Mathematica

It’s not surprising if you couldn’t find the words to describe how you tie your shoes. Most likely, you didn’t even try. Writing mathematics, that is, transcribing mental images with enough clarity and precision to allow others to understand and reproduce them, is an art.

would remain inexplicable. Nobody would know what to do with them, except the few folks who’d have received the “gift of vacuuming” (that is, they’d have accidently come up with a proper way to use them). The hidden sense of vacuum cleaners isn’t found in the manual. It’s a secret we pass on by word of mouth. What goes for vacuum cleaners also goes

mathematical definition is neither a commentary nor an explication: it is the exact assembly guide of a new mental image and the “birth certificate” of the new word chosen to designate it. (In practice, existing words are often reused, receiving new meanings that may have no direct relation to what these words mean in everyday life.)

Every definition is an approximation. The meaning of words is always fluid, ambiguous, changing. Nothing is ever clear-cut. Inside our head, the world is abstract and vague.

The only way to get there is to go beyond words. Replacing “the sum of whole numbers from 1 to 100” with “1 + 2 + 3 + . . . + 98 + 99 + 100” is a good start. You might have the impression of seeing the sum in a more tangible and concrete way. But that’s only ever an illusion. In reality, you’ll be missing most of the numbers, those hidden by the el

In an interview with the New York Times, Thurston summed up things as follows: “People don’t understand how I can visualize in four or five dimensions. Five-dimensional shapes are hard to visualize—but it doesn’t mean you can’t think about them. Thinking is really the same as seeing.”

doing it without ever noticing. We start with human language, shift to mathematical language for reasoning, and return to human language. We do this each time we formulate hypotheses and try to draw conclusions from them. This day-to-day activity

The lesson is very much in line with Thurston’s remark that “The product of mathematics is clarity and understanding. Not theorems, by themselves.”

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