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I still don't quite agree with the approach to mathematics as enumerating problems of a certain size, but to be honest I'm not prepared or motivated to keep this conversation going without turning to handwavy arguments based on my imperfect idea of what mathematics is and how it works. However, it's certainly given me something to think about, so it hasn't been completely pointless. Thank you for the discussion.

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No worries, and thank you as well.

By the way, if I'm not completely mistaken, Gödel's argument to show the incompleteness of mathematics relies on encoding all mathematical statements as numbers. So I'm certainly not being very original here.

FWIW I think the concept of "interesting" is so subjective, time-dependent, and nebulous, that to say there's "only a finite number of interesting" mathematics problems feels absurd to me. What's interesting in ten years time will depend on what other interesting things have been discovered in the intervening time. But you're right, I shouldn't have responded with a cheap joke.

Godel's incompleteness theorem technically relates to individual axiomatic theories (i.e. the set of facts that logically derive from a given set of axioms). The numerical encoding you refer to applies to logical statements within that theory. Arguably, the kind of mathematics that humans do is not constrained to a single theory, but is a more general form of reasoning that is often flexible about which axioms may or may not be assumed.

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