How Mathematicians Think (Book Discussion)

This discussion will explore the thesis that learning, research, and thinking in mathematics is suffused with ambiguity, the contradictory, and paradox. The thesis is presented in the 2007 book by William Byers "How Mathematicians Think: Using Ambiguity, Contradiction, and Paradox to Create Mathematics" (Princeton University Press's web page for Byers' book "How Mathematicians Think").

"If the modern world stands on a mathematical foundation, it behooves every thoughtful, educated person to attempt to gain some familiarity with the world of mathematics. Not only with some particular subject, but with the culture of mathematics, with the manner in which mathematicians think and the manner in which they see this world of their own creation."
--- William Byers

I led a discussion on this topic for the Greater Philadelphia Thinking Society last November (http://www.meetup.com/thinkingsociety/events/80974562/). I found the thesis so interesting, I wanted to discuss it again with a more mathematical audience.  No mathematical background nor familiarity with Byers' book is required.  But it would be nice if you could at least read the introduction that the publisher provides on-line for free (see my comments below).

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  • Martin C.

    Good lively discussion

    1 · February 12, 2013

  • CJ F.

    Great discussion. Thanks for everyone's input.

    January 26, 2013

  • Martin C.

    Related to this discussion, I strongly recommend reading what has come to be known as Lockhart's Lament - http://www.maa.org/devlin/LockhartsLament.pdf It provides another view of what mathematics is, with recommendations as to how it should be taught.

    January 21, 2013

    • CJ F.

      Marty, I'm looking forward to seeing you again on Saturday!

      Lockhart is sure strident and satirical in that piece. I didn't have the patience to finish reading his rant. I like his math is art argument. But he rants excessively (to my mind) about the math curriculum being appallingly bad. Saturday, with Byers' book, we will surely talk about math as an art. Byers thinks mathematics is a creative art. But his book does not address the shortcomings of our math ed system. So that subject would be somewhat off-topic.

      No need to finish the book, but as I said below chapters 1 and 2 are most important followed by 7 then 3 and 4. I really enjoyed Byers' book, I hope everyone else does too.

      January 21, 2013

    • CJ F.

      Marty, it was great seeing you again. I had these notes about a book and YouTube video on symmetry. Maybe we should discuss this at a future meetup? [I have not read the book nor watched the video. But it is on my TODO list.]

      Symmetry: A Journey into the Patterns of Nature by Marcus Du Sautoy
      Review: http://www.ams.org/no...­
      Video: Marcus du Sautoy - OUMNH 150th Anniversary Lectures
      Description: In this fascinating lecture, Professor Marcus du Sautoy talks about his quest to discover Symmetry in Mathematics, and in Life.
      http://www.youtube.co...­
      Video length: 51 minutes

      January 26, 2013

  • CJ F.

    Byers cites a wonderful piece written by William P. Thurston, one of the greatest modern geometers who recently died. Thurston's April 1994 essay in the Bulletin of the AMS discusses his personal experience working as a mathematician. The essay reveals deep truths about how mathematicians really think: http://www.ams.org/journals/bull/1994-30-02/S0273-0979-1994-00502-6/S0273-0979-1994-00502-6.pdf

    This essay was so good and so powerful, I plan to re-read it before Saturday.

    January 22, 2013

    • CJ F.

      I finished re-reading this article. It is fantastic. Thurston describes in 1994 how he refrained from proving theorems so that he could encourage the development of the mathematics around his "geometrization conjecture" of 1982 which led in 2003 to Grigori Perelman's "proof" and with that the solution to the Poincaré conjecture. It is a telling and very modern story of how ideas are more important than proofs in mathematics. Of course, both Byers and Thurston argue that proofs are very important. But mathematics is slowed by emphasizing proof over idea.

      January 25, 2013

  • Martin C.

    I am afraid I don't share the general enthusiam for the book. I found this MAA review that pretty well summarizes how I feel about it. http://mathdl.maa.org/mathDL/19/?pa=reviews&sa=viewBook&bookId=67881 The term ambiguity is used in so many ways that it can cover practically anything.

    January 23, 2013

    • CJ F.

      Thanks, I had not seen that review. I agree that there is some sloppy copyediting and I thought some of the redundancy was too much myself. I would have preferred a "tighter" argument.However, I thought the definition of ambiguity in chapter 1 was clear. In addition, I thought he transcended the postmodernist angst by accommodating logic and all of mathematics into his creative, curious, and dynamic vision of mathematics. I think Byers puts logic on an equal footing with ambiguity avoiding the postmodernist extremes.

      January 23, 2013

  • CJ F.

    If you don't have time by next Saturday to read William Byers' book "How Mathematicians Think: Using Ambiguity, Contradiction, and Paradox to Create Mathematics", it is OK (we will summarize the arguments during the discussion). The Introduction (the first 19 pages) of the book are on-line from the publisher at http://press.princeton.edu/chapters/i8386.html

    Also in PDF format: http://press.princeton.edu/chapters/i8386.pdf

    January 19, 2013

  • CJ F.

    Reuben Hersh's review of William Byers book is excellent: http://www.ams.org/notices/200711/tx071101496p.pdf

    Hersh concludes: "This book strikes me as profound, unpretentious, and courageous." I agree! I'm looking forward to discussing it on Sat Jan 26th (in a week and a half).

    January 16, 2013

  • January 3, 2013

  • CJ Fearnley removed the location. It was Capriccio Cafe & Espresso Bar

    January 3, 2013

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