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What is the difference between mathematics and philosophy of mathematics?
Socrates’ Meno is the connection. There is an explanation of what is going on at Math Forum - Ask Dr. Math. The way I look at it, philosphers and scientists are searching for some kind of truth. Most of modern science would be incomprehensible to an ancient philospher, but the same is not necessarily true for quite advanced mathematics. For example, a ball (cannonball shot or tennis ball thrown) travels in a parabola. That it does so is due to gravity, which as a force accelerates bodies towards each other. This can be described mathematically just as the doubling of the square in Meno can, but is mathematically the result of integrating a constant (g) twice to get a position (height). Mathematics is also used to solve the Schrödinger equation, which describes how matter behaves on very small scales. This is probably why modern philosphers prefer to argue about simpler things than physics, but causes lovely headaches for non-philosophers. As you said, logic is the main area of overlap, but another connection is Bertrand Russell, who together with Whitehead had a go at some interesting mathematical logic but also was an excellent philosopher.
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There is a reason that one of the big arguments of the modern analytic school of philosophy is the “duality of meaning” — it all depends on how you look at it. What are mathematicians and scientists doing looking at these results and saying they aren't about meaning? It is a logical fallacy called the “uncontroversial argument”, which is used to argue in favor of a position that doesn't hold up to any scrutiny. But if you look at a really obvious example of that kind of argument, we can't make sense of it; it's simply unanswerable. But sometimes you need to use this kind of simple logic. For example, in general, a contradiction is a situation in which both premises are true, but they contradict each other. That implies that you should not believe the one you are considering, but you should believe both. So, why aren't.
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