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

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...

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