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I am typically interested in finding out the interrelationships as regards knowledge per se and its applications. Typically, I see this interesting phenomenon of the difficulty of proving theorems in elementary Euclidean geometry, but its non-consideration in mainstream mathematics as non-useful in a larger perspective. What it shows is, according to me, that knowledge has regard in so far as it has some real-life applications, otherwise it is not so much valued in itself. What does this observation say about the stand which analytic philosophy have towards metaphysical philosophy?

  • A pragmatist might argue that the only genuine knowledge is know-how, i.e. "application" is all there is to it. there is no "knowledge per se", if by that you mean something with no practical "application". – user20153 Dec 31 '16 at 21:40
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I don't see your argument as compelling. Euclidian geometry is (arguably) no longer studied in mainstream mathematics, not because it's not-useful in "real-life", but because there are various fields which extend it and prove results far more general than simple euclidian geometry. It's not useful anymore because we can go beyond, not because it's useless in real life. Moreover, one could argue that Euclidian geometry is still studied, as the study of normed vector spaces etc. (but again, it's a generalization of said geometry) To show how real life applications don't matter, I can only point to the recent study of large cardinal and forcing axioms in mainstream set theory, or that of algebraization of logic, both of which are totally useless in real life (they're not even useful in computer science). So your observation isn't quite true.

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  • But where do you find such concepts as symmedians, ceva's theorem, menelaus theorem, brocard problem, butterfly theorem etc. in the generalizations and where do you have such synthetic power in mainstream mathematics. – vidyarthi Jan 1 '17 at 5:30
  • Simply in modern geometry. Geometry is still a large field of study, but now it has been generalized beyond the simple plane – Maxtimax Jan 14 '17 at 15:40

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