The Story of Proof: Logic and the History of Mathematics

The concept of proof has been essential to the development of mathematical knowledge, inspiring innovation and generating new ideas. John Stillwell's book explores how proof has evolved through critical episodes in its history, from Euclid to modern times, across all major mathematical disciplines. It demonstrates that proof is a mathematically vital concept, inspiring innovation and playing a critical role in generating knowledge.

Format: Hardback
Length: 456 pages
Publication date: 10 January 2023
Publisher: Princeton University Press

The evolution of the concept of proof has played a pivotal role in the development of mathematical knowledge throughout history. From the Pythagorean theorem to modern times, and across various mathematical disciplines, John Stillwell explores the significance of proof as a fundamental aspect of mathematical thought.

Stillwell begins his journey by examining the influence of Euclid on the development of geometry and the methods of proof employed in this discipline. Euclid's systematic approach to mathematics, characterized by its axiomatic and deductive nature, laid the foundation for subsequent mathematical developments. However, as Stillwell notes, algebra emerged as a separate discipline and eventually rivaled geometry in its mathematical impact.

One of the significant developments in the history of proof was the transition from axiomatic proofs to computational proofs in calculus. Initially, calculus was viewed as a form of "infinitesimal algebra," and its proofs were based on the principles of Euclid. However, as mathematicians continued to explore the realm of calculus, they realized that it could be used to prove mathematical theorems beyond the scope of Euclid's axiomatic system.

In the areas of number theory, non-Euclidean geometry, topology, and logic, Stillwell delves into the complex relationship between the concept of proof and arithmetic. He highlights the work of mathematicians such as Cantor, Gödel, Turing, and others who discovered that proof is ultimately rooted in arithmetic. This revelation imposed fundamental limits on what theorems can be proved and what problems can be solved within the realm of mathematics.

The Story of Proof offers a captivating new perspective on the power and progress of mathematics. By shedding light on the workings of mathematics at its most fundamental levels, Stillwell provides readers with a deeper understanding of the field's rich history and its ongoing evolution. This book is a valuable resource for students, mathematicians, and anyone interested in the development of mathematical knowledge.

Dimension: 235 x 156 (mm)
ISBN-13: 9780691234366