Mathematica

When you see things that others don’t yet perceive, sharing your vision requires finding a way to get others to re-create those things in their own heads. A mathematical definition serves this purpose. It provides detailed instructions allowing others, starting with things they are already able to see, to mentally construct those new things.

This is why I conceived of this book as a book of awakening and emancipation.

Fosbury wasn’t able to jump like others so he had to invent his own way of jumping. Thurston wasn’t able to see like others so he had to invent his own way of seeing. Fosbury’s approach allowed him to jump higher. Deliberately working to see better, as Thurston did, allows you to see further, more distinctly, down to the heart of things.

one that allows you to experience awe and then magic without getting burdened down with specialized techniques and language. In that respect, the discoveries of Georg Cantor (1845–1918) are perfect.

Real Magic Doesn’t Exist At the start of their careers, young mathematicians often feel like imposters. It’s a feeling I know well, and in my case it seemed entirely justified. The results contained in my PhD thesis were so obvious that it was almost like a trick. My theorems were always simple, and their proofs never contained any real difficultie

counterintuitive by nature. The things we’re told are “counterintuitive” or “paradoxical” are either false or poorly explained.

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 sleep

It’s surprising that Wittgenstein’s philosophy isn’t more well known outside of specialized circles, as it holds a very practical life lesson: we should accept going step by step, clarifying our language as we move forward, and being regularly surprised. It’s an effective antidote for paranoia.

If The Art of Fencing were published today, it might become an instant best seller, at least if the second part of the treatise lived up to its promise: “how you can always beat your opponent” if it’s a competition of “two people of the same size, same strength, and same weapons.” But here again, our modern perspective might make us completely miss

You know it’s true even though you don’t like being reminded of it. The quickest way to get on someone’s nerves is to make fun of their physique, but showing that their intuition is wrong is almost as good. In general, it provokes one of two defense mechanisms: people either say to themselves that they’re losers, develop an inferiority complex, and