QC @QiaochuYuan 2023-03-17
maybe one of the biggest things i got out of doing math was seeing how precise it’s possible for metaphors to be. in math it’s possible to have completely precise metaphors (isomorphisms). so i hunt for unusually precise metaphors in other parts of life, and often find them
here’s the simplest nontrivial example of a completely precise metaphor (isomorphism): the way {even, odd} add is the same as the way the numbers {1, -1} multiply. the isomorphism is given by x → (-1)^x. there’s a single underlying structure of which these two are instantiations
much of my thinking one way or another is about “structure” and math is an excellent place to spend a lot of time working with and internalizing what sort of thing a “structure” is, esp. the ways in which they are *independent of the words one chooses to describe them*
all of this overlaps in a very interesting way with gendlin focusing. the real meat of thinking about mathematics involves murky shuffling around of felt senses and i don’t believe any mathematician has described how this works in any amount of detail
http://previous.focusing.org/gendlin/docs/gol\_2160.html…
was talking about this with @aphercotropist - e.g. when i think about lie groups i usually visualize myself rotating a sphere in my mind, i sort of use the felt sense of rotation to guide my reasoning. something about reasoning about a general thing using an exemplar
have you seen the biographical work of grothendieck? I feel like you would vibe with him well; he says “most of my colleagues don’t know how to feel”
he goes into some detail about the mechanics of seeing
not me literally having just been thinking about how inversion is isomorphic
some good stuff in http://math.tau.ac.il/~milman/files/I%20am%2070%20today.pdf… and Gromov’s ergobrain/ergosystems
I have a little insight here, gained from analysing my own ways of thinking. A critical part of the reasoning process is to distil things down to their fundamental essence. Get rid of all the superfluous information.