t itself be a plane, the " p l a n e at infinity" of space.
E ' Advantages
and Limitations of the Elements at Infinity
P r o j e c t i v e G e o m e t r y e n j o y s a considerable a d v a n t a g e f r o m
the artifice w h i c h identifies the direction of a line with a
point a n d t h e o r i e n t a t i o n of a plane with a line. T h e propositions of p r o j e c t i v e g e o m e t r y a c q u i r e a simplicity and a gen-
THE LURE OF THE INFINITE
97
erality t h a t they c o u l d n o t o t h e r w i s e h a v e . M o r e o v e r , t h e
elements at infinity give t o p r o j e c t i v e g e o m e t r y a d e g r e e of
unification t h a t greatly facilitates t h e t h i n k i n g in this d o m a i n
a n d offers a suggestive i m a g e r y t h a t is very h e l p f u l in t h e
acquisition of results. O n t h e o t h e r h a n d , projective geo m e t r y stands r e a d y to a b a n d o n these fictions w h e n e v e r t h a t
seems desirable, a n d to express t h e c o r r e s p o n d i n g p r o p o s i tions in t e r m s of direction of a line a n d t h e o r i e n t a t i o n of a
plane, to the great benefit of the science of g e o m e t r y .
B u t the suggestive p o w e r of w o r d s is such that w e a r e
t e m p t e d t o f o r g e t the precise a n d severe limitations u n d e r
w h i c h the e l e m e n t s at infinity h a v e b e e n i n t r o d u c e d . W e a r e
p r o n e to ascribe to the e l e m e n t s at infinity o t h e r p r o p e r t i e s
of points, lines, a n d planes. T o t a k e but o n e e x a m p l e : o n e
might speculate on t h e implications of the f a c t that space
is limited by a plane. T o d e c l a r e space limitless a n d to p r o vide that limitless space with a b o u n d a r y is sheer c o n t r a d i c tion, at least in t e r m s . O n t o p of t h a t , to claim t h a t t h e
statement is justified m a t h e m a t i c a l l y is utterly u n f a i r . T h e
p l a n e at infinity of projective g e o m e t r y h a s n o p r o p e r t i e s ;
it is simply a figure of speech, a r o u n d - a b o u t w a y of saying that t h r o u g h a given point o n e a n d only o n e p l a n e
can be d r a w n parallel to a given plane. C o m p e t e n t m a t h e maticians d o not t a k e t h e " e l e m e n t s at infinity" of p r o jective g e o m e t r y f o r a n y t h i n g m o r e t h a n the convenient fiction that they are, w i t h i n t h e limits of applicability of these
elements, a n d d o not hesitate to f o r s a k e t h e m f o r s o m e t h i n g
else that m a y p r o v e t o b e m o r e c o n v e n i e n t u n d e r d i f f e r e n t
circumstances.
F ° Could Euclid Find Room in His Elements for Points at
Infinity?
A m u c h m o r e serious p r o b l e m arises w h e n it is
a t t e m p t e d to i n t r o d u c e the e l e m e n t s at infinity of p r o j e c t i v e
g e o m e t r y into the m e t r i c a l g e o m e t r y of Euclid. T h e p r o x i m ity of the t w o b r a n c h e s of g e o m e t r y acts as a p o w e r f u l
t e m p t a t i o n . If a line h a s a p o i n t at infinity in projective
geometry, w h y not in E u c l i d e a n g e o m e t r y ? I n d e e d , w h y did
not Euclid himself t h i n k of t h e trick? A little reflection will
show that the e l e m e n t s at infinity w o u l d w o r k h a v o c with
metrical g e o m e t r y . A t every p o i n t of a line a p e r p e n d i c u l a r
can be erected and only o n e . C a n a p e r p e n d i c u l a r be erected
at the point at infinity of t h e line? S u c h a p e r p e n d i c u l a r does
a o t exist, or at best is i n d e t e r m i n a t e , a n d the E u c l i d e a n
proposition considered loses its generality. T w o p e r p e n d i c ulars t o the s a m e fine a r e parallel. If t w o parallel lines h a v e
98
MATHEMATICS IN FUN AND IN EARNEST
a p o i n t at infinity in c o m m o n , this contradicts the f u n d a m e n t a l p r o p o s i t i o n t h a t f r o m a point outside a given line one
a n d only o n e p e r p e n d i c u l a r c a n be d r a w n to the line. T h e
points at infinity w o u l d ruin the entire theory of c o n g r u e n c e
of triangles. W h a t w o u l d be the distance between two points
o n the line at infinity? If the a n s w e r is to be infinity, t h e n
every p o i n t of the line w o u l d be equidistant f r o m all the
o t h e r points o n t h e line. A n d so on.
G * Do the Elements at Infinity "Enrich" Projective Geometry?
S o m e writers o n projective g e o m e t r y insist that the
straight line of projective g e o m e t r y is the Euclidean straight
line with a n e x t r a p o i n t a d d e d . T h e projective plane and p r o jective space are, in t u r n , the E u c l i d e a n plane a n d Euclidean
s p a c e e n r i c h e d , respectively, by an additional line a n d a n
additional plane, just as n o w a d a y s , let us say, bread is enriched by a d d e d vitamins. T h e s e statements, despite their
w i d e s p r e a d a c c e p t a n c e , are nevertheless misleading. T h e extra p o i n t w h i c h projective g e o m e t r y claims to add to the
E u c l i d e a n line is the w a y in w h i c h projective geometry acc o u n t s f o r the p r o p e r t y of the straight line which Euclidean
g e o m e t r y recognizes as the " d i r e c t i o n " of the line. T h e diff e r e n c e b e t w e e n the E u c l i d e a n line a n d the projective line is
p u r e l y verbal. T h e g e o m e t r i c c o n t e n t is the same.
E q u a l l y illusory is the difference b e t w e e n the Euclidean
p l a n e a n d t h e projective plane. T h e line at infinity of the
projective p l a n e is the w a y in which projective geometry inc o r p o r a t e s into its plane g e o m e t r y the parallelism of the
E u c l i d e a n p l a n e . T h e difference in verbiage m a y be striking
b u t the g e o m e t r i c s u b s t a n c e is the s a m e , a n d there is n o
justification f o r the claim of projective geometry that its
p l a n e is " r i c h e r " t h a n the E u c l i d e a n plane. T h e same considerations o b t a i n f o r the p l a n e infinity. Euclidean space has
its parallel planes. It suits the convenience of projective
g e o m e t r y t o c h a n g e t h e t e r m i n o l o g y a n d r e f e r to this parallelism of Euclid by s p e a k i n g of a plane at infinity; but such a
c h a n g e in n o m e n c l a t u r e does not constitute a n increase in
geometric content.
T h e claim of h a v i n g " e n r i c h e d " E u c l i d e a n space h a s not
led projective g e o m e t r y to m a k e any unjustifiable use of its
truly m a r v e l o u s e l e m e n t s at infinity. It is nevertheless desirable that w e dot o u r i's a n d k n o w precisely the origin and
relation of these e l e m e n t s in the t w o geometries. It m a k e s
f o r clearer thinking. It m a y also help dispel s o m e of the f o g
of m a t h e m a t i c a l mysticism.
THE LURE OF THE INFINITE
3
•
99
The Motionless Arrow
A • Arrows
In the lore of m a n k i n d the a r r o w occupies a
c o n s p i c u o u s place, a place of distinction. T h e r e is t h e
h e r i o c a r r o w with w h i c h the l e g e n d a r y William Tell, at the behest of a t y r a n t , shot a n apple off his o w n son's h e a d , t o
say n o t h i n g of the o t h e r a r r o w that Tell held in reserve f o r
the tyrant himself, in case his first aim should p r o v e t o o
low. T h e r e is the soaring a r r o w of H i a w a t h a t h a t w o u l d n o t
t o u c h the g r o u n d b e f o r e the t e n t h was u p in the air. T h e r e
is the universally f a m o u s r o m a n t i c a r r o w with w h i c h C u p i d
pierces the h e a r t s of his f a v o r i t e s — o r shall I say victims?
T h e r e is also an a r r o w t h a t is philosophical, or scientific,
or, better still, both. This f a m o u s " m o t i o n l e s s a r r o w , " as it
m a y best be called, h a s stirred the m i n d , excited the i m a g i n a tion, and s h a r p e n e d the wits of p r o f o u n d thinkers a n d e r u d i t e
scholars f o r well over t w o t h o u s a n d years.
B • The Arguments
of Zeno and Those of His
Imitators
Z e n o of Elea, w h o flourished in the fifth c e n t u r y B.C., c o n f r o n t e d his fellow p h i l o s o p h e r s a n d a n y b o d y else w h o was
willing to listen with the bold assertion that an a r r o w , the
swiftest object k n o w n to his c o n t e m p o r a r i e s , c a n n o t m o v e at
all.
A c c o r d i n g t o Aristotle, Z e n o ' s a r g u m e n t for, or proof of,
his e m b a r r a s s i n g proposition r a n as follows: " E v e r y t h i n g ,
w h e n in u n i f o r m state, is continually either at rest or in m o tion, and a b o d y m o v i n g in space is continually in t h e N o w
( i n s t a n t ) , h e n c e the a r r o w in flight is at r e s t . " S o m e six centuries later a n o t h e r G r e e k p h i l o s o p h e r offered a s o m e w h a t
clearer f o r m u l a t i o n of the a r g u m e n t : " T h a t w h i c h m o v e s
c a n neither m o v e in the place w h e r e it is, nor yet in t h e
place w h e r e it is n o t . " T h e r e f o r e , m o t i o n is impossible.
T h e " m o t i o n l e s s a r r o w " was not Zeno's only a r g u m e n t
of its kind. H e h a d others. Z e n o h a d Achilles engage in a
race with a tortoise a n d s h o w e d a priori that the "light-off o o t " Achilles could never o v e r t a k e the proverbially slow
turtle. In Aristotle's p r e s e n t a t i o n , h e r e is the a r g u m e n t : " I n
a race the faster c a n n o t o v e r t a k e the slower, f o r the p u r s u e r
m u s t always first arrive at the point f r o m w h i c h the o n e p u r sued h a s just d e p a r t e d , so that the slower is always a small
distance a h e a d . " A m o d e r n p h i l o s o p h e r states t h e a r g u m e n t
m o r e explicitly: "Achilles m u s t first reach the place f r o m
which the tortoise h a s started. By that t i m e the tortoise will
h a v e got o n a little w a y . Achilles m u s t t h e n t r a v e r s e t h a t ,
100
MATHEMATICS IN FUN AND IN EARNEST
a n d still t h e tortoise will be a h e a d . H e is always nearer, b u t
h e n e v e r m a k e s u p t o it."
A t h i r d a r g u m e n t of Z e n o ' s against m o t i o n is k n o w n as
t h e " D i c h o t o m y . " In Aristotle's w o r d s : " A thing m o v i n g in
space m u s t a r r i v e at the m i d - p o i n t b e f o r e it reaches the endp o i n t . " J. B u r n e t offers a m o r e elaborate presentation of
this a r g u m e n t :
Y o u c a n n o t t r a v e r s e a n infinite n u m b e r of points in a
finite time. Y o u m u s t traverse half a given distance bef o r e y o u t r a v e r s e the whole, a n d half of that again bef o r e you t r a v e r s e it. T h i s goes on ad infinitum, so that
(if s p a c e is m a d e u p of p o i n t s ) t h e r e a r e an infinite
n u m b e r in a n y given space, a n d it c a n n o t be traversed
in a finite time.
Z e n o h a d still o t h e r a r g u m e n t s of this kind. But I shall
r e f r a i n f r o m q u o t i n g t h e m , f o r b y n o w a goodly n u m b e r of
y o u h a v e n o d o u b t a l r e a d y b e g u n to w o n d e r w h a t this is
all a b o u t , w h a t it is s u p p o s e d t o m e a n , if anything, and h o w
seriously it is to b e t a k e n . Y o u r incredulity, y o u r skepticism,
reflect the intellectual climate in w h i c h y o u were b r o u g h t
u p a n d in w h i c h y o u c o n t i n u e to live. But that climate h a s
n o t always b e e n the s a m e . It h a s c h a n g e d m o r e t h a n o n c e
since the days of Z e n o .
T o t a k e a simple e x a m p l e . W e teach o u r children in o u r
schools t h a t t h e e a r t h is r o u n d , that it rotates about its axis,
a n d also t h a t it revolves a r o u n d the sun. T h e s e ideas are a n
integral p a r t of o u r intellectual e q u i p m e n t , a n d it seems to
us impossible to get a l o n g w i t h o u t t h e m , m u c h less to doubt
t h e m . A n d yet w h e n C o p e r n i c u s , o r M i k o l a j K o p e r n i k , as
the Poles call h i m , published his e p o c h - m a k i n g work barely
f o u r centuries ago, in 1543, the b o o k was b a n n e d as sinful.
Half a c e n t u r y later, in 1600, G i o r d a n o B r u n o was b u r n e d
at the stake in a public place in R o m e f o r a d h e r i n g to the
C o p e r n i c a n t h e o r y a n d o t h e r heresies. Galileo, o n e of the
f o u n d e r s of m o d e r n science, f o r p r o f e s s i n g the s a m e theories,
was in jail not m u c h m o r e t h a n three centuries ago.
W h a t Z e n o himself t h o u g h t of his a r g u m e n t s , f o r w h a t
reasons h e a d v a n c e d t h e m , w h a t p u r p o s e he w a n t e d to
achieve b y t h e m , c a n n o t b e told with a n y degree of certainty.
T h e d a t a c o n c e r n i n g his life a r e scant a n d unreliable. N o n e
of his writings are e x t a n t . Like the title c h a r a c t e r s of s o m e
m o d e r n novels such as Rebecca,
by D a p h n e d u M a u r i e r , or
Mr. Skeffington,
b y Elizabeth A r n i m Russell, Z e n o is k n o w n
only b y w h a t is told of h i m by others, chiefly his critics a n d
THE LURE OF THE INFINITE
101
detractors. T h e exact m e a n i n g of his a r g u m e n t s is n o t always
certain.
Z e n o m a y o r m a y not have b e e n m i s i n t e r p r e t e d . B u t h e
certainly h a s not b e e n neglected. S o m e writers even p a i d
h i m the highest possible c o m p l i m e n t — t h e y tried to imitate
h i m . T h u s t h e " D i c h o t o m y " suggested t o G i u s e p p e B i a n c a n i ,
of Bologna, in 1615 a " p r o o f " t h a t n o t w o lines c a n h a v e
a c o m m o n m e a s u r e . F o r t h e c o m m o n m e a s u r e , b e f o r e it
could be applied t o the w h o l e line, m u s t first be applied to
half the line, a n d so on. T h u s t h e m e a s u r e c a n n o t be applied
to either line, w h i c h proves that t w o lines a r e always i n c o m mensurable.
A fellow G r e e k , Sextus E m p i r i c u s , of the t h i r d c e n t u r y
A.D., taking the "motionless a r r o w " f o r his m o d e l , a r g u e d
that a m a n c a n n e v e r die, f o r if a m a n dies, it m u s t be
either at a t i m e w h e n he is alive or w h e n h e is d e a d , etc.
It m a y be of interest to m e n t i o n in this c o n n e c t i o n t h a t
the Chinese p h i l o s o p h e r H u i T z u a r g u e d t h a t a m o t h e r l e s s
colt never h a d a m o t h e r . W h e n it h a d a m o t h e r it was not
motherless a n d at every o t h e r m o m e n t of its life it h a d n o
mother.
C ' Aristotle's Arguments
about the Infinite Divisibility of
Both Time and Space
S o m e writers offered very e l a b o r a t e
interpretations of Zeno's a r g u m e n t s . T h e s e writers saw in the
c r e a t o r of these a r g u m e n t s a m a n of p r o f o u n d philosophical
insight a n d a logician of the first m a g n i t u d e . Such was the
attitude of I m m a n u e l K a n t a n d , a c e n t u r y later, of the
F r e n c h m a t h e m a t i c i a n Jules T a n n e r y ( 1 8 4 8 - 1 9 1 0 ) . T o Aristotle, w h o was b o r a a b o u t a c e n t u r y a f t e r Zeno, these a r g u m e n t s were just a n n o y i n g s o p h i s m s w h o s e h i d d e n fallacy it
was all the m o r e necessary to expose in view of the plausible
logical f o r m in w h i c h they are clothed. O t h e r writers displayed just as m u c h zeal in showing that Zeno's a r g u m e n t s
are irrefutable.
Aristotle's f u n d a m e n t a l a s s u m p t i o n s a r e t h a t b o t h t i m e
and space a r e c o n t i n u o u s , that is, " a l w a y s divisible into
divisible p a r t s . " H e f u r t h e r a d d s : " T h e c o n t i n u a l bisection
of a quantity is unlimited, so t h a t the u n l i m i t e d exists p o tentially, but it is never r e a c h e d . "
W i t h r e g a r d t o the " a r r o w " h e says:
A t h i n g is at rest
a n d still in a n o t h e r
m a i n i n g in t h e s a m e
rest in the N o w . . .
when
Now,
status.
. In
it is u n c h a n g e d
itself as well as
. . . T h e r e is n o
a t i m e interval,
in t h e N o w
its p a r t s remotion, nor
o n t h e con-
102
MATHEMATICS IN FUN AND IN EARNEST
t r a r y , it ( a v a r i a b l e ) c a n n o t exist in the same state of
rest, f o r o t h e r w i s e it w o u l d follow t h a t t h e t h i n g in m o t i o n
is at rest.
T h a t it is impossible to traverse a n unlimited n u m b e r of
h a l f - d i s t a n c e s ( t h e " D i c h o t o m y " ) , Aristotle r e f u t e s by pointing out that " t i m e h a s unlimitedly m a n y parts, in conseq u e n c e of w h i c h there is n o a b s u r d i t y in the consideration
t h a t in a n unlimited n u m b e r of t i m e intervals o n e passes
o v e r unlimited m a n y spaces." T h e a r g u m e n t Aristotle directs
against " A c h i l l e s " is as f o l l o w s :
If t i m e is c o n t i n u o u s , so is distance, f o r in half the
t i m e a t h i n g passes over half the distance, a n d , in
general, in the s m a l l e r time the smaller distance, f o r
t i m e a n d distance h a v e the s a m e divisions, and if one
of the t w o is unlimited, so is the other. F o r that reason
the a r g u m e n t of Z e n o a s s u m e s a n u n t r u t h , that one
unlimited c a n n o t travel over a n o t h e r unlimited along its
o w n parts, or t o u c h such a n unlimited, in a finite
t i m e ; f o r length as well as time a n d , in general, everything c o n t i n u o u s , m a y be considered unlimited in a
d o u b l e sense, n a m e l y a c c o r d i n g to the ( n u m b e r o f )
divisions o r a c c o r d i n g to the (distances between t h e )
o u t e r m o s t ends. 4
Aristotle seems to insist that as t h e distances between
Achilles a n d the tortoise keep on diminishing, the intervals of
t i m e necessary to cover these distances also diminish, a n d in
the s a m e p r o p o r t i o n .
T h e reasonings of Aristotle cut n o ice w h a t e v e r with the
F r e n c h p h i l o s o p h e r P i e r r e Bayle ( 1 6 4 7 - 1 7 0 6 ) , w h o in 1696
published his Dictionnaire
Historique
et Critique,
translated
into English in 1710. Bayle goes into a detailed discussion
of Zeno's a r g u m e n t s a n d is entirely on the side of Zeno. H e
categorically rejects the infinite divisibility of time.
Successive d u r a t i o n of things is c o m p o s e d of m o m e n t s , p r o p e r l y so called, each of w h i c h is simple a n d
indivisible, perfectly distinct f r o m the past a n d f u t u r e
a n d c o n t a i n s n o m o r e t h a n the present time. T h o s e
w h o d e n y this c o n s e q u e n c e m u s t be given u p to their
stupidity, o r their w a n t of sincerity, or to the unsurm o u n t a b l e p o w e r of their prejudices.
T h u s t h e " A r r o w " will n e v e r b u d g e .
103
THE LURE OF THE INFINITE
T h e philosophical discussion of the divisibility o r t h e n o n divisibility of t i m e a n d space c o n t i n u e s t h r o u g h the centuries.
A s late as the close of the past c e n t u r y Z e n o ' s a r g u m e n t s
based on this g r o u n d w e r e the topic of a very a n i m a t e d
discussion in the philosophical j o u r n a l s of F r a n c e .
D
'
The Potential Infinity
and the Actual
Infinity
A
m a t h e m a t i c a l a p p r o a c h to "Achilles" is d u e to G r e g o r y St.
V i n c e n t ( 1 5 8 4 - 1 6 6 7 ) , w h o in 1647 c o n s i d e r e d a s e g m e n t
$
S
&
®
@
»
& e®
Figure 8
A K on w h i c h h e c o n s t r u c t e d a n u n l i m i t e d n u m b e r of p o i n t s
B, C, D, . . . such that A B / A K = B C / B K = C D / C K = . . .
= r, (Fig. 8 ) w h e r e r is t h e ratio, say of t h e speed of the t o r toise to the speed of Achilles. H e t h u s o b t a i n s the infinite
geometric progression A B + B C + C D f . . . a n d , since t h i s
series is c o n v e r g e n t , Achilles does o v e r t a k e the elusive
tortoise.
Descartes solved the " A c h i l l e s " by t h e u s e of t h e geometric progression 1 / 1 0 + 1 / 1 0 0 + 1 / 1 0 0 0 + . . . = 1 / 9 . L a ter writers q u o t e d this advice o r rediscovered it time a n d
again. But this solution of the p r o b l e m raised b r a n d - n e w
questions.
St. V i n c e n t overlooked the i m p o r t a n t f a c t t h a t Achilles
will fail to o v e r t a k e the slow-moving tortoise a f t e r all, u n less the variable sum of the g e o m e t r i c progression actually
reaches its limit. N o w : D o e s a variable reach its limit, or
does it not? T h e question transcends, b y far, the "Achilles."
It was, f o r instance, hotly d e b a t e d in c o n n e c t i o n with t h e
then nascent differential a n d integral calculus. N e w t o n believed that his variables r e a c h e d their limits. D i d e r o t ( 1 7 1 3 1 7 8 4 ) , writing a c e n t u r y o r so later in the f a m o u s
Encyclopedic, is quite definite that a variable c a n n o t d o that, a n d s o
is A . D e M o r g a n ( 1 8 0 6 - 1 8 7 1 ) , in the Penny Cyclopedia
in
1846. Sadi C a r n o t ( 1 7 9 6 - 1 8 3 2 ) a n d A . L . C a u c h y ( 1 7 8 9 1 8 5 7 ) , like N e w t o n , h a v e n o objection to variables r e a c h i n g
their limits.
T h e o t h e r question that arises in c o n n e c t i o n with St. V i n cent's progression is: H o w m a n y t e r m s does the progression
have? T h e a n s w e r ordinarily given is that the n u m b e r is
infinite. This answer, h o w e v e r , m a y h a v e t w o different m e a n -
104
MATHEMATICS IN FUN AND IN EARNEST
ings. W e m a y m e a n to say t h a t w e c a n c o m p u t e as m a n y
t e r m s of this progression as we w a n t a n d , n o m a t t e r h o w
m a n y w e h a v e c o m p u t e d , we c a n still c o n t i n u e the process.
T h u s the n u m b e r of t e r m s of the progression is "potentially"
infinite. O n t h e o t h e r h a n d , we m a y imagine that all the
t e r m s h a v e b e e n c a l c u l a t e d a n d are ali, t h e r e f o r m i n g a n infinite collection. T h a t w 6 u l d m a k e a n " a c t u a l " infinity. A r e
t h e r e actually infinite collections in n a t u r e ? Obviously, collections as large "as the stars of t h e h e a v e n , a n d as the sand
which is u p o n t h e s e a s h o r e , " are nevertheless finite collections. 5
F r o m a q u o t a t i o n of Aristotle a l r e a d y given it w o u l d seem
t h a t h e did not believe in the actually infinite. Galileo, on the
o t h e r h a n d , a c c e p t e d the existence of actual infinity, although
h e saw clearly the difficulties involved. If the n u m b e r of
integers is not only potentially but actually infinite, then
there are as m a n y p e r f e c t squares as there are integers, since
f o r every integer t h e r e is a p e r f e c t s q u a r e a n d every p e r f e c t
s q u a r e h a s a s q u a r e root. 6 Galileo ( 1 5 6 4 - 1 6 4 2 ) tried to console himself by saying t h a t the difficulties a r e d u e to the f a c t
that o u r finite m i n d c a n n o t c o p e with the infinite. But D e
M o r g a n sees n o point to this a r g u m e n t , f o r , even admitting
the " f i n i t u d e " of o u r m i n d , "it is not necessary to have a
blue m i n d to c o n c e i v e of a pair of b l u e eyes."
A y o u n g e r c o n t e m p o r a r y of Galileo, the p r o m i n e n t E n g lish p h i l o s o p h e r T h o m a s H o b b e s ( 1 5 8 8 - 1 6 7 9 ) , could not accept Galileo's actual infinity, o n theological g r o u n d s . " W h o
thinks that the n u m b e r of even integers is equal to the n u m b e r of all integers is taking a w a y eternity f r o m the C r e a t o r . "
H o w e v e r , the very s a m e theological r e a s o n s led an illustrious
y o u n g e r c o n t e m p o r a r y of H o b b e s , n a m e l y , G . W. Leibniz
( 1 6 4 6 - 1 7 1 7 ) , to the firm belief that actual infinities exist in
nature pour mieux marquer les perfections
de son auteur.
T h e actual infinite w a s erected into a b o d y of doctrine by
G e o r g C a n t o r ( 1 8 4 5 - 1 9 1 8 ) in his t h e o r y of transfinite n u m bers. T h e o u t s t a n d i n g A m e r i c a n historian of m a t h e m a t i c s ,
F l o r i a n C a j o r i ( 1 8 5 9 - 1 9 3 0 ) , considers that this doctrine of
C a n t o r ' s p r o v i d e d a final and definite a n s w e r to Zeno's p a r a doxes a n d t h u s relegates t h e m to the status of " p r o b l e m s of
the past."
T o b i a s D a n t z i g in his Number,
the Language
of
Science,
is not quite so h a p p y a b o u t it, in view of the f a c t that the
w h o l e t h e o r y of C a n t o r ' s is of d o u b t f u l solidity.
E ' Motion
reasons that
and Dynamics
W h a t e v e r m a y h a v e been the
p r o m p t e d Z e n o t o p r o m u l g a t e his p a r a d o x e s ,
THE LURE OF THE INFINITE
105
h e certainly m u s t h a v e been a m a n of c o u r a g e if he d a r e d
to deny the existence of m o t i o n . W e learn of m o t i o n a n d
learn to a p p r e c i a t e it at a very, very early age; m o t i o n is
firmly i m b e d d e d in o u r daily existence a n d b e c o m e s a basic
element of o u r psychological m a k e - u p . It seems intolerable
to us that w e could be deprived of m o t i o n , even in a jest.
Nevertheless, the systematic study of m o t i o n is of fairly
recent origin. T h e ancient world k n e w a good deal a b o u t
Statics, as evidenced by the size a n d solidity of the structures that have survived to the p r e s e n t day. But they k n e w
next to n o t h i n g a b o u t D y n a m i c s , f o r the f o r m s of m o t i o n
with which they h a d a n y e x p e r i e n c e w e r e of very n a r r o w
scope. T h e i r m a c h i n e s w e r e of the c r u d e s t a n d very limited
in variety. Zeno's p a r a d o x e s of m o t i o n w e r e f o r the G r e e k
philosophers " p u r e l y a c a d e m i c " questions.
T h e a s t r o n o m e r s w e r e the first to m a k e systematic o b s e r vations of m o t i o n not d u e to m u s c u l a r f o r c e and to m a k e
deductions f r o m their observations. M a n studied m o t i o n in
the skies b e f o r e he busied himself with such studies on earth.
H o w difficult it was f o r the ancients to dissociate m o t i o n
f r o m m u s c u l a r effort is illustrated by the f a c t that Helios
(the s u n ) was said by the G r e e k s to h a v e a palace in t h e
east w h e n c e he was d r a w n daily across the sky in a fiery
chariot by f o u r white horses t o a palace in t h e west.
T h e f a m o u s e x p e r i m e n t s of G a l i l e o with falling bodies
are the beginning of m o d e r n D y n a m i c s . T h e great voyages
created a d e m a n d f o r reliable clocks, a n d the study of clock
m e c h a n i s m s a n d their m o t i o n engaged the attention of such
o u t s t a n d i n g scholars as H u y g e n s . N o small incentive f o r the
study of m o t i o n was provided by the needs of the developing
artillery. T h e g u n n e r s h a d to k n o w the trajectories of their
missiles. T h e theoretical studies of m o t i o n p r o m p t e d by
these and o t h e r technical d e v e l o p m e n t s w e r e in need of
a new m a t h e m a t i c a l tool to solve the newly arising p r o b l e m s ,
a n d calculus c a m e into being. 7
T h e infinite, the infinitesimal, limits a n d o t h e r notions that
were involved, p e r h a p s crudely, in the discussion of Zeno's
a r g u m e n t s w e r e also involved in this new b r a n c h of m a t h e matics. T h e s e notions were as hazy as they were essential.
Both N e w t o n a n d Leibniz c h a n g e d their views on these
points d u r i n g their lifetimes because of their own critical
a c u m e n as well as the s e a r c h i n g criticism of their c o n t e m poraries. But n e i t h e r of t h e m ever e n t e r t a i n e d the idea of
giving u p their precious find, f o r the good a n d sufficient
reason that this new and m a r v e l o u s tool gave t h e m the
solution of s o m e of the p r o b l e m s that h a d defied all the e f -
106
MATHEMATICS IN FUN AND IN EARNEST
f o r t s of m a t h e m a t i c i a n s of p r e c e d i n g generations. T h e succ e e d i n g c e n t u r y , t h e eighteenth, exploited to the u t m o s t this
n e w i n s t r u m e n t in its a p p l i c a t i o n t o the study of m o t i o n , a n d
b e f o r e t h e c e n t u r y was o v e r it t r i u m p h a n t l y presented to the
l e a r n e d w o r l d t w o m o n u m e n t a l w o r k s : t h e Mecanique
Analytique of J. L . L a g r a n g e ( 1 7 3 6 - 1 8 1 3 ) , a n d the
Mecanique
Celeste of P. S. L a p l a c e ( 1 7 4 9 - 1 8 2 7 ) .
T h e d e v e l o p m e n t of D y n a m i c s did n o t stop there. It k e p t
p a c e with t h e p h e n o m e n a l d e v e l o p m e n t of the experimental
sciences in t h e n i n e t e e n t h c e n t u r y . T h e s e theoretical studies
o n the o n e h a n d served as a basis f o r t h e creation of a
t e c h n o l o g y t h a t surpassed t h e wildest d r e a m s of past generations a n d o n the o t h e r h a n d c h a n g e d radically o u r attitude
t o w a r d m a n y of the p r o b l e m s of t h e p a s t ; they created a
n e w intellectual a t m o s p h e r e , a n e w "intellectual climate."
Z e n o ' s a r g u m e n t s , o r p a r a d o x e s , if y o u p r e f e r , deal with
t w o questions w h i c h in the discussions of these p a r a d o x e s
a r e very closely c o n n e c t e d , not to say m i x e d u p : W h a t is
m o t i o n , a n d h o w c a n m o t i o n be a c c o u n t e d f o r in a rational,
intellectual w a y ? By s e p a r a t i n g the t w o p a r t s of t h e p r o b l e m
we m a y be able t o c o m e m u c h closer to finding a satisfactory
a n s w e r to the question, in a c c o r d with t h e present-day intellectual o u t l o o k .
F ' Motion—An
"Undefined
Term"
T h e critical study of
t h e f o u n d a t i o n s of m a t h e m a t i c s d u r i n g the nineteenth cent u r y m a d e it a b u n d a n t l y clear that n o science a n d , m o r e
generally, n o intellectual discipline can define all the terms
it uses w i t h o u t c r e a t i n g a vicious circle. T o define a term
m e a n s t o r e d u c e it t o s o m e m o r e f a m i l i a r c o m p o n e n t parts.
Such a p r o c e d u r e obviously h a s a limit b e y o n d which it cann o t go. M o s t of us k n o w w h a t the color " r e d " is. W e can
discuss this color with e a c h o t h e r ; w e c a n w o n d e r h o w m u c h
t h e r e d color c o n t r i b u t e s t o the b e a u t y of the sunset; we c a n
m a k e use of this c o m m o n k n o w l e d g e of the red color f o r a
c o m m o n p u r p o s e , such as directing traffic. But w e c a n n o t
u n d e r t a k e to explain w h a t the red color is t o a person b o r n
color blind.
I n the science of D y n a m i c s m o t i o n is such a t e r m , such
a n " u n d e f i n e d " t e r m , t o use the technical expression f o r it. 8
D y n a m i c s does not p r o p o s e to explain w h a t n o t i o n is to
a n y o n e w h o does n o t k n o w that already. M o t i o n is o n e of
its starting points, o n e of its u n d e f i n e d , or primitive, terms.
T h i s is its a n s w e r to the q u e s t i o n : W h a t is m o t i o n ?
Y o u h a v e h e a r d m a n y stories a b o u t Diogenes ( 4 1 2 - 3 2 3
B.C.). H e lived in a b a r r e l . H e t h r e w a w a y his d r i n k i n g c u p
THE LURE OF THE INFINITE
107
w h e n he noticed a b o y d r i n k i n g out of the hollow of his
h a n d . H e told his visitor, A l e x a n d e r the G r e a t , that the only
f a v o r the m i g h t y c o n q u e r o r could possibly d o f o r h i m was
to step aside so as not to obstruct the sun f o r the philosopher. Well, there is also the story that w h e n Diogenes was
told of Zeno's a r g u m e n t s a b o u t the impossibility of m o t i o n ,
he arose f r o m the place w h e r e he was sitting on the g r o u n d
alongside his barrel, took a f e w steps, a n d r e t u r n e d to his
place at the b a r r e l w i t h o u t saying a single w o r d . T h i s w a s
the celebrated C y n i c p h i l o s o p h e r ' s " e l o q u e n t " w a y of saying
that motion is. A n d did he not also say at the s a m e t i m e
that motion is an " u n d e f i n e d t e r m " ?
St. A u g u s t i n e ( 3 5 4 - 4 5 0 ) used a n even m o r e c o n v i n c i n g
m e t h o d t o e m p h a s i z e the s a m e point. H e w r o t e :
W h e n the discourse ( o n m o t i o n ) was c o n c l u d e d , a
boy c a m e r u n n i n g f r o m the h o u s e to call f o r d i n n e r . I
then r e m a r k e d that this boy c o m p e l s us not only to define m o t i o n , but to see it b e f o r e o u r very eyes. So let us
go a n d pass f r o m one place t o a n o t h e r , f o r t h a t is, if I
a m not m i s t a k e n , n o t h i n g else t h a n m o t i o n .
T h e revered theologian seems to h a v e k n o w n , f r o m p e r sonal experience, that n o t h i n g is as likely t o set a m a n in
m o t i o n as a well-garnished table.
G ' Theory and Observation
Let us n o w t u r n to the second part involved in Zeno's p a r a d o x e s , n a m e l y , h o w to account f o r m o t i o n in a rational way. All science m a y be said
to be a n a t t e m p t to give a rational a c c o u n t of events in
nature, of the ways n a t u r a l p h e n o m e n a r u n their courses.
T h e scientific theories are a rational description of n a t u r e
that enables us to foresee and foretell the c o u r s e of n a t u r a l
events. This characteristic of scientific theories affords us a n
intellectual satisfaction, o n the o n e h a n d , a n d , o n the o t h e r
h a n d , shows us h o w to control n a t u r e f o r o u r benefit, to
serve o u r needs a n d c o m f o r t s . Prevoir
pour pouvoir,
to
quote H e n r i P o i n c a r e . A scientific theory, t h a t is, a rational
description of a sector of n a t u r e , is a c c e p t a b l e and accepted
only as long as its previsions agree with the facts of observation. T h e r e c a n be n o b a d theory. If a t h e o r y is b a d o r
goes b a d , it is modified o r it is t h r o w n o u t completely.
"Achilles" is a n a t t e m p t at a rational a c c o u n t of a race,
a theoretical i n t e r p r e t a t i o n of a physical p h e n o m e n o n . T h e
terrible thing is that Zeno's theory predicts o n e result, while
everybody in his senses k n o w s quite well that exactly the
108
MATHEMATICS IN FUN AND IN EARNEST
c o n t r a r y actually t a k e s place. Aristotle in his time a n d d a y
felt called u p o n t o use all his vast intellectual powers to ref u t e the p a r a d o x . O u r p r e s e n t intellectual climate imposes
n o such obligation u p o n us. If saying t h a t in o r d e r to overt a k e t h e tortoise Achilles m u s t first arrive at the point f r o m
w h i c h t h e tortoise started, etc., leads to t h e conclusion that
h e will never o v e r t a k e the creeping a n i m a l , w e simply i n f e r
t h a t Z e n o ' s t h e o r y of a race does n o t serve the p u r p o s e f o r
w h i c h it was created. W e d e c l a r e the s c h e m e to be u n w o r k able a n d p r o c e e d to evolve a n o t h e r t h e o r y which will render
a m o r e satisfactory a c c o u n t of the o u t c o m e of the race.
T h a t , of course, is a s s u m i n g that the t h e o r y of Z e n o was
o f f e r e d in good f a i t h . If it was not, then it is an idle plaything, v e r y a m u s i n g , p e r h a p s , very ingenious, if you like,
b u t not w o r t h y of a n y serious consideration. T h e r e are m o r e
w o r t h w h i l e w a y s of s p e n d i n g one's time t h a n in s h a d o w boxing. O u r indifferent attitude t o w a r d s Zeno's p a r a d o x e s is
p e r h a p s best m a n i f e s t e d b y the f a c t that t h e article " m o t i o n "
in the Britannica
does not m e n t i o n Zeno, w h e r e a s Einstein
is given c o n s i d e r a b l e a t t e n t i o n ; the Americana
dismisses " m o t i o n " w i t h the c u r t r e f e r e n c e "see M e c h a n i c s . "
C o n s i d e r a n elastic ball w h i c h r e b o u n d s f r o m the g r o u n d
t o 2 / 3 of the height f r o m w h i c h it fell. W h e n d r o p p e d f r o m
a height of 3 0 feet, h o w f a r will the ball have traveled b y
the t i m e it stops? A n y bright f r e s h m a n will immediately
raise the question w h e t h e r that ball will ever stop. O n the
o t h e r h a n d , t h a t s a m e f r e s h m a n k n o w s full well that a f t e r
a while the ball will quietly lie on the g r o u n d . Will we be
very m u c h w o r r i e d b y this c o n t r a d i c t i o n ? N o t at all. W e will
simply d r a w the conclusion t h a t the law of r e b o u n d i n g of
the ball, as described, is f a u l t y .
H ' Instantaneous
Velocity
T h e difficulties encountered in
c o n n e c t i o n with t h e question of a variable reaching o r not
r e a c h i n g a limit a r e of the s a m e k i n d a n d nature. T h e m o d e
of variation of a variable is either a description of a n a t u r a l
event or a creation of o u r imagination, w i t h o u t a n y physical
c o n n o t a t i o n . In t h e latter case, the law of variation of the
v a r i a b l e is p r e s c r i b e d by o u r f a n c y , and the variable is c o m pletely at o u r m e r c y . W e c a n m a k e it reach the limit or
keep it f r o m d o i n g so, as w e m a y see fit. In the f o r m e r case
it is the physical p h e n o m e n o n t h a t decides the question f o r
us.
T w o bicycle riders, 60 miles a p a r t , start t o w a r d s each
o t h e r , at the rate of 10 miles p e r h o u r . A t the m o m e n t w h e n
they start a fly takes off f r o m the r i m of t h e wheel of o n e
THE LURE OF THE INFINITE
109
rider a n d flies directly t o w a r d s the s e c o n d rider at the r a t e
of 15 miles per h o u r . A s soon as t h e fly r e a c h e s the second
rider it turns a r o u n d a n d flies t o w a r d t h e first, etc. W h a t is
the sum of the distances of the oscillations of the fly? I n
Zeno's p r e s e n t a t i o n the n u m b e r of these oscillations is infinite. But t h e flying time was exactly 3 h o u r s , a n d the fly
covered a distance of 4 5 miles. T h e v a r i a b l e s u m actually
r e a c h e d its limit.
T h e s e q u e n c e of n u m b e r s 1, Vi, Va, Vs, ^ . . . obviously
h a s f o r its limit zero. D o e s the s e q u e n c e r e a c h its limit? Let us
interpret this s e q u e n c e , s o m e w h a t facetiously, in the following m a n n e r . A rabbit hiding in a hollow log noticed a d o g
s t a n d i n g at the e n d n e a r him. T h e r a b b i t got scared a n d
with o n e leap was at the o t h e r e n d ; b u t t h e r e w a s a n o t h e r
dog. T h e rabbit got twice as scared, a n d in half the time h e
w a s b a c k at the first e n d ; but t h e r e was t h e first dog, so the
rabbit got twice as scared again, etc. If this s e q u e n c e reaches
its limit, the rabbit will e n d u p b y b e i n g at b o t h e n d s of
the log at the s a m e time.
Figure 9
If a point Q of a curve ( C ) moves t o w a r d a fixed p o i n t
P (Fig. 9 ) of the curve, the line P Q revolves a b o u t P. If
Q a p p r o a c h e s P as a limit, t h e line P Q obviously a p p r o a c h e s
as a limiting position the t a n g e n t to the c u r v e ( C ) at t h e
point P ; a n d if the point Q reaches the position P or, w h a t
is the s a m e thing, coincides with P , the line P Q will coincide
with the t a n g e n t t o ( C ) at P. 9
If s r e p r e s e n t s the distance traveled b y a m o v i n g point in
the time t, does the ratio s / t a p p r o a c h a limit w h e n t a p p r o a c h e s zero as a limit? In o t h e r words, does a m o v i n g object h a v e a n i n s t a n t a n e o u s velocity at a p o i n t of its course,
o r its t r a j e c t o r y ? Aristotle could n o t a n s w e r that question;
he p r o b a b l y could n o t m a k e a n y sense of t h e question. Aristotle agreed with Z e n o that t h e r e c a n be n o m o t i o n in the
N o w ( m o m e n t ) . B u t t o us t h e a n s w e r t o this question is n o t
110
MATHEMATICS IN FUN AND IN EARNEST
subject to a n y d o u b t w h a t e v e r : w e are too a c c u s t o m e d to
r e a d t h e i n s t a n t a n e o u s velocities o n the speedometers of o u r
cars.
T h e divisibility o r the nondivisibility of time a n d space
was a vital question to the G r e e k philosophers, and they had
n o criterion a c c o r d i n g to w h i c h they could settle the dispute. T o us t i m e a n d space are c o n s t r u c t s t h a t w e use to acc o u n t f o r physical p h e n o m e n a , constructs of o u r o w n
m a k i n g , a n d as such w e are f r e e to use t h e m in any m a n n e r
w e see fit. A l b e r t Einstein did not hesitate to mix u p the
t w o a n d m a k e of t h e m a space-time c o n t i n u u m w h e n he
f o u n d t h a t such a c o n s t r u c t is better a d a p t e d to a c c o u n t f o r
physical p h e n o m e n a a c c o r d i n g t o his theory of relativity.
I
•
A Modern
Answer
to Zeno's
Paradoxes
We have
dealt with the t w o p a r t s of Z e n o ' s p a r a d o x e s : the definition
of m o t i o n a n d the description of m o t i o n . T h e r e is, however,
a third e l e m e n t in these p a r a d o x e s , a n d it is this third elem e n t that is p r o b a b l y m o r e responsible f o r the interest that
these p a r a d o x e s held t h r o u g h o u t the centuries than those w e
have considered already. T h i s is the logical element.
T h a t Z e n o w a s d e f e n d i n g an indefensible cause was clear
to all those w h o tried to r e f u t e him. But h o w is it possible
to d e f e n d a false cause with a p p a r e n t l y s o u n d logic? This
is a very serious challenge. If s o u n d logic is not an absolute
g u a r a n t y t h a t the propositions d e f e n d e d by that m e t h o d are
valid, all o u r intellectual e n d e a v o r s are built on quicksand,
o u r c o u r t s of justice are meaningless p a n t o m i n e , etc.
Aristotle considered t h a t the f u n d a m e n t a l difficulty involved in Z e n o ' s a r g u m e n t against motion was the meaning
Z e n o a t t a c h e d t o his " N o w . " If the " N o w , " the m o m e n t , as
w e w o u l d say, does not represent any length of time but
only the durationless b o u n d r y b e t w e e n two a d j a c e n t intervals of time, as a point w i t h o u t length is the c o m m o n
b o u n d a r y of t w o a d j a c e n t segments of a line, then in such a
m o m e n t t h e r e c a n be n o m o t i o n ; the a r r o w is motionless.
Aristotle tried to r e f u t e Z e n o ' s denial of motion by pointing
out that it is w r o n g to say t h a t t i m e is m a d e u p of d u r a t i o n less m o m e n t s . B u t Aristotle was not very convincing, judging by the vitality of Z e n o ' s a r g u m e n t s .
O u r m o d e r n k n o w l e d g e of m o t i o n provided us with better
w a y s of m e e t i n g Zeno's p a r a d o x e s . W e can grant Z e n o both
the durationless " N o w " a n d the immobility of the object
in the " N o w " a n d still c o n t e n d that these t w o premises d o
not imply the immobility of the a r r o w . W h i l e the a r r o w does
n o t m o v e in the " N o w , " it conserves its capacity, its poten-
MATHESIS THE BEAUTIFUL!
Ill
tiality of m o t i o n . In o u r m o d e r n terminology, in t h e " N o w "
the a r r o w h a s an i n s t a n t a n e o u s velocity. This n o t i o n of instantaneous velocity is c o m m o n p l a c e with us; we r e a d it
"with o u r o w n eyes" on o u r s p e e d o m e t e r s every day. But it
was completely foreign to the ancients. T h u s Zeno's reasoning was f a u l t y because he did not k n o w e n o u g h a b o u t t h e
subject he was r e a s o n i n g about. 1 0
Zeno's a p p a r e n t l y unextinguishable p a r a d o x e s , as t h e y a r e
r e f e r r e d to by E . T . Bell (b. 1883) in a n article published
in Scripta Mathematica,
will not be put out of circulation b y
o u r r e m a r k s a b o u t t h e m . I have n o illusions a b o u t that;
neither d o I h a v e a n y such ambitions. T h e s e p a r a d o x e s h a v e
amused and excited countless generations, a n d they should
continue to d o so. W h y not?
FOOTNOTES
1
Cf.
Cf.
See
4
Cf.
5
See
2
3
8
Chapter
Chapter
Chapter
Chapter
Chapter
III, Section 3d.
III, Section 3.
I, Section 2d.
III, Section lc.
III, Section lb.
Ibid.
'Cf.
8
Cf.
"Cf.
"Cf.
Chapter
Chapter
Chapter
Chapter
II, Section
I, Section
II, Section
I, Section
IV
lc.
3f; Chapter V, Section lj.
1c; Chapter V, Section 2c.
3e.
MATHESIS T H E BEAUTIFUL!
1
•
Mathematics and Esthetics
A • Beauty in Mathematics
A c u l t u r e d person with literary proclivities once asked this writer w h e t h e r m a t h e m a t i cians see b e a u t y in their science. D u r i n g her school c a r e e r
she had heard h e r t e a c h e r of m a t h e m a t i c s , w h o s e subject,
by the way, she e n j o y e d very little, r e f e r t o a t h e o r e m as
being beautiful, and this statement seemed to h e r preposterous. In reply to h e r question o n e could h a v e q u o t e d those
great masters of m a t h e m a t i c a l t h o u g h t w h o spoke so eloquently on the subject. H e n r i P o i n c a r e ( 1 8 5 4 - 1 9 1 2 ) , one of
the greatest m a t h e m a t i c i a n s and o n e of the greatest m i n d s of
all times, said in this c o n n e c t i o n : " A b o v e all, adepts find in
m a t h e m a t i c s delights analogous to those that painting a n d
112
MATHEMATICS IN FUN AND IN EARNEST
m u s i c give. T h e y a d m i r e t h e delicate h a r m o n y of n u m b e r s
a n d of f o r m s ; t h e y a r e a m a z e d w h e n a n e w discovery discloses t o t h e m a n u n l o o k e d - f o r perspective, a n d the joy they
t h u s e x p e r i e n c e h a s it n o t t h e esthetic c h a r a c t e r , although
the senses t a k e n o p a r t in it? O n l y t h e privileged f e w are
called to e n j o y it fully, b u t is it not so with all the noblest
a r t s ? " O u r distinguished c o n t e m p o r a r y B e r t r a n d Russell (b.
1 8 7 2 ) s a i d : " M a t h e m a t i c s , rightly viewed, possesses not
o n l y t r u t h , b u t s u p r e m e b e a u t y — a b e a u t y cold a n d austere,
like t h a t of s c u l p t u r e . . . T h e t r u e spirit of delight, the exaltation, the sense of being m o r e t h a n a m a n w h i c h is the
t o u c h s t o n e of t h e highest excellence, is to be f o u n d in m a t h ematics as surely as in p o e t r y . " L e t m e a d d just one m o r e
quotation, f r o m an American clergyman and mathematician,
T h o m a s Hill ( 1 8 1 8 - 1 8 9 3 ) , " T h e m a t h e m a t i c s is usually considered as being the very a n t i p o d e s of poesy. Y e t mathesis
a n d p o e s y are of t h e closest k i n d r e d , f o r they are both w o r k s
of i m a g i n a t i o n . "
B • Mathematics
in Beauty
F o r the initiate m a t h e m a t i c s
h a s very m u c h in c o m m o n with the fine arts. O n the other
h a n d the fine arts are greatly indebted to m a t h e m a t i c s . T o
achieve verse r h y t h m the p o e t m u s t count the feet in his lines,
i.e. the regularly r e c u r r i n g metrical
units. T h e w o r d s in a
verse m u s t be p l a c e d in measured
and cadenced formation
so as to p r o d u c e a metrical effect.
A r t u r o A l d u n a t e Phillips, a Chilean essayist, economist,
p o e t , a n d engineer, in his Matematica
y Poesia (essayo y
entusiasmo)1
goes m u c h f a r t h e r . H e not only sees m a n y close
ties, b o t h intellectual a n d artistic, between m a t h e m a t i c s a n d
poetry, b u t he traces a close parallel between the two, in
their historical d e v e l o p m e n t , as well as in their role in the
history of culture.
T h e role of m a t h e m a t i c s in music is a quite intimate one.
Several centuries b e f o r e o u r present era P y t h a g o r a s observed
a l r e a d y that w h e n t h e m u s i c a l strings of equal length are
stretched b y weights h a v i n g the p r o p o r t i o n s of Vz, % , % ,
they p r o d u c e intervals w h i c h are a n octave, a fifth, a n d a
f o u r t h . E v e r since that t i m e m a t h e m a t i c i a n s have greatly contributed t o w a r d s the e l a b o r a t i o n of the t h e o r y of music. E u clid, t h e a u t h o r of the f a m o u s Elements,
wrote two books on
the t h e o r y of m u s i c . W h e n the music of the ancients, the
h o m o p h o n i c music, gave w a y to the p o l y p h o n i c music of the
M i d d l e Ages, m a t h e m a t i c i a n s have f u r t h e r e d its theoretical
d e v e l o p m e n t . T h e R e n a i s s a n c e has witnessed the birth of o u r
m o d e r n , h a r m o n i c music, a n d a m o n g those w h o contributed
MATHESIS THE BEAUTIFUL!
Ill
towards t h e study of its t h e o r y w e find s u c h n a m e s as K e p ler ( 1 5 7 1 - 1 6 3 0 ) , D e s c a r t e s ( 1 5 9 6 - 1 6 5 0 ) H u y g e n s ( 1 6 2 9 1695).
T h e close c o n n e c t i o n b e t w e e n m a t h e m a t i c s a n d m u s i c h a s
been expressed b y H . H e l m h o l t z ( 1 8 2 1 - 1 8 9 4 ) as follows:
" M a t h e m a t i c s a n d music, the m o s t s h a r p l y c o n t r a s t e d fields
of scientific activity, a n d yet related, s u p p o r t i n g each other,
as if to s h o w f o r t h the secret c o n n e c t i o n w h i c h ties together
all the activities of the m i n d , and w h i c h leads to surmise t h a t
the m a n i f e s t a t i o n s of the artist's genius are b u t u n c o n s c i o u s
expressions of a mysteriously acting rationality." Leibniz is
even m o r e specific: " M u s i c is a h i d d e n exercise in arithmetic,
of a m i n d u n c o n s c i o u s of dealing with n u m b e r s . " T h e love
of m a t h e m a t i c i a n s f o r m u s i c is a well established fact. T h e
great c o n t e m p o r a r y m a t h e m a t i c a l genius A l b e r t Einstein
( 1 8 7 8 - 1 9 5 5 ) was an excellent violinist.
Sculpture, a r c h i t e c t u r e , painting, a n d t h e g r a p h i c arts in
general, obviously involve geometric considerations. W h a t
geometric c o n s t r u c t i o n s artists h a v e used, consciously o r u n consciously, to achieve their esthetic effects h a s been well
analyzed a n d put clearly i n t o evidence. W e shall m e n t i o n that
o n e of the most telling esthetic effects is o b t a i n e d by the
so-called " G o l d e n S e c t i o n " a n d its derivatives, a n d this section is c o n n e c t e d with the q u a d r a t i c e q u a t i o n x 3 —ax—a z .~
It is f a r f r o m a m e r e c o i n c i d e n c e t h a t great artists like
L e o n a r d o da Vinci, R a p h a e l , M i c h a e l A n g e l o a n d A l b e r t
D u r e r felt a very great attraction f o r m a t h e m a t i c s . T h e great
a c c u m u l a t i o n of k n o w l e d g e of o u r o w n d a y m a k e s such
manifestations m o r e rare.
T h o s e not belonging to the f o r t u n a t e f e w w h o c a n discern
b e a u t y in m a t h e m a t i c s m a y p e r h a p s learn to perceive m a t h ematics in beauty.
2
s
Art and Mathematics
A ' Mathematics,
Logic,
Music
M a t h e m a t i c s is p r o u d ,
and justly p r o u d , of the logic of its p r o o f s . B u t these proofs
are not an integral part of the m a t h e m a t i c a l doctrine. T h e y
are tools the m a t h e m a t i c i a n uses in o r d e r to achieve his r e sults, they are the vehicle h e m o u n t s to cover the territory
he wants to survey, to c o n t e m p l a t e , to a d m i r e . T h e car you
travel in is not a part of the b e a u t i f u l scenery y o u r j o u r n e y
u n c o v e r s f o r you. T r u e , to the people w h o have m a d e only
a slight and passing a c q u a i n t a n c e with m a t h e m a t i c s , this
subject consists of n o t h i n g but p r o o f s . T h e y are in t h e s a m e
114
MATHEMATICS IN FUN AND IN EARNEST
position as t h e t r a v e l e r w h o h a s h a d t o e x p a n d so m u c h
effort t o m a k e his vehicle go f o r w a r d t h a t he has n o time
a n d n o eyes f o r t h e c o u n t r y s i d e . A n o t h e r p o p u l a r misconception is t h a t m a t h e m a t i c s consists of p r o p o s i t i o n s , of so
m a n y c h o p p e d u p sentences, of so m a n y little pigeon-holes.
T o the initiated, h o w e v e r , m a t h e m a t i c s is just as c o n n e c t e d ,
just as c o n t i n u o u s as a b e a u t i f u l landscape. T h e propositions
a r e singled out as points of orientation, as spots of particular
interest o r a t t r a c t i o n , as elevations f r o m which an unexpected, a n u n u s u a l l y pleasing view m a y be beheld. But
p r o p o s i t i o n s a r e not the w h o l e of m a t h e m a t i c s a n y m o r e t h a n
the elevations of the c o u n t r y s i d e constitute the w h o l e view.
A c o n t e m p o r a r y p h i l o s o p h e r , P r o f e s s o r Scott B u c h a n a n , said:
" T h e s t r u c t u r e s with w h i c h m a t h e m a t i c s deals are like lace,
like the leaves of trees, like the play of light and s h a d o w
on a m e a d o w , o r on a h u m a n f a c e . "
M a t h e m a t i c a l p r o o f s m a y not be a p a r t of the m a t h e matical s t r u c t u r e w h e n the latter is c o m p l e t e d , but they are
o n e of the c h a r m s c o n n e c t e d with the study of the subject.
T h e best m a t h e m a t i c a l p r o o f s are usually short, direct, a n d
penetrating. T h e " i n d e e d " of such a proof has sometimes
the m e l l o w n e s s of a c o n d e s c e n d i n g smile, sometimes the
swiftness of e p i g r a m m a t i c sarcasm a n d sometimes the surprise of a p o i n t e d , witty anecdote. A long m a t h e m a t i c a l
proof m a y lack the directness of a short one, but often it
m a k e s u p f o r it b y h a v i n g the swing and sway, the r h y t h m
of music. It even m a y have the very s t r u c t u r e of a musical
c o m p o s i t i o n . H e r e is a casual, seemingly irrelevant beginn i n g of a train of t h o u g h t carried to a certain point a n d
d r o p p e d , like a m u s i c a l t h e m e developed to a certain degree a n d a b a n d o n e d f o r a n o t h e r , a p p a r e n t l y unrelated t h e m e ;
a n o t h e r line of logical a r g u m e n t , a n d t h e n p e r h a p s also a
third is started, in the s a m e way. T h e n the separate logical
a r g u m e n t s , the muscial t h e m e s , begin to a p p r o a c h each
o t h e r , t o intermingle, to intertwine, t h e n they b e c o m e closely
knit t o g e t h e r , a n d finally burst out in a t r i u m p h a n t finale
of a c h i e v e m e n t .
If y o u g r a n t that t h e r e m a y be m u s i c in m a t h e m a t i c a l
logic, t h e n it is n o t h i n g b u t m a t h e m a t i c a l custom to inquire
w h e t h e r the c o n v e r s e of the proposition is not true, w h e t h e r
a musical c o m p o s i t i o n m a y not resemble a logical reasoning.
T h i s m a y be t o o f a r f e t c h e d , but I, f o r one, a m inclined to
believe that this c o n v e r s e proposition is true. T a k e any masterpiece of musical literature, a short piece, f o r the sake of
simplicity; say, S c h u b e r t ' s " A v e M a r i a , " if you will. F o r g e t
the w o r d s of this song, think of it as played on the violin. It
MATHESIS THE BEAUTIFUL!
Ill
starts o u t with a brief p h r a s e , a simple s t a t e m e n t . T h e n e x t
phrase is m o r e e m p h a t i c , b u t n o t yet sufficiently convincing.
Follow a series of musical sentences, b o u n d t o g e t h e r b y a n
indestructible tie of logical necessity, e a c h m o r e insistent,
each o v e r s h a d o w i n g a n d e n f o r c i n g t h e p r e c e d i n g o n e , until
the highest pitch is r e a c h e d — t h e irresistible a r g u m e n t of ecstasy, a f t e r w h i c h there r e t u r n s the m u s i c a l p h r a s e of t h e
b e g i n n i n g — t h e p r o p o s i t i o n is p r o v e d .
B • Opinions of Mathematicans
about Mathematics
What-
ever m a y be said f o r the b e a u t y of m a t h e m a t i c a l logic, w e
m a y be expected to say s o m e t h i n g a b o u t the interrelation of
m a t h e m a t i c s p r o p e r a n d b e a u t y . But, in the first place, is
there a n y such relation? Is there n o t r a t h e r direct opposition? W h a t , p r a y , c a n rigid, cold, calculating m a t h e m a t i c s
have in c o m m o n with subtle, creative, l o f t y , imaginative art?
This question f a i t h f u l l y m i r r o r s the state of m i n d of m o s t
people, even of m o s t e d u c a t e d people. B u t the great leaders
of m a t h e m a t i c a l t h o u g h t , the c r e a t o r s a n d builders of the
noble edifice of the oldest of sciences, h a v e f r e q u e n t l y a n d
repeatedly asserted t h a t the object of their p u r s u i t s is just as
m u c h a n art as it is a science, a n d p e r h a p s even outright
a fine art.
T h e b e a r e r of o n e of the greatest n a m e s a m o n g A m e r i c a n
m a t h e m a t i c i a n s at the beginning of the c e n t u r y , M a x i m e
B o c h e r ( 1 8 6 7 - 1 9 1 8 ) w r o t e : " I like t o look at m a t h e m a t i c s
almost m o r e as a n art t h a n as a science; f o r the activity of
the m a t h e m a t i c i a n , c o n s t a n t l y creating as h e is, guided alt h o u g h not controlled b y the external w o r l d of senses, bears a
resemblance, not f a n c i f u l , I believe, b u t real, to the activities
of the artist, of a painter, let us say. R i g o r o u s deductive
reasoning o n the part of the m a t h e m a t i c i a n m a y be likened
h e r e t o the technical skill in d r a w i n g on the p a r t of the
painter. Just as o n e c a n n o t b e c o m e a p a i n t e r w i t h o u t a certain a m o u n t of skill, so n o o n e c a n b e c o m e a m a t h e m a t i c i a n
without the p o w e r to r e a s o n a c c u r a t e l y u p to a certain
point. Y e t these qualities, f u n d a m e n t a l t h o u g h t h e y are, d o
not m a k e a p a i n t e r o r a m a t h e m a t i c i a n w o r t h y of the n a m e ,
n o r indeed are they t h e m o s t i m p o r t a n t f a c t o r s in t h e case.
O t h e r qualities of a f a r m o r e subtle sort, chief a m o n g w h i c h
in both cases is i m a g i n a t i o n , go to the m a k i n g of a good
artist o r a good m a t h e m a t i c i a n . "
J. J. Sylvester ( 1 8 1 4 - 1 8 9 7 ) , o n e of the greatest English
m a t h e m a t i c i a n s of t h e n i n e t e e n t h c e n t u r y a n d the first exponent of higher m a t h e m a t i c s in the U n i t e d States, goes so f a r
as to assign a definite place to m a t h e m a t i c s as a fine art. H e
116
MATHEMATICS IN FUN AND IN EARNEST
gives a g e o m e t r i c a l p i c t u r e of t h e m u t u a l relations of t h e
arts.
" S u r e l y , " h e says, " t h e c l a i m of m a t h e m a t i c s t o t a k e place
a m o n g t h e liberal arts m u s t n o w b e a d m i t t e d as fully m a d e
good. It seems t o m e t h a t the w h o l e of esthetics m a y be
r e g a r d e d as a s c h e m e h a v i n g f o u r centers, w h i c h m a y b e
treated as t h e apices of a t e t r a h e d r o n , n a m e l y Epic, Music,
Plastic, a n d M a t h e m a t i c s . T h e r e will b e f o u n d a c o m m o n
p l a n e to e v e r y t h r e e of these, outside of w h i c h lies the
f o u r t h , a n d t h r o u g h e v e r y t w o m a y be d r a w n a c o m m o n
axis opposite to the axis passing t h r o u g h the o t h e r two. So
f a r is d e m o n s t r a b l e . I t h i n k it also possible that there is a
c e n t e r of gravity to each set of three, a n d that the lines
joining e a c h c e n t e r to t h e outside a p e x will intersect in a
c o m m o n p o i n t — t h e c e n t e r of gravity of the w h o l e b o d y of
esthetic; b u t w h a t t h a t c e n t e r is o r m u s t b e I have not h a d
time to think out."
A c o n t e m p o r a r y m a t h e m a t i c i a n , D . P e d o e , reviewing a
b o o k o n t h e " F o u n d a t i o n s of A l g e b r a i c G e o m e t r y " says:
" T h e u l t i m a t e a i m of w o r k e r s o n t h e f o u n d a t i o n s of Algebraic g e o m e t r i e s is to erect an esthetically pleasing structure,
f r e e f r o m logical faults, o n w h i c h t h e m a n y o r n a m e n t s of
Italian g e o m e t r y c a n be tastefully displayed." 3
J u l i a n L . C o o l i d g e ( 1 8 7 3 - 1 9 5 4 ) c o n c l u d e s the p r e f a c e t o
his " T r e a t i s e o n A l g e b r a i c P l a n e C u r v e s " with the f o r t h r i g h t
d e c l a r a t i o n : " T h e p r e s e n t a u t h o r h u m b l y confesses that, to
h i m , g e o m e t r y is n o t h i n g at all, if not a b r a n c h of a r t . "
C • The Opinion of a Poet
P e r h a p s all this s o u n d s like the
p r e a c h i n g of professionals f o r the benefit of their o w n chapel,
as t h e F r e n c h p r o v e r b says. T o allay such suspicions I
could m e n t i o n to t h e s a m e effect the writings of E m e r s o n ,
T h o r e a u , a n d o t h e r s . I will limit myself to q u o t i n g a sonnet
f r o m the p e n of o n e of the most talented c o n t e m p o r a r y
A m e r i c a n poets, E d n a St. V i n c e n t Millay.
SONNET
E u c l i d alone h a s l o o k e d o n B e a u t y b a r e .
L e t all w h o p r a t e o n B e a u t y hold their peace.
A n d lay t h e m p r o n e u p o n t h e earth a n d cease
T o p o n d e r o n themselves, t h e while they stare
A t n o t h i n g , intricately d r a w n n o w h e r e
I n s h a p e of s h i f t i n g lineage; let geese
G a b b l e a n d hiss, b u t h e r o e s seek release
Ill
MATHESIS THE BEAUTIFUL!
F r o m dusty b o n d a g e i n t o l u m i n o u s air.
O blinding h o u r , O holy, terrible day,
W h e n first t h e s h a f t i n t o his vision s h o n e
Of light a n a t o m i z e d . E u c l i d a l o n e
H a s l o o k e d o n B e a u t y b a r e . F o r t u n a t e they
W h o , t h o u g h o n c e only a n d t h e n f r o m away,
H a v e h e a r d h e r massive s a n d a l set o n stone.
D • Mathematics—a
Creation of the Imagination
The ob-
jection m a y be m a d e t h a t , a f t e r all, these are t h e subjective
reactions of s o m e people, be they m a t h e m a t i c i a n s o r poets.
If m a t h e m a t i c s w a n t s to lay claim to being a n art, it m u s t ,
in a n objective way, s h o w t h a t it possesses a n d m a k e s use of
at least s o m e of t h e e l e m e n t s w h i c h go to m a k e u p the
things of b e a u t y . W h a t is t h e m o s t essential e l e m e n t of a n
art? Is it not i m a g i n a t i o n , creative i m a g i n a t i o n ? T h a t m a t h e matics exhibits a b u n d a n t l y this p r e c i o u s quality h a s b e e n
pointed out in t h e passage w h i c h I q u o t e d a while ago. B u t
it w o u l d be good to insist o n this subject a little m o r e .
L e t us t a k e a n y o b j e c t of g e o m e t r i c a l study, say, the
circle. T o the so-called " m a n in the s t r e e t " this is a r i m of a
wheel, with, p e r h a p s , spokes in it. E l e m e n t a r y g e o m e t r y has
c r o w d e d this simple figure with radii, diameters, c h o r d s ,
sectors, tangents, inscribed a n d c i r c u m s c r i b e d polygons, a
t r a n s c e n d e n t a l r a t i o of t h e c i r c u m f e r e n c e t o the d i a m e t e r ,
a n d so on. By a f u r t h e r flight of t h e i m a g i n a t i o n it h a s
w r a p p e d a r o u n d t h e circle all the lines a n d all the points of
t h e p l a n e in the f a n c i f u l t h e o r y of poles a n d polars. H e r e y o u
h a v e already a n entire g e o m e t r i c a l w o r l d c r e a t e d f r o m a
very r u d i m e n t a r y beginning. A s if this w e r e not e n o u g h ,
the m a t h e m a t i c i a n h a s b y a f u r t h e r effort discovered t w o
imaginary points o n every circle, the s a m e t w o points o n all
t h e circles of the plane, a n d , f o r the sake of good m e a s u r e ,
I suppose, he p l a c e d t h e s e t w o points o n the u n a t t a i n a b l e
b o u n d a r y of t h e plane, a n d christened t h e m " t h e circular
points at infinity." T h i s is u n d e n i a b l e proof of t h e c r e a tive p o w e r of t h e m a t h e m a t i c i a n , f o r n o m o r t a l eye h a s
ever beheld these points, o r ever will. But as t h o u g h to show
y o u that n o t h i n g c o u l d stop h i m , the m a t h e m a t i c i a n allows
the whole circle t o vanish, declares it t o be " i m a g i n a r y , "
a n d then keeps o n toying with his n e w c r e a t i o n in m u c h t h e
same w a y a n d with m u c h the s a m e gusto as he did with the
innocent little t h i n g y o u allowed h i m to start o u t with. A n d
all this, r e m e m b e r , please, is just e l e m e n t a r y plane geometry.
Starting with o r d i n a r y integers, tools of c o m m o n c o u n t i n g ,
118
MATHEMATICS IN FUN AND IN EARNEST
t h e m a t h e m a t i c i a n a d d s to t h e m , in rapid succession, f r a c tional n u m b e r s , irrational n u m b e r s , t r a n s c e n d e n t a l n u m b e r s .
If t h e s e n u m b e r s still h a v e s o m e t h i n g of the so-called " r e a l "
in t h e m , t h e m a t h e m a t i c a l i m a g i n a t i o n c a n treat you to complex n u m b e r s , a n d if t h a t be not e n o u g h , to ideal n u m b e r s .
B u t w i t h all that the m a t h e m a t i c i a n finds that this is too
h a m p e r i n g , t o o confining. H e cuts loose f r o m all n u m b e r s .
S u p r e m e l y c o n f i d e n t in his creative genius, he declares, like
a n Olympic g o d : " L e t t h e r e be c h a o s and let m e have a
g r o u p of objects of any kind w h a t e v e r in the chaotic w o r l d . "
T h e n he begins to p r o m u l g a t e laws by which these objects
( o r shall we say subjects?) shall be governed. H e calls these
laws relations, like reflexive, transitive, asymmetric, etc.
P r e t t y s o o n he h a s w h a t h e calls a t h e o r y of aggregates. F u r t h e r h e c r e a t e s such relations as " b e t w e e n n e s s , " " t o the
r i g h t , " etc., a n d , lo a n d b e h o l d , h e r e h e has b r o u g h t o r d e r
o u t of chaos, a n d o n e of these o r d e r s is the very n u m b e r
system he used to a m u s e himself with, w h e n the world was
y o u n g . But t h e r e is n o need f o r m e to go to these abstract
d o m a i n s to see t h e m a t h e m a t i c a l i m a g i n a t i o n at w o r k . It is
e n o u g h to t a k e c o g n i z a n c e of the i m m e n s e literature massed
a r o u n d t h e e l e m e n t a r y t h e o r y of n u m b e r s , w h i c h theory
deals only with the o r d i n a r y integers, in o r d e r to get an idea
of s o m e t h i n g w h i c h is incredible but true. T a k e the theory
of g r o u p s , t a k e the elusive a n d yet enticing theory of probabilities, a n d so on. T r u l y , the creative imagination displayed
b y the m a t h e m a t i c i a n h a s n o w h e r e been exceeded, not even
paralleled, a n d I w o u l d m a k e bold to say, not even closely
a p p r o a c h e d a n y w h e r e else.
E ' Further Analogies Between Mathematics and the Arts:
Symbolism, Condensation, Care in Execution, Etc. In many
w a y s m a t h e m a t i c s exhibits the s a m e e l e m e n t s of beauty that
are generally a c k n o w l e d g e d to be the essence of poetry. Let
us first c o n s i d e r a m i n o r point. T h e poet arranges his writings
o n t h e p a g e in verses. H i s p o e m first appeals to the eye bef o r e it r e a c h e s the e a r or the m i n d . Similarly the m a t h e m a t i c i a n lines u p his f o r m u l a s a n d his e q u a t i o n s so that their
f o r m m a y m a k e an esthetic impression.
S o m e m a t h e m a t i c i a n s are given to this love of arranging
a n d exhibiting their e q u a t i o n s to a degree which borders on
a fault. T r i g o n o m e t r y , a b r a n c h of e l e m e n t a r y m a t h e m a t i c s
p a r t i c u l a r l y rich in f o r m u l a s , offers s o m e curious groups of
t h e m , c u r i o u s in their s y m m e t r y a n d their a r r a n g e m e n t . H e r e
is o n e s u c h g r o u p .
Ill
MATHESIS THE BEAUTIFUL!
Sin
Cos
Sin
Cos
(a+b)=Sin
(a+b)=Cos
(a—b) =Sin
(a—b)=Cos
a Cos b + Cos
a Cos b—Sin
a Cos b—Cos
a Cos b + S i n
a
a
a
a
Sin
Sin
Sin
Sin
b
b
b
b
T h e superiority of p o e t r y over o t h e r f o r m s of v e r b a l expression lies first in the s y m b o l i s m of p o e t r y , a n d secondly
in its e x t r e m e c o n d e n s a t i o n , in its studied e c o n o m y of words.
T a k e a n y of y o u r f a v o r i t e p o e m s , o r let us c h o o s e o n e of
universally a c k n o w l e d g e d m e r i t , o n e w e all k n o w , say, Shelley's p o e m " t o N i g h t . " H e r e is the s e c o n d s t a n z a :
W r a p thy f o r m in a m a n t l e g r a y ,
Star—inwrought.
Blind with thine h a i r t h e eyes of D a y ;
Kiss h e r until she be w e a r i e d o u t ;
T h e n w a n d e r o ' e r city a n d sea a n d l a n d ,
T o u c h i n g all w i t h t h i n e opiate w a n d —
C o m e , long sought!
T a k e n literally all this is, of course, sheer nonsense, a n d
n o t h i n g else. N i g h t h a s n o hair, night does not w e a r any
clothes a n d night is not a n illicit p e d d l e r of narcotics. B u t is
there a n y b o d y b a l m y e n o u g h to t a k e the w o r d s of the p o e t
literally? T h e w o r d s h e r e are only c o m p a r i s o n s , only symbols.
F o r the sake of c o n d e n s a t i o n the p o e t omits to state t h a t
they are such a n d goes o n t o treat his s y m b o l s as realities.
This e c o n o m y of w o r d s is clearly a p p a r e n t in the line " s t a r
i n w r o u g h t . " Just think of all the sentences y o u w o u l d h a v e to
string out, if y o u w a n t e d to c o n v e y t h e s a m e idea, a n d state
explicitly all the similes involved.
T h e m a t h e m a t i c i a n does things precisely as does the poet.
T a k e the field of n u m b e r s . T o begin with, the very idea of a
n u m b e r is an abstraction, a symbol. W h e n you write the
figure 3, y o u h a v e c r e a t e d a s y m b o l f o r a symbol, a n d
w h e n you say, in algebra, t h a t a is a n u m b e r , you h a v e condensed all the symbols f o r all the n u m b e r s into o n e alle m b r a c i n g s y m b o l . W h e n y o u f u r t h e r w r i t e a m , the n u m b e r
m b e c o m e s a symbol o n c e r e m o v e d again. T h e n the m a t h e matician b e c o m e s so confident in his symbols that h e p u t s
t h e m to uses w h i c h h e n e v e r c o n t e m p l a t e d a n d he o f t e n c a n
not tell himself h o w f a r these s y m b o l s are r e m o v e d f r o m
their original m e a n i n g . T h i n k of the use of e x p o n e n t s as
o r d e r of differentiation, o r of derivatives of f r a c t i o n a l orders,
to t a k e relatively simple cases.
As a s y m b o l of a different d o m a i n , t a k e the circular
points at infinity r e f e r r e d t o b e f o r e . If y o u t a k e t h e state-
120
MATHEMATICS IN FUN AND IN EARNEST
m e n t s c o n c e r n i n g t h e m in a literal sense, they a r e obviously
a b s u r d . B u t t h e y are p e r f e c t l y intelligible as symbols. It
w o u l d t a k e us t o o f a r afield to exhibit their precise meaning.
Suffice it t o say t h a t they are a c o n d e n s e d , c o n c e n t r a t e d
w a y of stating a l o n g a n d c o m p l i c a t e d c h a i n of r a t h e r simple
g e o m e t r i c a l relations.
T h i s brings m e b a c k to the e c o n o m y of words, a virtue
t h e m a t h e m a t i c i a n h a s raised to t h e heights of a creed,
against w h i c h n o transgressions c a n be tolerated. H e went so
f a r as to d o a w a y with w o r d s altogether, or a p p r o x i m a t e l y
so, a n d to r e p l a c e t h e m by special n o t a t i o n s and symbols.
T h i s gives his s t a t e m e n t s a concision a n d a n elegance quite
inimitable. H e has, h o w e v e r , to p a y the penalty that his writings are i n c o m p r e h e n s i b l e to t h e non-initiate.
A n o t h e r side b y w h i c h m a t h e m a t i c s a p p r o a c h e s is the
c a r e it exercises in r e g a r d to t e c h n i q u e of execution. Y o u d o
n o t e n j o y a p o e m w h i c h is strained in the choice of words,
w h e r e t h e r h y m e s a r e f o r c e d , a p o e m w h i c h bears o n its
f a c e t h e m a r k s of t h e l a b o r of the poet. N o t that the poet is
e x p e c t e d t o p r o d u c e his p o e m w i t h o u t effort. F e w poets do,
a n d those only occasionally. W e all k n o w the stories a b o u t
t h e p a i n s poets of great r e n o w n h a v e t a k e n with their works,
with e a c h individual p o e m , with each line of the p o e m . B u t
t h e result m u s t be such t h a t those labors are hidden b e h i n d
a n a p p e a r a n c e of effortless ease. It is o n l y then that you
will g r a n t t h a t the p o e m is b e a u t i f u l . T h e s a m e is n o less
t r u e of a n y o t h e r art. In the p e r f o r m a n c e of music we go
so f a r as t o e n j o y a piece of r a t h e r m e d i o c r e quality, if the
piece presents c o n s i d e r a b l e technical difficulties a n d the perf o r m e r c a n s h o w t h a t t h e y cause h i m n o e m b a r r a s s m e n t .
M a t h e m a t i c i a n s are just as exacting with their t e c h n i q u e
of e x e c u t i o n as a n y p o e t or artist is. T h e y are constantly preo c c u p i e d with t h e elegance of their p r o o f s o r of the solutions
of their p r o b l e m s . A n y m a t h e m a t i c i a n will immediately assign t o the s c r a p h e a p a n y of his p r o o f s , if he c a n think of
a n o t h e r w a y t o get the s a m e result with less a p p a r a n t e f f o r t , with the accent on the w o r d " a p p a r e n t . " H e does not
hesitate to s p e n d a great deal of extra t i m e o n the solutions
h e h a s a l r e a d y , if he has any inkling that he m a y abbreviate
o r s i m p l i f y these solutions. A n d w h e n he succeeds, w h e n he
has f o u n d this simplicity, he h a s the esthetic satisfaction of
h a v i n g b r o u g h t f o r t h a n elegant solution. N o r is this effort
limited to the individual. M a t h e m a t i c i a n s as a profession are
always at w o r k m a k i n g the exposition of their science esthetically m o r e satisfying. T h e success t h e y achieve in this labor
MATHESIS THE BEAUTIFUL!
Ill
is o f t e n r e m a r k a b l e . S o m e of the results w h i c h t h e original
discoverers h a v e o b t a i n e d in a most l a b o r i o u s w a y , m a k i n g
use of the m o s t a d v a n c e d a n d c o m p l i c a t e d b r a n c h e s of t h e
science, m a y b e c o m e , within a g e n e r a t i o n o r two, very simple, very elegant, a n d b a s e d o n a l m o s t e l e m e n t a r y c o n s i d e r a tions. T h e b e a u t y of this new w a y of e x e c u t i o n b e c o m e s
t h e n the joy a n d the p r i d e of the p r o f e s s i o n .
T h e m e t h o d s of w o r k of the m a t h e m a t i c i a n a n d especially
of the geometrician are very m u c h like the m e t h o d s of w o r k
of the poet. A physicist, a biologist, n e e d s a l a b o r a t o r y to
c a r r y o n his w o r k . A p a i n t e r n e e d s b r u s h e s a n d paints, a n d
canvas, a n d a studio as well; a n architect is still m o r e exacting. But the geometrician, like t h e poet, needs n o t h i n g at
all f o r his w o r k , b e y o n d a s c r a p of p a p e r a n d a pencil to
help out his i m a g i n a t i o n by a r o u g h a n d f r a g m e n t a r y
sketch of the fleeting a n d c o m p l e x creations he allows his
f a n c y to play with. T h e geometrician, like t h e poet, is a
d r e a m e r , a n incorrigible d a y d r e a m e r . Y o u m a y accuse b o t h
of t h e m of a b s e n t - m i n d e d n e s s if you will, b u t neither of
t h e m w o u l d give u p his d a y d r e a m s f o r a n y t h i n g t h a t t h e
world could offer in e x c h a n g e . T h e s e solitary d r e a m s , these
soaring flights of the excited i m a g i n a t i o n , m a k e the geometrician, as they m a k e the poet, oblivious to e v e r y t h i n g
a r o u n d h i m , m a k e h i m forget his duties, his friends, his o w n
self. But they are to h i m the m o s t c h e r i s h e d h a p p e n i n g s ,
the most precious m o m e n t s of his life.
F ' "Movements"
in Mathematics
M a t h e m a t i c s resembles
imaginative literature a n d t h e fine arts in general in t h a t ,
taken historically, it has, like the arts, its " m o v e m e n t s . " D u r ing the s e c o n d q u a r t e r of the seventeenth c e n t u r y analytic
g e o m e t r y c a m e into being. W i t h i n a s h o r t space of time this
" m o d e r n " g e o m e t r y b e c a m e the height of f a s h i o n in the
m a t h e m a t i c a l world. T h e r e p u t e d l y poised, s o b e r - m i n d e d ,
m a t t e r - o f - f a c t m a t h e m a t i c i a n s b e c a m e so i n f a t u a t e d with
their newly a c q u i r e d plaything, the C a r t e s i a n g e o m e t r y , t h a t
they h a d n o t i m e a n d n o use f o r a n y t h i n g t h a t recalled the
synthetic " m e t h o d of t h e a n c i e n t s . " T h e splendid c o n t r i b u tions of Pascal, D e s a r g u e s , de la H i r e , w e r e consigned t o
a c c u m u l a t i n g dust f o r a c e n t u r y a n d a half.
This " m o v e m e n t " m a y p e r h a p s be called the " s y m b o l i c "
m o v e m e n t in g e o m e t r y f o r it m a r k s the period of i n t r o d u c t i o n
of algebraic symbols into the study of g e o m e t r y . T h e t u r n
between the eighteenth a n d the n i n e t e e n t h centuries m a r k s
the beginning of a n o t h e r " m o v e m e n t " in g e o m e t r y w h i c h
122
MATHEMATICS IN FUN AND IN EARNEST
m a y be called t h e " r o m a n t i c " m o v e m e n t , f o r it corresponds,
in t i m e , to the r o m a n t i c m o v e m e n t in literature. J. L. Coolidge, in a n a d d r e s s b e f o r e the A m e r i c a n M a t h e m a t i c a l Society, called a s o m e w h a t later phase of the s a m e m o v e m e n t
the " h e r o i c a g e " of g e o m e t r y . T h e p r e c u r s o r s of the movem e n t w e r e restored to their r i g h t f u l place, a n d the enthusia s m f o r t h e new, projective g e o m e t r y was just as great and
just as exclusive as in the case of the " s y m b o l i c " m o v e m e n t .
R u n n i n g t r u e to f o r m , the old school accused the new movem e n t of m a t h e m a t i c a l heresy a n d d e n o u n c e d bitterly P o n c e let's " p r i n c i p l e of c o n t i n u i t y . "
G ' Conclusion
W e h a v e insisted on the c o m m o n features
of m a t h e m a t i c s and p o e t r y . O n e might think that w e have
o v e r l o o k e d the differences. H e r e is the w a y P r o f . B u c h a n a n ,
to w h o m I h a v e a l r e a d y r e f e r r e d , f o r m u l a t e s these differences. " T h e m a t h e m a t i c i a n sees and deals in relations; the
p o e t sees a n d deals in qualities. M a t h e m a t i c s is analytic, seeing w h o l e s as systems of relations; poetry is synthetic, seeing
w h o l e s as s i m p l e qualities. T h e qualities that a poet sees
a r e d u e to relations, says t h e m a t h e m a t i c i a n . T h e y need
p u r g a t i o n . T h e relations t h a t t h e m a t h e m a t i c i a n sees are concrete a n d f a c t u a l , says the poet. T h e y need appreciation
and love."
It is o f t e n said that m a t h e m a t i c s should be studied for
its u s e f u l n e s s . T h i s is quite right. It s h o u l d be studied f o r the
philosophic insight it a f f o r d s , a n d m o r e so for the logic it
uses a n d creates. B u t p e r h a p s its m a i n claim to y o u r attention is based o n the intrinsic b e a u t y it reveals to those w h o
c a n see it. T h e poet s a i d : " B e a u t y is its o w n excuse f o r being." I w o u l d a d d t o t h i s : The cultivation
of beauty is its
own
reward.
FOOTNOTES
1
Matematica y Poesia (essayo y entusiasmo), Arturo Aldunate
Phillip? (Editiones Ercill, 1940).
2
Those interested in this subject may consult:
Caskey, L. D., Geometry of the Greek Vases.
Ghyka, Matilla C., Esthetique des proportions dans la nature
et dans les arts.
3
294.
Hamridge, Jay, Dynamic
Symmetry.
Review by D. Pedoe, Mathematical Gazette, Vol. 31, 1947, p.
V
MATHEMATICS AND T H E MATHEMATICIAN
1
•
A
Is Mathematics an Exact Science
* Mathematicians
Are
Human
A • A Definition
of Mathematics
T h e f a m o u s eighteenthcentury Encyclopedic
Methodique
gave the following definition of m a t h e m a t i c s : C'est la science qui a pour objet les
proprietes de la grandeur en tant quelle
est calculable ou
mesurable.
(It is the science which has f o r its object the p r o p erties of m a g n i t u d e i n a s m u c h as they arc calculable o r m e a s urable.) Precise, concise, definite, and simple. This was in
1787. Even t h o u g h this definition was a d e q u a t e f o r the time,
it was not destined to r e m a i n so very long. T w o decades
earlier G a s p a r M o n g e ( 1 7 4 6 - 1 8 1 8 ) h a d invented descriptive
geometry. H e did not publish his results until 1795 b e c a u s e
f o r over a q u a r t e r of a c e n t u r y the F r e n c h H i g h C o m m a n d
considered descriptive g e o m e t r y its o w n private military secret. M o n g e ' s invention led his pupils to the c r e a t i o n of p r o jective g e o m e t r y , a b r a n c h of m a t h e m a t i c s that does n o t
deal with magnitude. 1 T h e q u a n t i t a t i v e c o n c e p t i o n of m a t h e matics thus b e c a m e obsolete. M a n y efforts h a v e been m a d e
since to find a definition that w o u l d e m b r a c e all of m a t h e matics. T h e e n o r m o u s growth of the science d u r i n g the past
century and a half, and the inclusion of such b r a n c h e s as
the theory of groups, topology, a n d symbolic logic, r e n d e r e d
all such a t t e m p t s u n s a t i s f a c t o r y . T h e hopeless task w a s finally given u p in f a v o r of simply saying that m a t h e m a t i c s
is what m a t h e m a t i c i a n s are doing.
B ' Is the Mathematician
"Objective"?
H o w do mathematicians acquit themselves of the h e a v y responsibilities that
such a definition puts u p o n their shoulders? T h e y h a v e t h e
advantage that they start out with a great a m o u n t of credit.
T o the l a y m a n m a t h e m a t i c s is s y n o n y m o u s with exactness,
nay, with certainty. M a t h e m a t i c s is precise, m a t h e m a t i c s
proves all the assertions it m a k e s , all the propositions it a d vances. A n d b o o k s written by m a t h e m a t i c i a n s seem to b e a r
out the l a y m a n ' s opinion a b o u t the a u t h o r s . T h e s e b o o k s
seem to be written with c o m p l e t e d e t a c h m e n t a n d strict objectivity. T h e r e is not a single e x c l a m a t i o n point to be f o u n d
on any of their pages, except w h e n it is used as a symbol f o r
a factorial. But d o m a t h e m a t i c i a n s actually d o their w o r k
123
124
MATHEMATICS IN FUN AND IN EARNEST
w i t h t h a t O l y m p i c impartiality t h a t the final p r o d u c t seems
t o exhibit?
A s k t h e m a t h e m a t i c i a n w o r t h his chalk w h y he spends
so m u c h t i m e a n d effort o n his research, a n d he will almost
i n v a r i a b l y tell y o u — q u i t e t r u t h f u l l y s o — t h a t he does it because h e finds it very interesting, because he loves to d o it,
b e c a u s e to h i m it is a m o s t exciting a d v e n t u r e . Sentimental
r e a s o n s all. B u t is s e n t i m e n t a reliable p a r t n e r of objectivity?
C • Priority
Disputes
E v e r y active m a t h e m a t i c i a n will
readily agree t h a t he is t r y i n g b y his efforts to p r o m o t e a n d
a d v a n c e the science of his choice. T h e r e is n o doubt that he
is telling t h e t r u t h a n d t h a t h e is quite sincere a b o u t it. But is
it " t h e w h o l e t r u t h a n d n o t h i n g but the t r u t h ? " If it were,
if t h e m a t h e m a t i c i a n w e r e interested in the p r o m o t i o n of his
science in a p u r e l y objective way, it w o u l d m a k e n o difference to h i m w h e t h e r it was A o r B t h a t took a given forw a r d step, as long as the a d v a n c e h a d been accomplished.
B u t this is n o t t h e case, as is a b u n d a n t l y p r o v e d by the
historically f a m o u s , a n d disgraceful, controversies over prio r i t y rights of m a t h e m a t i c a l inventions. T h e N e w t o n - L e i b n i z
q u a r r e l o v e r the invention of t h e calculus was just as bitter
as it w a s h a r m f u l . It actually h i n d e r e d the progress of the
calculus in Britain f o r o v e r a century.
T h e dispute b e t w e e n P o n c e l e t a n d G e r g o n n e as to w h o
w a s the r i g h t f u l o w n e r of the title to the invention of the
p r i n c i p l e of duality m a y h a v e yielded in scope to the N e w t o n - L e i b n i z c o n t r o v e r s y , b u t it was fully as acrimonious, if
n o t worse. 2 O n e could cite the q u a r r e l between Descartes
a n d F e r m a t , b e t w e e n A . M . L e n g e n d r e ( 1 7 5 2 - 1 8 3 3 ) and
C . F . G a u s s ( 1 7 7 7 - 1 8 5 5 ) , a n d so o n a n d on, ad
nauseam.
C a r d a n ( 1 5 0 7 - 1 5 7 6 ) o b t a i n e d f r o m Tartaglia ( 1 5 0 0 - 1 5 5 7 )
the solution of the cubic e q u a t i o n u n d e r oath of secrecy a n d
t h e n n o t only p u b l i s h e d t h e solution but claimed it as his
o w n . O u r m e t h o d s m a y n o t be as c r u d e , but w e are as
jealous of o u r priority rights n o w as a n y b o d y ever was.
E d i t o r s seem to t h i n k that priority claims are established by
t h e date a given article reaches their desk, and publish this
d a t e as part of the article. P e r h a p s w h e t h e r it was A o r B
t h a t m a d e a c o n t r i b u t i o n m a y not be of so m u c h m o m e n t ,
b u t w h e t h e r it w a s I o r not 1 is of t r e m e n d o u s i m p o r t a n c e .
T h e s u b l i m e i n d i f f e r e n c e t o w a r d public acclaim exhibited by
a F e r m a t does not seem to be of this planet. It m a y be
a r g u e d that m a t h e m a t i c i a n s as a rule get little else f o r their
labors; they are t h e r e f o r e at least entitled to the h o n o r and
recognition their a c c o m p l i s h m e n t s c a n bring t h e m . This is
MATHEMATICS AND THE MATHEMATICIAN
125
t r u e e n o u g h . B u t it is a w e a k a r g u m e n t in f a v o r of t h e s u p posed d e t a c h m e n t a n d objectivity w i t h w h i c h m a t h e m a t i c i a n s
view their w o r k .
D ' Withholding
Results T h e r e are cases o n record w h e n
m a t h e m a t i c i a n s w e r e r e l u c t a n t t o publish t h e results of t h e i r
findings, their reticence m o t i v a t e d b y their solicitude f o r their
science. W h e n the researches of the P y t h a g o r e a n s b r o u g h t
t h e m f a c e to f a c e with irrational n u m b e r s , they w e r e overw h e l m e d b y their discovery. It c o n t r a d i c t e d the f u n d a m e n t a l
tenet of their p h i l o s o p h y t h a t everything is ( r a t i o n a l ) n u m b e r .
T h e surest w a y o u t w a s to m a k e of this t r o u b l e s o m e result a
professional secret a n d to i n d u c e the gods to destroy a n y o n e
w h o w o u l d d a r e to divulge to t h e lay c r o w d t h e exclusive
w i s d o m with w h i c h only the initiates c o u l d be trusted.
W e h a v e a similar e x a m p l e in m o d e r n times. G a u s s was
in possession of n o n - E u c l i d e a n g e o m e t r y a h e a d of b o t h L o b a cevskii a n d J a n o s Bolyai ( 1 8 0 2 - 1 8 6 0 ) , b u t he was l o a t h t o
publish his results. H e f e a r e d that such a n u n o r t h o d o x discovery m i g h t u n d e r m i n e the f a i t h of t h e y o u n g in t h e validity
of m a t h e m a t i c s in general.
T h e j u d g m e n t of b o t h the P y t h a g o r e a n s a n d of G a u s s as
to the effect of t h e i r discoveries u p o n the d e v e l o p m e n t of
m a t h e m a t i c s w a s totally w r o n g . B u t this is h e r e quite beside
the point. W h a t is i m p o r t a n t t o n o t e in this c o n n e c t i o n is t h a t
the concealing of the t r u t h is h a r d l y t h e p r o p e r m e t h o d t o
inspire c o n f i d e n c e in t h e exactness of the science o n e is t r y i n g
to p r o m o t e .
E • Mistakes of Mathematicians
F o r m e n w h o are supposedly dealing with an exact science, the n u m b e r of mistakes
m a t h e m a t i c i a n s m a k e is b o t h puzzling a n d disconcerting. T h e
Belgian m a t h e m a t i c i a n M a u r i c e L e c a t p u b l i s h e d a collection
of Erreurs des Mathematiciens.3
T h e list of n a m e s m e n t i o n e d
looks pretty m u c h like a " W h o ' s W h o in M a t h e m a t i c s . "
H e n r i P o i n c a r e w a s a w a r d e d a prize f o r a p a p e r t h a t h a d a
serious m i s t a k e in it. H e detected the e r r o r himself while his
p a p e r w a s in the process of being published, b u t it was t o o
late to r e m e d y t h e situation, a n d t h e K i n g of S w e d e n f o r mally c o n f e r r e d u p o n the a u t h o r a prize f o r a p a p e r t h a t
was w r o n g .
F • Disputes
Over Results Obtained
I n an exact science
it should be easy to evaluate the m e r i t s of a p a p e r , a n d
experts in the p r o f e s s i o n s h o u l d be able to decide w h i c h of
several solutions of the s a m e p r o b l e m is the c o r r e c t one. B u t
126
MATHEMATICS IN FUN AND IN EARNEST
this is only t o o o f t e n not t h e case. H e r e is o n e example,
of m a n y t h a t could b e q u o t e d . I n a p a p e r Fourier's
Series,
p u b l i s h e d b y the M a t h e m a t i c a l Association of A m e r i c a , R . E .
L a n g e r ( b . 1 8 9 4 ) relates a c o n t r o v e r s y participated in b y
d ' A l e m b e r t ( 1 7 1 7 - 1 7 9 3 ) , E u l e r a n d D a n i e l Bernoulli ( 1 7 0 0 1 7 8 2 ) . E a c h of t h e s e l u m i n a r i e s w r o t e a p a p e r o n the p r o b l e m of v i b r a t i n g strings. T h e t h r e e - c o r n e r e d polemic lasted
m o r e t h a n a d e c a d e . T h e only p o i n t of a g r e e m e n t that
e m e r g e d clearly was t h a t t h e r e was always a two-to-one
m a j o r i t y t h a t t h e t h i r d p a r t y was w r o n g . H u m a n , all t o o
h u m a n . B u t w h e r e does t h e exact science c o m e in?
B
'
Schools of Thought in Mathematics
G • The Quest for Rigor
I n spite of all these foibles,
m a t h e m a t i c i a n s m o u n t a vigilant a n d jealous guard over
t h e exactness of their science a n d a r e not a bit sparing of
o n e a n o t h e r w h e n the impeccability of t h a t science comes
into question.
T h e i n v e n t i o n of t h e calculus p r o v o k e d a flood of criticism as t o t h e m a t h e m a t i c a l a n d logical soundness of the
n e w d o c t r i n e . N e i t h e r N e w t o n n o r Leibniz was quite c o n v i n c e d t h a t the r e p r o a c h e s w e r e groundless, b u t they f o u n d
n o w a y of disposing of t h e m . 4 L e o n a r d E u l e r ( 1 7 0 7 - 8 3 ) , their
m o s t distinguished i m m e d i a t e c o n t i n u a t o r , paid still less att e n t i o n t o this c o n t r o v e r s y . H e used his great gifts to exp a n d a n d e n r i c h the w o r k of his illustrious mentors, and
his u n e r r i n g instinct f o r w h a t w a s right kept h i m firmly
o n t h e straight p a t h . H o w e v e r , L a g r a n g e ( 1 7 3 7 - 1 8 1 3 ) , a
y o u n g e r c o n t e m p o r a r y of E u l e r , did not s h a r e the faith of the
c o u r t i e r of t h e czars of Russia in the f o r m a l i s m of m a t h e m a t i c s . In L a g r a n g e ' s e s t i m a t i o n Euler's calculus "did not
m a k e sense."
T h e m a t h e m a t i c a l analysis b e q u e a t h e d by the eighteenth
c e n t u r y a p p e a r e d to the m a t h e m a t i c i a n s of the early ninet e e n t h c e n t u r y to be a s t r u c t u r e totally devoid of any f o u n d a t i o n . U n d e r the l e a d e r s h i p of A. L. C a u c h y ( 1 7 8 9 - 1 8 5 7 )
t h e y u n d e r t o o k to p r o v i d e analysis with u n d e r p i n n i n g s solid
e n o u g h t o r e n d e r this b r a n c h of m a t h e m a t i c s impervious to
t h e m o s t e x a c t i n g criticism a n d at the s a m e time to safeg u a r d the results of m a t h e m a t i c a l analysis f r o m all possible
errors.
T h u s c a m e into b e i n g the school of rigor of the first half
of t h e n i n e t e e n t h c e n t u r y . It a c c o m p l i s h e d a great deal, but
its a c h i e v e m e n t s w e r e a n y t h i n g but final. T h e second half of
the n i n e t e e n t h c e n t u r y set n e w goals f o r vigor. A n a t t e m p t
MATHEMATICS AND THE MATHEMATICIAN
127
was m a d e to " a r i t h m e t i z e " m a t h e m a t i c a l analysis. J. W . R .
D e d e k i n d ( 1 8 3 1 - 1 9 1 6 ) p r o d u c e d his t h e o r y of irrational
n u m b e r s , G e o r g C a n t o r the t h e o r y of p o i n t sets, a n d so o n .
A n d the quest f o r rigor is still on the m a r c h . W h a t satisfies
the m o s t rigid r e q u i r e m e n t s of o n e g e n e r a t i o n of m a t h e maticians seems totally i n a d e q u a t e f o r t h e next. E . H .
M o o r e ( 1 8 6 2 - 1 9 3 2 ) , f o r m a n y years p r o f e s s o r of analysis
at the University of C h i c a g o , expressed this in the apt
a d a p t a t i o n of a biblical p h r a s e : "Sufficient u n t o the day is
the rigor t h e r e o f . " It w o u l d seem, h o w e v e r , that m a t h e m a t i cal rigor is a very elusive thing. T h e h a r d e r it is p u r s u e d ,
the m o r e adroitly it evades the p u r s u e r . In spite of all t h e
advances that the nineteenth c e n t u r y c o n t r i b u t e d t o w a r d m a t h ematical rigor, the m a t h e m a t i c i a n s of the present g e n e r a t i o n
feel that they are m o r e " u p in the a i r " t h a n a n y o t h e r generation ever was.
H ' Euclid and the "Obvious" Foundations
of
Mathemat-
ics. A s a t e x t b o o k Euclid's Elements
h a s n o rival, not only
in m a t h e m a t i c s , but in a n y o t h e r subject. M o r e people over
m o r e centuries h a v e learned their g e o m e t r y f r o m that b o o k
than have l e a r n e d a n y o t h e r subject f r o m a n y o t h e r single
b o o k , with the exception of the Bible. A n d yet this is not
the greatest of the merits of the b o o k . T h e great role t h a t
this book played in the cultural history of m a n k i n d is d u e
to the f a c t that Euclid's Elements
was the first m o d e l of a
deductive science. Euclid begins by defining the entities he is
going to c o n s i d e r : point, line, angle, etc. T h e n he lines u p
his axioms a n d his postulates, i.e., those p r o p o s t i o n s that
he accepts as valid on a c c o u n t of their plausibility o r " o b viousness." All the propositions that follow are derived
f r o m those a s s u m e d b y p u r e reasoning, a c c o r d i n g to the
strict precepts of logic. F o r s o m e two t h o u s a n d years t h e r e
was n o t h i n g t h a t a p p r o a c h e d E u c l i d ' s m o d e l in p e r f e c t i o n . 5
I • Formalism
It is a q u e e r irony of o u r intellectual history that it is precisely this p e r f e c t i o n of Euclid's g e o m e t r y
that inspired the invention of n o n - E u c l i d e a n g e o m e t r y . All
t h r o u g h the ages students of g e o m e t r y felt that Euclid's p a r allel postulate was not sufficiently obvious. N o w a blemish
on the p e r f e c t w o r k of Euclid was an insufferable thing
which h a d to be r e m o v e d . T h e simplest and surest w a y t o
achieve this aim was to provide a proof f o r that postulate.
But the m a n y a n d various attempts to p r o v e it failed. In
the first half of the nineteenth c e n t u r y Lobacevskii and Bolyai,
following Euclid's m o d e l , e a c h i n d e p e n d e n t l y c o n s t r u c t e d a
128
MATHEMATICS IN FUN AND IN EARNEST
n o n - E u c l i d e a n g e o m e t r y b y a s s u m i n g t h a t Euclid's parallel
p o s t u l a t e is not valid. E a c h of t h e m p u s h e d his geometry
f a r e n o u g h a h e a d to c o n v i n c e the m o s t skeptical that their
systems are q u i t e c o h e r e n t a n d n o t likely to r u n i n t o inconsistencies. All d o u b t o n this score was finally dispelled
w h e n it w a s s h o w n t h a t the L o b a c e v s k i a n plane n o n - E u c l i d e a n g e o m e t r y m a y be i n t e r p r e t e d as E u c l i d e a n g e o m e t r y
on a pseudo-sphere.
T h e n o n - E u c l i d e a n geometries r e n d e r e d Euclid's parallel
postulate, if a n y t h i n g , even less obvious. Still Euclid succeeded in c o n s t r u c t i n g his elements in spite of this deficiency.
F r o m this t h e r e w a s only o n e step t o the conclusion that
the logical c o h e r e n c e of Euclid's Elements
is in n o wise dep e n d e n t u p o n the obviousness of its postulates, a n d that it
s h o u l d b e possible to build a consistent geometry with a set
of postulates t h a t w o u l d lay n o claim to obviousness w h a t ever.
T h e basic entities of E u c l i d ' s great w o r k f a r e d n o better
t h a n his axioms. It all started with the "principle of duality,"
to w h i c h allusion h a s a l r e a d y b e e n m a d e . T h i s principle asserts t h a t if in a n y valid proposition of plane projective geo m e t r y the w o r d s " p o i n t " a n d " l i n e " are interchanged, the
resulting p r o p o s i t i o n is also valid. T h i s a s t o u n d i n g discovery
inevitably led t o a s t r a n g e conclusion, n a m e l y , t h a t the n a t u r e of the basic entities t o w h i c h the basic postulates of a
d e d u c t i v e science a r e applied is quite immaterial. In fact,
these entities n e e d n o t h a v e a n y m e a n i n g of their own.
T h e i r relation to e a c h o t h e r is d e t e r m i n e d by the postulates
t h a t are a p p l i e d to t h e m , a n d that relation is all that m a t ters.
O n these f o u n d a t i o n s w a s built t h e " f o r m a l i s t school" of
m a t h e m a t i c s , of w h i c h D a v i d H i l b e r t ( 1 8 6 2 - 1 9 4 3 ) was the
leading e x p o n e n t , t h e high priest of the cult. T h e r e was,
h o w e v e r , a b o t h e r s o m e fly in the o i n t m e n t . In f a c t there
w e r e t w o such flies. If postulates f o r a m a t h e m a t i c a l science,
f o r e x a m p l e , g e o m e t r y , a r e set d o w n arbitrarily, a n d if the
entities to w h i c h they a r e applied are devoid of m e a n i n g ,
w h a t relation does s u c h a g e o m e t r y b e a r to the physical
w o r l d ? R i c h a r d C o u r a n t ( 1 8 8 8 - ) , a f o r m e r colleague of Hilbert, says in the p r e f a c e t o his "What
Is
Mathematics?"6
that such a d o c t r i n e "is a serious threat to the very life of
science," t h a t " s u c h M a t h e m a t i c s could not attract any intelligent p e r s o n . " T h e formalists, h o w e v e r , m a d e short shrift
of objections of this k i n d as long as they could feel that
their science r e m a i n e d logically w i t h o u t a blemish. On that
g r o u n d they w e r e u n d e n i a b l y right. But it was not so easy
MATHEMATICS AND THE MATHEMATICIAN
129
to kill t h e o t h e r fly, f o r n o t h i n g less w a s involved t h e r e
t h a n the logical f o u n d a t i o n of t h e f o r m a l i s t science. 7
T h e " o b v i o u s n e s s " of Euclid's basic p r o p o s i t i o n s r e f e r r e d
to the f a c t that these propositions are e x t r a c t e d f r o m o u r
daily experience a n d a r e realized, s o m e w h a t c r u d e l y , in
the w o r l d t h a t s u r r o u n d s u s : t h e y are t h u s consistent w i t h
o n e a n o t h e r . If t h e postulates are t a k e n arbitrarily, if they
h a v e n o intuitive c o n n o t a t i o n , w h a t g u a r a n t y is t h e r e t h a t
they a r e logically consistent? W i t h o u t a p r o o f of t h e c o n sistency of the postulates t h e w h o l e edifice is worthless. T h e
formalists realized t h a t n o less t h a n their bitterest critics.
Hilbert m a d e h e r o i c efforts t o find such a p r o o f . H e failed.
A n d t h e r e t h e m a t t e r rests, except t h a t it h a s b e e n p r o v e d
t o the satisfaction of t h o s e m o s t c o m p e t e n t t o j u d g e t h a t ,
within the f r a m e w o r k of a given f o r m a l i s t science, it is n o t
possible to find a proof t h a t science is consistent. If a proof
of consistency f o r a f o r m a l i s t science is t o be p r o d u c e d , it
m u s t c o m e f r o m o u t s i d e t h a t science. T h i s p r o p o s i t i o n is
d u e to K . G o e d e l . 8
J • Logicalism
T h e f o r m a l i s t school of t h o u g h t in m a t h e m a t i c s takes logic f o r granted. T o this logic it adds a n a r b i t a r y
set of e n t i t i e s — " u n d e f i n e d t e r m s " 9 a n d a n a r b i t r a r y set of
p o s t u l a t e s — " u n p r o v e d p r o p o s i t i o n s . " It is t h e n in possession
of all the necessary tools a n d materials f o r the b u i l d i n g of t h e
p r o p o s e d b r a n c h of m a t h e m a t i c s .
A n o t h e r school of t h o u g h t c a m e t o t h e conclusion t h a t
the formalists are e x t r a v a g a n t : they r e q u i r e t o o m u c h . Logic
alone is perfectly sufficient f o r t h e erection of t h e e n t i r e
edifice of m a t h e m a t i c s . N o t t h e old v e r b a l logic, b u t logic
r e d u c e d to a set of symbols, a f t e r t h e m a n n e r of algebra. By
m e a n s of this " s y m b o l i c logic," t o give it its p r o p e r n a m e ,
all m a t h e m a t i c a l entities, i n c l u d i n g t h e integers themselves,
c a n b e o b t a i n e d b y p u r e l y logical constructions. T h i s p h i losophy of m a t h e m a t i c s c u l m i n a t e d in t h e t h r e e - v o l u m e w o r k
Principia Mathematica
( 1 9 1 0 - 1 3 ) by A . N . W h i t e h e a d a n d
B e r t r a n d Russell. T h i s w a s a n e x t r e m e l y a m b i t i o u s u n d e r taking, u n d o u b t e d l y o n e of the greatest intellectual enterprises of all time. It was hailed with great e n t h u s i a s m in
E n g l a n d a n d in the U n i t e d States. H e l p i n g h a n d s c a m e f o r w a r d to r e n d e r the great w o r k still greater.
But the Principia b e g a n to suffer f r o m the s a m e malaise
as C a n t o r ' s t h e o r y of p o i n t sets, as the infinite processes p u t
to w o r k to provide a logical f o u n d a t i o n f o r the m a t h e m a t i cal c o n t i n u u m . P a r a d o x e s a n d a n t i n o m i e s c a m e to light t h a t
were very e m b a r r a s s i n g . S o m e of the f u n d a m e n t a l a s s u m p -
130
MATHEMATICS IN FUN AND IN EARNEST
tions of t h e Principia i n t r o d u c e d f o r the express purpose of
w a r d i n g off p a r a d o x e s w e r e f o u n d to be questionable and
finally rejected. It w a s not long b e f o r e the Principia
Mathematica was r e d u c e d to the status of o n e m o r e c o n t e n d e r
f o r t h e h o n o r of being the c u s t o d i a n of the f o u n d a t i o n s of
m a t h e m a t i c s , u n d e r the n a m e of "logicalism."
K ' Intuitionism
A m o n g the critics of the Principia
were
the F r e n c h intuitionists: E. Borel ( 1 8 7 1 - 1 9 5 6 ) , Lebesgue,
a n d others. B u t t h e greatest challenge of this work c a m e f r o m
m e m b e r s of the D u t c h school, called b y A r b a h a m A . F r a e n k e l
t h e " N e o i n t u i t i o n i s t s . " T h i s school, u n d e r the leadership
of L . E . J. B r o u w e r ( 1 8 8 2 - ) , put the Principia
upside
d o w n . N o t o n l y did they reject the idea that m a t h e m a t ics c a n be derived f r o m logic, t h e y denied logic a n y a u t o n o m o u s existence. Logic, a c c o r d i n g t o the intuitionists, is not a
science b u t a t e c h n i q u e derived f r o m science to facilitate
the study of t h e science. F u r t h e r m o r e , B r o u w e r boldly questions t h e validity of the basic processes of our generally acc e p t e d logic. H e rejects the law of the excluded middle, i.e.,
that a p r o p o s i t i o n is necessarily either t r u e or not true. It
m a y be neither, f o r t h e r e m a y be n o sufficient i n f o r m a t i o n
t o d e c i d e t h e question.
A s a n illustration of w h a t is m e a n t by Brouwer's negation
of the law of t h e excluded m i d d l e , let us consider the exa m p l e given by A b r a h a m A . F r a e n k e l . 1 0 T h e fractional part
of t h e n u m b e r 7r has b e e n c o m p u t e d f o r m a n y h u n d r e d s of
places, a n d m a n y m o r e such places could n o w be c o m p u t e d
with m u c h less l a b o r t h a n b e f o r e , by m e a n s of the new
electrical calculators. Is there a place in this long row of
n u m b e r s w h e r e the digit 7 occurs seven times in a row?
T h e r e is n o such place in t h a t p a r t of the f r a c t i o n that is
k n o w n at present, a n d we c a n n o t tell w h e t h e r it will o r will not
o c c u r if n e w digits of t h a t f r a c t i o n are c o m p u t e d .
N o w let us c o n s i d e r the real n u m b e r R which starts out
as 0 . 3 3 3 3 3 3 a n d every o t h e r digit of this d e c i m a l f r a c t i o n is
a 3, except t h a t if the n t h digit of the f r a c t i o n a l part of 7r
is a 7 f o l l o w e d by six m o r e digits 7, w e will take for the
n t h digit of R the digit 2, if n is o d d , a n d the digit 4, if n is
even. T h e digits of R are t h u s perfectly defined as f a r as
the digits of t h e f r a c t i o n a l part of 7r are k n o w n . But we cannot tell w h e t h e r R is e q u a l to 1 / 3 , smaller than 1 / 3 , or greater
than 1/3.
Is the f a m o u s saying " Y o u c a n n o t fool all the people all
the t i m e " t r u e o r false? P e r h a p s it is true. But it is conceivable t h a n a m a n publicly p e r p e t r a t e d a h o a x or a lie that
MATHEMATICS AND THE MATHEMATICIAN
131
r e m a i n e d u n d e t e c t e d d u r i n g his lifetime and that he t o o k his
secret with h i m into his grave. T h e n the proposition w o u l d ,
of course, be w r o n g , but w e w o u l d h a v e n o w a y of p r o v i n g
it. If the m a n w r o t e a c o n f e s s i o n , sealed it, a n d o r d e r e d his
heirs to open it on the o n e - h u n d r e d t h a n n i v e r s a r y of his
death, then w e shall find out o n that d a y that o u r proposition is false. But at present the p r o p o s i t i o n is n e i t h e r t r u e
n o r false. Hitler w a s quite certain that the proposition is
false. Witness his principle of " t h e big lie."
" F r a n c i s Bacon ( 1 5 6 1 - 1 6 2 6 ) is the a u t h o r of the so-called
S h a k e s p e a r e a n plays." Is t h e proposition t r u e o r false?
L • New Logics
T h i n g s did not b e c o m e any s m o o t h e r f o r
any of the c o n t e n d i n g schools of t h o u g h t w h e n the Polish
logician Lukasiewicz raised the question w h y logic should be
limited to only two alternatives, two values: t r u e and false.
H e p r o p o s e d a new logic w h i c h a d m i t s of t h r e e alternatives,
a three-valued logic. N o w ce n'est que le premier pas qui
coute. If logic can be three-valued, w h y can it not be f o u r valued, indeed, w h y not n-valued? T h e r e is n o reason, h o w ever, to stop there. W h y must the values of logic be a finite
w h o l e n u m b e r ? W e might as well h a v e a logic with a continu o u s n u m b e r of v a l u e s — s u c h proposals h a v e been a d v a n c e d .
T h a t u n s h a k a b l y solid rock of classical logic simply
slipped away f r o m u n d e r the m a t h e m a t i c a l edifice, a n d the
w h o l e structure is n o w "on the r o c k s . " A s m a t h e m a t i c i a n s
put it, their science is at p r e s e n t passing t h r o u g h a "crisis."
It has been in this state, roughly, since the beginning of
the present century. W h a t c o n n e c t i o n , if any, is t h e r e between this crisis a n d the social a n d political turmoil in
the throes of w h i c h suffering m a n k i n d has been l a b o r i n g d u r ing the same period of t i m e ? T h i s is not the time n o r the
place to consider this question, but so f a r as m a t h e m a t i c s is
c o n c e r n e d , o n e need not be overly a l a r m e d . M a t h e m a t i c s is
not going to t h e dogs.
M ' Conclusion
M a t h e m a t i c s h a s t w o aspects: O n t h e o n e
h a n d , it is a description of a segment of the world we live
in a n d it f u r n i s h e s tools f o r n o n - m a t h e m a t i c i a n s to describe
o t h e r segments of that world. This might be called t h e " f u n c tional" part of m a t h e m a t i c s . T h e o t h e r p a r t of m a t h e m a t i c s
deals with its f o u n d a t i o n s and m a y be said to be largely
philosophical. Of c o u r s e the two p a r t s are not unrelated. T h e
study of the f o u n d a t i o n s of m a t h e m a t i c s decides h o w f a r
the m a t h e m a t i c a l processes m a y be carried out b e f o r e they
r e a c h the limits of their validity. F o r t u n a t e l y , w h a t e v e r these
132
MATHEMATICS IN FUN AND IN EARNEST
limits m a y be, t h e r e is a m p l e r o o m f o r m a t h e m a t i c a l activity l o n g b e f o r e those limits a r e r e a c h e d . A s a m a t t e r of fact,
m o s t active m a t h e m a t i c i a n s a r e little c o n c e r n e d a b o u t those
f o u n d a t i o n s . A t least t h e y d o not allow those p r o b l e m s to
i n t e r f e r e w i t h their activities as investigators. M o r e t h a n
that, even t h o s e m a t h e m a t i c i a n s w h o t a k e a direct part in
t h e d e b a t e r e g a r d i n g the logical validity of their science m a n age to o b t a i n v e r y v a l u a b l e results in their own special field
of investigation t h a t h a v e little relation t o those theoretical
discussions.
B u t w h a t a b o u t t h e crisis itself? I t w o u l d , of course, b e
f o o l h a r d y f o r a n y o n e to try t o predict at present w h e r e t h e
crisis leads t o a n d h o w it will end. W h a t m a y be said, h o w ever, with p e r f e c t s a f e t y is that m a t h e m a t i c s will emerge
f r o m it e n r i c h e d a n d invigorated, t o c o n t i n u e the w o r k it
h a s b e e n so successfully c a r r y i n g o n u p to now.
2
•
Perplexities of a Potato-Pnsher
A ' Winning
a Prize
T h e p e a c e of m i n d of t h e reader
m a y p e r h a p s h a v e been disturbed by this title, f o r "potatop u s h e r " is n o t in the dictionary, not yet. If you are puzzled
as to w h a t a p o t a t o - p u s h e r m i g h t be, I m u s t hasten to p u t
you at ease b y supplying the a f o r e s a i d deficiency. Unlike
a p o t a t o - p e e l e r o r a p o t a t o - m a s h e r , a p o t a t o - p u s h e r is n o t a
k i t c h e n utensil b u t a person, a n d in t h e present circumstances the r e f e r e n c e is t o n o o t h e r b u t myself.
I a m quite s u r e that all of y o u will agree that I o u g h t
t o be p e r p l e x e d , f o r m a n y m o r e reasons than one. But
s o m e of y o u m i g h t w o n d e r o n w h a t g r o u n d I arrogate to
myself t h e h i g h - s o u n d i n g title of a potato-pusher. T h o s e ben i g h t e d individuals h a v e only themselves to b l a m e f o r their
ignorance. T h e y s h o u l d h a v e a t t e n d e d the p a r t y given by t h e
D e p a r t m e n t of M a t h e m a t i c s a n d A s t r o n o m y s o m e time ago
in the F a c u l t y C l u b . H a d they b e e n there, they w o u l d h a v e
witnessed, t h e y w o u l d h a v e seen with their own eyes the
prowess I displayed t h e n a n d t h e r e as a potato-pusher. W h y ,
I w a s the c h a m p i o n of the e v e n i n g a n d w o n the prize, the
only prize, m i n d y o u , that w a s a w a r d e d . W h e n all t h e nice
r i b b o n s w e r e untied a n d all the m u l t i t u d i n o u s pretty w r a p pings u n d o n e , t h e r e w a s the prize, f o r everyone to see:
twelve r o u n d , neatly p a c k e d , nice little potatoes. I a m afraid
t h a t s o m e e c h o e s of a malicious whispering c a m p a i g n
r e a c h e d y o u r ears that m i n e was the b o o b y prize. I a m
s u r e of t h a t c a m p a i g n , f o r I h e a r d it myself, all the way
MATHEMATICS AND THE MATHEMATICIAN
133
across, f r o m t h e o t h e r e n d of the r o o m . B u t never y o u
m i n d . Y o u k n o w h o w s o m e people a r e : envious, always r e a d y
to belittle a fellow, to deprive h i m of his just credit, of his
h a r d w o n dues. I a m t h e c h a m p i o n p o t a t o - p u s h e r w h e t h e r
they like it o r not.
B * Gambling
and Statistics
B u t I m u s t a d m i t that t h e
prize did not d o m e m u c h good. F o r it set m e a-thinking,
and as you k n o w t h i n k i n g is a weariness of the flesh. T h e
m o r e I t h o u g h t , the m o r e w o r r i e d , the m o r e perplexed I became. N o t that t h e r e is a n y t h i n g w r o n g in w i n n i n g a prize,
f r o m a n y point of view, least of all f r o m a m a t h e m a t i c a l
point of view, as I could readily prove to y o u by any n u m ber of examples. Let m e just tell y o u o n e story, a n excellent
story, even if it is a little better than t h r e e centuries old.
Chevalier de M e r e was b o t h a n o b l e m a n a n d a g a m b l e r .
H e h a d the good f o r t u n e to c o u n t Blaise Pascal ( 1 6 2 3 - 1 6 6 2 )
a m o n g his friends. T h e noble g a m b l e r o n c e asked his e r u d i t e
a n d r e s o u r c e f u l f r i e n d t o suggest a fair w a y out of a difficulty in which h e w a s involved. T o p u t the story on an impersonal basis, let us say that t w o players A and B, of equal
skill, agree to play a g a m e f o r a prize w h i c h is to go t o t h e
player w h o first wins t h r e e games. W h e n A h a d t w o g a m e s
t o his credit a n d B o n e game, the contest h a d to be given
u p . W h a t w o u l d be a n equitable w a y of dividing t h e prize
between A a n d B?
Pascal c o m m u n i c a t e d this question to F e r m a t , a n d between t h e m the t w o m a t h e m a t i c a l geniuses of the first half
of the seventeenth c e n t u r y evolved t w o solutions of the p r o b lem which were just as simple as they w e r e ingenious. L e t
us suppose that A a n d B play o n e m o r e g a m e a n d that B
wins it. W i t h t w o games to t h e credit of each player, t h e y
should divide the prize equally, w h i c h is t o say that half
the prize certainly belongs to A right n o w , b e f o r e the
hypothetical next g a m e is played, and that g a m e is played
only to decide w h a t to d o with the o t h e r half. N o w A h a s
as m u c h of a c h a n c e to win that g a m e as B does, h e n c e
that second half should be divided between t h e m equally.
T h u s o n e - f o u r t h of the prize should go to B a n d threef o u r t h s to A.
T h e second solution is even simpler t h a n this. B c a n win
the prize only if h e wins two games in succession, that is
to say, he h a s o n e c h a n c e in f o u r , like t h r o w i n g h e a d s with
a coin twice in a row. H e n c e o n e - f o u r t h of the prize should
go to him a n d the rest to A.
Little did the C h e v a l i e r suspect that the t r e n d of t h o u g h t
134
MATHEMATICS IN FUN AND IN EARNEST
p r o v o k e d b y his prize p r o b l e m w o u l d lay the f o u n d a t i o n
of the t h e o r y of probablities, a n d t h a t this, in t u r n , would
lead to the m i g h t y discipline n o w k n o w n u n d e r the n a m e
of M a t h e m a t i c a l Statistics. 1 1 T h e practitioners of this new
c r a f t are so p r o u d of their calling that they scorn the title of
M a t h e m a t i c i a n s w h o are engaged in the study of statistics.
T h e y insist t h a t t h e y are Statisticians w h o use m a t h e m a t i c s
as a tool, say, like physicists, o r engineers. In evidence
whereof they f o r m e d their own Statistical Society, separate
f r o m the M a t h e m a t i c a l Society, a n d have their o w n exclusive
Q u a r t e r l y of M a t h e m a t i c a l Statistics. This shows that the
Statisticians are even p r o u d e r t h a n the topologists. But f a r
be it f r o m m e to be p u t t i n g ideas i n t o the h e a d s of the
topologists.
C • Tit-tat-toe
Ancient
and Modern
All this mighty dev e l o p m e n t c a m e a b o u t because of Chevalier de Mere's
prize. But m y p o t a t o prize b r o u g h t m e n o t h i n g but perplexities. T h i s p o t a t o - p u s h i n g r e m i n d e d m e of o t h e r games I
used to play at o n e t i m e or a n o t h e r . T h e first that c a m e to
m y m i n d is o n e that s o m e of you m a y k n o w u n d e r the
n a m e of tit-tat-toe a n d I k n e w u n d e r an entirely different
n a m e . T h e e q u i p m e n t necessary f o r the g a m e consists of a
s q u a r e divided into nine smaller equal squares, or cells, and
two sets of t h r e e chips each. T h e t w o o p p o n e n t s m o v e their
chips in t u r n , o n e at a time, a n d the o n e w h o places his
t h r e e chips in a h o r i z o n t a l o r vertical r o w is the winner.
I used to play that f a s c i n a t i n g g a m e w h e n I was in the
grades. M y f a v o r i t e t i m e f o r the g a m e was d u r i n g school
h o u r s , especially d u r i n g the a r i t h m e t i c lessons, w h e n the
subject b e c a m e t o o repetitious a n d t o o boring. In the school
I a t t e n d e d the pupils h a d n o individual desks. W e were
seated o n long b e n c h e s , like c h u r c h benches. I h a d n o
t r o u b l e in i n d u c i n g a n e i g h b o r of m i n e t o play the g a m e
with m e . I used the s i m p l e device of bribing him with the
p r o m i s e to s h o w h i m m y solution of the next day's assignm e n t . W e m a n u f a c t u r e d the necessary e q u i p m e n t right o n
the spot. T w o pairs of m u t u a l l y p e r p e n d i c u l a r lines d r a w n
on a scrap of p a p e r served as the b o a r d , a n d the chips were
six bits of p a p e r , t h r e e m a r k e d with rings, and the o t h e r
t h r e e with bars. W e played to o u r h e a r t s ' c o n t e n t a n d h a d
the time of o u r life.
I h a v e quit playing tit-tat-toe a long, long time ago, and
I a m glad I did. F o r I h a v e f o u n d out that mine was "child's
p l a y . " Self-respecting people with p r o p e r mental e q u i p m e n t
do not play the g a m e t h e way I used to. F o r poise a n d
MATHEMATICS AND THE MATHEMATICIAN
135
dignity the g a m e is to be played in t h r e e dimensions. T h e
" b o a r d " of the g a m e is a c u b e sub-divided into twenty-seven
smaller a n d e q u a l cubes, o r cells. Of course, I k n e w n o t h i n g
of all t h a t in m y tit-tat-toe days. Besides, w h a t good could
that h a v e d o n e m e , h a d I k n o w n it? Y o u w o u l d agree that
such a device could h a r d l y h a v e e s c a p e d the b e n e v o l e n t
a n d vigilant eye of m y teacher, a n d m y perplexities w o u l d
have started right then. Besides, even this h i g h - b r o w style
of playing tit-tat-toe is obsolete, just as obsolete as t h e carriage of K i n g T u t - a n k h - a m e n . Y o u see, t w o m a t h e m a t i c i a n s
have gotten hold of that g a m e recently. T h e y f r e e d this pastime of all triviality a n d e n d o w e d it with the p r o p e r intellectual prestige b y elevating it to the f o u r t h d i m e n s i o n , n a y ,
to the n t h dimension. Yes, if y o u w a n t to keep y o u r selfrespect a n d keep u p with the times, y o u m u s t play y o u r tittat-toe at least in the f o u r t h dimension. 1 2 So f a r I h a v e not
played this hyperspatial tit-tat-toe. W h y ? T h e r e a s o n is
very simple. N o b o d y h a s yet tried to bribe m e into playing
f o u r dimensional tit-tat-toe b y offering to solve m y p r o b lems f o r me. I m e a n m y m a t h e m a t i c a l p r o b l e m s — m y o t h e r
problems, a n d especially m y financial problems, I k n o w f o r
certain to be insoluble—like t h e p r o b l e m of the duplication
of the cube, o r the solution of the nth degree e q u a t i o n .
D * New Checker Games for Old
M y t r i u m p h a n t exploit
in p o t a t o - p u s h i n g m a d e m e also think of m y c h e c k e r days,
that is, of the days w h e n I used to play checkers. C h r o n o logically that was a f t e r m y tit-tat-toe days. But I gave that
g a m e up, too. R o u g h l y speaking, that h a p p e n e d w h e n m y
m a t h e m a t i c a l p r o b l e m s b e c a m e tough e n o u g h a n d challenging enough so that I b e c a m e satisfied to grapple with t h e m
all by myself in the solitude a n d the silence of m y study. I
n o longer felt the n e e d of the stimulus w h i c h is p r o v i d e d by
the o p p o r t u n i t y to gloat over the demise of a d e f e a t e d o p p o n e n t , or, w h a t is the s a m e thing, the stimulus p r o v i d e d
by the gratification of m y ego in feeling superior to s o m e o n e else. But a m I right a b o u t that? N o w that I said it, I
a m afraid that, u p o n second t h o u g h t , I m a y h a v e to t a k e it
all back. W h a t a b o u t that nasty fellow with his m o c k i n g
grin on his repulsive f a c e w h o always peeps over y o u r shoulder at every w o r d you p u t d o w n on p a p e r , instantly r e a d y
to j u m p on you with his priority claims of having "got t h e r e
firstest, with the mostest a n d the bestest a r g u m e n t s ? " ( W i t h
apologies to G e n e r a l B e d f o r d F o r r e s t , of Civil W a r f a m e . )
I really love to beat that guy to the p u n c h . D o n ' t you?
But it m a t t e r s little w h a t the actual motive was that m a d e
136
MATHEMATICS IN FUN AND IN EARNEST
m e a b a n d o n c h e c k e r s . T h e point is, I a m glad I did. F o r I
f o u n d out t h a t this, too, is a n a n t i q u a t e d game, at least in
the f o r m I u s e d to play it. F o r o n e thing, there is n o good
r e a s o n w h y t h e g a m e of c h e c k e r s m u s t be played so that
t h e typical m o v e of t h e typical piece m u s t always be in a
straight line. T h e g a m e c o u l d be p l a y e d with t w o - d i m e n s i o n a l
moves. T h a t w o u l d h a v e the i m m e a s u r a b l e a n d enticing adv a n t a g e t h a t y o u could m o v e across a n edge of the cell,
o r t h r o u g h a c o r n e r , o r b o t h , if y o u are very ambitious.
Just f a n c y h o w m u c h c h e c k e r liberty w o u l d be yours, to
hold a n d t o cherish. H o w e v e r , with all these up-to-them i n u t e i m p r o v e m e n t s in y o u r checkers, you would still be
playing a p i k e r ' s g a m e . T h e real t h i n g is to play checkers in
three d i m e n s i o n s .
U s i n g a suitable f r a m e , several excellent c u b e c h e c k e r
g a m e s c a n be defined, with m a n y interesting new features.
T h e field of play m a y be a n e t w o r k of white a n d black
cells, o r a looser n e t w o r k of cells h o l d i n g together by their
c o r n e r s , or the entire f r a m e . Places of local safety, like the
f a m i l i a r d o u b l e c o r n e r , a n d o t h e r strategical features, app e a r in n e w f o r m s . T h e r e are m a n y possible kinds of cube
c h e c k e r g a m e s , p u r e , c o m b i n a t i o n , a n d hybrid games, multiple games, i n t e r f e r i n g games, cyclical games, a n d others.
T h e best h a v e already p r o v e d m o r e interesting than the
classical c h e c k e r g a m e .
I h o p e this does not m a k e y o u feel dizzy. If it does,
d o n ' t b l a m e m e . I have not invented it. N o r a m I reporting
a kind of fly-by-night s c h e m e . I h e a r d this three-dimensional
c h e c k e r g a m e e x p o u n d e d u n d e r the auspices of the A m e r i c a n M a t h e m a t i c a l Society at its m e e t i n g at Cornell U n i v e r sity. 1 3 a n d the A m e r i c a n M a t h e m a t i c a l Society, I w a n t you to
k n o w , is the largest, the richest, a n d most p o w e r f u l , the
m o s t influential, a n d t h e m o s t authoritative organization of
m a t h e m a t i c a l research w o r k e r s in t h e world today.
H o w e v e r , if y o u still h a v e a w e a k n e s s f o r the traditional
t h o u g h o u t m o d e d c h e c k e r g a m e , y o u m a y still h o p e f o r
a respite f o r s o m e time to c o m e . T h e exposition of the
t h r e e - d i m e n s i o n a l c h e c k e r t h e o r y was illustrated on an actual
m o d e l . M y i n n a t e s i m p l e m i n d e d n e s s p u s h e d m e to ask the
very n a i v e q u e s t i o n w h e r e such a progressive a n d up-tot h e - m i n u t e outfit could be secured. I was p r o m p t l y put in
m y place by the declaration of the s p e a k e r that as f a r as
he k n o w s t h e m o d e l b e f o r e h i m is t h e only o n e in existence at the present time. So the flatwitted checkers will
c o n t i n u e to flourish f o r s o m e time. B u t the m i l l e n n i u m of
progress is at h a n d .
MATHEMATICS AND THE MATHEMATICIAN
137
Y o u need not f e a r that the tit-tat-toe g a m e has a n y t h i n g
on checkers. T h e s p e a k e r was m a g n a n i m o u s to assure his
breathless a u d i e n c e t h a t checkers, too, could be played in
dimensions higher t h a n the third. T h e anxiety of all p r e s e n t
was visibly relieved. But, believe it o r not, this assertion
was not a c c o m p a n i e d by the exhibition of a m o d e l of s u c h
a g a m e in n dimensions. N o e x p l a n a t i o n f o r this omission
was offered. T h a t this was a grievous oversight was q u i t e
clear to m e right o n the spot, b u t I d a r e d not ask a n y m o r e
questions.
E ' Potato-pushing
a la Mode
B y n o w , I a m sure, y o u
realize a l r e a d y w h y the a f t e r m a t h of w i n n i n g a prize t u r n e d
out to be so full of perplexities f o r m e . It s u d d e n l y d a w n e d
u p o n m e that the time is ripe to generalize the p o t a t o pushing game. A n d w h o is t o d o it, if not the c h a m p i o n .
W h a t a w o n d e r f u l o p p o r t u n i t y ! W h a t an alluring vista! T h e
portals of i m m o r t a l i t y h a v e s u d d e n l y s w u n g wide o p e n right
in f r o n t of m e , b e c k o n i n g m e to e n t e r a n d join the illustrious a n d e n d u r i n g c o m p a n y of g e n e r a l i z e s w h i c h dwells
within. Small w o n d e r that the prospect t u r n e d m y head. I
also realized t h a t I h a v e n o time t o waste, f o r I m u s t m a k e
sure a n d r u n past that gate "firstest." Yes, but h o w d o y o u
generalize a p o t a t o - p u s h i n g g a m e ?
I u n d e r s t a n d that the present f o r m of the g a m e is alr e a d y the result of s o m e evolution. I n a p r e c e d i n g stage the
instrument with w h i c h the p o t a t o w a s p u s h e d was not a
stick, but a p a r t of the player's a n a t o m y , like the nose. Is t h a t
the a v e n u e of a p p r o a c h ? Clearly, that w o u l d be retrogression. Progress does not point in this direction. Besides, I d o
not think I w o u l d p a r t i c u l a r l y e n j o y the g a m e if I h a d t o
push the potato, say, with m y t o n g u e in m y cheek.
I h o p e d that s o m e solution will be suggested to m e by
y o u r w o m a n ' s intuition. I d o not m e a n m y wife's intuition,
I m e a n m y o w n . But n o t h i n g of the sort c a m e to relieve m e
of m y perplexities. I was t h e r e f o r e r e d u c e d t o the slow a n d
laborious m e t h o d of analyzing the p r o b l e m in detail, trusting that such a systematic p r o c e d u r e m a y yield s o m e salutary ideas. H o w can the g a m e of p u s h i n g a p o t a t o with a
stick be described in general a n d scientific language? S u p pose I say that the g a m e m a y ideally be conceived as consisting of the p r o p u l s i o n , over a plane, of an ellipsoid of
revolution by a straight line, along a p r e s c r i b e d p a t h w h i c h
is also a straight line. If this is a n a c c e p t a b l e way of looking
at the thing, I a m a b o u t to see a g l i m m e r of light. I could
give u p t h e prosaic, d o w n - t o - t h e - e a r t h straight line a n d
138
MATHEMATICS IN FUN AND IN EARNEST
m a k e the p o t a t o m o v e along, say, a spiral of A r c h i m e d e s
instead. Y o u w o u l d a d m i t that m y spiral, o r r a t h e r A r c h i m e d e s ' spiral, w o u l d m a k e the g a m e m u c h f a n c i e r , would
it not?
B u t I c o u l d n o t b e satisfied with that. T h e g a m e involves
f o u r g e o m e t r i c a l elements, a n d t h e r e is n o good reason w h y
o n e of t h e m s h o u l d be singled out, a n d the others neglected.
T h i s w o u l d be c o n t r a r y t o t h e d e m o c r a t i c spirit of the
times. So m y perplexities w o u l d c o n t i n u e , until I generalized
e a c h e l e m e n t in various w a y s a n d evolved a great m a n y
c o m b i n a t i o n s , t o o n u m e r o u s to m e n t i o n . T o give you an
idea of w h a t they w e r e like, I w o u l d say that we could roll
a p s e u d o s p h e r e a l o n g a geodesic c u r v e of a Frenel w a v e
s u r f a c e a n d use a witch of Agnesi to propel it. I felt quite
certain that if I o f f e r e d to the world a few dozen of such
i m p r o v e d p o t a t o - p u s h i n g g a m e s , I w o u l d c o n t r i b u t e powerf u l l y to the joy a n d h a p p i n e s s of m a n k i n d a n d e a r n thereby
all the acclaim a n d all the g r a t i t u d e a m a n c a n wish for.
B u t that blissful state of m i n d lasted only a short while.
M y perplexities r e t u r n e d to plague m e s o m e m o r e . I noticed t h a t all f o u r e l e m e n t s of m y g a m e , m u c h i m p r o v e d as
t h e y were, h a v e k e p t their original dimensions. T h a t m a k e s
m e a very p o o r generalizer, a n d I a m m u c h m o r e likely to
be l a u g h e d at t h a n c o m m e n d e d . A n d rightly so. W h y such
c o n s e r v a t i s m ? T h e r e r e m a i n e d the question, w h i c h of the
f o u r e l e m e n t s involved shall u n d e r g o a c h a n g e of d i m e n sions? T o m a k e a long a n d p a i n f u l story short, I will tell
y o u t h a t I c a m e to t h e inescapable conclusion t h a t in o r d e r
to d o the t h i n g p r o p e r l y I m u s t strike out boldly, take the
bull b y the h o r n s , a n d go the w h o l e hog. I m u s t assign to
the f o u r elements involved the d i m e n s i o n s p, q, r, s. T o
m a k e sure not to be o u t d o n e by a n y b o d y , I must allow
p, q, r, s to be a n y f o u r n u m b e r s w h a t e v e r , positive, negative, f r a c t i o n a l , t r a n s c e n d e n t a l . N o w I h a v e it. " E u r e k a . " I
rested on m y well-earned laurels.
B u t not f o r long. T h e r e is n o rest f o r potato-pushers, o r
r a t h e r generalizers. All of a s u d d e n I realized that in m y
t h i n k i n g I h a d b e e n visualizing E u c l i d e a n space. Such a limitation is absolutely intolerable. It is imperative that nonE u c l i d e a n spaces be b r o u g h t in, and n o n - A r c h i m e d e a n
spaces, a n d n o n - A r g u e s i a n spaces, too; and s o m e purely
topological spaces, like the B a n a c h space, must not be neglected either. W i t h such i m p r o v e m e n t s o u r potato-pushing
g a m e will d e f y all c o m p e t i t i o n .
B u t this feeling of h a v i n g r e a c h e d the ultimate did not
last. T h e g a m e involves m o t i o n . N o w m o t i o n is relative.
MATHEMATICS AND THE MATHEMATICIAN
139
H o w is o n e to tell which element of the g a m e is to r e m a i n
stationary and w h i c h is to m o v e ? T o c o m e back to the almost f o r g o t t e n p r o t o t y p e of o u r g a m e , w h y is it necessary
to m o v e the p o t a t o over the floor, w h e n the s a m e result
could be obtained if the p o t a t o were kept fixed and we
pushed the floor a b o u t . In the generalized g a m e the s a m e
a r g u m e n t m a y be applied to any o n e of the f o u r e l e m e n t s
involved. If w e carry this idea out to its logical limits, w h a t
a w o n d e r f u l g a m e we w o u l d have. But w h e r e a m I? D o
you k n o w ?
You m a y think that by n o w the g a m e is general e n o u g h .
But this is not so, not if you are a p o t a t o - p u s h e r w o r t h
y o u r salt. M y perplexities and m y worries w e r e back, w o r s e
than ever. At this stage it o c c u r r e d to m e that there is a
s h o r t c o m i n g that is c o m m o n to all the generalizations 1 a m
familiar with. It was quite evident t h a t I c a n n o t a f f o r d to
be c a u g h t in this kind of t r e a c h e r o u s t r a p myself. I h a v e
r e f e r e n c e to the patently n o t o r i o u s fact that all the generalize s have overlooked the player himself. T h e y left him invariant. This is u n w o r t h y of an honest-to-goodness generalizer,
let alone a p o t a t o - p u s h e r . This idea is not quite original with
me. If m a t h e m a t i c i a n s never t h o u g h t of it, thieves have actually practiced it since ancient times. Y o u k n o w of the
f a m o u s r o b b e r of ancient G r e e c e by the n a m e of P r o crustes w h o m a d e his victims fit the length of the bed he
kept in readiness f o r t h e m , either by stretching t h e m , o r
shortening t h e m , with an axe if need be. W e m i g h t say that
mathematically speaking, P r o c r u s t e s s u b m i t t e d those w h o m
he robbed to a linear t r a n s f o r m a t i o n . In all fairness this
t r a n s f o r m a t i o n should be called a " P r o c r u s t e a n T r a n s f o r m a t i o n . " W h e n it c o m e s to generalizing games, m a t h e m a ticians should look f o r inspiration to the r o b b e r s of antiquity.
T h e players of the g a m e should be subjected to a P r o c r u s tean t r a n s f o r m a t i o n which, of course, does not necessarily
have to be linear. T h e exigencies of the g a m e u n d e r consideration would decide that question of the details of t r a n s f o r m a tion to be used. In the particular case of the p o t a t o - p u s h i n g
g a m e I w o n d e r w h e t h e r the p u r p o s e s of the g a m e w o u l d be
better served if m y P r o c r u s t e a n t r a n s f o r m a t i o n should r e d u c e
the player to t w o dimensions, o r on the c o n t r a r y he should be
blown u p by that t r a n s f o r m a t i o n to f o u r dimensions, o r even
higher dimensions. This is still o n e of m y unresolved perplexities.
F • Conclusion
T h a t is as f a r as I got. H a v e I d o n e
everything that can honestly a n d p r o p e r l y be expected in the
140
MATHEMATICS IN FUN AND IN EARNEST
responsible task of generalizing the p o t a t o - p u s h i n g game? I
d o n o t k n o w a n d I a m perplexed. Y o u see, a f t e r having
p l a y e d his first a n d only g a m e of p o t a t o - p u s h i n g a fellow,
a l t h o u g h a c h a m p i o n , does n o t h a v e the p r o p e r perspective
n o r does h e h a v e t h e requisite insight to d o justice to the
g a m e in t h e w a y of i m p r o v i n g a n d generalizing it. I m a y
or m a y not gain i m m o r t a l i t y in the a t t e m p t , but I a m sure
t h a t I, too, c a n generalize all the f u n a n d all the joy out of
the g a m e , a n d every speck of c o m m o n sense along with it.
Y o u just give m e a c h a n c e .
3
•
Geometrical Magic
A ' A Point Fixation
R a t h e r inadvertently I f o u n d myself
n o t long a g o in a quite sophisticated gathering. T h e c o m p a n y w a s being e n t e r t a i n e d with a variety of tricks by a
skillful m a g i c i a n . F o r m o s t of the n u m b e r s of t h e puzzling
s h o w the p e r f o r m e r w a s enlisting the active participation of
s o m e m e m b e r s of his a u d i e n c e — t h e most successful of
his stunts, I s h o u l d judge.
A t o n e point of the spectacle the m a g i c i a n issued a call
f o r a n e w k i n d of h e l p : " W o u l d s o m e o n e assist by d r a w i n g
parallel lines?" A n u n e a s y silence fell a b r u p t l y u p o n the
a m u s e d c r o w d . N o b o d y b u d g e d , f o r w h a t seemed a long
time. T o save the situation f r o m b e c o m i n g t o o embarrassing I v o l u n t e e r e d , foolhardily. T h e magician sized m e u p with
a d i s a p p r o v i n g eye, b u t h e said nothing. T h e p o o r m a n h a d
n o b e t t e r choice.
I was a r m e d f o r m y task with t w o triangular rulers, in
addition to t h e pencil, a n d c o n f r o n t e d with a large triangle
A B C d r a w n on a sheet of p a p e r . T h e audience eagerly
c r o w d e d a r o u n d the big table, as t h o u g h expecting I d o not
k n o w w h a t miracle. T h e m a g i c i a n p l a n t e d himself right next
to m y c h a i r t o direct o p e r a t i o n s . T h e choice of the starting
p o i n t , say X , o n the base B C of the triangle was mine, but
t h e rest of the p r o c e d u r e w a s strictly prescribed. First I had
to d r a w a line X Y parallel to A B a n d t e r m i n a t e d on A C
at Y . T h e n I was bid t o d r a w a parallel Y Z to B C reaching A B in Z, a n d finally a parallel Z X ' to A C , t h u s r e t u r n i n g
t o the base B C , at t h e p o i n t X ' . (Fig. 10)
I was quite pleased with m y feat, a n d I was glad it was
over. T h a t feeling of relief, h o w e v e r , was short lived. T h e
m a g i c i a n d a r e d m e to d r a w a second sequence of three
lines, a n a l o g o u s to the first, but starting this time with m y
h a r d - w o n point X ' . I could not think of a n y good reason
MATHEMATICS AND THE MATHEMATICIAN
141
w h y I should decline, the m o r e so that I felt s a f e e n o u g h ,
since I k n e w w h a t was a h e a d . T o avoid a n a r g u m e n t , I
bravely d r e w the lines X ' Y ' , Y ' Z ' , a n d w a s r e a d y t o m a r k
t r i u m p h a n t l y m y t e r m i n a l point X ' o n B C , w h e n suddenly,
and almost involuntarily, I jerked m y pencil a w a y f r o m t h e
p a p e r : m y point X " fell so d a n g e r o u s l y close to the initial
point X that it was h a r d l y possible to tell t h e m a p a r t .
" M y parallels are not m u c h g o o d , " I said sheepishly,
looking u p to m y m e n t o r , o r better, t o r m e n t o r .
" O n the c o n t r a r y , sir," he said consolingly, " y o u r parallels are amazingly e x a c t . "
" T h e n h o w is it that I l a n d e d in o c c u p i e d territory? Is
there a n y t h i n g w r o n g w i t h the starting point I p i c k e d ? "
" Y o u w o u l d not w a n t m e to s u r m i s e , " h e said slyly, with
perceptible m o c k e r y in his voice, " t h a t y o u have a bad c o n science a b o u t that point a n d feel impelled to r e t u r n to t h e
place of y o u r original sin.
"But to be f r a n k with y o u , " he a d d e d a f t e r a b a r e l y n o ticeable pause, " y o u could not help c o m i n g b a c k to t h a t
point, sin o r n o sin. It is the effect of the m a g i c spell you are
u n d e r right now. It is a fixation with y o u . "
I u n d e r s t o o d quite well that this speech was m e a n t primarily f o r the benefit of the o n l o o k e r s w h o seemed to be
quite intrigued by m y d r a w i n g , o r p e r h a p s simply e n j o y e d
m y obvious discomfiture. Be that as it m a y , the choice between a guilty conscience and a m a g i c spell to j u s t i f y a
geometrical figure was f o r m e too h a r r o w i n g , t o o upsetting
a n experience. T h e magician seemed to h a v e sensed t h a t
by his magic, I p r e s u m e , or he m a y h a v e read it in m y
face. F o r a f t e r a little while he addressed m e again, in a
most c o n d e s c e n d i n g m a n n e r .
142
MATHEMATICS IN FUN AND IN EARNEST
"If y o u are t r o u b l e d i n y o u r m i n d , sir, b y the d o u b l e
dose of parallel lines I i m p o s e d u p o n you, I shall s h o w you
m y m a g i c p o w e r s o v e r y o u in a simpler w a y . Y o u m a y
still c h o o s e the starting p o i n t X at will a n d t h e n proceed
to d r a w only the first set of t h r e e parallel lines. B u t b e f o r e
y o u d r a w those lines I will m a r k the p o i n t X ' o n B C w h e r e
you will l a n d . "
By t h a t t i m e I felt that I h a d n o t h i n g m o r e to lose in
t h e w a y of dignity, o r prestige, a n d I might p e r h a p s gain
s o m e insight into the trickery of the magician. I accepted
the offer. I tried the t h i n g twice, o n c e taking X on BC,
a n d the o t h e r time t a k i n g X o n B C extended. 1 d r e w m y
parallels very c a r e f u l l y , b u t n o m a t t e r h o w h a r d I tried,
the c o n f o u n d e d m a g i c i a n c a m e out on t o p : I ended u p
each time m i g h t y close to t h e spot he designated b e f o r e hand.
I c o u l d get n o t h i n g m o r e o u t of m y magician. I left the
place feeling a dire n e e d f o r s o m e m o r e light. M a g i c spell—
piffle. But w h a t else w a s t h e r e b e h i n d his u n c a n n y ability to
foretell w h e r e m y parallels w o u l d lead m e ? It would m a k e
a tedious story if I a t t e m p t e d to tell you of all the schemes
I resorted to in m y efforts to b r e a k o p e n that irritating puzzle. Suffice it to say that I w a s definitely d e t e r m i n e d to find
a n a n s w e r , a n d find o n e I did.
W h i l e I w a s p r o u d and h a p p y to succeed in tearing to
shreds the veil of the m a g i c i a n ' s secret, there really is not
m u c h to c r o w a b o u t n o w , as I look b a c k on it. Y o u just
notice that t h e figure as d r a w n includes two parallelograms
B X Y Z a n d C X Z Y , so t h a t we have
BX=YZ=CX'.
If y o u b e a r this in m i n d it is p e r f e c t l y easy to write out the
f o l l o w i n g equalities:
BX'=BC-CX'=:BC-BX
=CX.
T h e r e is n o t h i n g m o r e to the "baffling m y s t e r y " of the magician t h a n that B X ' = C X . W h e n he saw h o w f a r f r o m C m y
point X was, he m a r k e d his point X ' at the s a m e distance
f r o m B. O r , to p u t it in o t h e r w o r d s , t h e points X , X ' are
equidistant f r o m the m i d - p o i n t A ' of the base BC. In this
light I see n o w that I could h a v e put a t r o u b l e s o m e c r i m p
into the w o r k s of the m a g i c i a n , h a d I chosen f o r m y point
X the m i d - p o i n t A ' of B C as I had learned in m y plane
g e o m e t r y . I wish I could h a v e t h o u g h t of it then!
W h a t is still m o r e curious, the observation that X , X '
are s y m m e t r i c with respect to A ' dispels also the " m a g i c " of
MATHEMATICS AND THE MATHEMATICIAN
143
the d o u b l e set of parallels. I n d e e d , if I start with t h e p o i n t
X ' a n d d r a w the additional set of parallels X T , . . . I h a v e
to end up, a c c o r d i n g to t h a t observation, at a p o i n t o n
the o t h e r side of the m i d - p o i n t A ' of B C , at a distance
equal to X ' A ' , a n d this is precisely the point X . P e r f e c t l y
w o n d e r f u l , is it not, a n d in spite of its simplicity o r p e r h a p s
just o n a c c o u n t of it?
Surely, there is n o r o o m a n d n o excuse f o r r e a d i n g m a g i c
into such an innocent figure. Y e t I a m s o m e h o w inclined t o
c o m m e n d the magician f o r m a k i n g this piece a p a r t of his
p r o g r a m . I a m r e a d y to agree t h a t t h e r e is m o r e " w o n d r o u s
m a g i c " in this o n e geometrical p r o b l e m t h a n in all t h e rest
of the trickster's repertoire. B u t you h a v e to learn to a p preciate its e n c h a n t m e n t , I guess, just as o n e h a s to learn to
appreciate the taste of coffee, or the s m o k i n g of a pipe.
B ' A Square Deal
A m o n g m y a c q u a i n t a n c e s of m o r e
recent vintage t h e r e is a m a n with a r e p u t a t i o n of being
s o m e t h i n g of a "big-shot" as a m a t h e m a t i c i a n . Let us call
h i m Null, f o r short. T h e o t h e r d a y , while visiting h i m , I
could not resist the t e m p t a t i o n to spring o n h i m the p r o b l e m
of m y magician. 1 4 I was put out quite a bit w h e n N u l l c l a i m e d
to be f a m i l i a r with the p r o b l e m a n d to k n o w the proof
thereof. But w h e n I told him h o w I c a m e into possession of
this piece of m a t h e m a t i c a l learning, t h e r e was a look of surprise in his wide open eyes. " T h a t ' s c u r i o u s , " he r e m a r k e d
calmly, "I was not a w a r e that m a g i c i a n s m a k e use of geometrical p r o b l e m s in plying their tricky trade. But w h e n I
c o m e to think of it," he a d d e d a f t e r a brief pause, " t h e r e
is no reason w h y they should not. M a n y a p r o b l e m in
geometry, and in o t h e r b r a n c h e s of M a t h e m a t i c s , f o r t h a t
matter, m a y easily be dressed u p so as to serve their p u z zling p u r p o s e s . "
N o w it was m y turn to be surprised. " C o u l d you p e r c h a n c e think of an e x a m p l e ? " I asked s o m e w h a t dubiously.
" W e l l , " N u l l said slowly, obviously playing f o r time, " t o
f u r n i s h such a n e x a m p l e it is necessary to think of f a m i l i a r
things in an u n u s u a l way, in a w a y different f r o m the customary one."
H e got up, p a c e d his study f o r t h a n d b a c k several times,
evidently p r e o c c u p i e d . Suddenly he stopped right in f r o n t of
m e : " W o u l d you be willing to act as m y ' m e d i u m ' too,
n o w that you are an experienced h a n d in the business?"
"I shall be glad to be of help, if 1 c a n , " I said as casually
as I could, ignoring his irony.
"This time the trick will be quite d i f f e r e n t : you will h a v e
144
MATHEMATICS IN FUN AND IN EARNEST
t o d r a w p e r p e n d i c u l a r s , " h e j o k e d with growing good h u m o r .
U n d e r N u l l ' s direction, I m a r k e d f o u r points A , P, Q, R
in a r o w , entirely of m y o w n choice, with the sole restriction t h a t t h e s e g m e n t s A P a n d Q R h a d t o be equal. A t the
p o i n t s P, Q, R , w e erected p e r p e n d i c u l a r s to the line A P Q R
a n d o n t h e m w e m a r k e d t h e points B, C , D , so that P B = A Q ,
Q D = Q R , R C = P Q , t a k i n g c a r e t h a t the points B, C be o n
t h e s a m e side of t h e line A P Q R , a n d the p o i n t D o n the
o p p o s i t e side of it ( F i g . 1 1 ) .
" N o w w e a r e r e a d y f o r the kill, I m e a n f o r the finishing
t o u c h , " N u l l a n n o u n c e d , evidently satisfied with
my
h a n d i w o r k . " L e t ' s join A to B a n d D , a n d again C to B
and D."
I m u s t h a v e l o o k e d q u i t e puzzled c o n t e m p l a t i n g the c o m pleted d r a w i n g , f o r N u l l asked m e with ill concealed amusement: "What's wrong?"
Figure 11
" I t just looks to m e that I h a v e unwittingly d r a w n a
s q u a r e . In fact, I a m r e a d y to swear u p o n the beard of
P y t h a g o r a s himself that A B C D is as good a s q u a r e as I
MATHEMATICS AND THE MATHEMATICIAN
145
h a v e ever d r a w n b e f o r e . It could n o t b e a n accident, could
it?"
W i t h strained reserve N u l l r e p l i e d : " I w o u l d r a t h e r t h a t
you find the a n s w e r to y o u r q u e s t i o n y o u r s e l f , if y o u d o
not m i n d . "
I h a d to grant, in m y o w n m i n d , t h a t his w a s a r e a s o n able attitude, the m o r e so that it suited m e quite well. So I
went to w o r k . I observed that the only e l e m e n t of latitude
in the d a t a of the p r o b l e m was t h e s p a c i n g of the f o u r
points on the line. So I varied t h e m all I could. But the u b i quitous s q u a r e was t h e r e every time, in all its p r o v o c a t i v e
and challenging exactness.
"Of c o u r s e , " N u l l c o m m e n t e d w h e n I declared myself
satisfied as to the inevitability of the s q u a r e , " y o u k n e w bef o r e h a n d that this w o u l d be the case. O t h e r w i s e m y g a m e
would h a v e been pointless."
"Yes, M r . N u l l , I realized all along t h a t I could not w i n , "
was m y h u m b l e admission, " b u t the d r a w i n g h a d a g o o d
effect u p o n m y peace of m i n d , the p r o f e s s i o n a l s ' p r e j u d i c e
against graphical p r o o f s not w i t h s t a n d i n g . "
" Y o u r u n p r o f e s s i o n a l b e h a v i o r is f a r m o r e excusable t h a n
m y o w n , " intervened N u l l in good h u m o r , " a n d I t r a n s gressed in m o r e w a y s t h a n one. I n the first place, p r o f e s s ional c u s t o m , not to say p r o f e s s i o n a l ethics, w o u l d r e q u i r e t h a t
I tell y o u w h a t the o u t c o m e of t h e c h a i n of o p e r a t i o n s will
be, leaving open the question as to the proof of t h a t statement."
" A n d the t e m p t a t i o n to p r o d u c e a m o r e telling effect m a d e
you deviate f r o m the n a r r o w p a t h of v i r t u e , " I interposed
with m a r k e d irony.
"Quite so, quite so." N u l l l a u g h e d . " B u t also b e c a u s e I
worked u n d e r pressure. Y o u r m a g i c i a n did not tell y o u
what the climax of the p e r f o r m a n c e w o u l d be. I h a d t o
meet the c o m p e t i t i o n , h a d I n o t ? "
" T h e statement of this p r o b l e m is w o r d e d p o o r l y , " N u l l
continued w i t h o u t waiting f o r m e to g r a n t h i m his p a r d o n .
" M o r e o v e r , not all the conditions m e n t i o n e d are necessary
to obtain the square. T h e p r o b l e m could be stated as follows: C o n s t r u c t a s q u a r e ( A B C D ) given o n e vertex ( A ) a n d
the projections ( P , Q ) of the t w o a d j a c e n t vertices (B, D )
u p o n a line ( A P Q ) passing t h r o u g h the given vertex ( A ) . " 1 5
"This is a m u c h m o r e t r a n s p a r e n t w a y of putting it," I
gladly c o n c e d e d , " w h y then resort to the o t h e r ? "
"Professional m a t h e m a t i c i a n s are h u m a n , too, believe it
or n o t , " N u l l replied, smiling indulgently. " T h e y are not al-
146
MATHEMATICS IN FUN AND IN EARNEST
w a y s o u t t o enlighten their c o n f r e r e s . S o m e t i m e s they would
r a t h e r dazzle a n d puzzle their brothers-in-arms. So they
c a m o u f l a g e the s o u r c e of their i n f o r m a t i o n the best they can
a n d invite o r challenge the others to try to u n e a r t h their
secrets."
" I r e a d s o m e w h e r e , " I said, trying to parallel his argum e n t , " t h a t a f o x w a l k i n g on snow-covered g r o u n d uses his
tail to obliterate his f o o t p r i n t s , in o r d e r to keep the h u n t e r
f r o m getting o n his trail. M a t h e m a t i c i a n s sometimes try to
be f o x y , t o o . "
" N o w , n o w , m y y o u n g f r i e n d , I a m a f r a i d you are waxing s o m e w h a t cynical," N u l l objected, his smiling eyes cont r a d i c t i n g the mildly r e p r o a c h f u l tone of his voice.
FOOTNOTES
1
Cf. Chapter III, Section 2b.
Cf. Chapter I, Section 4.
Erreurs des Mathematiciens,
Maurice Lecat (Brussels, 1935).
4
Cf. Chapter III, Section 3e.
6
Cf. Chapter I, Section Id.
* What Is Mathematics?, by Richard Courant and Herbert Robbins (1941).
7
Cf. Chapter I, Section Id; Section 3f; Section 4b.
8
Ibid., Section 4b.
" C f . Chapter I, Section 3f; Chapter III, Section 3f.
10
Scripta Mathematica,
13, Nos. 1-2, 1947.
11
See Chapter II, Section Id.
12
Funkenbush, William, and Eagle, Edwin, "Hyper-Spatial TitTat-Toe" or "Tit-Tat-Toe in Four Dimension," National
Mathematics Magazine, Vol. 20, No. 3, December 1944, pp. 119-122.
13
Walker, S. M., "Games of the Checkers Family in Line,
Plane and Space," Bulletin of the American Mathematical
Society,
Vol. 52, No. 9, September 1946, p. 825, art. 325.
14
See Chapter V, Section 3a.
16
The vertices B, D can be constructed as before. From the
two congruent right triangles ABP, ADQ we have:
AB=AD,
BAD = BAP + P A D = 90 •
hence A, B, D are the three vertices of a square.
The fourth vertex is the symmetric C of A with respect to the
mid-point of the diagonal BD.
Observe that if R is the projection of C upon the line APQ, we
have QR = AP, for the two segments are the projections of two
equal and parallel segments CD, AB upon the same line APQ.
Cf Educational Times, Reprints, series 3, Vol. 5, 1918, p 72, Q.
18570.
2
3
VI
MATHEMATICAL ASIDES
1
•
Mathematical Asides
A * "It Is Obvious That..
." It would be n o exaggeration to say that in the writing on n o other subject do the
authors have as often recourse to the phrase, "it is obvious
that," as is the case in mathematics. Y o u have been f r e quently annoyed, n o doubt, by this reference to the obviousness of certain statements which to you seem anything but
obvious. Does that m e a n that the mathematical writer is so
m u c h smarter than his reader? Is the a u t h o r indulging in a
sadistic pleasure, or is he trying to poke f u n at his reader?
Perhaps a few examples picked m o r e o r less at r a n d o m ,
might shed some light on this sore spot.
B • Four Examples
A student missed one of the f o u r quizzes given during the term. T h e instructor c o m p u t e d the student's average on the three quizzes taken. W h a t grade did
the instructor, by this procedure, grant the student f o r the
quiz missed?
If a, b, c, are the grades m a d e by the student, and x the
grade the instructor granted f o r the quiz missed, we have
(a + b+c)/2> = {a +
b+c+x)/4-(a+b+c)/<\+x/4
hence
x=(a+b+c)/3
Thus the instructor granted the student, f o r the quiz missed,
a grade equal to the average of the grades the student m a d e
on the quizzes he took.
A f t e r this result has been stated explicitly m a n y a reader
is likely to feel that the recourse to calculations was unnecessary, for this answer seemed obvious. This is actually the case.
All that is needed is to observe that since the " g r a n t e d " grade
does not alter the average, it must be equal to that average,
or may differ f r o m that average by a couple of points, at
most, in either direction, if fractions are to be taken into consideration.
As a second example consider the proposition: If two
perpendicular lines are drawn in the plane of a square, the
segment intercepted by a pair of opposite sides of the square
147
148
MATHEMATICS IN FUN AND IN EARNEST
on one of the two lines is equal to the segment which the
other pair of opposite
sides intercepts
on the other
given
line.
L e t A BCD b e t h e given s q u a r e a n d let the t w o pairs of
opposite sides AD, BC a n d AB, CD intercept the segments
EF, HG, respectively, o n t h e t w o given lines (Fig. 1 2 ) . W e
are to s h o w t h a t those t w o segments a r e equal.
T h e p r o p o s i t i o n m a y b e p r o v e d in a n u m b e r of ways of
v a r y i n g degrees of c o m p l e x i t y . B u t actually the proposition is
Figure 12
p r a c t i c a l l y obvious. I n d e e d , if w e leave the line EF in its
place a n d i m a g i n e t h a t w e spin the square, a n d the line GH
with it, a b o u t the c e n t e r O of the s q u a r e counter-clockwise b y
a n angle of 9 0 ° , the sides DA, AB, BC, CD will o c c u p y the
p r e s e n t positions of t h e sides AB, BC, CD, DA, respectively,
a n d t h e line HC w h i c h is p e r p e n d i c u l a r t o EF, by a s s u m p tion, will, o n a c c o u n t of the r o t a t i o n , b e c o m e parallel to
EF, a n d t h e r e f o r e
HG-EF.
A s a t h i r d e x a m p l e c o n s i d e r the p r o b l e m : A rigid ellipse
moves so that it constantly
remains tangent to the
coordinate
axes. Find the locus of the center of the elipse.
I n general a p r o b l e m of this t y p e offers considerable
difficulty a n d its solution m a y be l o n g a n d laborious. In the
p r e s e n t case, h o w e v e r , t h e solution is " o b v i o u s . "
I n d e e d , t h e c o o r d i n a t e axes are a pair of r e c t a n g u l a r
t a n g e n t s d r a w n t o t h e ellipse, h e n c e the origin O lies o n the
Monge circle ( a l s o called the orthoptic circle of t h e director
circle) of the ellipse. T h u s t h e d i s t a n c e of t h e origin f r o m the
c e n t e r C of t h e ellipse is e q u a l to the radius of the M o n g e
circle, h e n c e t h e locus of C is a circle equal to the Morige
circle of the ellipse, a n d h a v i n g O f o r center.
In o r d e r to c o m e closer to the " o b v i o u s n e s s " of an a u t h o r
MATHEMATICAL ASIDES
149
we shall consider n o w o n e m o r e e x a m p l e , the last, t a k e n
f r o m the writer's o w n experience.
G i v e n a t e t r a h e d r o n ( T ) = A B C D a n d a p o i n t M, the f o u r
planes passing t h r o u g h the vertices A, B, C, D, a n d p e r p e n dicular to the lines AM, BM, CM, DM, respectively, f o r m a
t e t r a h e d r o n called the antipedal t e t r a h e d r o n ( 5 ) of ( T ) f o r
the point M. N o w let A', B', C', D', be t h e points in w h i c h the
lines MA, MB, MC, MD, m e e t again the c i r c u m s p h e r e ( O )
of ( T ) , a n d let ( S ' ) be the antipedal t e t r a h e d r o n , f o r the
point M, of the t e t r a h e d r o n (T')
=A'B'C'D'.
Considering poles a n d p o l a r planes the writer arrived at
the surprising conclusion t h a t the lines joining the c o r r e sponding vertices of the t w o antipedal t e t r a h e d r o n s ( 5 ) , ( S ' )
m e e t in the center O of the s p h e r e ( O ) a n d bisect each o t h e r .
T h i s seemed incredible, f o r t h e p r o p e r t y does n o t involve
the point M, while both t e t r a h e d r o n s ( 5 ) a n d ( £ ' ) d e p e n d o n
that point. T h e writer did not follow u p this result.
Studying the s a m e figure f r o m a n entirely n e w angle t h e
writer s t u m b l e d again u p o n the very s a m e result. It w a s not
possible to d o u b t it a n y m o r e . But if the result is valid, its
simplicity w o u l d suggest that t h e r e m u s t be a m o r e direct
a p p r o a c h to it t h a n either of the t w o m e t h o d s used h i t h e r t o .
F u r t h e r reflection b r o u g h t the realization that the t w o f a c e s
of the t w o t e t r a h e d r o n s ( 5 ) , ( S ' ) w h i c h are p e r p e n d i c u l a r to
the c h o r d AM A' of the s p h e r e ( O ) at the points A, A' are
symmetrical with respect t o the m e d i a t o r (i.e., the p e r p e n d i c ular bisecting p l a n e ) of this c h o r d . N o w this m e d i a t o r passes
the center O of the s p h e r e ( 0 ) , h e n c e the t w o planes a r e
symmetrical with respect to the center O. T h e s a m e holds
f o r any o t h e r pair of c o r r e s p o n d i n g f a c e s of t h e t w o t e t r a h e d r o n s ( S ) , (S")- T h u s the t w o t e t r a h e d r o n s are s y m m e t rical with respect to the c e n t e r O, h e n c e the p r o p o s i t i o n ,
which thus b e c o m e s " a l m o s t o b v i o u s " a priori.
C • An Explanation
A n attentive scrutiny of t h e e x a m p l e s
given reveals s o m e very interesting f e a t u r e s of w h a t lies behind that so f r e q u e n t l y t r o u b l e s o m e s t a t e m e n t "it is obvious
t h a t . " T o s o m e o n e w h o is not f a m i l i a r with the M o n g e circle
of the ellipse the solution p o i n t e d out in c o n n e c t i o n with the
p r o b l e m of the ellipse is, of course, not obvious. T h e solver,
b e f o r e he arrives at a n answer, will h a v e to discover the
M o n g e circle f o r himself, even if only in a r o u n d - a b o u t w a y ,
and in not a very explicit f o r m . But even to a person to
w h o m the M o n g e circle is not a novelty, the solution indicated m a y not o c c u r very readily. T h e circle is not m e n t i o n e d
in the question. T h e pair of r e c t a n g u l a r tangents constituted
150
MATHEMATICS IN FUN AND IN EARNEST
b y the t w o c o o r d i n a t e axes is the only hint at that circle.
F o r success in t h e solution of the p r o b l e m this slight hint
m u s t be sufficient to e v o k e in the m i n d of the solver the
i m a g e of t h e M o n g e circle a n d its relation to the problem.
W h e t h e r that hint will suffice o r not m a y depend u p o n a
n u m b e r of c i r c u m s t a n c e s , like the degree of alertness of
the solver at t h e m o m e n t , a n d o t h e r such conditions, but
m a i n l y u p o n t h e d e g r e e to w h i c h the M o n g e circle is fresh
in the solver's m i n d .
T h i s seems to p o i n t to the f a c t that there are what might
be called degrees to w h i c h w e m a y k n o w a given fact.
S h o u l d y o u ask a f r i e n d w h e n C o l u m b u s discovered A m e r i c a
he m a y n o t b e able t o supply the date. But he m a y nevertheless be able to recall the last two digits of that date if you
w o u l d m e n t i o n t h e first two digits, 1, 4. Should y o u r friend
not be able to d o that either, a n d you w o u l d q u o t e the full
d a t e 1492, he m a y agree with you very readily and assure you
that he h a s k n o w n that d a t e f o r m a n y years. T h e situation
s e e m s to be a k i n to w h a t the psychologists call the "threshold
of sensitivity." T h e question is h o w m u c h stimulus is necessary in o r d e r t o m a k e the solver a w a r e of, to bring to the
f o r e f r o n t of his m i n d , and t o m a k e available f o r use, a
piece of i n f o r m a t i o n w h i c h is lying d o r m a n t s o m e w h e r e in
his m e m o r y , in his s t o r e r o o m of k n o w l e d g e . T h e f r e s h e r this
i n f o r m a t i o n is the less stimulus will be necessary. T h u s , if
the solver dealt with t h e M o n g e circle very recently, he is
m u c h m o r e likely to perceive the c o n n e c t i o n between that
circle a n d the p r o b l e m at h a n d than if he had no occasion
to r e f e r to it f o r a long time. If the circle is fresh in his
m i n d a n d he is well g r o u n d e d in its use, the solution is "obvious." E x a m p l e o n e illustrates the s a m e idea, perhaps, in
a smaller w a y . T o s o m e o n e w h o deals with averages and
statistical d a t a the question m a y b o r d e r on triviality, which
to s o m e o n e else the a n s w e r to the question becomes obvious
only post f a c t u m .
W h e n o n e has been w o r k i n g on a given subject a n y length
of time o n e h a s the o p p o r t u n i t y to see it f r o m different angles
a n d o n e b e c o m e s f a m i l i a r with its various ramifications. T h e
i n t e r c o n n e c t i o n s b e t w e e n the various parts b e c o m e as natural, say, as the c o n n e c t i o n b e t w e e n t w o parts of the same
f a m i l i a r m e l o d y . O n e gets the feeling that these interrelations
c a n n o t possibly be missed o r overlooked by a n y b o d y , so that
it is quite s u p e r f l u o u s to be pointing t h e m out, and a statem e n t like "it is o b v i o u s t h a t " will be sufficient to put any
r e a d e r on the right track. T h e good faith of the a u t h o r using
such an expression need not be questioned, but his optimism
MATHEMATICAL ASIDES
151
m a y h a v e less secure f o u n d a t i o n s t h a n he thinks. T o t h e
r e a d e r the c o n n e c t i o n s b e t w e e n the p a r t s of t h e subject m a y
not b e c o m e clear as r a p i d l y as the a u t h o r expects, a n d t h e
a u t h o r ' s "it is obvious t h a t " m a y s o u n d like m o c k e r y , o r at
least like a n u n d e s e r v e d r e p r o a c h . T h e r e a d e r m a y , if h e
w a n t s to, find consolation in the f a c t that, as t i m e goes o n ,
the subject m a y "cool o f f " in the m i n d of the a u t h o r , the
relations that were so vivid at the t i m e of writing m a y f a d e
away, a n d w h a t was " o b v i o u s " m a y b e c o m e to h i m i n c o m prehensible a n d even i m p e n e t r a b l e . It is n o u n c o m m o n o c c u r r e n c e to witness scholars of r e n o w n discussing their o w n
contributions a n d being " s t u m p e d " b y t h e " o b v i o u s " in
their o w n writings. W h i l e this m a y offer s o m e solace t o
the harassed reader, it does not d o a w a y with the difficulty.
T h e a u t h o r is always c o n f r o n t e d with the task of deciding
what he should explain in detail, a n d w h a t he can leave to
the erudition of the r e a d e r . U n f o r t u n a t e l y t h e r e can be n o
definite a n s w e r to this question. S u p e r f l u o u s verbosity m a y
obscure the subject just as effectively as u n d u e reticence.
T h e e x a m p l e with the antipedal t e t r a h e d r o n s illustrates a n o t h e r aspect of the " o b v i o u s . " It h a p p e n s quite o f t e n that w e
obtain propositions by m o r e o r less laborious m e t h o d s , only to
discover that the result m a y be established by very simple
reflections that are quite obvious, o r nearly so. W h y is it easier to arrive at these results by the devious r o u t e r a t h e r t h a n
by the simple o n e , the direct o n e ? T h e r e a s o n is p r o b a b l y
purely psychological. W e just d o not e x p e c t to get " s o m e thing f o r n o t h i n g , " a n d apply to o u r p r o b l e m s o u r s t a n d a r d
tools in which w e h a v e c o n f i d e n c e and w h i c h we k n o w h o w
to h a n d l e with s o m e skill. If it h a p p e n s that the simplicity
of the result obtained does not seem to be in k e e p i n g with
the heavy m a c h i n e r y we used to derive it, we m a y look
a r o u n d , discover a n a p p r o p r i a t e l y direct m e t h o d , a n d end
u p by declaring a posteriori
that the proposition was obvious a priori. M o r e o f t e n t h a n not this is d o n e without a n y
mention being m a d e of the original m e t h o d by w h i c h the
proposition was actually derived in the first place. T h e exa m p l e a b o u t the t e t r a h e d r o n is "telling out of s c h o o l . "
D ' Analogy as a Useful Guide to Discovery
The finding
of new properties, of new propositions is a t r o u b l e s o m e
u n d e r t a k i n g w h i c h e v e r way you m a y look at it. T h e r e is n o
standard path at the e n d of which is the r a i n b o w . This r o a d
of discovery, just as f o r b i d d i n g as it is alluring, is negotiated
mainly by groping, clumsily and blunderingly. Of the m e a g e r
sources of light available p r o b a b l y the best is provided by
152
MATHEMATICS IN FUN AND IN EARNEST
a n a l o g y . I n t h e t h e o r y of f u n c t i o n s of t w o or m o r e variables
w e are g u i d e d b y t h e ideas a n d results w h i c h p r o v e d successf u l in t h e s t u d y of f u n c t i o n s of o n e variable, to m e n t i o n a
simple a n d f a m i l i a r e x a m p l e . W h i l e studying t h r e e - d i m e n sional g e o m e t r y w e k e e p a n eye o n the results available in the
p l a n e . T h i s is p a r t i c u l a r l y t r u e w h e n properties of the tetrah e d r o n a n d t h e s p h e r e are sought. It is an excellent i n d o o r
s p o r t to e x t e n d to space k n o w n p r o p e r t i e s of the triangle a n d
the circle. T h e efforts exerted in this direction are f r e q u e n t l y
r e w a r d e d v e r y readily. N u m e r o u s e x a m p l e s of such extensions of p r o p e r t i e s of t h e p l a n e m a y be f o u n d in the p r o b lem d e p a r t m e n t s of t h e American
Mathematical
Monthly
and
the Mathematics
Magazine.
In m a n y cases the generalization
is so n a t u r a l , so close at h a n d , that o n e w o n d e r s h o w the a u t h o r of t h e p r o p e r t y in the p l a n e failed to think of the
t h r e e - d i m e n s i o n a l case. B u t let only those of us cast the
stone of r e p r o a c h w h o feel themselves w i t h o u t guilt. H a v e
you t h o u g h t of e x t e n d i n g to space the p r o p e r t y of the moving ellipse w e considered a while b a c k ? T h e proposition applies nevertheless, a n d with only obvious modifications of
the t e r m s involved, to a rigid ellipsoid w h i c h moves so as
to r e m a i n c o n s t a n t l y t a n g e n t to the c o o r d i n a t e planes, and
the locus of t h e c e n t e r of the ellipse is a sphere having the
origin f o r c e n t e r a n d equal to the M o n g e s p h e r e of the ellipsoid.
E ' Limitations
of That Method
V a l u a b l e as such analogies m a y be, o n e m u s t not place too m u c h reliance on them,
f o r they are o f t e n misleading. H e r e is a n illustration. F o u r
c o p l a n a r , n o n - c o n c y c l i c points, A, B, C, D, d e t e r m i n e f o u r
circles, ABC, BCD, CD A, DAB. N o w it is readily s h o w n b y
the use of inversion, that if a n y two of these f o u r circles are
o r t h o g o n a l , the r e m a i n i n g t w o circles a r e also o r t h o g o n a l to
e a c h o t h e r . T h e extension to space is obvious. Five nonc o s p h e r i c a l points A, B, C, D, E, d e t e r m i n e five spheres
ABCD,
BCDE, CDEA,
DEAB,
EABC.
If two of these five
s p h e r e s a r e o r t h o g o n a l , does it f o l l o w that the r e m a i n i n g
t h r e e a r e m u t u a l l y o r t h o g o n a l ? P e r h a p s this premise is t o o
w e a k , a n d to be o n the s a f e side it w o u l d be better to assume
t h a t t h r e e of the five spheres are m u t u a l l y o r t h o g o n a l and
d r a w the c o n c l u s i o n that the r e m a i n i n g two are o r t h o g o n a l to
each other. Well, as a m a t t e r of f a c t the proposition is false
in either case. M o r e t h a n that. N o t only does the o r t h o g o n ality of t h r e e of the spheres not imply the orthogonality of the
r e m a i n i n g two, but, o n the c o n t r a r y , if t h r e e of the spheres
MATHEMATICAL ASIDES
153
are o r t h o g o n a l , the r e m a i n i n g t w o cannot be orthogonal
to
each o t h e r .
T h e t h r e e projections, u p o n t h e sides of a triangle, of a
point o n the c i r c u m c i r c l e of t h a t triangle a r e collinear ( t h e
Simson l i n e ) , a n d this is only t r u e of t h e points o n t h e circumcircle. It is easy to f o r m u l a t e the a n a l o g o u s p r o p o s i t i o n
in space, b u t the analogy is distressingly misleading. T h e r e
are points in space w h o s e f o u r projections u p o n t h e f a c e s of
the t e t r a h e d r o n are c o p l a n a r , b u t these points h a v e n o t h i n g
to d o with the c i r c u m s p h e r e of t h e t e t r a h e d r o n . T h e locus of
those points is a cubic s u r f a c e passing t h r o u g h t h e edges of
the t e t r a h e d r o n a n d t h r o u g h the q u a d r i t a n g e n t c e n t e r s of t h e
tetrahedron.
But o n e does not h a v e to go so f a r afield t o find striking examples w h e n the analogy b e t w e e n the p l a n e a n d s p a c e
fails to h o l d in places w h e r e o n e w o u l d confidently expect it
t o be t h e case. T h a t the c u b e in space is the a n a l o g u e of
the s q u a r e in the p l a n e requires n o a r g u m e n t . L e t u s f o l l o w
u p this analogy b y c o n s i d e r i n g t h e d i a g o n a l s of the t w o figures. T h e diagonals of a s q u a r e a r e e q u a l . So are t h e d i a g o nals of a cube. T h e diagonals of a s q u a r e bisect e a c h o t h e r .
T h e s a m e holds f o r the diagonals of a c u b e . T h e diagonals of
a s q u a r e a r e p e r p e n d i c u l a r . I s t h a t t r u e of the d i a g o n a l s of
a cube? N o , it is not. T h e diagonals of a c u b e c a n n o t b e
m u t u a l l y p e r p e n d i c u l a r , f o r if t h e y w e r e w e w o u l d h a v e at
their c o m m o n p o i n t f o u r m u t u a l l y p e r p e n d i c u l a r lines w h i c h ,
in a t h r e e - d i m e n s i o n a l space, w o u l d be q u i t e a spectacle t o
behold.
T h e b r e a k i n g d o w n of the a n a l o g y b e t w e e n the p l a n e a n d
space is n o t the only s h o r t c o m i n g of this s o u r c e of inspiration
in o u r quest f o r new p r o p o s i t i o n s in space. T h r e e - d i m e n sional space is m u c h richer in relations t h a n is the t w o - d i m e n sional plane. It has p r o p e r t i e s w h i c h h a v e n o c o u n t e r p a r t
in the plane. T h e a n a l o g of c u r v e s in t h e p l a n e a r e s u r f a c e s
in space. B u t in addition t o t h e s u r f a c e s w e also study
curves in space, b o t h p l a n e a n d skew. If w e w o u l d rely t o o
closely u p o n t h e plane, specific p r o p e r t i e s of space m i g h t be
overlooked. W e shall n o w consider an e x a m p l e in w h i c h b o t h
the help a n d the limitations of t h e p l a n e c o n s i d e r a t i o n c o m e
into play.
O u r school b o o k s o n g e o m e t r y told us t h a t a triangle h a s
three altitudes a n d that those t h r e e lines h a v e a point in c o m m o n . C o n t r a r y to w h a t is implicitly believed, this p r o p o s i t i o n
is not to be f o u n d in Eucilid's Elements.
B u t it w o u l d b e erroneous to c o n c l u d e f r o m this absence t h a t Euclid did n o t
k n o w this p r o p e r t y . W e are c e r t a i n t h a t A r c h i m e d e s , w h o
lived soon a f t e r Euclid, was a w a r e of this p r o p e r t y of t h e
154
MATHEMATICS IN FUN AND IN EARNEST
triangle. H e r e f e r s to it in his writings as to s o m e t h i n g his
r e a d e r s are expected t o b e f a m i l i a r with.
F • The Altitudes of a Triangle and of a Tetrahedron
A
triangle is a p l a n e figure f o r m e d b y three n o n - c o n c u r r e n t
fines. Its a n a l o g in t h r e e d i m e n s i o n a l space is a figure determ i n e d b y f o u r p l a n e s h a v i n g n o p o i n t in c o m m o n , that is, a
t e t r a h e d r o n , o r w h a t is t h e s a m e thing, a t r i a n g u l a r p y r a m i d .
T h i s solid w a s f a m i l i a r t o the G r e e k s f r o m the earliest times.
T h e y l e a r n e d of it f r o m the E g y p t i a n s w h o were building
p y r a m i d s m a n y centuries b e f o r e the G r e e k s a p p e a r e d o n the
scene of history. T h e G r e e k s considered the altitudes of the
t e t r a h e d r o n in c o n n e c t i o n with t h e f o r m u l a f o r the v o l u m e of
t h a t solid.
A t e t r a h e d r o n h a s f o u r altitudes. T h e analogy with the
triangle suggests the obvious q u e s t i o n : " D o the f o u r altitudes
of a t e t r a h e d r o n m e e t in a p o i n t ? " It w o u l d take a h a r d y
soul, indeed, to m a i n t a i n t h a t this simple idea never o c c u r r e d
to t h e inquisitive m i n d s of t h e G r e e k geometers. O n the
o t h e r h a n d , in all the m a t h e m a t i c a l writings of the G r e e k s
w h i c h c a m e d o w n to us in a n y f o r m , t h e r e is not the slightest h i n t t h a t the question of t h e c o n c u r r e n c e of the altitudes
of a t e t r a h e d r o n has ever p r e o c c u p i e d those imaginative
scholars. M o r e o v e r , this question w a s ignored, with a persiste n c e w o r t h y of a b e t t e r cause, by the M i d d l e Ages, the R e n aissance, a n d clear d o w n to t h e nineteenth century.
It is, h o w e v e r , possible to find s o m e attenuating circumstances f o r this c u r i o u s silence. If o n e is asked to answer
o u r question b y a " Y e s " o r " N o " o n e is f a c e d with the same
u n c o m f o r t a b l e situation as w h e n c o n f r o n t e d with t h e question : "Will y o u stop beating y o u r w i f e ? " O n e w o u l d be in the
w r o n g n o m a t t e r w h i c h of the t w o alternatives o n e decides
to espouse.
T h e secret of this puzzling situation is simple. T h e r e are
types of t e t r a h e d r o n s w h o s e altitudes meet in a point, a n d
t h e r e are types f o r w h i c h this is not the case. If the altitudes
of a t e t r a h e d r o n m e e t in a point, that entails t h e p r o p e r t y
t h a t e a c h edge of the t e t r a h e d r o n m a k e s a right angle with
the opposite edge. Conversely, if a t e t r a h e d r o n has the latter
p r o p e r t y , it also h a s the f o r m e r . T h i s type of a t e t r a h e d r o n ,
c o m m o n l y r e f e r r e d to as " o r t h o c e n t r i c , " has a good m a n y
o t h e r p r o p e r t i e s w h i c h a r e close analogs of properties of the
triangle.
T h r e e m u t u a l l y o r t h o g o n a l planes, like, f o r instance, the
floor a n d t w o a d j a c e n t walls in a r o o m , a n d a n y f o u r t h plane
f o r m a " t r i r e c t a n g u l a r " t e t r a h e d r o n . T h e line of intersection
MATHEMATICAL ASIDES
155
of a n y t w o of t h e first t h r e e planes c o n s i d e r e d is p e r p e n d i c u lar to the third plane a n d is t h e r e f o r e a n altitude of t h e tetrah e d r o n . T h u s the f o u r altitudes of a t r i r e c t a n g u l a r t e t r a h e d r o n
all pass t h r o u g h the vertex of t h a t t e t r a h e d r o n c o m m o n to
the three m u t u a l l y o r t h o g o n a l f a c e s of t h a t solid.
A t e t r a h e d r o n A B C D m a y h a v e only o n e pair of m u t u a l l y
o r t h o g o n a l opposite edges, say, A B a n d C D . I n such a " s e m i o r t h o c e n t r i c " t e t r a h e d r o n the altitudes issued f r o m A a n d B
have a point in c o m m o n , a n d the s a m e h o l d s f o r the altitudes
issued f r o m C a n d D.
T h e t h r e e types of t e t r a h e d r o n s c o n s i d e r e d a b o v e is all
that we c a n get out of the a n a l o g y b e t w e e n the altitudes of
the triangle a n d the t e t r a h e d r o n . T h i s , h o w e v e r , does n o t exhaust the topic, f o r there are t e t r a h e d r o n s in w h i c h n o edge
is p e r p e n d i c u l a r to the opposite edge, a n d t h e r e f o r e n o n e of
the f o u r altitudes meets a n o t h e r altitude, t h a t is, t h e f o u r altitudes are f o u r m u t u a l l y skew lines. D o e s t h a t m e a n t h a t in
such a case the f o u r altitudes are f o u r totally u n r e l a t e d lines?
N o , the f o u r altitudes are n o t total strangers to each o t h e r ,
but their relation to o n e a n o t h e r is not a n analogy of a p r o p erty of the altitudes of a triangle.
In o r d e r to show w h a t t h a t relation is w e h a v e to c o n sider s o m e preliminaries. A given p o i n t M a n d t w o skew
lines a, b, in space d e t e r m i n e t w o planes ( P ) = M — a , (£?) =
M—b w h i c h h a v e a line u in c o m m o n ( F i g . 1 3 ) . T h e point M
lies on their c o m m o n line u. F u r t h e r m o r e , the t w o lines u
Figure 13
a n d a lie in the p l a n e ( P ) , h e n c e they h a v e a p o i n t in c o m m o n
(we neglect the special case of p a r a l l e l i s m ) , a n d t h e s a m e
holds f o r the lines u a n d b, f o r similar reasons. T h e u p s h o t of
156
MATHEMATICS IN FUN AND IN EARNEST
the story is t h a t w e h a v e constructed a line u passing t h r o u g h
the p o i n t M a n d intersecting t h e lines a a n d b.
S u p p o s e n o w t h a t t h r o u g h t h e p o i n t M w e d r a w a line c
s k e w t o the lines a a n d b. If w e should treat o t h e r points of
the line c t h e s a m e w a y as w e p r o c e e d e d with the point M
we m a y c o n s t r u c t an infinite n u m b e r of lines m e e t i n g the
t h r e e skew lines a, b, c.
L e t d be a n y line in space skew to each of the lines a, b, c.
H o w m a n y of the infinite n u m b e r of lines m e e t i n g the latter
t h r e e lines m e e t also the line dl It is p r o v e d that the
a n s w e r t o this q u e r y is: not m o r e t h a n t w o lines, as a rule.
A n d t h e latter qualifier "as a r u l e " is the c r u x of the matter.
T h e r e m a y be exceptions t o the rule. A n d if it should h a p p e n
t h a t of t h a t infinity of lines three lines should m e e t the line
d, all t h e rest of t h e m will d o likewise. T h u s the f o u r m u t u ally s k e w lines a, b, c, d will be m e t b y a n infinite n u m b e r
of straight lines. T h i s is usually stated m o r e succinctly by saying t h a t t h e lines a, b, c, d f o r m a " h y p e r b o l i c g r o u p . "
L e t us n o w r e t u r n to o u r f o u r altitudes. W e are n o w
r e a d y to state the p r o p o s i t i o n : " T h e f o u r altitudes of a tetr a h e d r o n a r e f o u r skew lines such t h a t a line which m e e t s
a n y t h r e e of t h e m also m e e t s the f o u r t h . " T h e proposition
m a y be stated m o r e briefly: " T h e f o u r altitudes of a tetrahedron f o r m a hyperbolic group."
T h i s p r o p o s i t i o n was first f o r m u l a t e d d u r i n g the third deca d e of t h e n i n e t e e n t h c e n t u r y .
2
•
"The Figure of the Bride"
A • Historical
Data
E a r l y in the n i n e t e e n t h century the
West b e c a m e a c q u a i n t e d with t h e m a t h e m a t i c a l writings of
B h a s k a r a ( 1 1 1 4 - 1 1 8 5 ? ) of India. In English a first glimpse
of these w o r k s w a s p r o v i d e d b y C h a r l e s H u t t o n ( 1 7 3 7 - 1 8 2 3 ) ,
p r o f e s s o r of m a t h e m a t i c s at t h e R o y a l Military A c a d e m y ,
W o o l w i c h . I n 1812 H u t t o n published in L o n d o n a three
v o l u m e collection of " T r a c t s o n M a t h e m a t i c a l a n d Philosophical S u b j e c t s . " T r a c t N o . 33, Vol. II, pp. 143-305 deals with
" T h e H i s t o r y of A l g e b r a of all N a t i o n s . " In particular, pages
151-179 a r e d e v o t e d t o " I n d i a n A l g e b r a . " T h e revelations
a b o u t I n d i a n m a t h e m a t i c s m a d e therein must h a v e created
quite a stir at that time, considering that shortly a f t e r H u t ton's w o r k a p p e a r e d , a p r o f e s s o r of m a t h e m a t i c s at the Royal
Schools of Artillery on the o t h e r side of the English C h a n nel, n a m e l y O . T e r q u e m ( 1 7 8 2 - 1 8 6 2 ) , translated the part of
MATHEMATICAL ASIDES
157
H u t t o n ' s " H i s t o r y " relating t o I n d i a a n d p u b l i s h e d it in H a c h -
ette's Correspondence sur I'Ecole Poly technique?
B •
The Theorem
of Pythagoras in India
One of the
w o r k s of B h a s k a r a w h i c h H u t t o n q u o t e s a n d c o m m e n t s u p o n
extensively is entitled " L i l a v a t i " a n d is d e v o t e d largely to
A r i t h m e t i c . I n it there is t h e p a s s a g e : " I n t h e m a r g i n of t h e
original, as h e r e a n n e x e d , is d r a w n a figure of f o u r e q u a l
right triangles joined in t h e m a n n e r indicated ( F i g . 14) exhibiting a n e w a n d obvious proof of the 4 7 t h proposition
of Euclid I ( t h a t is, the P y t h a g o r e a n t h e o r e m ) : f o r h e r e a r e
the f o u r right triangles, w h i c h are e q u a l to twice the r e c t a n gle of their t w o p e r p e n d i c u l a r sides, a n d w h i c h t o g e t h e r with
the small s q u a r e in t h e middle, b e i n g t h e s q u a r e of t h e difference of those t w o sides, m a k e u p the large s q u a r e o n t h e
h y p o t e n u s e ( I n m o d e r n n o t a t i o n : if a, b are the p e r p e n d i c u l a r
sides of o n e of the triangles, t h e a r e a of the big s q u a r e is
equal to 4 * a b / 2 + ( a — b ) 2 = a 2 + b 2 ) . T h e r e f o r e the s q u a r e
158
MATHEMATICS IN FUN AND IN EARNEST
o n t h e h y p o t e n u s e is e q u a l t o the s u m of t h e squares o n the
o t h e r t w o sides."
" A n d this m a y b e c o n s i d e r e d the I n d i a n d e m o n s t r a t i o n
of t h e c e l e b r a t e d p r o p e r t y of t h e sides of a right-angle triangle; a p r o p e r t y so m u c h e m p l o y e d b y their geometricians,
a n d so o f t e n r e f e r r e d to in their writings by the n a m e of 'the
figure of t h e b r i d e ' a n d 'the figure of the bride's c h a i r ' a n d
' t h e figure of t h e w e d d i n g c h a i r ' , epithets w h i c h we m a y
c o n j e c t u r e h a v e b e e n suggested by the a b o v e figure bearing
s o m e r e s e m b l a n c e to a p a l a n q u i n o r a s e d a n chair, in which
it is t h e usual practice, in that c o u n t r y , f o r the bride to be
c a r r i e d h o m e to h e r h u s b a n d ' s h o u s e . "
Is it quite c e r t a i n t h a t a " c e l e b r a t e d " proposition necessarily h a s to h a v e a n i c k n a m e ? B u t if it be assumed t h a t it
should, t h e n it w o u l d seem that a n epithet like " t h e figure of
the w e d d i n g c h a i r " s h o u l d be fully as acceptable as, say, the
well k n o w n " P o n s A s i n o r u m " ( b r i d g e of asses), a n a m e o f t e n
q u o t e d in c o n n e c t i o n w i t h a n o t h e r proposition of Euclid's
Elements.
C ' The Story of Lilavati
B h a s k a r a ' s b o o k Lilavati ( m e a n i n g : the b e a u t i f u l ) w a s translated by F y z i into Persian, "by
o r d e r of the k i n g . " In his p r e f a c e to the b o o k , Fyzi narrates
a story c o n n e c t e d with the origin of t h e b o o k . H u t t o n finds
this a c c o u n t t o be " v e r y curious, a n d containing s o m e useful
p a r t i c u l a r s " a n d t h e r e f o r e h e includes it "as a postscript" at
the e n d of his o w n n a r r a t i v e . " I t is said that the c o m p o s i n g of
t h e Lilavati w a s o c c a s i o n e d b y the following circumstance.
Lilavati w a s the n a m e of the a u t h o r ' s ( B h a s k a r a ' s ) daughter,
c o n c e r n i n g w h o m it a p p e a r e d , f r o m the qualities of the Asc e n d a n t at h e r birth, t h a t she was destined to pass h e r life
u n m a r r i e d , a n d t o r e m a i n w i t h o u t children. T h e f a t h e r asc e r t a i n e d a lucky h o u r f o r c o n t r a c t i n g h e r in marriage,
t h a t she m i g h t b e firmly c o n n e c t e d a n d h a v e children. It is
said t h a t w h e n t h a t h o u r a p p r o a c h e d , he b r o u g h t his d a u g h t e r
a n d his i n t e n d e d son n e a r him. H e left the h o u r c u p on the
vessel of w a t e r , a n d k e p t in a t t e n d a n c e a time-knowing astrologer, in o r d e r t h a t w h e n the c u p should subside in the
w a t e r , those t w o precious jewels should be united. But, as
t h e i n t e n d e d a r r a n g e m e n t w a s n o t a c c o r d i n g to destiny, it
h a p p e n e d t h a t t h e girl, f r o m a curiosity natural to children,
l o o k e d into t h e c u p , t o observe the w a t e r c o m i n g in at the
hole; w h e n b y c h a n c e a pearl s e p a r a t e d f r o m her bridal dress,
fell into the c u p , a n d , rolling d o w n to the hole, stopped the
influx of the w a t e r . So the astrologer waited in expectation
"f the p r o m i s e d h o u r . W h e n the o p e r a t i o n of the c u p h a d
MATHEMATICAL ASIDES
159
t h u s b e e n delayed b e y o n d all m o d e r a t e time, t h e f a t h e r w a s
in c o n s t e r n a t i o n , a n d e x a m i n i n g , he f o u n d that a small p e a r l
h a d stopped the c o u r s e of the w a t e r , a n d that t h e longexpected h o u r was passed. In short, the f a t h e r , t h u s disappointed, said to his u n f o r t u n a t e d a u g h t e r , " I will write a b o o k of
y o u r n a m e , w h i c h shall r e m a i n t o the latest t i m e s — f o r a g o o d
n a m e is a s e c o n d life, a n d the g r o u n d w o r k of eternal existence."
B h a s k a r a t h u s w r o t e his b o o k Lilavati in fulfillment of a
promise given his b e a u t i f u l d a u g h t e r w h e n h e f o u n d out b y
the stars t h a t she was fated to s p i n s t e r h o o d . T h e a c c o u n t m a y
be history, o r it m a y be legend. M o s t likely, it is a m i x t u r e
of both t r u t h a n d f a n c y , in u n k n o w n percentages. W h a t is
certain, h o w e v e r , is that the w o r k B h a s k a r a p r o d u c e d u n d e r
the title Lilavati will " r e m a i n to the latest times," as a d o c u m e n t in the history of c u l t u r e .
3
•
Running Around in Circles
W h e n w a t c h i n g the p o p u l a r g a m e of " P i n n i n g t h e tail o n
the d o n k e y " w e are o f t e n a m u s e d , a n d not a little surprised,
to see the b l i n d f o l d e d p e r f o r m e r s instead of m a k i n g straight
f o r the object sought, w a n d e r off to o n e side or the other.
H o w e v e r , these defenseless victims of o u r derision d o n o
worse t h a n they could be expected to. In f a c t they w o u l d go
m u c h f a r t h e r astray, if the " d o n k e y " w e r e placed a t a
greater distance f r o m the p o i n t w h e r e the chase begins.
T h e b e a u t i f u l San M a r c o c a t h e d r a l in V e n i c e is a b o u t ninety
yards wide, a n d the s q u a r e in t h e f r o n t of it ( P i a z z a San
M a r c o ) is n e a r l y t w o h u n d r e d y a r d s long. T h o s e w h o attempt to reach t h e c a t h e d r a l , b l i n d f o l d e d , starting f r o m the
end of the s q u a r e directly opposite the building, find t h e m selves at either the right side o r the left side of the s q u a r e
N o n e of t h e m ever r e a c h e s the c a t h e d r a l .
T h e R u s s i a n m a t h e m a t i c i a n , Y . I. P e r e l m a n , tells of o n e
h u n d r e d aviation cadets w h o w e r e lined up, b l i n d f o l d e d , at
the edge of a n airfield a n d o r d e r e d to walk straight a h e a d .
T h e y o u n g m e n started o u t as they w e r e bid, b u t they c o u l d
not keep it up. A f t e r a short while they began to t u r n to the
side, s o m e to the right, s o m e to the left. T h e y actually w a l k e d
in circles, each of t h e m repeatedly crossing his o w n tracks.
M o t h e r N a t u r e n u m b e r s in h e r vast arsenal of tricks quite
a few b l i n d f o l d i n g devices: pitch d a r k nights, dense fogs,
blinding s n o w s t o r m s , thick forests, trackless o p e n s p a c e s —
deserts, large bodies of water, etc. W h e n a hapless traveler,
160
MATHEMATICS IN FUN AND IN EARNEST
t r a p p e d b y t h e merciless elements, is deprived b y t h e m of
his sense of vision, he is u n a b l e t o f o l l o w a n y fixed direction,
a n d " r u n s a r o u n d in circles."
F r i g h t m a y h a v e a n equally disastrous effect u p o n a m a n ' s
ability t o o r i e n t himself. W h e n fleeing f r o m his pursuers, a
t r a c k e d m a n , believing himself to be r u n n i n g straight a h e a d
a n d a w a y f r o m d a n g e r , actually r u n s in circles.
T h e s e f a c t s h a v e b e e n k n o w n f o r a long time. T h e y h a v e
f r e q u e n t l y b e e n exploited b y writers of fiction. L e o Tolstoy,
t h o r o u g h l y f a m i l i a r w i t h the s n o w s t o r m s of the vast R u s s i a n
plains, as well as with t h e f o l k l o r e c o n n e c t e d with t h e m , has
m o r e t h a n o n c e described t h e aimless w a n d e r i n g s of people lost in t h e snowy deserts.
Stories of this k i n d f o r m a p a r t of t h e lore of the A m e r i can cowboy.2
T h e late A m e r i c a n p l a y w r i g h t , E u g e n e O'Neill, in his powe r f u l d r a m a Emperor Jones, describes the flight of the h o r r o r stricken " e m p e r o r " t h r o u g h a f o r e s t , at night. T h e r e n o w n e d
a u t h o r bases t h e c l i m a x of his play o n the f a c t that a f t e r
a n i g h t of f r a n t i c r u n n i n g t h e m a d d e n e d fugitive is overt a k e n at t h e spot w h e r e he e n t e r e d t h e forest the evening bef o r e . O n e of the c h a r a c t e r s of the play r e m a r k s , knowingly,
in good c o c k n e y dialect: "If 'e lost 'is w a y in these stinkin'
w o o d s , 'e'd likely t u r n in a circle w i t h o u t 'is k n o w i n g it.
T h e y all d o e s . "
T h e t e n d e n c y to m o v e in a circle o r circles, w h e n t h e
c o n t r o l l i n g a c t i o n of t h e eye is inoperative, is not a n exclusive characteristic of m a n . A n i m a l s b e h a v e likewise. W h e n a
c h i c k e n loses its h e a d , literally, it r u n s a r o u n d in circles,
as t h e p r o v e r b i a l saying h a s it. A b l i n d f o l d e d d o g swims in
circles. Blind birds fly in circles.
H u n t e d a n i m a l s w h e n consistently p u r s u e d , e n d u p b y
r u n n i n g in circles. A s reliable a n o b s e r v e r as R o y C h a p m a n
A n d r e w s , of the N e w Y o r k M u s e u m of N a t u r a l History,
in his article, " T h e L u r e of t h e M o n g o l i a n Plains," 3 testifies to
" t h e f a t a l desire (of the a n t e l o p e ) to t u r n in a circle about
the pursuer."
T o w a r d t h e e n d of t h e last c e n t u r y t h e N o r w e g i a n biologist, F . O. G u l d b e r g devoted considerable attention to the
question of circular m o t i o n in m a n a n d animals. H e collected
a good deal of a u t h e n t i c a t e d m a t e r i a l b e a r i n g u p o n the subject.
H e tells of t h r e e travelers w h o d u r i n g a snowy night left
the shelter of a w o o d m a n ' s h u t in an a t t e m p t to reach their
h o m e , located on t h e opposite side of a valley, about three
miles wide. T h e y started o u t in the p r o p e r direction, but a f t e r
MATHEMATICAL ASIDES
161
a while they deviated f r o m it, w i t h o u t realizing t h e c h a n g e .
By t h e t i m e t h e y e s t i m a t e d t h a t t h e y s h o u l d h a v e r e a c h e d
their destination, they discovered t h a t t h e y w e r e o n c e again
close t o the very h u t t h e y so i m p r u d e n t l y h a d a b a n d o n e d .
U n d a u n t e d b y this d i s a p p o i n t m e n t t h e y started o u t a g a i n —
w i t h t h e s a m e u n f o r t u n a t e result. T h e t h i r d a n d the f o u r t h
a t t e m p t s both h a d t h e s a m e d i s h e a r t e n i n g o u t c o m e . T h e y
c a m e b a c k t o the very s a m e h u t , as t h o u g h u n d e r s o m e m a g i c
spell, as t h o u g h tied to it b y a n invisible c h a i n . W h e n even
the fifth try b r o u g h t n o b e t t e r luck, o u r tired travelers arrived
at t h e conclusion t h a t it m i g h t b e the b e t t e r p a r t of v a l o r t o
wait f o r the light of day.
G u l d b e r g h a s similar well substantiated stories a b o u t r o w ers in the o p e n sea w h o try t o r e a c h a p o i n t o n the s h o r e
d u r i n g a d a r k starless night o r d u r i n g a fog. T h u s r o w e r s
w h o u n d e r t o o k to cross a s o u n d t h r e e miles wide d u r i n g
f o g g y w e a t h e r , never succeeded in r e a c h i n g their goal. W i t h out k n o w i n g it they described t w o circles. W h e n t h e y finally
c a m e a s h o r e they discovered to their great a m a z e m e n t t h a t
it was t h e spot t h e y started f r o m .
O n the strength of s u c h i n f o r m a t i o n G u l d b e r g c o n t r i b uted a n article to a biological m a g a z i n e in w h i c h h e discussed
the t o p i c : " C i r c u l a r M o t i o n as t h e Basic M o t i o n of A n i mals." 4
W h e n d a d d y c r o u c h e s d o w n o n t h e floor in o r d e r t o
wind u p j u n i o r ' s m e c h a n i c a l a u t o m o b i l e , f o r the a m u s e m e n t
of the boy, a n d n o less his o w n , t h e e n t e r t a i n i n g a n d p e r verse plaything seldom chooses to f o l l o w the straight a n d
n a r r o w p a t h lying directly a h e a d , b u t instead describes s o m e
k i n d of arc, a w a y f r o m the line of virtue. T h e s e extravagances of junior's t o y m a y seem s t r a n g e a n d capricious, if
n o t vicious. But a little reflection will readily explain the
puzzle of the little vehicle's b e h a v i o r .
I n o r d e r t h a t the propelled t o y r u n along a straight line,
it is necessary t h a t the wheels o n t h e t w o sides of it shall be
strictly of equal size. If t h e y are not, the little a u t o m o b i l e
will t u r n t o the side of the smaller wheels. T h e r e is n o r e a s o n
f o r suspecting the p l a y t h i n g of willful m i s c o n d u c t . B u t does
not its behavior, w h e t h e r w i c k e d o r not, offer a clue to t h e
mystifying stories of h u m a n m i s a d v e n t u r e s which w e h a v e
described?
U n d e r o r d i n a r y c i r c u m s t a n c e s a m a n , while walking,
"watches his step," a n d "looks w h e r e he is going." H e n e e d s
the help of his senses, principally his eyes, t o get t o t h e
point he intends to r e a c h . But w h e n these controls are n o t
available, the p e d e s t r i a n will follow t h e direction in w h i c h
162
MATHEMATICS IN FUN AND IN EARNEST
he started o u t , only if the length of the step he takes with
o n e f o o t is exactly e q u a l to the length of the step he takes
with the other. Is this equality of the steps a thing that
m a y be t a k e n f o r g r a n t e d ? In the vast m a j o r i t y of people
the m u s c u l a r d e v e l o p m e n t of the two legs is not the same,
it is t h e r e f o r e to be expected that the steps will be uneven,
r a t h e r t h a n the c o n t r a r y . T o be sure, w e are not a w a r e of
this difference, f o r the good r e a s o n that it usually a m o u n t s
to very little. B u t small as it m a y be, it brings a b o u t s o m e
very striking c o n s e q u e n c e s .
If the right a n d left steps w e r e strictly equal, the tracks of
the t w o feet w o u l d lie on t w o parallel lines, a certain distance, say w, a p a r t . But suppose there is a difference, say d,
b e t w e e n the length of the right a n d the left step; let us ass u m e that the difference is very small, say d a m o u n t s to n o
m o r e t h a n 1 / 2 0 0 of an inch. A f t e r twenty t h o u s a n d steps
with each f o o t the difference of the distances traveled by the
t w o f e e t will a m o u n t to 100 inches, w h i c h is nearly three
yards.
N o w , if the t w o feet are m o v i n g along t w o parallel lines,
such a n o u t c o m e is patently a b s u r d : o n e f o o t c a n n o t remain
three y a r d s b e h i n d the other. T h e difficulty vanishes on the
a s s u m p t i o n t h a t the two feet m o v e on two c o n c e n t r i c circles.
T h e d i f f e r e n c e b e t w e e n the radii of the two concentric
circles is the distance, w, b e t w e e n the tracks of the t w o feet.
T h u s if the smaller circle h a s a radius R, the larger circle has
a r a d i u s R + w . T h e lengths of the two c i r c u m f e r e n c e s of the
t w o circles are, respectively, 27rR a n d 2 7 i ( R + w ) according to
a well k n o w n f o r m u l a . T h e difference between the total distances traveled by the t w o feet while describing the two
circles is t h u s 2v"w.
If a p e d e s t r i a n m o v e s in a circle having a radius equal to
o n e mile ( a s was a p p r o x i m a t e l y the case with the three travelers w h o tried to cross the v a l l e y ) , h o w m u c h difference is
t h e r e in the steps of his feet? T h e length of the c i r c u m f e r ence of the circle is 27r. 12.5280 inches. If w e take the length
s of o n e step to be 27 inches, the pedestrian m a d e in all
27T. 1 2 . 5 2 8 0 / 2 7 steps. W i t h each f o o t he m a d e 2it. 12.5280/
2.27 steps. If in the expression 2ww w e take w = 4 in., we c o m e
to the conclusion that in 27T.12.5280/2.27 steps one foot covered a distance of 27T.4 inches longer than the other. If
w e divide the latter n u m b e r by the f o r m e r we obtain the
difference b e t w e e n the lengths of the steps of the two feet
of the p e d e s t r i a n . T h e actual c o m p u t a t i o n yields the surprising result of less t h a n 0.01 part of a third of an inch. A n d
this trifling d i f f e r e n c e was e n o u g h to keep o u r intrepid a n d
u n l u c k y travelers out of their h o m e !
MATHEMATICAL ASIDES
163
A n analogous a r g u m e n t m a y enable us to establish a relation between the difference, d, of the steps a n d the length,
R, of the radius of the circle w h i c h the pedestrian will describe. T h e length of the c i r c u m f e r e n c e of radius R is 27TR.
If s is the length of a step, the pedestrian will m a k e 2 7 r R / s
steps all told. W i t h one f o o t he will m a k e 2 7 t R / 2 s steps. If
d is the difference b e t w e e n the steps, the f o o t m a k i n g the
longer step will cover an additional distance of 2<7rRd/2s. T h i s
additional distance is equal to 2ttw, as w e h a v e seen b e f o r e .
W e have t h u s the e q u a t i o n
2 7 r R d / 2 s = 27rW
Rd=2sw.
If w e put s = 2 7 inches a n d w = 4 inches, w e h a v e
Rd=216,
w h e r e b o t h R a n d d are to be given in inches. T h i s f o r m u l a
shows that R a n d d are inversely p r o p o r t i o n a l . F u r t h e r m o r e ,
it enables us to c o m p u t e either R or d, if the o t h e r is given.
T h e d e v e l o p m e n t of the muscles in a m a n ' s two a r m s is not
a n y m o r e the s a m e than that of his t w o legs, h e n c e his strokes,
w h e n he is rowing, are of u n e q u a l efficacy, a n d his b o a t
will m o v e in a circle w h e n he is u n a b l e to control his c o u r s e
with the help of his sight.
Similarly f o r the strength of the wings of a bird, a n d so
on. This takes the mystery out of our story.
But we have not c o m e to the end of that story. A b o u t a
q u a r t e r of a century ago an A m e r i c a n biologist, the late A s a
A . Schaeffer, at the time p r o f e s s o r of zoology at the U n i v e r sity of Kansas, i m p a r t e d a new twist to our p r o b l e m .
G u l d b e r g had already noticed that the repeated circles of
o u r travelers in the snowy valley and of o t h e r a n a l o g o u s
cases fall into a p a t t e r n which looks like a clock-spring spiral.
N o w Schaeffer believed t h a t lower organisms, like a m e b a s ,
which m o v e in three dimensions, are governed in their m o tion by a m e c h a n i s m which m a k e s t h e m travel along a
helical spiral. This m e c h a n i s m , he believed, survived in
higher animals, including m a n . H e n c e b l i n d f o l d e d persons
walk, run, swim, row, and drive a u t o m o b i l e s in clock-spring
spiral paths, of greater o r less regularity, w h e n a t t e m p t i n g a
straight-away.
Schaeffer c o n d u c t e d a large n u m b e r of e x p e r i m e n t s involving m a n y people. H e r e c o r d e d his results carefully, with
all the refinements of m o d e r n e x p e r i m e n t a l technique. H e
164
MATHEMATICS IN FUN AND IN EARNEST
p u b l i s h e d his findings in a p a p e r m o r e t h a n a h u n d r e d
pages long, in the Journal of Morphology
and
Physiology,5
u n d e r the title: "Spiral M o t i o n in M a n . "
S c h a e f f e r rejected the o l d e r t h e o r y a n d its "simian simplicity." B u t even if the correctness of Schaeffer's own t h e o r y
be g r a n t e d , it does not seem t h a t this necessarily invalidates G u l d b e r g ' s e x p l a n a t i o n . T h e two causes m a y complem e n t e a c h o t h e r a n d m a y be o p e r a t i n g simultaneously.
4
•
Too Many?
W h e n C a s p e r e n t e r e d the den of his m a s t e r o n that bright
a n d c h e e r f u l m o r n i n g , he was a bit surprised to notice an
u n f a m i l i a r object o n the d e s k — a n elegant box of small
size. M r . P u r e f o y m u s t h a v e b r o u g h t it w h e n he c a m e h o m e
late last n i g h t ; o r was it a surprise gift f r o m M r s . P.? O r . . .
but w h e r e v e r it c a m e f r o m C a s p e r w o u l d be amiss in his
duties as a butler, if he did not e x a m i n e the u n f a m i l i a r object
very c a r e f u l l y . A n d t h e n open it. T h e latter o p e r a t i o n
t u r n e d out to be less difficult t h a n could be anticipated.
T h e b o x c o n t a i n e d a n e w b r a n d of cigarettes, p a c k e d very
carefully.
Of course, the butler's first impulse w a s to have a puff
at o n e of those n e w f a n g l e d things. But, o n second thought,
C a s p e r hesitated. T h e t o p of the box consisted of a layer of
t w e n t y cigarettes neatly and tightly placed one alongside the
other. T h e a b s e n c e of o n e of t h e m w o u l d be all too conspicuous. A f t e r e a r n e s t deliberation C a s p e r arrived at the conclusion that the tasting of new b r a n d s of cigarettes did not
fall within the scope of his duties. H e left the cigarettes w h e r e
they were, closed the b o x , a n d r e t u r n e d to the e x a m i n a t i o n
of the o u t l a n d i s h figures o n the little box.
C a s p e r ' s d e e p a b s o r p t i o n in his esthetic c o n t e m p l a t i o n s
w a s b r o u g h t to an a b r u p t e n d w h e n s u d d e n l y the lid of the
b o x s p r a n g o p e n , a n d m o s t of the cigarettes f o u n d t h e m selves on t o p of the desk, a n d s o m e u n d e r n e a t h . W h a t the
b u t l e r m u t t e r e d u n d e r his b r e a t h at this o c c u r r e n c e m a y
o r m a y not h a v e been fit to print, but w h a t e v e r it was it
h a d n o a p p r e c i a b l e effect u p o n the situation. A n d something h a d to be d o n e !
C r o s s a n d disguntled, the butler e m p t i e d the r e m a i n i n g
c o n t e n t s of the b o x u p o n the table, picked the cigarettes off
the floor, a n d w e n t to w o r k . H e quickly put twenty cigarettes
in a r o w , o n e beside the o t h e r , on the b o t t o m of the box.
T h i s c o v e r e d the b o t t o m completely. O n the top of this
MATHEMATICAL ASIDES
165
layer he placed a n o t h e r layer of t w e n t y cigarettes. C a s p e r
w o r k e d diligently. H e was c a r e f u l to keep the layers s m o o t h
and even. W h e n h e c o m p l e t e d the eighth layer, he b r e a t h e d
a sigh of relief; he was pleased with himself and his h a n d i w o r k . T h e b o x was full a n d looked exactly as w h e n he first
o p e n e d it. " Y o u could not see a n y d i f f e r e n c e t o save y o u r
life," he flattered himself.
C a s p e r lowered the lid u p o n the b o x a n d w a s r e a d y t o
put it in an a p p r o p r i a t e place, w h e n he noticed o n the desk,
b e h i n d the box, several cigarettes. H e blinked. " A m I seeing d o u b l e ? " he asked himself, bewildered. But the cigarettes
were real. T h e r e was n o " m a y b e " a b o u t it. H e c o u n t e d
t h e m . Sixteen cigarettes, of the s a m e b r a n d as those in the
box. H e c o u n t e d again, he c o u n t e d t h e m o n c e m o r e — t h e
same sixteen, n o m o r e and n o less. H e h a l f - h e a r t e d l y b e n t
d o w n to look again u n d e r the desk, lay d o w n o n t h e floor
to peep u n d e r the o t h e r pieces of f u r n i t u r e — a l l in vain;
n o o t h e r cigarettes. Just sixteen. N o t e n o u g h f o r a n o t h e r
layer. "But if I had a full c o m p l e m e n t of t w e n t y , " he a r g u e d
with himself, " t h a t w o u l d d o n o good, either. T h e r e just
is n o r o o m in the box f o r a n o t h e r r o w . " C a s p e r h a d as
full a box as he should have, a n d sixteen cigarettes o n t o p
of that. It should not have been so, but it was. C a s p e r m a y
never b e f o r e have experienced embarras
des richesses,
but
he did that m o r n i n g a n d it w a s not to his liking.
In the evening of the s a m e day, a f t e r d i n n e r , w h e n M r .
P u r e f o y retired to his d e n , t h e r e was a k n o c k on the door.
" A n y t h i n g very u r g e n t ? " asked the boss impatiently, w h e n
the butler stepped into the r o o m .
" T h o s e extra cigarettes, Sir. T h e y are in the side d r a w e r
of y o u r desk," C a s p e r r e p o r t e d in a s u b d u e d and u n u s u a l l y
meek voice.
M r . P. looked at his b u t l e r in a s t o n i s h m e n t . " W h a t a r e
you talking about, C a s p e r ? I b r o u g h t h o m e a b o x of cigarettes, this box in f a c t . " H e pointed to the box so f a m i l i a r to
Casper. It was s t a n d i n g there with the lid raised, but o t h e r wise u n d i s t u r b e d . "But I b r o u g h t n o extras of any k i n d , t h a t
I know."
" T h a t is quite correct, M r . P u r e f o y . B u t w h e n I was arranging y o u r den this m o r n i n g I s o m e h o w upset this box,
and the cigarettes fell out. I r e p a c k e d the box a n d filled it to
capacity; there were sixteen cigarettes left f o r which there
was n o r o o m in the b o x . "
T h e boss b r o k e out in a h e a r t y laugh. " Y o u s h o u l d h a v e
tried the trick again, C a s p e r , " he said. "If with each r e p a c k -
166
MATHEMATICS IN FUN AND IN EARNEST
ing y o u c o u l d save sixteen cigarettes, y o u . . . , " a n d h e
started l a u g h i n g again.
" I r e p a c k e d that b o x t h r e e times in a r o w , Sir, but n o such
t h i n g h a p p e n e d again. I got in eight rows of twenty cigarettes e a c h time, a n d n o m o r e , n o less. T h e thing has h a d
m e w o r r i e d all d a y long. I a m not a superstitious m a n , but
these extra sixteen cigarettes give m e the creeps, Sir."
By this t i m e it d a w n e d u p o n M r . P u r e f o y that to the m a n
in f r o n t of h i m the accident with t h e cigarettes was n o
laughing matter.
"Well, C a s p e r , " h e said reassuringly, "right now, as you
see, I h a v e b e f o r e m e s o m e u r g e n t p a p e r s that I must go
over. B u t t o m o r r o w m o r n i n g , if you e x a m i n e this box carefully, y o u m a y p e r h a p s find an answer to the question that
h a s been b o t h e r i n g you. B u t be s u r e , " he a d m o n i s h e d the
b u t l e r w h e n h e was o n the w a y out, " t o leave the cigarettes in
the b o x in t h e same o r d e r y o u find t h e m t h e r e . "
C a s p e r ' s sleep that night was quite disturbed. H e d r e a m e d
of boxes, large a n d small, carried by raging flood waters, of
b u r n i n g stacks of white logs bellowing with dense s m o k e . . .
MATHEMATICAL ASIDES
167
H e was himself trying t o rescue those logs b y p u s h i n g t h e m
into the boxes carried b y the flood. . . .
N e x t m o r n i n g he g r a b b e d the very first o p p o r t u n i t y t o
get close to that c o n f o u n d e d box. W h e n he raised its lid,
there was the r o w of t w e n t y cigarettes, as t h o u g h nothing
ever h a p p e n e d . H e r e m o v e d t h a t layer, being very c a r e f u l
not to disturb the cigarettes u n d e r n e a t h .
W h e n C a s p e r put the t w e n t y cigarettes aside a n d t o o k a n o t h e r look at the box, the scenery was entirely different.
T h e next r o w h a d only nineteen cigarettes, and they were
placed in the grooves f o r m e d by the a d j a c e n t cigarettes of
the layer below, which layer consisted of twenty cigarettes.
"I'll be d a r n e d , " said C a s p e r a l o u d , a n d as t h o u g h in spite
of himself. H e c o n t i n u e d to r e m o v e layer a f t e r layer. By
the time he r e a c h e d the b o t t o m of the box he h a d c o u n t e d
u p f o u r layers of nineteen cigarettes e a c h , s a n d w i c h e d in
between five layers of t w e n t y cigarettes each.
" F o r once I a m f o r c e d to a d m i t that the boss is right,"
Casper pensively m u r m u r e d to himself. " T h e y are all h e r e ,
d o w n to the very last of t h e o n e h u n d r e d a n d seventy six of
them."
T h a t evening, w h e n M r . P. h a p p e n e d to c o m e u p o n his
butler, he asked h i m , with a perceptible t o u c h of m o c k e r y in
his voice, " A n d those extra cigarettes, C a s p e r , w h a t a b o u t
them?"
" T h e r e is n o t h i n g extra a b o u t t h e m a n y longer, Sir. T h e y
are all alike now, snug in the s a m e b o x . "
" Y o u m a y p e r h a p s be interested to k n o w , " was t h e p a r t ing dart M r . P. t h r e w over his s h o u l d e r at his butler, " t h a t
the box with all the cigarettes in, is not quite as full as
w h e n you p a c k e d it y o u r w a y , leaving out the 'extras.' "
E x p l a n a t i o n . If the f r o n t wall of the box were t r a n s p a r ent, the cigarettes would a p p e a r to us as little circles, e a c h
tangent to all the a d j a c e n t circles.
If the radius of such a circle is a, the distance A B b e t w e e n
the centers A , B of the first t w o cigarettes in the lowest
layer is equal to 2a. ( F i g . 15)
T h e distance B C f r o m B to the center C of the first cigarette in the third layer u p is equal to a + 2 a + a = 4 a . H e n c e
( f r o m the right triangle A B C ) the vertical distance C A =
V(4a)2-(2a)2=2a + V3
T h e f o u r distances between the five twenty-cigarette rows
are thus together equal to 8 a \ / 3 , a n d the total height of the
stack is a + a + 8 a \ / 3 = 15.9a.
T h e eight layers of twenty cigarettes e a c h f o r m a stack
16a high.
168
MATHEMATICS IN FUN AND IN EARNEST
FOOTNOTES
1
Vol. m , No. 3, January 1816, pp. 259-283.
See, for instance, Holling, Holling C., The Book of Cowboys
(New York, 1936), Chapter 32.
3
"The Lure of the Mongolian Plains" Roy Chapman Andrews,
2
Harper's Magazine, Vol. 141, 1920.
4
"Circular Motion as the Basic Motion of Animals,"
fuer Biologie, Vol. 35, 1897, pp. 419-458.
6
Vol. 45, 1928, pp. 293-298.
VII
Zeitschrift
MATHEMATICS AS RECREATION
1
•
Mathematical Folklore
Introduction
T e a c h e r s m a y flatter themselves t h a t the task
of t e a c h i n g school is their exclusive privilege. But this is f a r
f r o m being t h e case. A considerable a m o u n t of teaching is
d o n e by t h e pupils. F o r better or worse, w e learn a good deal
f r o m o u r s c h o o l m a t e s . Occasionally w e learn f r o m t h e m
even s o m e m a t h e m a t i c s , a k i n d of m a t h e m a t i c s f o r which
the teachers h a v e n o time a n d n o patience, n o t to say n o use.
T h i s k i n d of m a t h e m a t i c s usually consists of riddles, w h i c h
are very simple in their s t a t e m e n t . T h e i r solution calls f o r alm o s t n o learning, n o e r u d i t i o n , but requires of the solver a
considerable effort of i m a g i n a t i o n a n d quite a bit of ingenuity.
T h e pupil w h o brings such a riddle to class usually learned
it himself b y w o r d of m o u t h f r o m s o m e o n e w h o in t u r n
learned it the s a m e w a y , so that it m a y quite appropriately
be said t o b e m a t h e m a t i c a l folklore. O t h e r reasons f o r the
use of this appellation m a y be gleaned f r o m the text that is
to follow.
A ' "River Crossing" Problems
T h e m o s t striking of those
riddles, the o n e that a p p e a l e d to m e m o s t those m a n y years
b a c k , has t o d o with the w o l f , the goat a n d the cabbage. T h e
story r u n s s o m e t h i n g like this. A b o a t m a n u n d e r t a k e s to f e r r y
a wolf, a goat, a n d a basket of c a b b a g e s across a river. His
b o a t is so small t h a t t h e r e is r o o m f o r himself a n d either the
wolf, o r t h e goat, or the basket of cabbages, but n o m o r e .
H o w is h e to a c c o m p l i s h his task w i t h o u t loss o r d a m a g e
to the p r o p e r t y t h a t was intrusted to h i m ?
In this riddle the m o s t t r o u b l e s o m e passenger, f r o m the
b o a t m a n ' s p o i n t of view, the o n e that "gets his goat", is,
of course, the goat. If the m a n s h o u l d start by taking across
the basket of cabbages, t h e r e m a y not be m u c h left of the
goat w h e n h e c o m e s b a c k . Should he t a k e the wolf first, he
MATHEMATICS AS RECREATION
169
is likely t o find u p o n his r e t u r n that the g o a t h a s d o n e c o n siderable d a m a g e t o the supply of cabbages.
O n e m a y be inclined to i n q u i r e w h o first i m a g i n e d this
very original puzzle. It is just as n a t u r a l t o ask this question
as it is difficult to a n s w e r it. W h a t is c e r t a i n is that the riddle is h o a r y with age. It was k n o w n in the Orient long
b e f o r e the Christian era. In the West it m a y be traced as
f a r back as t h e eighth c e n t u r y , t o a b o o k written b y A l cuin or F l a c c u s Albinus (c. 7 3 5 - 8 0 4 , ) , a n English e d u c a t o r
a n d ecclesiastic w h o lived at the c o u r t of C h a r l e m a g n e
a n d was in c h a r g e of e d u c a t i o n in this ruler's vast F r a n k i s h
empire.
T h e simplest, in f a c t the only w a y out, is t o t a k e along
the goat first. B u t w h a t next? W h e t h e r h e t a k e s next the wolf
o r the basket of c a b b a g e s h e will be in exactly the s a m e
p r e d i c a m e n t as he was b e f o r e , b y the t i m e he attempts to
return f o r the third item of his load. It is h e r e that the boatm a n h a d a .brilliant idea. W h e n h e brings t h e wolf across as
item n u m b e r two, he takes the t r o u b l e s o m e goat b a c k with
h i m to the first shore, leaves h i m t h e r e all b y himself a n d
ferries the c a b b a g e s across; t h e n h e c o m e s b a c k f o r the
goat, and the job is d o n e .
T h o s e w h o busied themselves with this ancient puzzle
d e e m e d the b o a t m a n ' s idea of f e r r y i n g t h e goat f o r t h a n d
back so striking t h a t they paid h i m t h e highest possible c o m p l i m e n t : they tried to imitate h i m . T h e y tried to m a k e u p
puzzles the solution of w h i c h involved the s a m e idea. H e r e
are some examples.
A g r o u p of soldiers wish t o cross a river. T h e y spy a b o a t
with t w o boys in it. E i t h e r b o y c a n o p e r a t e the boat. B u t t h e
b o a t is so small that it can c a r r y at m o s t o n e soldier o r the
t w o boys. T h e soldiers got across. H o w did they m a n a g e it?
In the Smith f a m i l y f a t h e r a n d m o t h e r weigh in the neighb o r h o o d of 160 lbs. each, while J o h n a n d M a r y tip the
scale at half that weight. O n a n excursion the w h o l e f a m i l y
a n d their spaniel dog, T o m , weighing a b o u t a d o z e n p o u n d s ,
have to cross a river in a b o a t a b o u t w h i c h they w e r e
w a r n e d that w h e n l o a d e d b e y o n d 160 lbs. it b e c o m e s definitely u n s a f e . J o h n , w h o was a bright boy a n d , besides,
k n e w the story a b o u t the wolf, the goat, a n d the cabbages,
f o u n d a w a y out of the difficult situation. Of course, it occ u r r e d to n o m e m b e r of the f a m i l y t o t h r o w T o m into the
water a n d let h i m get across u n d e r his o w n s t e a m : the
p o o r thing m i g h t catch a cold in t h e process.
T w o jealous h u s b a n d s a n d their wives m u s t cross a river
in a b o a t that holds only t w o persons. H o w can it be d o n e
so that a wife is never left with the o t h e r w o m a n ' s h u s b a n d
unless h e r o w n h u s b a n d is present?
170
MATHEMATICS IN FUN AND IN EARNEST
T h e a m b i t i o u s r e a d e r is not likely to have any m o r e
t r o u b l e with this p r o b l e m t h a n with t h e last two m e n t i o n e d
b e f o r e . H e will find t h a t the crossing c a n be accomplished in
five steps.
T h e p r o b l e m b e c o m e s m u c h m o r e complicated w h e n there
a r e t h r e e couples, a n d the third h u s b a n d is just as jealous as
e a c h of the first two. T h e task m a y be accomplished in the
following manner.
T w o of the t h r e e w o m e n go across, o n e returns, and takes
across the third one. W h e n o n e of t h e m returns, she rem a i n s with her h u s b a n d , while the o t h e r two men go
across to their wives. N e x t o n e of the t w o couples returns,
the w i f e r e m a i n s , a n d the t w o m e n go across. T h e only
w o m a n that is t h e r e goes across to bring with her one of the
wives, a n d then goes back again to bring the third w o m a n ,
b u t it w o u l d be m o r e c h i v a l r o u s f o r the h u s b a n d of that
third w o m a n to go across to b r i n g his wife over.
T h e successive steps m a y be a r r a n g e d in the following
table, w h e r e A , B, C r e p r e s e n t the h u s b a n d s , and X , Y , Z
their respective wives.
1°
2°
3°
4°
5°
6°
7°
8°
9°
A,
A,
A,
A,
A,
A,
A,
First bank
B, C; X , Y, Z
B, C; X
B, C; X , Y
B, C;
B, C; X,
Second b a n k
nobody
Y, Z
z
X, Y, z
Y, z
B, C
Y, z
x,
B,
10°
11° n o b o d y
; X, Y
X, Y ,
X , Y, Z
z
X,
Z
A,
A,
A,
A,
A,
B,
B,
B,
B,
B,
z
C
C
z
C
C X, Y
C
Y
C X, Y, z
Being in possession of the solution f o r three couples, it
m i g h t be suggested to the r e a d e r to try it with f o u r couples.
S u c h a challenge, h o w e v e r , w o u l d be n o t h i n g less than a
sadistic pleasure. F o r if you u n d e r t a k e the job and find a
solution, y o u r solution will be w r o n g . H o w e v e r , this would
not be m u c h of a h u m i l i a t i o n , f o r you w o u l d be in good c o m p a n y . A r e n o w n e d Italian m a t h e m a t i c i a n of the 16th century,
N . Tartiglia ( 1 5 0 0 - 1 5 5 7 ) , also f o u n d a solution, and the
solution was w r o n g . Yes, great m a t h e m a t i c i a n s also m a k e
mistakes. 1 T h e secret in the m a t t e r is that the p r o b l e m with
f o u r couples h a s n o solution. M a u r i c e Kraitchik, in his book,
Mathematical
Recreations
( N e w Y o r k , 1 9 4 2 ) , has considered
the p r o b l e m f o r a n y n u m b e r of couples. H e shows that with
MATHEMATICS AS RECREATION
171
a boat a c c o m m o d a t i n g three persons the p r o b l e m c a n b e
solved f o r five couples, but not f o r six o r m o r e .
B e f o r e w e quit this topic it m a y n o t be out of place t o
ask: is this m a t h e m a t i c s ? If y o u r a n s w e r is Yes, t h e n it is
c o n t r a r y to the c o m m o n c o n c e p t i o n t h a t M a t h e m a t i c s consists
in figuring, in long a n d involved c o m p u t a t i o n . If y o u r a n s w e r
is N o , then h o w is o n e to a c c o u n t f o r the f a c t that those
questions a t t r a c t e d and intrigued m a t h e m a t i c i a n s primarily,
even e m i n e n t ones a m o n g t h e m ; a n d the solutions of t h e
p r o b l e m s were f u r n i s h e d by m a t h e m a t i c i a n s ? 2
B ' Multiplication
Performed on the Fingers
Among the
most prized pieces of i n f o r m a t i o n g a t h e r e d b y t h e writer via
the folklore r o u t e is the secret of a m e c h a n i c a l multiplication
table, o r to be m o r e precise, of the m o r e a d v a n c e d , the m o r e
difficult part of t h a t table. T h e secret is t h e m o r e surprising in
that one always has with him the necessary tools to m a k e u s e
of that m e c h a n i c a l table. I n d e e d , all t h e requisite m a c h i n e r y
consists of a c o m p l e t e , u n a b r i d g e d set of o n e ' s fingers, t h u m b s
included. 3 T h e p r e l i m i n a r y mastery of the multiplication t a b l e
of n u m b e r s not exceeding five w o u l d be of help.
W e assign t h e n u m b e r 6 to the little finger o n each h a n d , 7 —
to the ring finger, 8 — t o the m i d d l e finger, 9 — t o t h e pointer,
a n d 1 0 — t o the t h u m b . W e are n o w set u p in business.
If you w a n t t o multiply, say, seven b y nine, put y o u r
two h a n d s b e f o r e y o u , p a l m s in, a n d put the tip of the ring
finger, value 7, of o n e h a n d , say, the left h a n d , against the tip
of the pointer, value 9, of the o t h e r h a n d . T h e t w o fingers
thus joined a n d those below t h e m are six in n u m b e r a n d they
count f o r sixty t o w a r d s the final result. A b o v e the t w o joined
fingers r e m a i n t h r e e fingers on the left h a n d a n d one finger o n
the right h a n d . Multiply those t w o n u m b e r s a n d add t h e
p r o d u c t three to the value sixty w e h a v e already, a n d y o u
have the required result. T h u s : 7 x 9 = 6 x 1 0 + 3 x 1 = 6 3 .
Let us d o it o n c e m o r e , to m a k e sure. T o multiply, say,
6 by 8 put the little finger on the left h a n d against t h e m i d d l e
finger of the right h a n d . T h e t w o joined fingers a n d t h o s e
below t h e m are f o u r fingers, a n d they c o u n t f o r f o r t y t o w a r d s
the final result. A b o v e the two joined fingers t h e r e are f o u r fingers on the left h a n d and two on the right h a n d ; multiply those
two n u m b e r s . T h e final result is: 6 x 8 = 4 x 1 0 + 4 x 2 = 4 8 .
With a little p r a c t i c e it is possible t o r e a d the result almost instantly.
Besides its arithmetical uses, this clever trick m a y also
serve, with telling effect, to e n h a n c e the prestige of an a m bitious g r a n d f a t h e r in the eyes of a bright f o u r t h - g r a d e g r a n d son. Strange to say, this s e c o n d a r y virtue of the ingenious
172
MATHEMATICS IN FUN AND IN EARNEST
a r i t h m e t i c a l device completely escaped m y notice w h e n I
first b e c a m e a c q u a i n t e d , folklorewise, with the p r i m a r y p u r pose of t h e artifice.
T h i s r e m a r k a b l e s c h e m e is a relic of r e m o t e antiquity. It
is a p a r t of a very e l a b o r a t e m e t h o d of digital c o m p u t a t i o n
d e v e l o p e d in the Orient p r o b a b l y b e f o r e the invention of
w r i t i n g a n d extensively used in classical antiquity. T h e m e t h o d
is f r e q u e n t l y alluded to in the writings of the latter period.
O n the o t h e r h a n d , this p a r t i c u l a r m e t h o d of multiplication h a s survived until the p r e s e n t day. C o m p e t e n t observers
r e p o r t that it is still resorted to by the Wallachian peasants
of s o u t h e r n R u m a n i a . 4
T h i s tricky m e t h o d of multiplication is, of course, a
p u r e l y e m p i r i c a l discovery. T h e m a t h e m a t i c a l basis of its puzzling success is the f a c t that the e q u a l i t y :
(p)
(5+x) ( 5 + y ) = 1 0 ( x + y ) + ( 5 - x ) ( 5 - y )
is a n identity.
T h i s identity m a y also b e p u t in the f o r m :
(q)
(5 + x ) ( 5 + y ) = 5 ( x + y ) + x y ) + 5 2 ,
w h i c h m a y p e r h a p s be s i m p l e r but does not exhibit
clearly its digital origin.
as
If in ( p ) w e replace 5 b y a, w h e r e a is any n u m b e r , we
o b t a i n the i d e n t i t y :
(r)
(a+x)(a+y)=2a(x+y) + (a-x)(a-y),
w h i c h m a y also be written in the f o r m :
(s)
(a+x) (a+y)=a(x+y)
+ xy+a2.
If in the identity ( r ) w e replace a by 10, w e obtain the
formula:
(t)
(10 + x ) ( 1 0 + y ) = 2 0 ( x + y )
w h i c h m a y also be written as:
(u)
+(10-x)(10-y),
(10+x)(10+y) = 10(x-fy)+xy+102.
If we interpret ( t ) in a m a n n e r analogous to the interp r e t a t i o n w e h a v e f o r ( p ) , we m a y use ( t ) for the multiplication of n u m b e r s within the r a n g e f r o m 11 to 15.
T h e process m a y be c o n t i n u e d by replacing a in ( r ) successively by 15, 20, 25, . . . a n d using the resulting identities
in a m a n n e r a n a l o g o u s to the way w e use ( p ) and ( t ) .
T h a t the masters of digital c o m p u t a t i o n ever used these
generalizations is quite unlikely.
C ' "Pouring" Problems ' The "Robot" Method
Another
MATHEMATICS AS RECREATION
173
type of p r o b l e m w h i c h I recall h a v i n g l e a r n e d f r o m m y
schoolmates is the following. T h e c o n t e n t s of a cask filled
with 8 quarts of wine is to be divided into two e q u a l p a r t s
using only the cask a n d t w o e m p t y jugs with capacities of
5 quarts and 3 q u a r t s respectively. 5
This, too, is a riddle m a n y , m a n y c e n t u r i e s old, exactly
h o w old is difficult to say. A solution m a y be arrived at b y
trial a n d e r r o r . T h e n u m b e r of a t t e m p t s necessary will be
considerably r e d u c e d if a record is k e p t of the trials att e m p t e d in the f o r m of, say, 3, 5, 0, w h i c h w o u l d m e a n t h a t
f r o m the 8 q u a r t cask w e filled the 5 q u a r t jug, a n d so o n .
Vessel
Stages
8 q u a r t cask
5 q u a r t jug
3 quart jug
Vessel
Stages
8 q u a r t cask
5 q u a r t jug
3 q u a r t jug
First solution
A m o u n t of wine in each vessel, by stages
1
2
3
4
5
6
7
8
9
8
5
5
2
2
7
7
4
4
0
0
3
3
5
0
1
1
4
0
3
0
3
1
1
0
3
0
Second solution
A m o u n t of wine in each vessel
1
2
3
4
5
6
7
8
8
3
3
6
6
1
1
4
0
5
2
2
0
5
4
4
0
0
3
0
2
2
3
0
9
In the N e w Y o r k q u a r t e r l y Scripta Mathematical
a British
m a t h e m a t i c i a n , D r . W . W . Sawyer, gives a clever description
of this kind of puzzle. H e arrives at a general rule of p r o cedure f o r their solution. S u p p o s e w e c o n s i d e r the case just
discussed (8, 5, 3 ) . R u l e 1. If the jug 5 is e m p t y , fill it f r o m
cask 8. Rule 2. If jug 5 is not e m p t y , t h e r e are t w o possibilities: a. If jug 3 is not full, fill it f r o m jug 5; b. if jug
3 is full, e m p t y it into the cask 8. R e p e a t e d application of
this p r o c e d u r e leads to the desired result. T h e second solution above c o n f o r m s to this rule. It m a y be tried on situations
like (24, 17, 7 ) , ( 1 2 , 7, 5 ) , a n d m a n y others that c a n be
imagined.
T h e R u s s i a n m a t h e m a t i c i a n Y . I. P e r e l m a n p r e f e r s to
h a v e his w o r k d o n e f o r h i m by a r o b o t which he affectionately calls his "clever little ball." T h e j o b of the robot is to r u n
e r r a n d s on a billiard table having the usual f o r m of a
parallelogram with a 60° angle. H e r e is h o w it works.
In o r d e r to solve the p r o b l e m , c o n s i d e r e d above, with
the three vessels a = 5 , b = 3 , c = 8 w e c o n s t r u c t a parallelog r a m with sides O A = 5 , O B = 3 a n d m a k i n g a n angle of 6 0 °
(Fig. 1 6 ) . T h e r o b o t r u n s his e r r a n d s o n this "billiard t a b l e , "
174
MATHEMATICS IN FUN AND IN EARNEST
s c r u p u l o u s l y observing t h e law of reflection, that is, the angle of reflection is to be e q u a l t o t h e angle of incidence (Fig.
1 7 ) . N o w s u p p o s e t h e r o b o t is l a u n c h e d f r o m the point O
a l o n g t h e line OB. H i s first call p o i n t is B = b 3 . F r o m there,
o b e y i n g t h e a f o r e m e n t i o n e d law, h e will go to the p o i n t a3.
Figure 16
F o r t h e line O B strikes B C at an angle of 6 0 ° , h e n c e the robot will h a v e to c o n t i n u e o n a line which also m a k e s with
B C a n angle of 6 0 ° . T h i s r e q u i r e d line coincides with the
d i a g o n a l of the little r h o m b u s a d j a c e n t t o the lines OB, BC,
since this d i a g o n a l bisects t h e angle O B C = 1 2 0 ° of that r h o m -
bus, by a k n o w n p r o p o s i t i o n of p l a n e geometry. T h u s the
r o b o t will follow the line Ba 3 .
F r o m the point as t h e r o b o t , b y the s a m e rule, will go,
successively, to t h e p o i n t s :
C3 di b j ai Ci a*
L e t us n o w allow o u r r o b o t to rest at the point a», f o r a
while, a n d ask ourselves w h a t his j o u r n e y h a s to d o with the
MATHEMATICS AS RECREATION
175
business of p o u r i n g we are s u p p o s e d to e n g a g e in? T o a n s w e r
the question let us e x a m i n e the location of each point r e a c h e d
by the robot, with r e f e r e n c e to the basic lines O A , OB. T o get
t o the point, say c 3 f r o m the point O o n e could travel a l o n g
the line O A three units, to the point a 3 , t h e n three units u p
the parallel to the line OB. Let us r e c o r d this result in the
f r o m c 3 (3, 3 ) . In professional m a t h e m a t i c a l p a r l a n c e
the n u m b e r s 3, 3 are said to be the c o o r d i n a t e s of the
point c 3 with r e f e r e n c e to the basic lines, o r axis O A , O B .
Let us n o w rewrite the successive points of call of the
robot, with the c o o r d i n a t e s associated with each point. T h u s :
b 3 ( 0 , 3 ) a»(3,0) c a ( 3 , 3 ) d , ( 5 , l ) b , ( 0 , l ) a , ( 1,0) c , ( l , 3 ) a<(4,0.
N o w we are ready to p o u r . T h e two n u m b e r s alongside
the point b 3 signify that the r o b o t suggests that you s h o u l d
have zero in the vessel a a n d three units in vessel b. F o r t h e
point a 3 the robot w a n t s you to h a v e 3 in the vessel a a n d
nothing in the vessel b. F o r c 3 the robot's advice is: three
units in each of the vessels a, b. A n d so on. If the r e a d e r
will go to the trouble of c o m p a r i n g the list of n u m b e r s we
are dealing with right n o w with t h e first solution of this
problem already given b e f o r e , he m a y be surprised to find
that the two sets of n u m b e r s are identical, o r in o t h e r words,
the d u m b ( ? ) c r e a t u r e has duplicated o u r solution. T h e r e a d e r
w h o would a m u s e himself trying to w o r k the same p r o b lem but launching the r o b o t f r o m O along the axis O A , h a s
a n o t h e r surprise in store f o r him. T h e surprise is ( o r p e r h a p s
it is no surprise at all) that by his effort he will have f o u n d
the second of the two solutions given b e f o r e .
It is h e l p f u l to observe that the r o b o t in its travels m o v e s
on two kinds of lines: 1°. lines parallel to the basic lines O A ,
OB; 2°. on those diagonals of the little r h o m b u s e s which bisect the angles of 120°. A s a general rule those t w o kinds of
lines are followed by the robot alternately. A n exceptional
case will be pointed out later.
T h e terminal point of the r o u t e solving a given p r o b l e m
m a y be m a r k e d b e f o r e h a n d . F o r instance, in the p r o b l e m
considered we w a n t to have f o u r gallons in each of two vessels, hence the p r o b l e m will be solved w h e n a contains t h a t
a m o u n t , that is, w h e n the robot gets to the point a 4 ( 4 , 0 ) .
It m a y also be observed that the robot is c a p a b l e of providing two solutions of each p r o b l e m : o n e w h e n l a u n c h e d along
the axis O A , a n d the o t h e r w h e n l a u n c h e d along the axis O B .
T h e r e a d e r m a y try to work the p r o b l e m : a = 7, b = 5 ,
c = 1 2 ; a = 9, b = 7, c = 1 6 . In the first p r o b l e m the r o b o t
launched along the axis O A passes successively t h r o u g h t h e
points:
a 7 c 2 a 2 b 2 d 2 c 4 , a 4 b, d 4 c 6 a 6 ;
176
MATHEMATICS IN FUN AND IN EARNEST
W h i l e f o l l o w i n g t h e p e r e g r i n a t i o n s of o u r r o b o t w e tacitly
a s s u m e d t h a t the agile r u n n e r is p r o v i d e d with a m e c h a n i s m
w h i c h stops this servant w h e n the preassigned p o i n t is
r e a c h e d . T h e question arises: w h a t will h a p p e n if that b r a k ing m e c h a n i s m s h o u l d get out of o r d e r a n d fail to stop the
r o b o t , say, at the p o i n t a* in the s e c o n d solution of the p r o b l e m a = 5 , b = 3, c = 8 ? Well, the f a i t h f u l servant will simply
c o n t i n u e to r u n in its u n f a i l i n g obedience to the prescribed
law a n d r e t u r n b a c k h o m e , that is, to the starting point O,
a f t e r h a v i n g f o l l o w e d the s u p p l e m e n t a r y p a t h :
a 4 Ci ai bi di c 3 as b 3 O.
If w e e x a m i n e the r o u n d trip of the r o b o t w e notice that
t h e p a t h c o v e r e d involves the points a t a 2 a 3 a 4 a 5 . T h a t is to
say, the vessel a at o n e t i m e o r a n o t h e r contained the a m o u n t s
of liquid 1, 2, 3, 4, 5. T h i s is the very i m p o r t a n t discovery,
indeed. F o r it s h o w s t h a t w e could solve the p r o b l e m , n o
m a t t e r w h a t a m o u n t of liquid it w o u l d be required to have in
t h e vessel a, not exceeding 5, of course. W e thus enlarge the
scope of o u r p r o b l e m s a n d e n h a n c e the value of the " r o b o t
m e t h o d " of solving t h e m . T h u s in the suggested p r o b l e m a = 7 ,
b = 5, c = 1 2 w e m a y ask to h a v e in the vessel a three gallons
of liquid. If we c o n t i n u e the list of " r o b o t points," given
above, b e y o n d the point a 6 w e find:
Ci ai bi di c 3 a 3 b 3 d 3 c 5 b 5 d s O.
T h u s t h e desired p o i n t a 3 has been r e a c h e d and o u r dem a n d is satisfied. O n the o t h e r h a n d if w e w a n t e d f o u r units
in the vessel a w e w o u l d n o t h a v e to go even as f a r as a 6 . Let
us notice, in passing, that h e r e , too, we can have in a a n y
a m o u n t between 1 a n d 7.
But is that always the case? C o n s i d e r the p r o b l e m : a = 6 ,
b = 4 , c = 1 0 . In the usual w a y w e obtain the following table:
a6
a=6
b=0
c=4
c2
2
4
4
a2
2
0
8
b2
0
2
8
d2
6
2
2
c4
4
4
2
a4 b4
4 0
0 4
6 6
0
0
0
10
T h i s table shows that in n o n e of t h e t h r e e vessels can we
h a v e an o d d n u m b e r of gallons, a n d this includes the crucial
n u m b e r 5 w h i c h w o u l d divide the contents of c into two
e q u a l parts.
T h e e x a m p l e s w h i c h w e h a v e considered u p t o n o w h a v e
a c o m m o n feature, n a m e l y that c—a + b. If c is greater
t h a n a+b, o u r "robot"' m e t h o d of solving the p r o b l e m s rem a i n s applicable. T h e "billiard t a b l e " remains unaltered,
that is, its d i m e n s i o n s r e m a i n a, b, as b e f o r e .
A s an illustration let us solve the p r o b l e m : a = 6 , b = 4 ,
c = 1 2 . O u r f a i t h f u l r o b o t m a y surprise us by following the
MATHEMATICS AS RECREATION
177
s a m e itinerary as in t h e p r e c e d i n g p r o b l e m a — 6 , b = 4 ,
c = 1 0 , with the s a m e dire c o n s e q u e n c e s , as the r e a d e r m a y
readily verify.
Is o u r r o b o t going to play o n us t h e s a m e trick w h e n e v e r
c is greater t h a n a + b ? T o find the a n s w e r t a k e the case w h e n
a = 6 , b = 4 , c = l l . It t u r n s o u t t h a t t h e first t h r e e lines of
solution of the p r o b l e m a = 6 , b = 4 , c = 10 given a b o v e rem a i n valid f o r o u r p r e s e n t case, with t h e s a m e limitations
f o r the vessel a ( a n d b ) as in that case. B u t t h e f o u r t h line
c o m e s to the rescue. T h a t line is n o w :
c=5 5 9 9 3 3 7 7
T h u s in the present case w e c a n p o u r off n o t only a n y even
a m o u n t but any o d d a m o u n t as well. A n y such o d d a m o u n t
is n o n e the w o r s e f o r being in the vessel c t h a n it w o u l d b e
in either of the t w o o t h e r vessels. E v e n t h e a m o u n t of 1
gallon c a n be h a d in c b y filling t h e o t h e r t w o vessels.
Figure 18
T h e r e a d e r m a y investigate the cases w h e n
b=4,
a n d c = 1 3 , 14, 15, . . . a n d try t o f o r m u l a t e s o m e r u l e a b o u t
the results.
W h e n c is smaller t h a n a + b t h e situation changes c o n s i d e r ably. W e m a y still h a v e t h e services of t h e r o b o t , b u t o u r billiard table has to be altered, n a m e l y , it h a s t o b e s u b j e c t e d to a n
a m p u t a t i o n of a c o r n e r . C o n s i d e r , f o r instance, t h e case w h e n
a = 6 , b = 4 , c = 7 . In o r d e r to h a v e o u r table we d r a w , as bef o r e the axis O A = 6 , G B = 4 at a n angle of 6 0 ° . N o w o n t h e
parallel to O A t h r o u g h B lay off B C = 7 — 4 = 3 , a n d o n t h e
parallel t o O B t h r o u g h A lay off O D = 7 - 6 = l . T h e " w a l l "
a r o u n d o u r table is t h e b r o k e n line O A D C B (Fig. 1 8 ) .
If we l a u n c h o u r r o b o t f r o m O a l o n g t h e axis O A , w e
o b t a i n t h e following a r r a y :
178
MATHEMATICS IN FUN AND IN EARNEST
a« c 2 a 2 b 2 d 2 a 5 Ci ai b t di c 3 a 3 b 3 d 3 a 4 b 4
a=6 2 2 0 5 5 1
1 0 6 3 3 0 4 4 0
b=0 4 0 2 2 0 4 0 1
1 4 0 3 3 0 4
c=l
1 5 5 0 2 2 6 6 0 0 4 4 0 3 3
0
0
0
7
A t the p o i n t d 2 this table presents a novel feature. T h e line
b 2 d 2 strikes t h e p a r t D C of the " w a l l " at a n angle of 60°. T h e
o t h e r line w h i c h m a k e s with D C at that point an angle of
6 0 ° is t h e line d 2 a s , a n d that is the p a t h o u r law abiding robot
follows, albeit reluctantly, f o r it is that servant's w o n t to alt e r n a t e a line parallel to a n axis with a diagonal direction.
T h i s is the e x c e p t i o n a l situation alluded to before.
N o t i c e that a c c o r d i n g to o u r table we are able to p o u r off
i n t o t h e vessel a a n y a m o u n t between I a n d 6.
T h e r e a d e r m a y consider the p r o b l e m : a = 9 ,
b=7,
c = 1 2 . C o n s t r u c t i n g , with the help of the untiring robot, the
c o r r e s p o n d i n g table the r e a d e r m a y c o n v i n c e himself that
he m a y h a v e in the vessel a any a m o u n t between 1 and 9,
with the c o n s p i c u o u s exception of 6, precisely the a m o u n t w e
w o u l d n e e d if w e w a n t e d to divide the contents of c into
equal parts. Sheer spite!
I n the case a = 6 , b = 3, c = 9, c a n we h a v e in b o n e gallon
o r t w o gallons? If you ask the robot, the answer will be: N o .
M a n y o t h e r intriguing p r o b l e m s of this sort m a y be
solved by a n interested r e a d e r , with t h e help of the robot.
W i t h a little p r a c t i c e this m a y b e c o m e as good a g a m e of
solitaire as a n y .
D ' The "False Coin" Problem
T o the three " f o l k l o r e "
p r o b l e m s w e h a v e c o n s i d e r e d a good m a n y m o r e could be
a d d e d , just as clever, just as ingenious. T h e y all are a legacy
of the past, a p r e c i o u s i n h e r i t a n c e b e q u e a t h e d to us by the
centuries gone by. But w h a t a b o u t the present c e n t u r y ? H a v e
the g e n e r a t i o n s that have c o m e b e f o r e us been so m u c h superior that they left to us a gift which we c a n enjoy but c a n n o t
duplicate, not to say rival? T h i s is not the case. O u r m a t h e m a t i c a l j o u r n a l s o f t e n publish questions fully as enticing as
those w e h a v e learned f r o m the folklore of the past.
Of the m a n y e x a m p l e s t h a t could be given let us take o n e
that originated a b o u t t h e m i d d l e of the present century, in
the U n i t e d States.
In the issue f o r J a n u a r y 1945 the American
Mathematical Monthly
p r o p o s e d the following q u e s t i o n : 7 Y o u have
eight similar coins a n d a b e a m b a l a n c e ( w i t h o u t w e i g h t s ) .
A t m o s t , o n e coin is c o u n t e r f e i t a n d t h e r e f o r e lighter. H o w
c a n y o u d e t e r m i n e w h e t h e r t h e r e is a n u n d e r w e i g h t coin, a n d
if so, which one, using the b a l a n c e only twice?
T h e difficulty lies in the restrictive p h r a s e , " u s i n g " the bal-
MATHEMATICS AS RECREATION
179
a n c e only twice." O t h e r w i s e all o n e h a s t o d o is to p u t o n e coin
o n the b a l a n c e a n d w e i g h against it e a c h of t h e r e m a i n i n g
coins, o n e at a time. If l u c k is against y o u , y o u m a y h a v e to d o
it seven times b e f o r e y o u arrive at the r e q u i r e d answer. Being
restricted to t w o weighings m a k e s the difficulty. B u t t h e s a m e
is the case, say, with t h e w o l f - g o a t - c a b b a g e riddle. 8 T h e r e
w o u l d h a v e been n o p r o b l e m if t h e b o a t could a c c o m m o d a t e
all three, or even t w o of t h e m . It is t h e restriction of " o n e
passenger at a t i m e " t h a t creates the p r o b l e m .
O n e w h o w o u l d give this coin p r o b l e m a little t h o u g h t is
likely t o agree t h a t it is f u l l y as challenging as a n y of t h e
river crossing p r o b l e m s .
Eight m o n t h s a f t e r t h e question a p p e a r e d in the " M o n t h ly" this periodical p u b l i s h e d a solution 9 w h i c h m a y be stated
as follows: W e i g h a n y t h r e e of the given coins against a n y
o t h e r t h r e e of t h e m . If t h e t w o sets b a l a n c e , weigh t h e rem a i n i n g two coins against each other, a n d the lighter of t h e
two is the w r o n g coin. If the first t w o sets d o n o t b a l a n c e ,
the lighter of the t w o sets includes t h e suspect. B a l a n c e
a n y t w o of those t h r e e coins against e a c h o t h e r , a n d k e e p
the third o n e in y o u r h a n d . If they d o n o t b a l a n c e , the
lighter coin is the o n e sought, a n d if t h e y d o b a l a n c e , y o u a r e
holding the culprit in y o u r h a n d .
Soon a f t e r the a p p e a r a n c e of this solution in the
Ameri-
can Mathematical Monthly, the quarterly Scripta Mathematica
published the following m o r e general p r o b l e m : 1 0 H o w , by balancing coins on a scale only t h r e e times c a n o n e detect w h i c h
o n e of 12 a p p a r e n t l y equal coins differs slightly in weight,
without even k n o w i n g w h e t h e r it is u n d e r - o r over- w e i g h t ? "
T w o solutions a c c o m p a n y the p r o b l e m .
A f e w m o n t h s later the American
Mathematical
Monthly
p r o p o s e d the s a m e p r o b l e m t o its r e a d e r s . 1 1 T w o solutions
were offered later. 1 2
T h e solution given b e l o w follows closely t h e s e c o n d solution in Scripta Mathematica
m e n t i o n e d above.
T h r o u g h o u t the discussion of the p r o b l e m it is essential
to b e a r in m i n d t h a t only one of the coins is a b n o r m a l . C o n sequently, if t w o of the 12 coins are of e q u a l weight, both
coins are n o r m a l . M o r e t h a n that, if t w o g r o u p s of coins
equal in n u m b e r are also equal in weight, all the coins in
both groups are normal.
T o facilitate the solution of t h e p r o b l e m it is necessary t o
m a k e the coins distinguishable f r o m o n e a n o t h e r by s o m e
kind of m a r k o r m a r k s . W e shall a s s u m e that they h a v e been
n u m b e r e d f r o m 1 to 12.
Let us begin b y p u t t i n g f o u r of t h e coins, say, 1234 in
180
mathematics in f u n and in earnest
one pan of the scales, and f o u r more coins, say 5678 in the
other pan. Of this first weighing we consider the two alternatives: I : 1 2 3 4 = 5 6 7 8 ; 1 2 3 4 ^ 5 6 7 8 .
In case I all the eight coins on the scale are normal,
and the suspect is one of the coins 9, 10, 11, 12. Put any
three of the latter coins, say, 9, 10, 11 in a pan, and in
the other pan put any three coins we already know to be
normal, say, 123. In this second weighing we consider the
two possibilities:
l a : 123 = 9, 10, 11; l b : 1 2 3 ^ 9 , 10, 11.
In case l a the coin 12 is obviously the abnormal one,
and a weighing, the third of the series, of the coin 12
against any one of the normal eleven coins will determine
whether 12 is under- or overweight.
In case l b one of the three coins 9, 10, 11, is abnormal,
and is light or heavy according as the pan containing 9, 10,
11 is lighter or heavier than 123. In order to identify the
wrong coin we take any two of the three coins under suspicion, say, 9, 10, and put them on the two different pans
of the scale. If, in this third weighing of the series, it turns
out that 9 = 1 0 , the abnormal coin is 11, and whether light or
heavy we know already f r o m the previous remark. If, however, 9=^=10, one of these two is the culprit, and if it is the
lighter of the two, since we are looking for a lighter coin.
This completes the discussion of case I.
Let us turn to case II. W e observe, in the first place,
that the abnormal coin is on the scale, and therefore the
coins 9, 10, 11, 12 are normal. Furthermore, any one of the
f o u r coins 1234 may be either standard or heavy, but none
of them is too light, while any one of the coins 5678 is
either standard or light, but none of them is heavy.
F o r our second weighing let us put into one pan any
three of the f o u r standard coins, say, 9, 10, 11 and add to
them one of the possibly lighter coins, say, 5. On the other
pan let us put two of the possibly heavy coins, say, 34, and
let us add to them two of the remaining three possibly light
coins, say 67. In this second weighing we consider the three
possibilities:
H a : 9,10,11,5 = 3467; l i b : 9 , 1 0 , 1 1 , 5 > 3 4 6 7 :
lie: 9,10,11,5<3567
In case H a all the coins on the scale are standard, and so
is the coin 12. Of the remaining coins 8 may be light, or
of the coins 1, 2 one may be heavy. T o decide the issue we
put the latter two coins on the two pans of the scale. This
third weighing may present us with one of the two possibilities 1 = 2, 1 ^ 2 . In the first case 8 would be light. In the second case the heavier of the two coins is overweight.
mathematics as r e c r e a t i o n
181
lib. That second weighing shows that 5 is necessarily
standard and that one of the remaining two lightweight suspects 6, 7 is the guilty party. The issue can be decided by
putting the two coins in the two pans of the scale (third
weighing). T h e lighter of the two is the coin sought.
Case lie could be brought about by 5 actually being
light, or by one of the two coins 3, 4, both open to the suspicion of being heavy, actually being overweight. Put those
two coins in the pans on the scale (third weighing). If they
balance, then 5 is light; if they do not, the heavier is overweight.
The entire discussion is condensed in the chart which
follows:
Case I
Weighing 1
Weighing 2
Weighing 3
f 123=9,10,11
12
1234=5678
9=10
11
1 2 3 ^ 9 , 1 0 , 1 1 9=^=10
Case II
f 1=2
8 is light
a. 9,10,11,5 = 3467 J 1=^2 The heavier is
i
overweight
1243=^5678
8 is light
r 6 ==7
b. 9,10,11,5=^3467 < 6^=7 T h e lighter of the
1
two is light
t
f 3=4
5 is light
c. 9,10,11,5=7^3467 < 3=^4 The heavier of the
I
two is overweight
The interest in the "bad coin" problem did not remain
confined to this continent. Right f r o m the start the problem
attracted the attention of the Mathematical
Gazette of London. As early as 1945 this periodical published a note on
it. 13 Two other notes were published by the Gazette the
following year, 1946, 14 and in 1947 the Gazette devoted to
the question a nine page article. 15 A m o n g the questions discussed is the following: We have seen that the twelve coin
problem has been solved in several different ways. T h e
question may therefore be asked: Given a number of coins
and the number of weighings necessary for the detection
of the counterfeit coin, in how m a n y different ways may
this problem be solved?
There are many other ramifications that have been suggested or considered in connection with this problem; what
has been said may perhaps have given an idea in what way
182
mathematics in f u n and in earnest
and with what speed a riddle of this sort expands and develops nowadays.
H a s this problem been exhausted? One of the three contributors to the volume of the Scripta Mathematica
for the
year 1946, to which reference was made, seemed to be quite
sure that this is the case, judging by the fact that he entitled
his article: Epitaph on the coin problem. It would seem, however, that h e spoke out of turn. F o r the same Scripta Mathematica in its issue for M a r c h — J u n e 1950 published an article
under the heading: "Those twelve coins again."
The well-springs of wisdom of the h u m a n race have not
gone dry. They are more active, more vigorous than they
have ever been. This manifests itself just as surely in mathematics on the recreational level as on the most abstract
heights that mathematics can reach.
2 * Famous Problems
Introduction
Some problems are famous and deserve special
attention. Felix Klein's ( 1 8 4 9 - 1 9 2 5 ) : Lectures on Famous
Problems of Elementary
Geometry, namely the trisection of
an angle, the squaring of the circle, and the duplication of
the cube 16 has been a classic for many years. Other such
collections have been published more recently. 17
N o w suppose we agree that there are famous problems
in mathematics. W h a t then is a famous problem? It is by no
means easy to answer this question. But we can make a
few guesses and see where they may land us.
Is a " f a m o u s " problem one of long standing, an ancient
problem? Well, yes. There are such problems, and the problems of antiquity which were just mentioned above are the
best examples of the kind. There are problems that are
even older than those, as for example, the "river crossing"
problem discussed in the preceding section. 18
A ' Morley's Problem
But venerable age is by no means
an indispensable characteristic, or, as we say in mathematics,
a "necessary" condition for a " f a m o u s " problem. There
are problems as f a m o u s as any and which originated in the
present century. One of them is the late Frank Morley's
trisector problem. Morley (1860-1937) was not concerned
with the methods of trisecting an angle. H e took that part
of it for granted. H e trisected all three angles of a triangle
and took the points of intersection of the pairs of trisectors
adjacent to the three sides of the triangles. The triangle
formed by the three points thus obtained is, according to
Morley, equilateral.
Morley arrived at this property, while studying some higher
mathematics as r e c r e a t i o n
183
plane curves, during the first decade of the present century.
The problem became f a m o u s almost overnight. It ended
up by attracting the attention of m e n of the stature of
Henri Lebesgue (1875-1941).
The proposition was proved by m a n y writers in a variety
of ways. Moreover, Morley's contemporaries exhibited a
great deal of ingenuity in generalizing his simple theorem.
Those mathematicians first considered all the points of intersection of the six trisectors, picking out more equilateral
triangles, then they replaced the trisectors by n-sectors of
the angles, etc. The original prototype is completely snowed
under, and the problem as a whole has by now an enormous bibliography. You will grant that the problem is entitled to be considered famous.
B ' The Problem of Apollonius
Does a " f a m o u s " problem have to be attached to a famous name? Certainly, this
is a valuable asset f o r a problem. If famous names in the
world of the movies or the world of sports may be used
to tell you which cigarettes you should smoke, what beer
to drink, what soap to wash your hands with, why would
not a famous name in the mathematical world be a good recommendation for a problem in mathematics? There are
many such problems. Thus the problem of drawing a circle
tangent to three given circles is known as the problem of
Apollonius, who is supposed to have solved it. But his solution did not come down to us. Francois Viete (1540-1603)
was the first to produce a solution. Rene Descartes (15961650) and Isaac Newton (1642-1727) worked on this problem. Is this not enough to entitle any problem to fame?
The problem rose again to the heights of prominence
when in the early nineteenth century the newly developed
or discovered theories of the radical axis, centers of similitude, etc., were applied to its solution. It figured prominently in the competition between the analytic and synthetic
methods in geometry of that time. The problem was used as a
proof that constructions to which analytical considerations give
rise may be as simple as those obtained by purely geometrical
reasoning. These arguments, not to say quarrels, involved men
as illustrious as Poncelet, Gergonne, and others.
C ' "Fermat's Last Theorem"
N o n e of the famous problems outranks "Fermat's last theorem," namely the proposition that the equation
xN+yN=zN
has integral solutions in x, y, z for n = l , 2, and for no other
number greater than 2. An incredible amount of time and
184
mathematics in f u n and in earnest
effort on the part of mathematicians, f r o m the lowliest to
the most renowned, has been devoted to this problem. Some
mathematicians, like H . S. Vandiver ( 1 8 8 2 — ) , made this
problem the center of their activities in research. During
the second half of the nineteenth century a prize was established to be awarded to anyone who could prove or
disprove Fermat's last theorem. The prize was no trifle, for
it amounted to one hundred thousand G e r m a n marks, the
equivalent at the time of $25,000. At the time when the
prize was established most college professors, both in Europe and in America, could not expect to earn much more
in a dozen years of teaching. It is reasonable to assume that
m a n y an ambitious soul had his eyes fixed on this carrot.
But you need not worry about it. Whether you try or not,
you cannot get that prize any longer. Not that somebody
got ahead of you. The inflation and deflation which followed the first world war rendered the bequest worthless.
T h e problem remains unsolved. It is the more tantalizing
due to the claim of F e r m a t that he had a very short and
simple proof for it, but he kept it to himself, so that others
could have the pleasure of discovering it in their turn.
D • Goldbach's
Conjecture
Christian Goldbach
(16901764) was a contemporary of Euler. H e was Russian and
lived in St. Petersburg (now Leningrad). One day he made
a shrewd guess—in dignified scientific language it is called
a conjecture—that every even number is the sum of two
prime numbers. Thus 1 0 = 3 + 7, 2 4 = 1 1 + 13 = 7 + 1 7 .
Goldbach communicated his guess to his illustrious friend
Leonhard Euler (1707-1783), who was quite impressed. The
surmise seemed to him to be a true proposition. But his
sustained efforts to prove it were all in vain. A n d so were
the efforts of all those followers of his who tried their hand
at it, up to the present time, or almost.
In the late thirties of the present century the Russian
mathematician I. M. Vinogradov proved that any odd number is the sum of three prime numbers. Thus 2 1 = 3 + 7 + 1 1 ,
3 5 = 5 + 7 + 2 3 = 7 + 11 + 17. In connection with Vinogradov's proof E. T. Bell ( 1 8 8 3 — ) remarks that the work of
the Russian inspires sympathy for Euler, in his failure to
prove Goldbach's surmise. In the wake of Vinogradov two
other Russian mathematicians, Linnik and Tchudakov, produced other proofs of Vinogradov's theorem.
Does Vinogradov's theorem prove Goldbach's surmise?
Sufficiently so to warrant the change of the name from
Goldbach's surmise to the Goldbach-Vinogradov theorem,
but not enough to consider the question as quite settled.
mathematics as r e c r e a t i o n
185
A great m a n y other famous problems associated with
great names may be added. Nevertheless this is not a necessary requirement for a problem to be famous. The problems
of antiquity may again serve as an appropriate example.
The "coin" problem 1 9 if its f a m e should turn out to be of the
lasting variety, is likely to continue to be k n o w n just as the
"coin" problem, and nothing more.
E * The Problem of the Tangent. The f a m o u s problems
that we have considered so far are tainted with a degree of
glamor f o r one reason or another. But it cannot be said
that glamor is an indispensable attribute of a famous problem. There are famous problems which can hardly lay claim
to being spectacular, glamorous, as, for instance, the problem of drawing a tangent to a curve at a point on the
curve. On the other hand, the whole history of mathematics
is, in a way, reflected in the history of this problem.
The Greek definition of a tangent is: a line passing
through the given point and such that no other line drawn
through the point can lie between the tangent and the
curve. It is a far cry f r o m this definition to the definition
according to which a tangent is the limiting position of a
secant, the definition which is now standard in our textbooks.
The various ways in which the solution of the problem
of drawing a tangent to a curve is solved are intimately connected with the way we define the term "curve." "Tell me
what a curve is, and I will tell you what a tangent is." If
we consider, with Roberval ( 1 6 0 2 - 1 6 7 5 ) , whose actual
name, by the way, was Gilles Personnier, that a curve is
the path of a moving point, the tangent is determined by
the parallelogram of velocities. If you define a curve as a
graph of an analytic equation, as we do now when we use
analytic methods, the direction of the tangent to the curve
is determined by the derivative of the function considered.
In projective geometry the definition of the tangent is an
immediate consequence of the mode of generation of the
curve considered. 2 0
The tangent, or rather its absence, came again prominently to the fore when in 1861 Karl Weierstrass (18151897) made the extraordinary and unbelievable discovery
that a continuous (that is, a smooth, u n b r o k e n ) curve may
not admit of a tangent at any of its points. The story of the
problem of the tangent is neatly summarized by Paul Serret
in his little book "Des Method.es en
Geometrie.21
F ' The Recurrence of "Famous" Problems. We have seen
that a famous problem may be very old, or more recent,
186
mathematics in f u n and in earnest
and even very recent; it may or may not be attached to the
name of a great mathematician, that it may or may not
be of intrinsic importance in the history of mathematics.
There are, however, some characteristics which seem to be
fairly c o m m o n to all of them. All of them are simple in
their statement, readily make a picture in your mind, and
can therefore be carried in your memory without effort. In
brief, a f a m o u s problem has a "simple formulation."
But the best definition that could be given of a famous problem is the following: " A f a m o u s problem is one that nobody
ever heard of, least of all those who busy themselves with it."
This sounds paradoxical, but it is the inevitable conclusion
one arrives at when one examines the history of such problems,
and the more f a m o u s the problem, the more applicable is the
paradox. But it is not so strange, after all, if you think about
it. It is inherent in the very nature of those problems.
Since those problems are simple in their conceptions, they
occur to many individuals independently, without knowledge
of any previous efforts by others in connection with any one
of those problems. In past centuries this was aggravated by
the paucity of means of communication between individual
mathematicians. The same effect is produced in modern
times by the multiplicity of mathematical journals published
in m a n y different countries, in a number of different languages too large for comfort.
G • Conclusion
One may raise the question: Of what use
is all this time and effort spent so lavishly on these questions?
W h o will ever have any need for the answers provided?
Well, these are tough questions. One may have no qualms
in telling you that nobody is expected to make any "use" of
the results obtained in most of these problems.
Mathematics is useful. It is the practical need for mathematics that accounts for its origin. The continuous development of mathematics is due to the constant growing need
for answers to questions that mathematics can supply.
It is nevertheless true that the part of mathematics that
has practical, bread and butter applications is a small part
of the whole body of the science. Some of this residue may
yet become useful one day. But the bulk of it may never be.
D o you think that the so called "perfect numbers" could
have great practical value? W h o will argue that the two new
perfect numbers which were recently computed are destined
to play a great role in h u m a n affairs, in spite of the hundreds of digits it would take to display each of them in its
full magnificence? But if they are not of any use, why waste
the wonderful computing machines on such futilities? The
answer is, because it is interesting, because the riddle of the
mathematics as r e c r e a t i o n
187
perfect numbers is a challenge which we would like to meet.
Most of mathematics is in the same boat. Y o u may say
that the inquiring genius of m a n has erected the stately
edifice which is mathematics as a m o n u m e n t to the greatness of his intellectual power, as a permanent proof that
"man cannot live by bread alone." If you are not given to
self-glorification and "high-sticularious" language, you may
simply say that a man is curious and is willing to pay a high
price, in time and in effort, to find an answer to what strikes
his fancy. H e likes to exercise his inventiveness and to display it before others, namely before those who can get as
excited about it as he does himself.
That is the best that can be said in defense of most of
mathematics. If one cannot derive a world of satisfaction
from the solution of a problem, if one cannot be charmed
by the result obtained, regardless of the ulterior values that
result may or may not have, one is not a mathematician.
This is the outstanding fact exhibited by the famous problems. They offer a challenge, they present an opportunity
for a display and an exercise of cleverness, of intellectual
prowess. May those enjoy it who can.
3 • Without the Benefit of Paper and Pencil
A ' Mathematics
and Computation
In the minds of a
great many people the terms "mathematics" and "calculations" are synonymous. To be a "great mathematician" is to
be a rapid computer. This idea is superficial. It is true that
some of the great mathematicians were also skillful and accurate computers. But those were the exceptions. As a rule
rather the contrary is the case. T o take but one example,
let us quote a statement made by Henri Poincare, one of
the greatest mathematicians that ever lived: "Quant a moi,
je suis oblige de I'avouer, je suis absolument
incapable de
faire une addition sans faute." (As to myself, I am obliged
to own I am absolutely unable to perform an addition without making a mistake.) 2 2
In some branches of mathematics calculations play a very
minor, even negligible role. Any high school student, past or
present, who was exposed to a course in plane geometry
found that out for himself. But even when calculations are
used in a mathematical problem, they are seldom the core
of the matter. A n artist in the process of producing a painting
has to mix his paints. Some painters acquired a considerable
renown for having devised very effective shadows of some colors, like the "Titian red." But it does not occur to anybody to
identify the profession of an artist with paintmixing.
If a mathematician has to answer a question which calls
for a number, he may have to do some computation to obtain
the required result. However, the essential part of the solu-
188
mathematics in f u n and in earnest
tion of the problem is not the computations, but the
reasoning process which enables the solver to choose the
appropriate computations. It is this intellectual effort of analyzing the situation that constitutes the mathematical character of the problem. The wolf-goat-cabbage riddle and its
generalizations 2 3 are a good example very m u c h to the point.
As an illustration less ancient but even more forceful one
may quote the two elegant solutions devised by Pascal and
F e r m a t of the problem proposed by the Chevalier de Mere. 24
The "eight coin" problem 2 5 requires analysis and imagination, or wit and shrewdness, if you prefer simpler words, but
no writing is needed. It would be difficult to handle the
"twelve coin" problem 2 6 completely without writing, on account of the considerable n u m b e r of different cases involved
that it is necessary to keep track of. But the writing is not
what makes for the added interest of the problem as compared with its simpler prototype. Of course, one may object
that those are riddles rather than mathematical problems in
the usual sense of the word. But this is rather in favor of the
contention presented here than against it. The fact that such
things appear in professional mathematical journals and are
grappled with by the readers of those publications goes to
show where the mathematician feels his meat is.
The problem of the suburban traveler which we considered before 2 7 is more of the standard type and requires
some calculations, but those are of the kind that can be performed mentally. The achievement of the solver does not
lie in the arthmetical skill shown, but in the intellectual
acumen used to grasp the interrelations involved. The following problems are other examples of this kind. The reader may
find satisfaction in attempting to solve them and check his solutions against those given below. Some of those problems may
serve to intrigue and enliven m a n y social gathering.
B • Problems
a. If each boy at a picnic were given three
apples f r o m the available supply, one of the boys would have
to be satisfied with two apples. But if each boy were given
two apples, eight apples would remain. How large was the
supply of apples?
b. T w o trains start at 7 A.M., one f r o m A going to B
and the other f r o m B going to A. The first train makes the
trip in 8 hours and the second in 12 hours. At what hour
of the day will the two trains pass each other?
c. Three brothers, Tom, Dick, and Harry, stopped at an
inn and ordered a dinner. When at the end of the meal no
dessert was served, they asked the innkeeper to stew some
prunes for them. While waiting in their comfortable chairs
mathematics as r e c r e a t i o n
189
for the prunes, all three of them fell asleep. A f t e r a while
Tom woke up and, finding a bowl of prunes on the table,
ate his share and went back to sleep. W h e n Dick woke a
little later, he ate what he thought was his share and he, too,
fell asleep again. When H a r r y awoke he proceeded the same
way. When T o m awoke for the second time, he aroused the
two younger brothers, and after a little discussion the whole
story was cleared up. The remaining eight prunes were divided equitably between Dick and H a r r y . H o w many prunes
did each of them get?
d. A steamer plying between two river ports A and B
makes the trip f r o m A to B in 12 hours and the return trip
in 18 hours. H o w long will it take a log thrown into the
water at A to reach B?
e. Find two numbers whose difference and whose quotient
are both equal to three.
f. A commuter ordinarily reaches the railroad station
nearest his home at 5 P.M., where he is met by his wife, in
the family car. One day he unexpectedly arrived at the station
at 4 P.M. and instead of waiting for his car at the station he
started out for home, on foot. A f t e r a certain length of time
he meets his wife and makes the rest of the way home in
the car, as usual. H e reached home sixteen minutes ahead
of the usual time. How long did he walk?
g. Find a number such that if one sixth part of it multiplied
by one eighth part of it, the result is equal to the number.
h. A summer camper rowed one mile up-stream when his
hat blew off into the water beside him. As it was an old hat
he decided to let it go. Ten minutes later he remembered
that he put his return ticket under the hatband. Rowing at
the same rate as before, he reached the hat (and the ticket)
at the same point where he started out in the boat. H o w
fast was the stream flowing?
k. What is the smallest number of cuts that would divide
a cube of wood 3 inches on the edge into cubes one inch on
the edge? 28
m. A courier rode f r o m the rear of a column of marching
soldiers to the front and returned forthwith to the rear of
the column. H e kept his horse jogging along exactly three
times as fast as the column itself was advancing. Where on
the road, with reference to the original position of the vanguard, did he complete his journey?
C ' Solutions
a. Each boy will have two apples if we take
back one apple f r o m each lot of three apples planned originally, and we will thus accumulate eight apples. Hence the
supply of apples consists of 8 x 3 + 2 = 2 6 apples.
b. First solution. The speeds of the two trains are inversely
190
mathematics in f u n and in earnest
proportional to the time it takes them to cover the distance
AB, hence the ratio of those speeds is equal to 12 : 8 = 3:2,
and the distance the two trains cover in the same length of
time are proportional to their speeds, that is 3 : 2 . At the
time the two trains meet the train f r o m A and the train
f r o m B will have covered, respectively, 3 / 5 and 2 / 5 of the
distance AB. T h e time the train f r o m A traveled to reach
the meeting point is 3 / 5 of 8 hours, that is, 4 hours and
48 minutes, so that the trains meet at 11.48 A.M.
Second solution. T h e trains cover, respectively, 1 / 8 and
1 / 1 2 of the distance AB, per hour. Hence they approach
each other by a 1 / 8 + 1 / 1 2 = 5 / 2 4 part of the distance AB
per hour. Thus they will meet after 2 4 / 5 hours of travel.
c. First solution. T o m left 2 / 3 of the number of prunes
he f o u n d on the table. Dick left 2 / 3 of the prunes he found,
which was 2 / 3 x 2 / 3 = 4 / 9 of the number of the prunes the
innkeeper served. Finally H a r r y left 4 / 9 x 2 / 3 = 8 / 2 7 of the
original n u m b e r of prunes, and this amounted to eight
prunes. H e n c e the innkeeper served originally 27 prunes in
all. Of those T o m ate 9, Dick ate 6, and Harry 4. Thus of
the eight prunes remaining Dick is entitled to 9 — 6 = 3 and
H a r r y to 9 — 4 = 5 .
Second solution. F o r this "frontal attack" upon the problem we may substitute a "back door" solution. Harry left
8 prunes for his two brothers, hence he ate himself four
prunes, that is, he found on the table 8 + 4 = 1 2 prunes.
T h a t was the n u m b e r H a r r y left for the other two brothers,
hence he f o u n d on the table 18 prunes, left by Tom, etc.
Notice that the arithmetic is about fourth grade level,
but the problem is not.
d. The hourly rate of the steamer going up-stream is less
than the hourly speed going down-stream by two hourly
speeds of the current. A f t e r having traveled 12 hours upstream the boat is therefore 2 x 1 2 = 2 4 hourly speeds of the
current away f r o m its destination A. This distance the boat is
expected to cover in 1 8 — 1 2 = 6 hours, hence, when going upstream, the boat covers per hour a distance equal to 2 4 : 6 = 4
times the hourly speed of the current, and in 18 hours the
boat covers 1 8 x 4 = 7 2 such distances, which is thus the
n u m b e r of hours it will take the log to cover the distance AB.
e. T h e quotient of the two numbers being 3, the difference between the larger n u m b e r and the smaller number is
equal to twice the small number. On the other hand, this
difference is equal to three, hence the smaller number is
equal to 3 / 2 , and the larger n u m b e r to 3 / 2 + 3 = 9 / 2 .
f. Instead of worrying about the man, it is more to the
point to consider the role of the wife (As usual: Cherchez
mathematics as r e c r e a t i o n
191
la femme!).
She, too, saved 16 minutes on her usual trip.
This time is made up of eight minutes saved by not going
f r o m the meeting place to the station and of eight minutes
saved on the return trip. But the wife expected to be at the
station at 5 P.M., hence the meeting took place at 4.52
P.M., and " h u b b y " thus exercised for 52 minutes.
Notice that the lady in the car travels 5 2 / 8 = 1 3 / 2 times
faster than her husband travels on foot.
g. If instead of multiplying 1 / 6 of the required n u m b e r
by 1/8 of it we would multiply that n u m b e r by itself, our
result would be 8 x 6 = 4 8 times larger than expected, that is, it
would be equal 48 times the required number. Thus multiplying the number by itself produces the same effect as multiplying it by 48, hence 48 is equal to the required number.
h. The important circumstance to notice in the situation
is that when the camper rowed away f r o m the hat and
against the current, his rate of separation f r o m the hat was
equal to his rate of rowing in still water, and this is also the
rate of approach to the hat when he turns around and tries
to catch up with his hat. It follows that since he was rowing
away f r o m the hat for ten minutes, it will take him ten
minutes to catch up with that precious hat (and the ticket in
it). Thus the hat was in the water a total of twenty minutes
and in that length of time covered a distance of one mile,
hence the hourly rate of the current is three miles.
k. Assume that the given cube lies on a horizontal floor.
We divide the top face of the cube into nine equal squares
by two pairs of parallel lines. By four vertical cuts along
those four lines we divide the cube into nine equal columns
which can be divided into twenty seven equal cubes by two
horizontal cuts. The assigned task has thus been accomplished
by six cuts. Could it be done with a smaller number of cuts?
TTie answer is: No. This becomes clear when we consider
the small cube which occupied the center of the given cube:
each of its six faces had to be obtained by a separate cut.
m. Since the courier travels at a speed equal to three times
the speed of the marching column, he approaches the vanguard of the column on the first part of his journey with a
speed equal to twice the speed of the column (the vanguard
is moving away f r o m h i m ) . Hence during the time the courier
traveled f r o m the rear to the head of the column, the
column moved on a distance equal to half the length of the
column.
On his return trip the courier approaches the rear of the
column at a speed equal to ( 3 + 1 = 4 ) f o u r times as great as
the speed of the column (the rear is coming to meet h i m ) .
Hence by the time the courier reached the rear, the column
192
mathematics in f u n and in earnest
traveled one f o u r t h of its own length. Thus the courier returns to the rear of the column at a point on the road
which is at a distance equal to V2-\-Va — % the length of the
column, and this point is also VA of the length of the column
behind the point occupied by the vanguard when the courier
started out on his journey.
FOOTNOTES
*Cf. Chapter V, Section le.
2
Cf. Chapter VII, Section 3.
8
Cf. Chapter II, Section la.
4
"Digital Reckoning Among the Ancients," Leon I. Richardson, American Mathematical Monthly, Vol. 23, No. 1, January,
1916, pp. 7-13.
6
Cf. Chapter I, Section 3c.
8
Scripta Mathematica, March-June 1950.
''American Mathematical Monthly, January 1945, p. 42.
8
Cf. Chapter VII, Section la.
0
American Mathematical Monthly, August-September 1945,
p. 397.
10
Scripta Mathematica, Vol. 11, Nos. 3-4, Sept.-Dec., 1945,
p. 360.
11
American Mathematical Monthly, Vol. 53, No. 3, March
1946, p. 156.
12
American Mathematical Monthly, Vol. 54, No. 1, January
1947, pp. 46-48.
18
Mathematical Gazette (London, 1945), p. 227.
14
Ibid., 1946, p. 23 Iff.
16
Ibid., 1947, pp. 31-39.
18
Lectures on Famous Problems of Elementary Geometry, Felix
Klein, translated by W. W. Beman and D. E. Smith (sec. ed.),
revised by R. C. Archibald (New York, 1930).
17
Doerrie, Henrich, Triumph der Mathematik, the subtitle of
which reads as follows: Hundert beruemte Probleme aus zwei
Jahrtausenden mathematischer Kultur (one hundred famous problems from the mathematical culture of the last two thousand
years) (Breslau, 1933). Callandreau, Edouard Celebres problemes des Mathematiques (Paris, 1949).
18
Chapter VII, Section la.
10
Ibid., Section Id.
20
Cf. Chapter HI, Section 3h.
21
Des methodes en geometrie, Paul Serret (Paris, 1855).
22
Science et Methode (Paris, 1909), p. 46.
28
See Chapter VII, Section la.
24
Cf. Chapter V, Section 2b.
25
Chapter VI, Sec. Id.
28
Cf. Chapter VII, Section Id.
27
Chapter I, Section 3c.
28
Cf. Chapter I, Section 3c.