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

3.73  ·  Rating details ·  77 Ratings  ·  14 Reviews

To many outsiders, mathematicians appear to think like computers, grimly grinding away with a strict formal logic and moving methodically--even algorithmically--from one black-and-white deduction to another. Yet mathematicians often describe their most important breakthroughs as creative, intuitive responses to ambiguity, contradiction, and paradox. A unique examination of

Hardcover, 424 pages
Published May 27th 2007 by Princeton University Press (first published January 1st 2007)
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Jonathan Peto
Dec 27, 2010 rated it it was amazing  ·  review of another edition
Shelves: nonfiction, math
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
Greg Linster
Oct 27, 2011 rated it it was amazing
Perhaps, like me, you've wondered about what mathematicians really do. Describing what mathematicians do, and also how they think about what they do, is precisely the subject of William Byers excellent book How Mathematicians Think . In short, this book helped me wrap my mind around what mathematics really is.

I have always had a love/hate relationship with mathematics. Throughout my formal education, I found math to be intimidating, especially in my undergraduate and graduate studies. After rea
J Scott Shipman
Nov 14, 2011 rated it liked it
I read this title originally in 2007, and at the time enjoyed Byers unique treatment of ambiguity. From his work, I concluded that there was good ambiguity (where you "know" there is a problem and have a hunch at a solution) and bad ambiguity (where cluelessness prevails). Byers treatment of contradiction, paradox, and patterns went largely over my head. At the time it was an ok forward to this year. Last month I finished Howard Margolis' Patterns, Thinking, and Cognition A Theory o ...more
Prithvi Shams
Sep 23, 2015 rated it liked it
Mathematics is often seen as a rigid system where all "true" statements are generated from a priori assumptions via a series of logical operations. In this "formalist" view of mathematics, ambiguity and contradiction are anathema. The writer argues to the contrary that ambiguity and contradictions are an essential part of doing mathematics, and intuition more often than not drives mathematical innovation. To the question "Will computers ever be able to replace mathematicians?", this book makes a ...more
Nov 20, 2013 rated it really liked it
Shelves: philosophy
I expected, from the title, a book about the working habits of mathematicians with biographical sketches, but I was surprised to discover a solid work on the philosophy of mathematics instead. The math described is quite accessible so this would be a good book for the layman interested in questions about the limits of artificial intelligence, the relationship between mathematics and objective truth, et cetera.
Nov 08, 2008 rated it it was ok
I gave up on this book after about 200 pages. It was too philosophical for me. The only thing I got out of the first 200 pages was the quote: "To grapple with infinity is one of the bravest and extraordinary endeavors that human beings have ever undertaken". I can't convince myself that's grammatically correct, but it's cool anyway, and who am I to judge?
Manish Katyal
Jan 26, 2017 rated it it was amazing
A combination of philosophy and hands-on mathematics where you walk through interesting proofs [cantor set theory, infinity, zenos paradox, greek approach to math, etc.]. Very interesting book.
Nov 21, 2015 rated it it was amazing
This was an eye-opener. Mathematics is actually a natural science, but driven by brain's capability to handle numbers. Hypotheses come from intuition, results from proof. Outside of brain these are environment and experiments.

Great stuff, recommended for anyone interested in mathematics, its differences in science from other branches, human logic and soul beneath it, too.
Kung Ying xiang
Sep 23, 2014 rated it really liked it
Since this is one of the first few books I have read on mathematics in a long time, I think it does provide exposure into the philosophy of Mathematics, and how mathematical ideas are created. However, it does tend to get draggy or repetitive at some points
Sue Wilson
Nov 20, 2012 rated it it was amazing
I am really enjoying how this book is making me think about mathematics and the teaching of math. It would be a great addition to a math education course.
Sep 24, 2012 rated it it was ok
Shelves: mathematics
Uses more words than necessary to explain his ideas. I kind of understand what he's trying to say but not really. I'm sure there is a more eloquent way to convey his ideas.
Chris Brownell
May 29, 2012 rated it it was amazing
Great read! I highly recommend it to all my friends.
Kris Langman
Feb 10, 2012 rated it really liked it
Great book on the creative side of mathematics.
Paul Rowe
Feb 21, 2016 rated it really liked it  ·  review of another edition
I really wanted to give this book 5 stars. It has so many important insights that I will revisit for years. However, it could be about 40% shorter. It really needed a better editor.
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