Peter Scholze on definitions

The following quote is taken from a recent post of Michael Harris, where he quotes Scholze directly (with permission)

“What I care most about are definitions. For one thing, humans describe mathematics through language, and, as always, we need sharp words in order to articulate our ideas clearly. (For example, for a long time, I had some idea of the concept of diamonds. But only when I came up with a good name could I really start to think about it, let alone communicate it to others. Finding the name took several months (or even a year?). Then it took another two or three years to finally write down the correct definition (among many close variants). The essential difficulty in writing “Etale cohomology of diamonds” was (by far) not giving the proofs, but finding the definitions.) But even beyond mere language, we perceive mathematical nature through the lenses given by definitions, and it is critical that the definitions put the essential points into focus.

Unfortunately, it is impossible to find the right definitions by pure thought; one needs to detect the correct problems where progress will require the isolation of a new key concept.”

Harris was asked to write a draft for New Scientist about Scholze’s work, but they were after something far too low-brow in the end. I have only seen a couple of good articles in there about serious mathematical topics (one on Woodin’s “Ultimate L” was reasonable, but gave really no decent mathematical information about the topic) and wouldn’t have expected Harris, not an easy writer even for mathematicians, to be the best fit for their intended audience.

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