penguin press science philip j davis reuben hersh the mathematical experience penguin books ltd 1990

The
Mathematical
Experience


The
Mathematical
Experience
Philip J. Davis
Reuben Hersh
With an Introduction by Gian-Carlo Rota

HOVGHTON MIFFLIN COMPANY BOSTON


CopyrigJII

© 1981 by Birkhliuser Boston

All rights rcser\'ed. Nil I'"rl of Ihis work ma)' bc reproduced
or I ransmilled in lUI), form or b)' an)' mcans, cieci runic 01'
mechanical, including photocopying lllld rccnrding, or by
allY informmion siorage or rClric\'l.1 S)·slelll. excepl as may
he expressly permillcd h)' Ihc 1!l76 Cop)Tiglli ACI or in
,..-riling from Ihc publishcr. Requests for permission should
he lHldl'cssed in "Tiling 10 I-)oughlon ~[irnin COJ1lpany.
2 "ark Sireet, Boston, Massachusells 1J21OH.
Ubrary rif C/JIIKmt.l Cntalogillg ill I'/llilicalioll Data

navis, Philip J.. date
The malhematical experience.
Reprint. Originally puhlishcd: BnSlOn: Uirkhauser.

19111.
niblingraphy: p.
Includes index.
I. ~Ialhemal ics -Philosoph)'. 2. ~I alhemlllics- H iswl'y.
:t ~lathematirs-Slll(ly :lIId leaching. I. Hcrsh. Rcuben.
date. J I. Titlc,
QAHA.1>37 1982
5)()
RI·203(H
ISBi\ O·:l9:'·!~2131·X (pbk.)
AACR2
"rillled in the Uniled Sillies of ,\mericli
ALIO 9 8 7 6 5

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3 2 I

Reprinled by arrangemcnl wilh nirkhliuser BnSlO1l
Houghton Mimi" COJ1lP:IIlY 1':.pcl'hilck [982


For my parents,
Mildred and Philip Hersh

****

For my brother,
Hyman R. Davis


Contents
Preface
Acknowledgements
Introduction
Overture
1. The Mathematical Landscape
What is Mathematics?
Where is Mathematics?
The Mathematical Community
The Tools of the Trade
How Much Mathematics is Now Known?
Ulam's Dilemma
How Much Mathematics Can There Be?
Appendix A-Brief Chronological Table to
1910
Appendix B-The Classification of Mathematics. 1868 and 1979 Compared
2. Varieties of Mathematical Experience
The Current Individual and Collective ConsCIousness
The Ideal Mathematician
A Physicist Looks at Mathematics
I. R. Shafarevitch and the New Neo~~n~m

Unorthodoxies
The Individual and the Culture
3. Outer Issues
Why Mathematics Works: A Conventionalist

Answer

XI
XIII
XVII

6
8

9
13
17

20
24

26
29

32
34

44
~

55
60

68

Contents

Mathematical Models
Utility

1. Varieties of Mathematical Uses
2. On the Utility of Mathematics to
Mathematics
3. On the Utilil)' of Mathematics to Other
Scientific or Technological Fields
4. Pure vs. Applied Mathematics
5. From Hardyism to Mathematical Maoism
Underneath the Fig Leaf
1.
2.
3.
4.
5.
6.

Mathematics in the Marketplace
Mathematics and War
Number Mysticism
He17fletic Geometry
Astrology
Religion

Abstraction and Scholastic Theology

ii
i9
i9
80

83
85
8i
89
89
93
911

100
101

IOH
II :~

4. Inner Issues
Symbols
Abstraction
Generalization
Formalization
Mathematical Objects and Structures; Existence
Proof
Infinity, or the Miraculous Jar of
Mathematics
The Stretched String
The Coin of Tyche

The Aesthetic Component
Pattern, Order, and Chaos
Algorithmic vs. Dialectic Mathematics
The Drive to Generality and Abstraction
The Chinese Remainder Theorem: A
Case Study
Mathematics as Enigma
Unity within Diversity
5. Selected Topics in Mathematics
Group Theory and the Classification of
Finite Simple Groups

19 '>
1211
134
131)
~.

140
14i
152
158
163
168

I-<}
,.

180

18i
196
19H

203


Contents
The Prime Number Theorem
Non-Euclidean Geometry
Non-Cantorian Set Theon',
Appendix A
Nonstandard Analysis
Fourier Analysis
6. Teaching and Learning
Confessions of a Prep School Math
Teacher
The Classic Classroom Crisis of Understanding and Pedagogy
P6lya's Craft of Discovery
The Creation of New Mathematics: An
Application of the Lakatos Heuristic
Comparative Aesthetics
l'\onanalytic Aspects of Mathematics
7. From Certainty to Fallibility
Platonism, Formalism, Constructivism
The Philosophical Plight of the Working
Mathematician
The Euclid ~lyth
Foundations, Found and Lost
The Formalist Philosophy of Mathematics

Lakatos and the Philosophy of Dubitability
8. Mathematical Reality
The Riemann Hypothesis
17" and ir
Mathematical Models, Computers, and
Platonism
Why Should I Believe a Computer?
Classification of Finite Simple Groups
Intuition
Four-Dimensional Intuition
True Facts AboUl Imaginary Objects
Glossary
Bibliography
Index

209
217
223
237
237

255

272
274

285
291

298

301
318
321
322
330

339
345

%3
369

375
380

387
391
400
406
412
417

435


Preface
H E. OL DEST MAT I-I E;"IAT I CA L tab lets we
ha\"c dale from 2400 II. C., but there is no reaso n
1.0 suppose that t.he urge 1.0 create a nd usc 111;'1l l1 cma t.its is not coex te nsive \I·jlb the whol e o f civili1.;ll lo n . I n lour OJ" five mil lenni a:l vast. bod y of p nlC l.iccs concepts knOII"n as mathema tics h

linked in a varic L), of I\'(!),s \I' ith ou r day-to-cla )· life. What
is the n aUl rc o r mathcm ,lIics? What is its m eanin g? Wha t
arc its co ncerns? What is ils meth odology? H oI\' is il
c rea t.ed? How is ilused ? J-Iow docs it fit in wit h the va r ieties
o f hu m a n expe rience? Wh a t be ne fits flow from it ? What
harm ? What impoI"IT hese d irficult question s arc no t made ca siel' by (he fact
tha t the amo unt or material is so large int erl inki ng is so extensive th at it is sim p ly no t poss ible fo r
all )' o ne person (() com pl"c hen d it all, lei a lo ne sum it up
and co mpress Ihe Sli III III a I"y between the cove l"S o f an avel"ag-e-sizc.::d book " LeSI we hc cowcd b )' Ihi s ,"aSt amo unt of
mater ial, let us thi nk of m ;1I hem ;u ics in another wa )," Math e matics has been a hum an activit }, for thou sands of" years"
T o some small exte nt , evcl"ybod y is a math e matician and
docs mathelllatics co nscio usly" To bu y at the ma rket, 10
mea sure a st rip of I\'all p aper or 10 decorate a ce ra rn ic po t
with a regu lar pall crn is doing malhematics" Further,
eve rybody is to somc sm all cx tc n l a p hiloso phc l" or lllaLh eIll ,liics" Let him only exclaim on OCCilsio n : "Bu l fig ul"cS
C;IIl 't lie !" a nd he joins lh e ran ks of Plato and of Lakatos"
In addit ion to the VO il a modest scale, I here a l"e a smal l IlUmbCI" of peo ple who
,Ire p rofessio nal ma li lemalician s" They praClice ma themat -

T

XI


Preface
ics, foster it, teach it, create it, and use it in a wide variety of
situations. It should be possible to explain to nonprofessionalsjust what these people are doing, what they say ther

are doing, and why the rest of the world should support
them at it. This, in brief, is the task we have set for ourselves. The book is not intended to present a systematic,
self-contained discussion of a specific corpus of mathematical material, either recent or classical. It is intended rather
to capture the inexhaustible variety presented by the mathematical experience. The major strands of our exposition
will be the substance of mathematics, its history. its philosophy, and how mathematical knowledge is elicited. The
book should be regarded not as a compression but rather
as an impression. It is not a mathematics book; it is a book
about mathematics. Inevitably it must contain some mathematics. Similarly, it is not a history or a philosophy book,
but it will discuss mathematical history and philosophy. It
follows that the reader must bring to it some slight prior
knowledge of these things and a seed of interest to plant
and water. The general reader with this hackground
should have no difficulty in getting through the mitior portion of the book. But there are a number of places where
we have brought in specialized material and directed our
exposition to the professional who uses or produces mathematics. Here the reader may feel like a guest who has
been invited to a family dinner. After polite general conversation, the family turns to narrow family concerns, its
delights and its worries, and the guest is left up in the air,
but fascinated. At such places the reader should judiciously
and lightheartedly push on.
For the most part, the essays in this book can be read independently of each other.
Some comment is necessary abollt the use of the word
"I" in a book written by two people. In some instances it
will be obvious which of the authors wrote the "I." In any
case, mistaken identity can lead to no great damage, for
each author agrees, in a general way, with the opinions of
his colleague.

XII

Acknowledgements

S

OME OF THE MATERIAL of this book was excerpted from published articles. Several of these
have joint authorship: "Non-Cantorian Set Theory" by Paul Cohen and Reuben Hersh and "NonStandard Analysis" by Martin Davis and Reuben Hersh
both appeared in the Scientific American. "Nonanalytic Aspects of Mathematics" by Philip J. Davis and James A. Anderson appeared in the SIAM Review. To Professors Anderson, Cohen, and M. Davis and to these publishers, we
extend our grateful acknowledgement for permission to
include their work here.
Individual articles by the authors excerpted here include
"Number," "Numerical Analysis," and "Mathematics by
Fiat?" by Philip J. Davis which appeared in the Scientific
American, "The Mathematical Sciences," M.l.T. Press. and
the Two Year College Mathematical Jounwl respectively;
"Some Proposals for Reviving the Philosophy of Mathematics" and "Introducing Imre Lakatos" by Reuben
Hersh, which appeared in Advances in Mathematics and the
Mathematical lntelligencer, respectively.
We appreciate the courtesy of the following organizations and individuals who allowed us to reproduce material
in this book: The Academy of Sciences at GOttingen.
Ambix. Dover Publishers, Mathematics of Computation,
M.I.T. Press, New Yorker Magazine. Professor A. H. Schoenfeld, and John Wiley and Sons.
The section on Fourier analrsis was written by Reuben
Hersh and Phyllis Hersh. In critical discussions of philosophical questions, in patient and careful editing of rough
drafts, and in her unfailing moral support of this
X III


Acknowledgements

project, Phyllis Hersh made essential contributions which it

is a pleasure to acknowledge.
The following individuals and institutions generously allowed us to reproduce graphic and artistic material: Professors Thomas Banchoff and Charles Strauss, the Brown
University Library, the Museum of Modern Art, The
Lummus Company, Professor Ron Resch, Routledge and
Kegan Paul, Professor A. J. Sachs, the Univt'rsily of Chicago Press, the Whitworth Art Gallery, the University of
Manchester, the University of Utah, Department of Computer Science, the Yale University Press.
We wish to thank Professors Peter Lax and Gian-Carlo
Rota for encouragement and suggestions. Pl'Ofessor Gabriel Stoltzenberg engaged us in a lively and productive
correspondence on some of the issues discussed here. Professor Lawrence D. Kugler read the manuscript and made
many valuable criticisms. Professor Jose Luis Abreu's participation in a Seminar on the Philosophy of Mathematics
at the University of New Mexico is greatly appreciated.
The participants in the Seminar on Philosophical Issues
in Mathematics, held at Brown University, as well as the
students in courses given at the University of New Mexico
and at Brown, helped us crystallize our views and this help
is gratefully acknowledged. The assistance of Professor
Igor Najfeld was particularly welcome.
We should like to express our appreciation to our colleagues in the History of Mathematics Department at
Brown University. In the course of many years of shared
lunches, Professors David Pingree, Otto Neugebauer, A. J.
Sachs, and Gerald Toomer supplied us with the "three I's":
information, insight, and inspiration. Thanks go to Professor Din-Yu Hsieh for information about the history of Chinese mathematics.
Special thanks to Eleanor Addison for many line drawings. We are grateful to Edith Lazear for her careful and
critical reading of Chapters 7 and 8 and her editorial comments.
We wish to thank Katrina Avery, Frances Beagan, Joseph M. Davis, Ezoura Fonseca, and Frances G
XIV


!

lhei~' efficient help in the preparation and handling of the
maquscripl. Ms. Avery also helped us with a number of
classical references.
i
!

P.
R.

J.

DAVIS

HERSH

xv


Introdtlctiol1
DEDICATED TO ~IARK KAC

"oh philosoplzie alimeulaire!"
-Sarin'

THE TURN OF THE CENTURY, the Swiss historian Jakob Burckhardt, who, unlike most
historians, was fond of guessing the future, once
confided to his friend Friedrich Nietzsche the
prediction that the Twentieth Century would be "the age
of oversimplification".

Burckhardt's prediction has proved frighteningly accurate. Dictators and demagogues of all colors have captured
the trust of (he masses by promising a life of bread and
bliss, to come right after the war (0 end all wars. Philosophers have proposed daring reductions of the complexity
of existence to the mechanics of elastic billiard balls;
others, more sophisticated, have held that life is language,
and that language is in turn nothing but strings of marblelike units held together by the catchy connectives of Fregean logic. Artists who dished out in all seriollsness checkerboard patterns in red, white, and blue are now fetching
the highest bids at Sotheby's. The use of slich words as
"mechanically" "automatically" and "immediately" is now
accepted by the wizards of Madison Avenue as the first law
of advertising.
I\ot even the best minds of Science have been immune to
the lure of oversimplification. Physics has been driven by
the search f()J' one, only one law which one day, just
around the corner, will unify all forces: gravitation and

K

XVII


Introduction
electricity and strong and weak interactions and what not.
Biologists are now mesmerized by the prospect that the secret of life may be gleaned from a double helix dOlled with
large molecules. Psychologists have prescribed in turn sexual release, wonder drugs and primal screams as the cure
for common depression, while preachers would counter
with the less expensive offer to join the hosannahing chorus of the born-again.
It goes to the credit of mathematicians to have been the
slowest to join this movement. Mathematics, like theology
and all free creations of the Mind, obeys the inexorable
laws of the imaginary, and the Pollyannas of the day are of

little help in establishing the truth of a conjecture. One
may pay lip service to Descartes and Grothendieck when
they wish that geometry be reduced to algebra. or to Russell and Gentzen when they command that mathematics
become logic. but we know that some mathematicians are
more endowed with the talent of drawing pictures, others
with that ofjuggling symbols and yet others with the ability
of picking the flaw in an argument.
Nonetheless, some mathematicians have given in to the
simplistics of our day when it comes to the understanding
of the nature of their activity and of the standing of mathematics in the world at large. With good reason. nobody
likes to be told what he is really doing or to have his intimate working habits analyzed and written up. What might
Senator Proxmire say if he were to set his eyes upon such
an account? It might be more rewarding to slip into the
Senator's hands the textbook for Philosophy of Science
301, where the author, an ambitious young member of the
Philosophy Department. depicts with im peccable clarity
the ideal mathematician ideally working in an ideal world.
We often hear that mathematics consists mainly in
"proving theorems". Is a writer's job mainly that of
"writing sentences"? A mathematician's work is mostly a
tangle of guesswork, analogy. wishful thinking and frustration, and proof, far from being the core of discovery. is
more often than not a way of making sure that our minds
are not playing tricks. Few people, if any. had dared write
this out loud before Davis and Hersh. Theorems are not to
XVIII


Introduction
mathematics what successful courses are to a meal. The nutritional analogy is misleading. To master mathematics is to
master an intangible view, it is to acquire the skill of the virtliOSO who cannot pin his performance on criteria. The

theorems of gcometry are not related to the field of Geometry as elements are to a set. The relationship is more subtle, and Davis and Hersh give a rare honest description of
this relationship.
After Davis and Hersh, it will he hard to uphold the GlasJ)(,rlenspiel view of mathematics. The mystery of mathematics, in the authors' amply documented account, is that conclusions originating in the play of the mind do find sl riking
practical applications. Davis and Hersh have chosen to describe the mystery rather than explain it away.
l\·laking mathematics accessible to the educated layman,
while keeping high scientific standards, has always been
considered a treacherous navigation between the Scylla of
professionalt:ontempt and the Charybdis of public misunderstanding. Da\'is and Hersh have sailed across the Strait
under full sail. They have opened a discussion of the mathematical experience that is inevitable for survival. Watching from the stern of their ship. we breathe a sigh of relief
as the vortex of oversimplification recedes into I he distance.
ClAN-CARLO ROTA

Augll.!19. 1980

XIX


"The knowledge at which geometry aims is the knowledge
of the eternal."
Pl.ATO, REPUBLIC.

VII. 527

"That sometimes clear . . . and sometimes vague
. . which is . . . mathematics."

stl~ff.

IMRE LAKA'roS.

1922-1974

"What is laid down, ordered, factual, is never enough to
embrace the whole truth: life always spills over the rim of
every cup."
BORIS PASTERNAK,

1890-1960


Overture

P TILL ABOUT five years ago, I was a normal
mathematician. I didn't do risky and unorthodox things, like writing a book such as this. I had
my "field"-partial differential equations-and
I stayed in it, or at most wandered across its borders into an
acHacent field. My serious thinking, my real intellectual life,
used categories and evaluative modes that I had absorbed
years before, in my training as a graduate student. Because
I did not stray far from these modes and categories, I was
only dimly conscious of them. They were part of the way I
saw the world, not part of the world I was looking at.
My advancement was dependent on my research and
publication in my field. That is to say, there were important rewards for mastering the outlook and ways of
thought shared by those whose training was similar to
mine, the other workers in the field. Their judgment
would decide the value of what I did. No one else would be
qualified to do so; and it is very doubtful that anyone else
would have been interested in doing so. To liberate myself
from this outlook-that is, to recognize it, to become

aware that it was only one of many possible ways of looking
at the world, to be able to put it on or off by choice, to compare it and evaluate it with other ways of looking at the
world-none of this was required by my job or my career.
On the contrary, such unorthodox and dubious adven-

U


OlJl,,.1 11 re

lUres would ha\'e seemed at best a foolish waste of precious
time-at worst, a disreputable dabbling with !-hady and
suspect ventures such as psychology, sociology, 01' philosophy.
The fact is, though, that I have come to a point where
m)' wonderment and fascination wit h the meaning and
purpose, if an)" of this st range activity we call mat hcmatics
is equal to, sometimes even stronger than, my fascination
with actually doing mathematics. I find mathematics an infinitely complex and mysterious world; exploring it is an addiction from which I hope never to be cured. In this, I am
a mathematician like all others, But in addition, I have de\'e1oped a second half, an Other, who watches this mathematician with amazement, and is even Illore fascinated that
such a strange creature and such a strange activity have
come into the world, and persisted for thousands of years.
I trace its beginnings to the day when I came al last to
teach a course called Foulldations of I\lathematics. This is a
course intended primarily for mathematics majors. at the
upper divisioll (junior or senior) level. My purpose in
teaching this course, as in the others I had taught over the
years, was t.o learn the material myself. At that lime I knew
that there was a history of contro\'ersy about the foundations. I knew that there had been three major "schools";
the logicists associated with Bertrand Russell, the formalists led by David Hilbert, and the constructivist srhool of L.
E. J. Brouwer. I had a general idea of the teachillg of each

of these three schools. But I had no idea which one I
agreed with, if any, and I had only it vague idea of what
had become of the three schools in the half century since
their founders were active.
I hoped that by teaching the course I would have t he opportunity In rcad and study about the founclalions of
mathematics, and ultimately to clarify my OWII views of
those pans which were controversial. I did not expect to
bccome a researcher in the foundations of mathematics,
an)' more than I became a number theorist after tcaching
number theory.
Since my interest in thc foundations was philosophical
rather than technical, I tried to plan the courst: so Ihat it


Ova/un'
could be aflended by illterested students with no special requirements or prerequisites; in particular. I hoped to attract philosophy Sl udents. and mathematics ed lIcation students. As it happened there were a few such students;
thcre were also students from electrieal engineering. from
computer science. and other fields. Still. the mathematics
students were thc m~ority. I found a couple of good-looking textbooks. and plungcd in.
In standing beforc a mixed class of mathematics. educalioll, alld philosophy Sl udents, to lecture on thc foundations of mathematics. I found myself in a new and strangc
siwation. I had been teaching mathematics I()r some 15
years. at all levels and in lllaIl), different wpics. but in all
my other courses the job was not to talk about mathematics. it was to do it. Here m)' purpose was not to do il. but to
talk about it. It was different and frightening.
As the semester progressed. it became clear to me that
lhis time it was going to be a different story. The course
was a success in one sense, I(ll" there was a lot of interesting
material. lots of chalKes for stimulating discussions and independent study. lots of things for l11e to learn that I had
never looked at before. But in another sense, I saw that my
pn~ject was hopeless.

111 an ordinary mathematics class. the program is fairly
clear cut. We have problems to solve. or a method of calculation to explain. or a theorem to prove. The main work 10
he done will he in writing. usually on the blackboard. If the
problems arc solved, the theorems proved. or the calculations completed. then teacher and class know that they
have cOlllpleted the daily task. Of course. even in this ordinary mathematical setting. there is always the possibility or
likelihood of something unexpected happening. An UIlf()rescen difficulty. an unexpecled queslion from a student. can cause the progress of the class to deviate from
what the instructor had intended. Still. one knew where
one was supposed to be going; one also knew that the main
thing was what you wrote down. As 10 spoken words. either
from the class or from the teacher. they were important insofar as they helped to communicate the import of what
was written.
3


Overture

In opening my course on the foundations of mathematics, I formulated the questions which I believed were central, and which I hoped we could answer or at least clarify
by the end of the semester.
What is a number? What is a set? What is a proof? What
do we know in mathematics, and how do we know it? What
is "mathematical rigor"? What is "mathematical intuition"?
As I formulated these questions, I realized that I didn't
know the answers. Of course, this was not surprising, for
such vague questions, "philosophical" questions, should
not be expected to have clearcut answers of the kind we
look for in mathematics. There will always be differences
of opinion about questions such as these.
But what bothered me was that I didn't know what my
own opinion was. What was worse, I didn't have a basis, a
criterion on which to evaluate different opinions, to advocate or attack one view point or another.

I started to talk to other mathematicians about proof.
knowledge, and reality in mathematics and I found that
my situation of confused uncertainty was typical. But I also
found a remarkable thirst for conversation and discussion
about our private experiences and inner beliefs.
This book is part of the outcome of these years of pondering, listening, and arguing.

4


1
THE
MATHEMATICAL
LANDSCAPE


What is
Mathematics?

A

NAIVE DEFINITION, adequate for the dictionary and for an initial understanding, is that
mathematics is the science of quantity and spaa. Expanding this definition a bit, one might add that
mathematics also deals with the symbolism relating to
quantity and to space.
This definition certainly has a historical basis and will
serve us for a start, but it is one of the purposes of this
work to modify and amplify it in a way that reflects the
growth of the subject over the past several centuries and
indicates the visions of various schools of mathematics as to

what the subject ought to be.
The sciences of quantity and of space in their simpler
forms are known as arithmetic and geometry. Arithmetic, as
taught in grade school, is concerned with numbers of various sorts, and the rules for operations with numbers-addition, subtraction, and so forth. And it deals with situations in daily life where these operations are used.
Geometry is taught in the later grades. It is concerned in
part with questions of spatial measurements. If I draw such
a line and another such line, how far apart will their end
points be? How many square inches are there in a rectangle 4 inches long and 8 inches wide? Geometry is also concerned with aspects of space that have a strong aest hetic
appeal or a surprise element. For example, it tells us that in
any parallelogram whatsoever, the diagonals bisect one another; in any triangle whatsoever, the three medians intersect in a common point. It teaches us that a floor can be
6


What is Mathematics?

tiled with equilateral triangles or hexagons, but not with
regular pentagons.
But geometry, if taught according to the arrangement
laid out by Euclid in 300 B.C., has another vitally significant
aspect. This is its presentation as a deductive science. Beginning with a number of elementary ideas which are assumed to be self-evident, and on the basis of a few definite
rules of mathematical and logical manipulation, Euclidean
geometry builds up a fabric of deductions of increasing
complexity.
What is stressed in the teaching of elementary geometry
is not only the spatial or visual aspect of the subject but the
melhodology wherein hypothesis leads to conclusion. This
deductive process is known as proof Euclidean geometry is
the first example of a formalized deductive system and has
become the model for all such systems. Geometry has been
the great practice field for logical thinking, and the study

of geometry has been held (rightly or wrongly) to provide
the student with a basic training in such thinking.
Although the deductive aspects of arithmetic were clear
to ancient mathematicians, these were not stressed either
in teaching or in the creation of new mathematics until the
1800s. Indeed, as late as the 1950s one heard statements
from secondary school teachers, reeling under the impact
of the "new math," to the effect that they had always
thought geometry had "proof" while arithmetic and algebra did not.
With the increased emphasis placed on the deductive aspects of all branches of mathematics, C. S. Peirce in the
middle of the nineteenth century, announced that "mathematics is the science of making necessary conclusions."
Conclusions about what? About quantity? About space?
The content of mathematics is not defined by this definition; mathematics could be "about" anything as long as it is
a subject that exhibits the pattern of assumption-deduction-conclusion. Sherlock Holmes remarks to Watson in
The Sign of Four that "Detection is, or ought to be, an exact
science and should be treated in the same cold and unemotional manner. You have attempted to tinge it with romanticism. which produces much the same effect as if you
7