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
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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.)
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"rillled in the Uniled Sillies of ,\mericli
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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
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