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

by William Byers

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 this less-familiar aspect of mathematics, How Mathematicians Think reveals that mathematics is a profoundly creative activity and not just a body of formalized rules and results.


Nonlogical qualities, William Byers shows, play an essential role in mathematics. Ambiguities, contradictions, and paradoxes can arise when ideas developed in different contexts come into contact. Uncertainties and conflicts do not impede but rather spur the development of mathematics. Creativity often means bringing apparently incompatible perspectives together as complementary aspects of a new, more subtle theory. The secret of mathematics is not to be found only in its logical structure.


The creative dimensions of mathematical work have great implications for our notions of mathematical and scientific truth, and How Mathematicians Think provides a novel approach to many fundamental questions. Is mathematics objectively true? Is it discovered or invented? And is there such a thing as a "final" scientific theory?


Ultimately, How Mathematicians Think shows that the nature of mathematical thinking can teach us a great deal about the human condition itself.

Genres: Mathematics, Philosophy, Science, Nonfiction

424 pages, Hardcover

First published January 1, 2007

Reviews

[Jonathan Peto · Rating 5 out of 5]

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.

[Prithvi Shams · Rating 3 out of 5]

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 an elaborate negative response, by emphasizing the human dimension of mathematics.

[J Scott · Rating 3 out of 5]

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 read....fast forward to this year. Last month I finished Howard Margolis' Patterns, Thinking, and Cognition A Theory of Judgment. While writing, I grabbed Byers from the shelf---and before I knew, four hours passed and I'd reread most of the book. Some math specialists have panned the book as simplistic, but I'm not a math specialist----I found Byers' insights have profound analogous impacts in areas other than math---like decision making and cognition. This is a very good book for the math review---and there is a fair amount---but viewed in the macro, Byers offers another perspective, another view into how we reason. Highly recommended.

[Julian · Rating 4 out of 5]

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.

[Antti · Rating 5 out of 5]

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.

[r · Rating 4 out of 5]

"Mathematization involves more than just the practical uses of arithmetic, geometry, statistics, and so on. It involves what can only be called culture, a way of looking at the world.
[...]
In this regard let us take note of what the famous musician, Leonard Bernstein, had to say: 'ambiguity... is one of art's most potent aesthetic functions. The more ambiguous, the more expressive.' His words apply not only to music and art, but surprisingly also to science and mathematics. In mathematics, we could amend his remarks by saying , 'the more ambiguous, the more potentially original and creative.'"

[Nick · Rating 2 out of 5]

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?

[Bob · Rating 5 out of 5]

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.

[Fernando Pestana da Costa · Rating 4 out of 5]

Here we have one of those important books we should approach with some care and about which I am somewhat divided (or, should I say, ambiguous...). The main argument of the book, as the subtitle clearly points out, is how do people (and, in particular, mathematicians) use "ambiguity, contradiction, and paradox to create mathematics". Written by an active research mathematician (hence, by someone how knowns what is he talking about!) every research mathematician will certainly recognize the truth of attributing to those non-logical elements a central role in the production of new mathematics, that is, in the creative aspects of the field, both when producing new results and when trying to understand some body of existing mathematics. The stress of the argument is, thus, in these non-logical components, and, although the author repeatedly points out the importance of the logical component for the overall mathematical enterprise, I am afraid the point will be lost by most readers without a solid mathematical education, since most of the main examples are rather sophisticated (about the level of first year undergraduates). This can have as a consequence that the reading of this book by people from the humanities, with no mathematical training but with a propensity for post-modernist thinking, can result in a misrepresentation of what is mathematics and how mathematicians work that could be more off the mark then, say, their (ab)use of Gödel's incompleteness theorem. In short: an excellent book that should be required reading for someone with an undergraduate mathematical education, but that should require the same mathematical education as a prerequisite to be read: a necessary and sufficient condition...

[Sreyas Sivaanand · Rating 5 out of 5]

Fascinating

[Garrett Ruley · Rating 5 out of 5]

How Mathematicians Think is so far my favorite philosophy of mathematics book, hands down. Byers tackles classic questions like What is Mathematics? and dives into discussions on famous mathematical problems, but does so from a skeptical and sociologically-inclined standpoint that humanizes mathematics rather than abstracting it.

Byers threads together two main arguments: one, that mathematics is created by humans when we encounter difficult problems or as-yet-unexplained phenomena, and two, that true mathematics retains the ambiguity and paradox of its birthplace and is not a sterile domain of pure logic and deduction. It's thus a flawed - albeit sometimes useful - but ultimately flawed - idea to try and remove all traces of contradiction and uncertainty from mathematics, because mathematics is fundamentally ambiguous and uncertain. Most importantly for me, Byers connects his argument to education, arguing - correctly in my opinion - that if you want to teach math effectively, you have to teach it in all its ambiguity and strangeness, and not pretend that it all "makes sense" or is a perfect realm of uncorrupted logical thought.

He also names his own theory of what mathematics is the "Rainbow Theory of Mathematical Truth", and although I'm 99% sure it isn't an intentional Pride reference, it's still a great theory name for this month.

Byers could perhaps write a little more concisely - a lot of ground is recovered in each chapter as Byers builds his argument step by step. I would also have been very interested for him to apply his ideas a little more to understanding the effects of mathematical thinking on society - he merely states that there is a connection. Overall, though, How Mathematicians Think is an electrifying and through-provoking read.

[Kung · Rating 4 out of 5]

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

[Kris Langman · Rating 4 out of 5]

Great book on the creative side of mathematics.

[Chris Brownell · Rating 5 out of 5]

Great read! I highly recommend it to all my friends.

[Anthony · Rating 2 out of 5]

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.

[Sue Wilson · Rating 5 out of 5]

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.

[Paul Rowe · Rating 4 out of 5]

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.