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...

Well, Jake, it is flattering that you put the philosophers on this site on the same level as the famous philosophers of the past! But I don't think we quite deserve that . And maybe, indeed, what separates us from those all-time greats is some quality of mind that we're never going to acquire, however hard we work -- something innate, if you like. But let's not worry about trying to emulate the great: let's take your question to be one about how to get modestly competent at philosophy (like your average philosophy professor here). The short answer is: by reading, and thinking, and discussing. A lot. It is no good reading without thinking hard about what you are reading, trying your best to understand and critically evaluate the arguments as you go. It is no good just thinking without reading, or you will almost inevitably just re-invent various tempting views that are now well known to be horribly problematic: you need all the help you can get to avoid the pitfalls. And even if you read and...

I can't agree that it sounds a cool class. Indeed, if this is going on in a school or college, it is an intellectual disgrace. "Teachings" of any kind are only worth the arguments that support them: and refusal to engage in critical argument by their proponents deprives the teachings of any call on our respect. Sure, if this is an extra-curricular, non-school, class that you are finding culturally interesting or emotionally satisfying, carry on with it. Get what you can out of it. But whatever is going on doesn't sound like philosophy to me.