The best books on thinking, creativity, and mathematics

William Byers Author Of How Mathematicians Think: Using Ambiguity, Contradiction, and Paradox to Create Mathematics
By William Byers

Who am I?

I'm a mathematician but an unusual one because I am interested in how mathematics is created and how it is learned. From an early age, I loved mathematics because of the beauty of its concepts and the precision of its organization and reasoning. When I started to do research I realized that things were not so simple. To create something new you had to suspend or go beyond your rational mind for a while. I realized that the learning and creating of math have non-logical features. This was my eureka moment. It turned the conventional wisdom (about what math is and how it is done) on its head.

I wrote...

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

By William Byers,

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

What is my book about?

If you love mathematics then this book will show you where the beauty and profundity that you love comes from. Most people mistakenly think that mathematics is nothing but logic, something like an AI program.  This book demonstrates that something very different is going on. Mathematics makes use of non-logical features like ambiguity, contradiction, and paradox. It is precisely these non-logical features that make math profound. The book demonstrates this with fascinating examples from all levels of math.

Profundity comes from being able to look at an idea from more than one point of view. Profound ideas often come from resolving situations of conflict, for example, zero resolves the conflict of having something that stands for nothing. Maybe I should have called the book, “mathematics beyond logic”.

The books I picked & why

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What Is Mathematics, Really?

By Reuben Hersh,

Book cover of What Is Mathematics, Really?

Why this book?

Reuben Hersh is responsible for a revolution in the way we look at mathematics. His main idea is very simple: mathematics is something that is created by human beings. Isn’t that obvious, you say? Not if you believe that mathematics is there even before life itself, that it is built into the nature of reality in some way. In philosophy, this view is called Platonism. Hersh had the radical but obvious idea that if we want to understand what mathematics is we should look at what mathematicians actually do when they create mathematics. Like all great ideas it can be stated very simply but the implications are enormous.  His ideas are what got me started writing my own books about math and science.

Proofs and Refutations

By Imre Lakatos,

Book cover of Proofs and Refutations

Why this book?

Lots of people have a priori ideas about what mathematics is all about but Lakatos had the brilliant idea of looking at what actually happened. His book is all about one famous theorem: “for all regular polyhedra, V – E + F =2, where V is the number of vertices, E is the number of edges, and F is the number of faces.  Think of a cube where V=8, E = 12, F = 6.  

We tend to think that mathematics proceeds from a well-defined hypothesis to conclusion. But that is only the finishing step. Along the way the definitions keep changing as do the hypotheses and even the conclusion. Everything is moving! This is what makes doing mathematics so exciting!

The Origin of Concepts

By Susan Carey,

Book cover of The Origin of Concepts

Why this book?

I’m interested in how mathematicians create mathematics but this book made me realize that learning mathematics is also a form of creativity. Each of us has created our understanding of mathematics as we were growing up. We are all creative!  

What is amazing about this book is that even children as young as six months possess rudimentary mathematical concepts, in particular, the concept of number. (Actually, Carey shows children have two distinct ways of thinking about numbers). The concept of number is built-in. That’s amazing to me! The mastery of counting numbers, 1,2,3,… is a great creative leap in the development of the child. This leap is followed by a series of further amazing accomplishments, for example, the insight that a fraction like 2/3, is a completely new kind of number (and not just a problem in division). How do kids manage to accomplish such radical changes in their concept of number? If we could answer this we might be able to say what creativity is.

The Philosophical Baby: What Children's Minds Tell Us about Truth, Love, and the Meaning of Life

By Alison Gopnik,

Book cover of The Philosophical Baby: What Children's Minds Tell Us about Truth, Love, and the Meaning of Life

Why this book?

This is another book about the new research into how babies think. I am excited about this research because of its implications for how people learn mathematics and how researchers create math. This book taught me something important about how we all think. Gopnik distinguishes between what she calls flashlight consciousness and lantern consciousness. Flashlight is the way adults think. You focus on one thing at a time and give it your full attention. But babies, she claims, use their minds differently. Their lantern consciousness is unfocused and is aware of the big picture all at once.  

So what happens to lantern consciousness when you grow up? The answer is that creative individuals use it and alternate between lantern and flashlight consciousness. When we are creating or learning something new, we have to drop back to lantern consciousness. Logic comes from flashlight consciousness and, by itself, will never produce anything original. So creativity in math and elsewhere comes from using your mind in a way that seems new but is actually the way we all thought when we were babies.

The Palliative Society: Pain Today

By Byung-Chul Han, Daniel Steuer (translator),

Book cover of The Palliative Society: Pain Today

Why this book?

It’s a little weird that this book should find a place on my list. It’s a book about how society has become resistant to anything that is difficult and painful and the kinds of people that we have become as a result. But mathematics is difficult! To understand mathematics you have to think hard, sometimes for a long time. Moreover understanding something hard is discontinuous, it requires a leap to a new way of thinking. You have to start with a problem and this problem might be an ambiguity or a contradiction. A is true and B is true but A and B seem to contradict one another. When you sort out this problem you will have learned something.

The moral here is to embrace things that are difficult if you want to learn significant new things. “No pain, no gain.” You don’t have to worry about some super AI program taking over the world and making you redundant. It takes human beings to work with situations that don’t seem to make sense, find a new way to think about the situation and, as a result, expand mathematical knowledge or personal understanding.

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