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How Mathematicians Think

This discussion will explore the surprising 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"). The discussion will involve only elementary mathematics such as the number zero, simple equations like 1+1=2, and irrational numbers. The most advanced mathematics discussed will be the paradoxes of infinity, but even here we will discuss it in simple terms that should be understandable by anyone regardless of their mathematical background.

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

Some questions that might be addressed during this meetup follow:

  • Is there really ambiguity in mathematics? How can that be, I thought mathematics was certain and definitive? Has mathematics lost the "holy grail" of certainty?
  • Is zero "the nothingness that is"? If so, why is zero such a powerful number?
  • What is irrational about irrational numbers? What is complex about complex numbers? What is imaginary about imaginary numbers?
  • Are equations inherently ambiguous?
  • Is mathematics the art of getting precise about ambiguous situations? Is that why it is difficult to learn?
  • Would it help students if the ambiguity, contradictory, and paradox in mathematics was discussed by teachers instead of the current style that emphasizes certainty and ignores the ambiguous?
  • Are great ideas built on deep paradoxes?
  • Does creativity in mathematics really get its source from ambiguity, contradiction, and paradox?
  • Is it unfortunate that mathematical proofs emphasize the logic instead of the ideas with all their ambiguities, contradictions and paradoxes in plain view?
  • Could mathematical writing be improved by embracing ambiguity, the contradictory and paradox?
  • What are the main philosophies of mathematics? How does ambiguity, contradiction, and paradox relate to each of these philosophies?

We are seeking volunteers to lead future Ben Franklin Thinking Society meetups. If you have a topic of interest, please send an e-mail to [masked] outlining your subject.

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  • CJ F.

    On the bike ride home, I was contemplating Hermann Weyl's quote "Mathematics is the science of the infinite". I said to myself (and to Jeannie): No, that seems too limiting. And then we laughed: how can the infinite be too limiting?

    Byers' thesis fascinates me: if ambiguity, the contradictory and paradox suffuse mathematical thinking, then these are primordial concepts for all thinking. In that case his book is a profound philosophical watershed. Great ideas may very well be related to just how subtly we can dance on the terra firma in and around concepts which are contradictory or paradoxical. As the group rightly expressed: it is a controversial thesis. But it may be true! Something to think about more.

    November 11, 2012

  • CJ F.

    One of Byers' key arguments is that ideas are more important in mathematics than its logic and its proofs. He illustrates this with a discussion on William P. Thurston, one of the greatest modern geometers. Thurston wrote a gorgeous, accessible response to a 1993 paper that criticized his work on the "geometrization theorem" by calling it an "unredeemed claim [which] became a roadblock rather than an inspiration." It should be noted that Thurston's (by all accounts) brilliant insights were finally proved in 2003 and led to the positive resolution of one of the most vexing problems in mathematics, the Poincaré conjecture. I read Thurston's wonderfully insightful paper and it strongly demonstrates the importance of ideas in mathematics work. Reading this 17 page paper will give a deep insight into the nature of "How Mathematicians Think":

    November 10, 2012

  • CJ F.

    Here is a quick outline of Byers' thesis on ambiguity. First he looks at the Oxford English Dictionary on ambiguity: "admitting more than one interpretation or explanation: having a double meaning or reference". Then he cites Arthur Koestler's 1964 book "The Act of Creation": "Ambiguity involves a single situation or idea that is perceived in two self-consistent but mutually incompatible frames of reference." Byers admits the definition is too static for such a dynamic idea, but you have to start somewhere.

    At the end of Chapter 1 he concludes in part: "The ambiguity is 'resolved' by the creation of a larger meaning that contains the original meanings and reduces to them in special cases. This process requires a creative act of understanding or insight. Thus ambiguity can be the doorway to understanding, the doorway to creativity."

    We will explore this argument in depth on Sunday with elementary mathematics examples to show ambiguity under the strictest logical control.

    November 9, 2012

  • CJ F.

    The Introduction (the first 19 pages) of William Byers' book "How Mathematicians Think: Using Ambiguity, Contradiction, and Paradox to Create Mathematics" is available on-line from the publisher:

    Also in PDF format:

    November 8, 2012

  • CJ F.

    Reuben Hersh's review of William Byers book, the focus of this Sunday's discussion, provides an excellent summary of the book (and so provides a good basis for our discussion):

    There are a few difficult sentences in the review (the AMS is a professional society for mathematicians); however, the review ends profoundly so I recommend skipping the one technical paragraph in the middle and a few other sentences so that you can read the whole review.

    November 7, 2012

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