Mathematics

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.

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First, I'm going to bristle. Logicians are mathematicians, even ifmost mathematics departments nowadays don't seem to want them. Work inlogic is often driven by profound philosophical concerns, and in thebest such work—Solomon Feferman's would be an example—mathematics andphilosophy are so intertwined that it would be pointless to try todisentangle them. But I'll stop bristling now and assume that thequestion concerned how foundational work might affect non-foundationalwork.

Onemajor question is to what extent incompleteness, of the sort thatinterested Gödel, is of serious (non-foundational) mathematicalconcern. It's been known for some time now that most of the centralresults of non-foundational mathematics can be proven in tiny fragmentsof Zermelo set theory, let alone Zermelo-Frankel set theory. Much workin set theory, on the other hand, concerns extensions of ZF that makethe first inaccessible cardinal—that being the smallest cardinal whoseexistence cannot be proved in ZFC (if it is consistent!)—lookminiscule. There are two sorts of reactions one might have to thisfact. One would be that set theory has come loose from its mathematicalroots and, as a result, doesn't really supply a proper foundation fornon-foundational mathematics. Feferman seems to have a view along theselines.

A different options has been pursued by Harvey Friedman, who argues that there arequestions that are of non-foundational significance whose resolutionwould require powerful set-theoretic axioms. Friedman's method is todevelop a mathematical theory in the conventional way, proving a seriesof results, and then use that theory to motivate certain naturalconjectures that amount to generalizations of basic results. He thenproves that the conjectures imply strong set-theoretic axioms. Somepeople find Friedman's results more compelling than others do, but itis at least an interesting approach.

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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