Proof and the Art of Mathematics

A coming-of-age book for mathematicians aspiring to write proofs.

  • J. D. Hamkins, Proof and the Art of Mathematics, MIT Press, 2020.
    author = {Joel David Hamkins},
    title = {Proof and the {Art} of {Mathematics}},
    publisher = {MIT Press},
    year = {2020},
    isbn = {978-0-262-53979-1},
    keywords = {book},
    url = {},

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From the Preface:

This is a mathematical coming-of-age book, for students on the cusp, who are maturing into mathematicians, aspiring to communicate mathematical truths to other mathematicians in the currency of mathematics, which is: proof. This is a book for students who are learning—perhaps for the first time in a serious way—how to write a mathematical proof. I hope to show how a mathematician makes an argument establishing a mathematical truth.

Proofs tell us not only that a mathematical statement is true, but also why it is true, and they communicate this truth. The best proofs give us insight into the nature of mathematical reality. They lead us to those sublime yet elusive Aha! moments, a joyous experience for any mathematician, occurring when a previously opaque, confounding issue becomes transparent and our mathematical gaze suddenly penetrates completely through it, grasping it all in one take. So let us learn together how to write proofs well, producing clear and correct mathematical arguments that logically establish their conclusions, with whatever insight and elegance we can muster. We shall do so in the context of the diverse mathematical topics that I have gathered together here in this book for the purpose.

6 thoughts on “Proof and the Art of Mathematics

    • The Habits are brief mathematical habits of mind or pieces of advice that I believe help guide one toward being an effective mathematician. Each chapter ends with a handful of these Habits, usually connected in some way with the methods or ideas that were considered in that chapter.

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