Can philosophy of mathematics influence mathematics, or it is just an abstraction of what actually works?

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Three examples to think about. First, Frege's invention of the predicate calculus was driven by philosophical reflection on the nature of quantified propositions, and led in turn to modern mathematical logic. Second, the so-called Hilbert programme was driven in part by more philosophical reflection, this time on the limits of what we can directly "intuit" to be mathematically correct; that programme led in turn to the development of modern proof theory. Third, Kurt Gödel's philosophically driven work on set theory was mathematically hugely important. [Sorry, those reference links are inevitably to material that quickly gets mathematically heavy!]So, it surely is the case that specific philosophical ideas -- philosophical reflections on foundational matters -- have influenced the development of mathematics. And one might say too that a more general set of philosophical ideas about the proper nature of mathematics drove the whole Bourbaki project which has been so influential in the development of modern mathematics. On the other hand, it has to be said that a lot of the obsessions of contemporary philosophers of mathematics do leave working mathematicians stone cold! For example, a "hot topic" among the philosophers is the pros and cons of "fictionalism" about mathematics -- the idea that, strictly speaking, it isn't really true that 2 + 2 = 4 , any more than it is strictly speaking true that Sherlock Holmes lived at 222B Baker Street. There aren't really numbers sitting in some Platonic heaven, any more than there is a detective living in Baker Street. Strictly speaking, we should say that according to Conan Doyle's fiction, Sherlock Holmes lived at 222B Baker Street. Likewise, the argument goes, we should say that according to the mathematical fiction, 2 + 2 = 4. But try that on your local friendly number-theorists and they will glaze over, or think you are joking!

As Peter Smith's examples make clear, sometimes "the philosophy of mathematics" appears in other than philosophy journals and is done by other than people in philosophy departments. Another instance of this is the long current of constructivism in mathematics. The development by mathematicians of constructivist mathematics (most notably, intuitionistic mathematics) is often motivated by their -- for want of a better term -- philosophical reflection on the nature of mathematics.

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