Suddenly, the fog lifts . . .

Andrew Wiles, prover of Wiles’ Theorem (aka Fermat’s Last Theorem), on the doing of mathematics:

Perhaps I could best describe my experience of doing mathematics in terms of entering a dark mansion. One goes into the first room, and it’s dark, completely dark. One stumbles around bumping into the furniture, and gradually, you learn where each piece of furniture is, and finally, after six months or so, you find the light switch. You turn it on, and suddenly, it’s all illuminated. You can see exactly where you were.

This describes my experience, over shorter time-frames, in studying pure mathematics as an undergraduate, with each new topic covered: epsilon-delta arguments in analysis; point-set topology; axiomatic set theory; functional analysis; measure theory; group theory; algebraic topology; category theory; statistical decision theory; integral geometry; etc.    A very similar process happens in learning a new language, whether a natural (human) language or a programming language.     Likewise, similar words describe the experience of entering a new organization (either as an employee or as a management consultant), and trying to understand how the organization works, who has the real power, what are the social relationships and dynamics within the organization, etc, something I have previously described here.
One encounters a new discipline or social organization, one studies it and thinks about it from as many angles and perspectives as one can, and eventually, if one is clever and diligent, or just lucky, a light goes on and all is illuminated.    Like visiting a new city and learning its layout by walking through it, frequently getting lost and finding one’s way again,  enlightenment requires work.  Over time, one learns not to be afraid in encountering a new subject, but rather to relish the state of inchoateness and confusion in the period between starting and enlightenment.  The pleasure and wonder of the enlightenment is so great, that it all the prior pain is forgotten.
Andrew Wiles [1996],  speaking in Fermat’s Last Theorem, a BBC documentary by S. Singh and John Lynch: Horizon, BBC 1996,  cited in Frans Oort [2011 ]:  Did earlier thoughts inspire Grothendieck? (Hat tip).

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