Mathematica

“An attitude of curiosity that excludes all fear” Outside of mathematics, the level of certainty required by Descartes turns out to be impossible to attain for profound reasons that relate to how language and our brains function, and that have only recently been understood (we’ll come back to this later). That takes nothing away from the strength a

Our Incredible Intuition You’re the only person capable of seeing what’s in your head. Even if it’s painful, it’s only by making the effort of rigorously translating your vision into words and symbols that you can share it with others. And it’s also the only way to make sure that your intuition is right. Because sometimes your intuition is wrong.

At a profound level, math is the only successful attempt by humanity to speak with precision about things that we can’t point to with our fingers. This is one of the central themes of this book and we’ll come back to it a number of times. In a math book, the most important passages aren’t the theorems or the proofs: they are the definitions. Mathem

Yes, plasticity is a slow and silent mechanism that occurs without any real effort on our part, provided we’re exposed to the right mental images. Yes, we learn precisely when we force ourselves to imagine things that we don’t yet understand, which unfortunately is the same exact thing that most people run away from.

Grothendieck did it differently. He knew that it was worthless to gather information about things that you can’t yet see. Instead, he allowed himself to imagine the things right away, without waiting, even when he was well aware that it

Official math can be found in textbooks, where it is presented in a logical and structured manner, in an esoteric language that relies on indecipherable symbols. Secret math, also known as mathematical intuition, can be found in the heads of mathematicians. It consists of mental representations and abstract sensations, often visual, that are for th

Fear of mistakes and fear of the truth is one and the same thing. The person who fears being wrong is powerless to discover anything new. It’s when we fear making a mistake that the error which is inside of us becomes immovable as a rock.

Mathematical intuition is so banal, simple, and stupid, that you need a lot of self-confidence not to throw it in the trash. When you’re no longer a child, you have only one wish, and that’s to reduce your false intuitions to silence. That’s what almost happened to me when I thought I had to get rid of my stupid ideas and parasitical thoughts.

My basic technique hadn’t changed: lending an ear to the dissonance between my intuition and logic. This technique remained my instrument for exploring the world, just like Ben Underwood’s tongue clicks.

best by six inches. It was his first real success in a school competition. It was then that he knew he was on to something. But his coaches didn’t believe him. For years, they continued to try to convince him to jump the “right” way. Fosbury himself didn’t really have any counterarguments for them. He just said that maybe his technique wasn’t right