One of my favorite books, one of the few that it would break my heart to give away, is Categories for the Working Mathematician by Saunders Mac Lane. Every time I come across it, I smile to myself. This book, first published in 1971, remains a reference in category theory, the revolutionary way of seeing and thinking about mathematical structures

It’s through imitation that we learn how to tie our shoes, use a toaster, ride a bike. We may not get it right on the first try, but watching others do it gives us an idea how. We know more or less what a shoelace or toaster or bike is for, and we know more or less how to use them. But math, because it relies on unseen actions, can’t be learned

Thurston’s response offers a radical change of perspective: The product of mathematics is clarity and understanding. Not theorems, by themselves. The world does not suffer from an oversupply of clarity and understanding (to put it mildly). The real satisfaction from mathematics is in learning from others and sharing with others. All of us have

I was fortunate enough to be able to rely on the writings of Descartes, Grothendieck, and Thurston.

an adult, I’ve developed my own way of making use of the special state of mind just before falling asleep. Rather than focusing on subjects that preoccupy me, I’ve learned to simply let myself be filled with them. The nuance is subtle but fundamental. Focusing is thinking intensely, in search of solutions. It never works and it keeps you from

“My candour will be appreciated by everyone” What the Rules didn’t make explicit, however, was that Descartes was confronted with a major problem. This problem is central to his philosophy, and yet he was never able to resolve it: he had tried this method himself, he knew that it worked, but he could never explain why. It’s a problem that all

Bill Thurston It’s not only math books. There are other books that no one ever reads. Have you ever read the user manual for your toaster?

Mathematicians are human beings. They can understand mathematical objects only in a perceptual manner, via false human interpretations, approximations, translations from mathematical vocabulary into human language.

My intuition has the mental age of a two-year-old—it has no inhibitions and always wants to learn. If you stop mistreating your own, you’ll see that it’s exactly like mine, only asking to be allowed to grow.