The Mathematical Experience

by Davis, Philip J.

• ISBN13: 9780395929681
• ISBN10: 0395929687
• Edition: Reprint
• Format: Paperback
• Copyright: 1999-01-14
• Publisher: Lightning Source Inc

Summary

This is the classic introduction for the educated lay reader to the richly diverse world of mathematics: its history, philosophy, principles, and personalities.

Author Biography

Philip J. Davis is a professor of applied mathematics at Brown University. Reuben Hersh is a professor of mathematics at the University of New Mexico in Albuquerque.

Table of Contents

Preface	xi	(2)
Acknowledgements	xiii	(4)
Introduction	xvii
Overture	1	(5)
1. The Mathematical Landscape	6	(26)
What is Mathematics?	6	(2)
Where is Mathematics?	8	(1)
The Mathematical Community	9	(4)
The Tools of the Trade	13	(4)
How Much Mathematics is Now Known?	17	(3)
Ulam's Dilemma	20	(4)
How Much Mathematics Can There Be?	24	(2)
Appendix A -- Brief Chronological Table to 1910	26	(3)
Appendix B -- The Classification of Mathematics, 1868 and 1979 Compared	29	(3)
2. Varieties of Mathematical Experience	32	(36)
The Current Individual and Collective Consciousness	32	(2)
The Ideal Mathematician	34	(10)
A Physicist Looks at Mathematics	44	(8)
I. R. Shafarevitch and the New Neoplatonism	52	(3)
Unorthodoxies	55	(5)
The Individual and the Culture	60	(8)
3. Outer Issues	68	(54)
Why Mathematics Works: A Conventionalist Answer	68	(9)
Mathematical Models	77	(2)
Utility	79	(10)
1. Varieties of Mathematical Uses	79	(1)
2. On the Utility of Mathematics to Mathematics	80	(3)
3. On the Utility of Mathematics to Other Scientific or Technological Fields	83	(2)
4. Pure vs. Applied Mathematics	85	(2)
5. From Hardyism to Mathematical Maoism	87	(2)
Underneath the Fig Leaf	89	(24)
1. Mathematics in the Marketplace	89	(4)
2. Mathematics and War	93	(3)
3. Number Mysticism	96	(4)
4. Hermetic Geometry	100	(1)
5. Astrology	101	(7)
6. Religion	108	(5)
Abstraction and Scholastic Theology	113	(9)
4. Inner Issues	122	(81)
Symbols	122	(4)
Abstraction	126	(8)
Generalization	134	(2)
Formalization	136	(4)
Mathematical Objects and Structures; Existence	140	(7)
Proof	147	(5)
Infinity, or the Miraculous Jar of Mathematics	152	(6)
The Stretched String	158	(5)
The Coin of Tyche	163	(5)
The Aesthetic Component	168	(4)
Pattern, Order, and Chaos	172	(8)
Algorithmic vs. Dialectic Mathematics	180	(7)
The Drive to Generality and Abstraction The Chinese Remainder Theorem: A Case Study	187	(9)
Mathematics as Enigma	196	(2)
Unity within Diversity	198	(5)
5. Selected Topics in Mathematics	203	(69)
Group Theory and the Classification of Finite Simple Groups	203	(6)
The Prime Number Theorem	209	(8)
Non-Euclidean Geometry	217	(6)
Non-Cantorian Set Theory	223	(14)
Appendix A	237	(1)
Nonstandard Analysis	237	(18)
Fourier Analysis	255	(17)
6. Teaching and Learning	272	(46)
Confessions of a Prep School Math Teacher	272	(2)
The Classic Classroom Crisis of Understanding and Pedagogy	274	(11)
Polya's Craft of Discovery	285	(6)
The Creation of New Mathematics: An Application of the Lakatos Heuristic	291	(7)
Comparative Aesthetics	298	(3)
Nonanalytic Aspects of Mathematics	301	(17)
7. From Certainty to Fallibility	318	(45)
Platonism, Formalism, Constructivism	318	(3)
The Philosophical Plight of the Working Mathematician	321	(1)
The Euclid Myth	322	(8)
Foundations, Found and Lost	330	(9)
The Formalist Philosophy of Mathematics	339	(6)
Lakatos and the Philosophy of Dubitability	345	(18)
8. Mathematical Reality	363	(49)
The Riemann Hypothesis	363	(6)
Pie and Pie	369	(6)
Mathematical Models, Computers, and Platonism	375	(5)
Why Should I Believe a Computer?	380	(7)
Classification of Finite Simple Groups	387	(4)
Intuition	391	(9)
Four-Dimensional Intuition	400	(6)
True Facts About Imaginary Objects	406	(6)
Glossary	412	(5)
Bibliography	417	(18)
Index	435