Basically, what exactly is Philosophy of Mathematics? Why not simply call it mathematics? At what point does a mathematician practice the philosophy of mathematics?
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The philosophy of mathematics is definitely not mathematics; one can learn a lot of mathematics without knowing any philosophy - consider it as a carpenter who learns how to use his tools; so there is an artisanal quality to it; and a good carpenter knows how to use his tools well.
Modern mathematics has many different disciplines but they're natural outgrowths of each other or related in an organic fashion; and the relations change in time; plus, of course there are relations with other subjects; hence one shouldn't think of it like a tree but Rhizomatically.
The philosophy of mathematics really grew out of the foundational crisis of mathematics in the early twentieth century when there was a large programme (pushed by Hilbert, but I imagine there was antecedents) to place the whole of mathematics in exactly a tree-like organisation; with its root in logic - mathematical logic; I suppose one could go further back and suggest it was also the criticisms by Bishop Berkeley (and no doubt others) of the lack of foundations in calculus; and then implicitly Newtonian Mechanics.
This programme is still continuing - with for example various efforts to place QFT, or the path integral on a amthematically precise bearing.
The main question of mathematical philosophy is the ontological status of mathematical objects - numbers, sets, algebras etc; hence the three main schools: intuitionism, platonism, and formalism; the first says they're human constructions, the second says they're objectively real; the third says anything goes so long as it has significance and we stay within the rules of logic.
note: sometimes the word philosophy in mathematics is used in a non-philosophical manner; for example the 'philosophy' of the Langlands programme; in this sense its better thought of strategy and tactics; in other words the mathematical landscape hasn't been fixed so one requires guidance of how to look at it, and how to navigate it.
For example the Langlands programme can be called non-abelian class field theory; which sounds impressively intimidating; but if you know that abelian just means it matters not one whit which order you do things.
For instance it doesn't matter if you put on your left shoe first, and then your right; or vice-versa; but it does matter when you unlock a door and turn the handle.
Then consider that abelian class field theory is an already well-established field; then one can see that the Langlands programme is adding 'a twist' to this theory; but why add 'twists'? Well, because Quantum Mechanics, considered mathematically, is non-abelian or 'twisty'.
answered Jul 31, 2015 at 15:34
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It's worth pointing out that while interest in the philosophy of mathematics has increased since the work of Frege, it's not as if pre-Fregeans ignored the philosophy of math altogether. Basically every major philosopher (and plenty of minor ones) since at least Plato (probably before him too - I'm less familiar with the pre-Socratics) has had something to say about the nature of mathematics. So it's not as if phil. math is a particularly new discipline, despite the sophistication that the last ~100 years has seen develop in the field. Commented Aug 1, 2015 at 0:34
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@possibleworld: the Pythagoreans come to mind; but their orientation is different from today. Commented Aug 1, 2015 at 11:39
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An interesting aside to this may prove worth researching. Sir Thomas Heath, considered to be the greatest Maths Historian in his collected work The History of Mathematics maintains that mathematics in its original state contained no numbers. It was not until the pre-Socratics that numbers became a feature. This has serious implications for any Philosophy of Mathematics which asks whether Maths was ;created' or discovered'. In 'To Discern Divinity' there is a 'mythological story which attempts to recreate the birth of Maths. Available at Academia.com and Amazon. Hope this is not too tangential– user37981Commented Mar 28, 2019 at 14:18
Typically (though not always), philosophies of X, where X is some discipline or domain of enquiry, ask questions about the epistemological, metaphysical, or conceptual foundations of X. These are distinct from questions investigated within X.
For instance, mathematicians study things like numbers and their properties, mathematical structures, sets, categories, and so on. These types of questions are distinct from questions about our knowledge of mathematical truths, or about the metaphysical status of mathematical objects. Philosophy of mathematics concerns itself with questions of the second kind. This also gives us a neat answer to your second question: we don't call philosophy of mathematics 'mathematics' because the two are distinct disciplines, concerned with different questions.
In principle a mathematician could successfully go about her career without considering questions of the second kind. She can produce interesting results without considering either the nature of her knowledge of mathematics, or the nature of the objects with which she works. None of this requires knowing of the conceptual/metaphysical/epistemological foundations of her discipline (though it couldn't hurt!). Similarly, a philosopher of mathematics could successfully do philosophy of mathematics without doing actual mathematics. This suggests an answer to your third question: a mathematician practices the philosophy of mathematics when she investigates philosophical, and not mathematical, problems.
