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Publisher:

Mariner Books

Publication Date:

1999

Number of Pages:

464

Format:

Paperback

Price:

21.95

ISBN:

0395929687

Category:

General

The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by , on ]

Allen Stenger

06/28/2010

This is a kaleidoscopic look at the philosophy of mathematics, aimed at readers who are neither philosophers nor mathematicians. It can best be thought of as an anthropological study of mathematicians. We learn all about their sacred beliefs and rituals, the historical reasons for these things, and some of the accomplishments of the mathematical culture. But we never learn what it’s like to *be* a mathematician, because, despite the importance of the topics studied here, mathematicians spend almost no time thinking about them. They just do them.

The book appears to have been assembled from a lot of individual articles dealing with these issues, and has abrupt changes of subject. This fragmentation, and the diversity of viewpoints, can be disorienting. The book’s underlying theme, to the extent that there is one, is an attempt to undermine the popular idea that mathematics is certain and infallible. This is done by showing the diversity of opinion among mathematicians about what actually constitutes a proof, by looking at different philosophies of mathematical existence and proof (Platonism, formalism, intuitionism), and by discussing some extremely long and complex proofs and the possibilities for error therein. The book, published in 1981, is a little dated today in that it emphasizes some then-recent developments in mathematics whose novelty has now faded; these include Appel and Haken’s 1976 computer-aided proof of the four-color theorem and Imre Lakatos’s 1976 book *Proofs and Refutations*.

This book is probably not a good pick for a math appreciation course (even though many students would enjoy it), because it does not really get at the nature of mathematics as it is practiced. For that, it would be better to pick a book where students experience being a mathematician, even if in a very modest way, such as Burger and Starbird’s *The Heart of Mathematics* and Courant and Robbins and Stewart’s *What is Mathematics?.*

See also the review of the Study Edition.

Allen Stenger is a math hobbyist and retired software developer. He is webmaster and newsletter editor for the MAA Southwestern Section and is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis. He volunteers in his spare time at MathNerds.com, a math help site that fosters inquiry learning.

- Preface
- Acknowledgements
- Introduction
- Overture

- 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

- Varieties of Mathematical Experience
- The Current Individual and Collective Consciousness
- The Ideal Mathematician
- A Physicist Looks at Mathematics
- I. R. Shafarevitch and the New Neoplatonism
- Unorthodoxies
- The Individual and the Culture

- Outer Issues
- Why Mathematics Works: A Conventionalist Answer
- Mathematical Models
- Utility
- Varieties of Mathematical Uses
- On the Utility of Mathematics to Mathematics
- On the Utility of Mathematics to Other Scientific or Technological Fields
- Pure vs. Applied Mathematics
- From Hardyism to Mathematical Maoism

- Underneath the Fig Leaf
- Mathematics in the Marketplace
- Mathematics and War
- Number Mysticism
- Hermetic Geometry
- Astrology
- Religion

- Abstraction and Scholastic Theology

- 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

- Selected Topics in Mathematics
- Group Theory and the Classification of Finite Simple Groups
- The Prime Number Theorem
- Non-Euclidean Geometry
- Non-Cantorian Set Theory
- Appendix A: Cantor's Diagonal Process
- Nonstandard Analysis
- Fourier Analysis

- Teaching and Learning
- Confessions of a Prep School Math Teacher
- The Classic Classroom Crisis of Understanding and Pedagogy
- Pólya's Craft of Discovery
- The Creation of New Mathematics: An Application of the Lakatos Heuristic
- Comparative Aesthetics
- Nonanalytic Aspects of Mathematics

- From Certainty to Fallibility
- Platonism, Formalism, Constructivism
- The Philosophical Plight of the Working Mathematician
- The Euclid Myth
- Foundations, Found and Lost
- The Formalist Philosophy of Mathematics
- Lakatos and the Philosophy of Dubitability

- Mathematical Reality
- The Riemann Hypothesis
- π and π'
- Mathematical Models, Computers, and Platonism
- Why Should I Believe a Computer?
- Classification of Finite Simple Groups
- Intuition
- Four-Dimensional Intuition
- True Facts About Imaginary Objects

- Glossary
- Bibliography
- Index

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