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The Story of Proof: Logic and the History of Mathematics

by John Stillwell (Author)
4.5 4.5 out of 5 stars 52 ratings
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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.

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  1. ISBN-10
    0691234361
  2. ISBN-13
    978-0691234366
  3. Publisher
    Princeton University Press
  4. Publication date
    November 15, 2022
  5. Language
    English
  6. Dimensions
    6.5 x 1 x 9.5 inches
  7. Print length
    456 pages
  8. See all details
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Editorial Reviews

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

Review

“I am a great admirer of Stillwell’s writing, and this book does not disappoint. Ranging broadly and authoritatively over the history of mathematics, he takes the reader into those places where proofs have been innovative and have played a critical role.”―David M. Bressoud, author of Calculus Reordered: A History of the Big Ideas



"This is a lively story of the way that mathematics develops new concepts and ideas in order to solve hard problems. It is, at the same time, a story of the way that logic and the theory of computability emerged organically from reflection on mathematical method and thought. This book will be highly valuable to anyone interested in contemporary mathematics and how it came to be."―Jeremy Avigad, Carnegie Mellon University

“The Story of Proof―a book about mathematics as proof, and proof as mathematics, through the ages―is unique and enjoyable."―Anil Nerode, Cornell University

Product details

  • Publisher ‏ : ‎ Princeton University Press (November 15, 2022)
  • Language ‏ : ‎ English
  • Hardcover ‏ : ‎ 456 pages
  • ISBN-10 ‏ : ‎ 0691234361
  • ISBN-13 ‏ : ‎ 978-0691234366
  • Item Weight ‏ : ‎ 2.31 pounds
  • Dimensions ‏ : ‎ 6.5 x 1 x 9.5 inches
  • Customer Reviews:
    4.5 4.5 out of 5 stars 52 ratings

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John Stillwell

John Stillwell

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Customer reviews

4.5 out of 5 stars
4.5 out of 5
52 global ratings

Top reviews from the United States

Bryan Runck
5.0 out of 5 stars Essential reading for those with a love of mathematics
Reviewed in the United States on February 10, 2023
Verified Purchase
I am not a trained mathematician, but do have advanced STEM degrees. This means that some of the book was over my head, but I couldn’t have cared in the slightest because it’s so well written, and the summary sections put the details in a much broader context. It instead inspired me to dig deeper into the subfields I’m unfamiliar with.

What I appreciate most about the book was how Stillwell relates the motivating problem of infinity in mathematics (and physics) to the many different branches of the field, and what these fields considered to be acceptable proof for a given mathematical idea.

If you have a love of maths, and are curious about the inner workings of the field, this is a great book for you.
28 people found this helpful
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A M
4.0 out of 5 stars A readable short history of mathematics
Reviewed in the United States on May 10, 2023
This book is fun to read for those who have some background in Mathematics and Logic. The book cannot be read on the Kindle device, due to its graphics.

A readable short history of mathematics
9 people found this helpful
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Kwok Ka Keung
5.0 out of 5 stars Good.
Reviewed in the United States on February 22, 2023
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Good.
One person found this helpful
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Dallas Entrepreneur
5.0 out of 5 stars Great mathematical history of how various proofs evolved through time.
Reviewed in the United States on December 6, 2022
Verified Purchase
Clear exposition and love for subject.
5 people found this helpful
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Keith Harbaugh
5.0 out of 5 stars I did not have a problem with text readability
Reviewed in the United States on April 23, 2024
Some other reviewers have complained about "light grey text throughout the book that was impossible to read without a great deal of eye strain".
I bought my copy directly from the publisher, Princeton University Press, and did not see this problem at all.

In general:
This is a really excellent and much-needed book, cross-cutting between, and showing the inter-relations between, the various fields of mathematics which are usually taught (for good reason) in isolation from each other in the academic curriculum.
There is one omission I want to point out:
There is an excellent elementary introduction to category theory, assuming nothing more than high school math:
"Conceptual Mathematics" by F. William Lawvere and Stephen Schanuel.
So, Stillwell is overstating things when he writes in his Section 10.9
"category theory [necessarily?] involves a high level of abstraction, and background too advanced for a book such as this".
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Linda Redding Bird
1.0 out of 5 stars Light grey text impossible to read
Reviewed in the United States on November 27, 2022
Verified Purchase
I was looking forward to this book until I received, opened it, and found light grey text throughout the book that was impossible to read without a great deal of eye strain after only a few minutes. Why do publishers choose not to print books in black text, more easily read? I suspect it is a matter of saving on cost, not concern for readers. At least inform us of font color before we purchase. I am returning book and will be certain not to purchase from this publisher again.
49 people found this helpful
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Top reviews from other countries

Me
4.0 out of 5 stars Very technical very quickly
Reviewed in the United Kingdom on June 24, 2023
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I am sure this is a very good book. But beware if like me you are wanting an introduction to the subject as it becomes very very mathematical and heavy very quickly and is beyond my feeble knowledge.
2 people found this helpful
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Unger
5.0 out of 5 stars A fascinating history of msthematics
Reviewed in Australia on December 31, 2023
Verified Purchase
I am enthralled by this history if mathematical proof. Context, I am a mathematician so I am not stretching to follow the maths in the exposition. The history fascinates me and explains the terminologies I knew but didn't understand the reason for.
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