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...

How can I achieve the level of philosophical know-how and thinking ability that the philosophers of this site and the famous philosophers throughout history have had? Must I simply read many works of philosophers of the past, does the philosophical mind come with becoming more educated, or is there something else - a thought process or state of mind I must develop? Or is it an innate quality that people get at birth? I am eager to know because I have been reading much philosophical work lately and I very strongly desire to be a philosopher. and I want to write a book. Thanks, Jake - 15 yr. old

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...

What is the difference between philosophy and religion? I am attending a weekly "Philosophy Class" but the group does not study any of the Philosophers or their works nor do we "philosophize", i.e. pick holes in philosophical statements, etc. We are read lines from the Upanishads and from the "teachings" of an Indian teacher and we are expected to accept these "teachings" as fact. The "teachings" include stories such as that of a woman had reached a high level on the path to realisation (after many re-incarnations of living spiritual lives) and had reached a state where fear was eliminated and she lived a life of bliss. Surely it would not be practical or possible to live a life where fear is eliminated. Is the emotion of fear not essential for survival? Also is life not a series of experiences of pain and pleasure? In any case if we question the teacher we are given his explanation of the story which we are expected to accept. Is this a Religion Class as opposed to a Philosophy class? We meditate...

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.

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