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How Mathematicians Think: Using Ambiguity, Contradiction, and Paradox to Create Mathematics

How Mathematicians Think: Using Ambiguity, Contradiction, and Paradox to Create Mathematics - William Byers This book is hard for me to rate. It is not perfect, but I'm going to go ahead and give it five stars, because I think Byers is onto something. His ultimate argument is that mathematics, at its heart, is a creative activity. I don't think that should be a radical thesis, but apparently it is.

What does Byers do? He undercuts the notion that math is purely logical, completely rational. He mines the history of mathematics for its great ideas and uses them as examples of how ambiguity, contradiction and paradox are used by great mathematicians to create new mathematics. Logic and proofs codify the ideas but do not capture the process. This is one aspect of his argument where my lack of knowledge was a shortcoming, perhaps. I think he was saying that the logical structure of the proofs do not tell the whole story. If it is not fully convincing, I'd like to hear from someone about that.

I am an elementary school teacher. In our lessons, many kindergarten and elementary school teachers teach math by giving the children experiences that will hopefully replicate the creative experience Byers describes. The children "construct" the knowledge from the ground up. To be honest, I am not so sure that many high school teachers aim for that effect. If they did, I believe more of us would have stuck with math. Byers stated at one point (p.363) that university teachers did not teach "constructively" either.

My advanced math knowledge is limited. At times, the writing seemed convoluted and/or redundant, especially when the topic is essentially philosophical.

I'd love to hear what math specialists and teachers of advanced math think of this book.