The story was circulating in early 2000s, so presumably it happened in 1990s. Kontsevich, it goes, opened a lecture course on mirror symmetry with:"This course is about two categories. One I will not define, the other nobody knows how to define, and we will be proving that they are equivalent".

I like the story because it highlights the informal side of mathematics, and counters the impression that what mathematicians do is logically deduce theorems from ready made definitions and axioms. But did it actually happen? There is no doubt that Kontsevich was giving lectures on mirror symmetry with categories in them, categorical mirror symmetry is his field after all. But did he say anything resembling the quote, or was it embellished like Eric Temple Bell embellished Cole's "talk without saying a word" about Mersenne numbers.

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    $\begingroup$ I don't know about that one, but I sure saw a talk by Takeshi Saito which started by "there are two possible definitions of the discriminant in this context, the first is well-defined and does not satisfy the desirable properties and the second has the dual properties." Exact date and location of the talk available upon request. $\endgroup$
    – Olivier
    May 19, 2015 at 17:56
  • $\begingroup$ Having never seen Kontsevich speak, I could not dream of answering your question, but I will note that this has the distinct whiff of truth. On the rare occasions that I see a mathematical physicist talk, I tend to hear something about how the Fukaya category has not really been defined. See some comments on the subject here mathoverflow.net/questions/2905/is-the-fukaya-category-defined $\endgroup$
    – stankewicz
    Nov 14, 2015 at 18:43
  • $\begingroup$ Was he talking about left and right? $\endgroup$ Jul 27, 2021 at 6:09
  • $\begingroup$ It is strange that the OP did not ask Kontsevich himself. His email address is easy to find. $\endgroup$
    – markvs
    Oct 10, 2021 at 0:32


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