As an aside, it's amusing that this conversation is a re-statement of a main point in TFA:

> However, math problems are really there to solve a real world problem.

> That theory might be inspired by the real world, but the problem itself is purely theoretical.

From TFA:

> In my essay The Two Cultures of Mathematics a quarter of a century ago, and which can be summarized by saying that there is a spectrum of attitudes in mathematics to the relationship between problem-solving and conceptual understanding.

The author thinks this letter was choosing only one of them as the "right" approach whereas the better stance is "porque no los dos?"

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BalinKing an hour ago | parent [-]

I only just skimmed the referenced essay, but a priori I don’t think “problem-solving” as Gowers uses it has anything to do with real-world practicality. The problems under consideration are entirely theoretical, regardless of which “culture” a mathematician belongs to.

	▲	keeda 33 minutes ago \| parent [-]
		You're right, but I also may have quoted poorly to give the impression that the first post was only about real-world problems. It goes on to point those out as an infinite source of theoretical problems, which sounded to me like an emphasis on the problem-solving culture. The reply to that seems to say there are theoretical problems not necessarily connected to real-world problems, which I interpreted as an emphasis on the conceptual understanding aspect. I may have misinterpreted either or both of them though!