Joel David Hamkins: preference

Transcript context

…Maybe to take a little bit of a tangent, can you speak… You’ve written a wonderful book about proofs and the art of mathematics. So what can you say about proving stuff in mathematics? What is the process of proof? What are the tools? What is the art? What is the science of proving things in mathematics? This is something that I find so wonderful to teach young mathematicians who are learning how to become mathematicians and learning about proof, and I wrote that book when I was teaching such a proof-writing class in New York. Many universities have such a course, the proof-writing course, which is usually taken by students who have learned some mathematics. Usually, they’ve completed maybe the calculus sequence and are making the kind of transition to higher mathematics, which tends to involve much more proof, and it’s a kind of challenging step for them. So many math departments have this kind of course on proof-writing where the students would get exposed to how to write proofs. I wasn’t happy with most of the other books that exist for those kind of courses, and the reason was that they were so often so dull because they would concentrate on these totally uninteresting parts of what it’s like to write a proof, these kind of mechanistic procedures about how to write a proof. You know, if you’re going to prove an implication, then you assume the hypothesis and argue for the conclusion, and so on. All of that is true and fine, and that’s good to know, except if that’s all that you’re saying about the nature of proof, then I don’t think you’re really learning very much. So I felt that it was possible to have a much better kind of book, one that was much more interesting and that had interesting theorems in it that still admitted elementary proof. So I wrote this book and tried to fill it with all of the compelling mathematical statements with very elementary proofs that exhibited lots of different proof styles in it. So, I found that the students appreciated it a lot. We should say, “We dedicate the book to my students, may all their theorems be true, proved by elegant arguments that flow effortlessly from hypothesis to conclusion while revealing fantastical mathematical beauty.” Is there some interesting proofs that maybe illustrate, for people outside of mathematics or for people who just take math classes……

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