How the concept of proof has enabled the creation of mathematical knowledge

The Story of Proof investigates the evolution of the concept of proof―one of the most significant and defining features of mathematical thought―through critical episodes in its history. From the Pythagorean theorem to modern times, and across all major mathematical disciplines, John Stillwell demonstrates that proof is a mathematically vital concept, inspiring innovation and playing a critical role in generating knowledge.

Stillwell begins with Euclid and his influence on the development of geometry and its methods of proof, followed by algebra, which began as a self-contained discipline but later came to rival geometry in its mathematical impact. In particular, the infinite processes of calculus were at first viewed as “infinitesimal algebra,” and calculus became an arena for algebraic, computational proofs rather than axiomatic proofs in the style of Euclid. Stillwell proceeds to the areas of number theory, non-Euclidean geometry, topology, and logic, and peers into the deep chasm between natural number arithmetic and the real numbers. In its depths, Cantor, Gödel, Turing, and others found that the concept of proof is ultimately part of arithmetic. This startling fact imposes fundamental limits on what theorems can be proved and what problems can be solved.

Shedding light on the workings of mathematics at its most fundamental levels, The Story of Proof offers a compelling new perspective on the field’s power and progress.

Table of Contents

1. Before Euclid

2. Euclid

3. After Euclid

4. Algebra

5. Algebraic Geometry

6. Calculus

7. Number Theory

8. The Fundamental Theorem of Algebra

9. Non-Euclidean Geometry

10. Topology

11. Arithmetization

12. Set Theory

13. Axioms for Numbers, Geometry, and Sets

14. The Axiom of Choice

15. Logic and Computation

16. Incompleteness

Review

"This book could well serve as a history of mathematics. … [Stillwell] has done an amazing job of collecting and categorizing many of the most important ideas in this area."---Jim Stein, New Books in Mathematics

"Stillwell’s [The Story of Proof] joins his two other Princeton University Press books in having my highest recommendation. I just wish they had been around when I was a student."---George Hacken, Computing Reviews

"I hugely enjoyed this book."---Jonathan Shock, Mathemafrica

"This book would be perfect for any keen undergraduate, keen amateur, or indeed a teacher of mathematics, who wants a book to dip into to use for the classroom."---Jonathan Shock, Mathemafrica

"

A well-crafted, thought-provoking meditation on the concept of proof in mathematics. . . .It is a substantive book that deserves to be read and reflected upon.

"---Tommy Murphy, Irish Mathematical Society Bulletin

"This is a work that mathematicians, historians, and philosophers will find especially engaging, as will anyone with a serious interest in mathematics and the limits of certainty that it is constantly probing."---J.W. Dauben, Choice

"Stillwell's book is fundamentally an ode to mathematical progress and a retrospective on how far we have come in developing a mathematically sound theory of proof itself."---Alexi Block Gorman, The Mathematical Intelligencer

About the Author

John Stillwell is emeritus professor of mathematics at the University of San Francisco. His many books include Elements of Mathematics and Reverse Mathematics (both Princeton).