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What are the major open questions of mathematical philosophy? Of these, which are mathematically significant, if any? By "mathematically significant," I mean "would affect the way mathematicians work." For example, the question of whether mathematics is created or discovered has no impact on working mathematicians. On the other hand, studies into the foundations of Math were certainly mathematically significant, and although one could argue that that was more Math than Phil, we can give Phil some credit. But that question is now closed, as far as mathematicians are concerned.

You write that "the question of whether mathematics is created or discovered has no impact on working mathematicians", but this doesn't seem so to me. If that question is a vivid way of asking whether intuitionistic logic or rather classical logic is correct, then the answer to the question has great consequences for how mathematicians work. For instance, if intuitionism captures the inferences that are really sound, then mathematicians will have to curtail the use of reductio ad absurdum arguments. (For more on this form of reasoning, see Question 121 .) Classical mathematicians do not hesitate to infer "P" from the derivation of a contradiction from the assumption "not-P". But intuitonists believe that this inference is not in general correct and so should be avoided. The German mathematician David Hilbert thought this had such great "impact" that it was like depriving the boxer of the use of his fists! (For some more on intuitionism, see Question 168 .)

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