Once mathematics began dealing properly with infinite objects it was no longer about the reality, but rather about abstract ideas.
Our "natural" intuitions (i.e. those we have from a pre-mathematical education time) are often very wrong about the infinite, to list some examples:
- The rationals are countable;
- The real numbers are uncountable;
- There are uncountably many ways (up to isomorphism) to well-order a countable set;
- Hilbert's Grand Hotel.
The list itself is infinite. It gets even larger if you wish to consider it in early 1900's eyes where the axiom of choice were still researched thoroughly.
However mathematics no longer deals solely with describing the real world, it deals with deductions from assumptions. Once accepting that it seems that a lot of the problems with infinities dissipate, as they follow from definition.
There comes a new problem with foundations of mathematics, the independence of claims, in particular the set theoretical ones. How can a set be countable in one model and uncountable in another? Let me use, once again, my usual analogies from field theory.
Suppose $F$ is a field (of characteristics $0$ if you prefer). What is the size of $\{x\in F\mid\exists n\in\mathbb N^+: x^n=1\}$
In the rational numbers the answer is $2$, in the rational numbers adjoined by a complex unit root of order $3$ the answer is $4$; in the algebraic closure of the rationals the answer is countably infinite. In the complex numbers you don't increase the size of this set, but you find a lot more transcendental elements on the unit circle which you can't even describe so nicely.
Note that field theory cannot express in a single formula the notion of being a unit root; but it can express the notion of being a unit root of order, say, $72$ or less. This should give us enough examples ($\mathbb Q$ still has only two; different extensions have four, five, etc.) of a specific definable set which changes in size between the models.
Why does no one complain when they are told that "in this field there are more unit roots than in that field"? My guess is that we are being educated to accept that "all numbers live in $\mathbb C$", so some are rationals, some are algebraic, etc. and thus different fields would have different amount of unit roots.
But set theory deals with sets, is this a surprise that different models of set theory would have different sets and if we pass from one model to a smaller model we may lose some of the information? No. If you study some axiomatic set theory you find out it's not surprising at all. It's what you'd expect, much like the way you may lose some unit roots in passing to a smaller field.
Now you are probably thinking, "he must be cheating me somewhere, because I feel completely fine with the unit root example, but it's impossible for sets to be countable here and uncountable there!". Well, sticking to first-order logic, you have to ask yourself what is the language that you use to describe the axioms and the model. In field theory you essentially describe the operations and the polynomials which have a solution in the field. In set theory you only have $\in$, but you describe a more complicated creature.
Is it a surprise that we have computers and an amoeba have only one cell? No, we are a far more complicated creature. Set theory is far more complicated, as a theory, than field theory. It should not be a surprising understanding that some of the things it can say about objects in the universe are more complicated. Since those are complicated it often seems that there should be some "canonical answer", but so far there is none. Whether it is good or bad, I can't tell. I hope there won't be a canonical answer because I enjoy the plethora of models, much like (I suppose) people studying measure theory enjoy the plethora of measures and spaces attached to those.
I will finish with one last point, Skolem tried to show in his paradox not that there is an inherent problem with set theory describing the world but rather that there is an inherent problem with using first-order logic to describe set theory. As it happens to be, he actually made clear the distinction between "internal" and "external" points of view in logic.
Yes, and no.
It helps because you will see proofs, you will see careful statements and you will learn, even if not directly more examples that can be used later on to study mathematical logic better.
But on the other hand, rarely anyone mentions formal logic in a course about analysis. You don't think about the inference rules, or what sort of statement you're writing, and so on. This might come up in set theory, or in model theory courses, but even then, you might be surprised how little thought we give these processes.
I am teaching a course about naive set theory (well, I'm the TA, but I'm giving a minicourse about extended topics under the guise of an exercise session). In the first class I wrote that if a set is not empty we can pick an arbitrary element of that set.
I've used existential instantiation, to move from $\exists x(x\in X)$ to an actual element $x$ from $X$; of course I didn't mention this. I hoped that they will let it slide, and they did.
But sure enough, when we discussed the axiom of choice, some six weeks later, they asked why can we choose from one non-empty set? How can we be sure that this is doable? And I explained that we have been doing that since the first class. The reason is the rules of logic, which we didn't mention to them -- because it's a naive set theory course, and these rules are applied naively.
To sum up, yes, taking math courses can help (at least those that deal with proofs) and it is very important to learn some basics as well, algebraic structures make excellent examples for theories and models in logic; but don't expect any mention of logic in those courses. You still have to learn it properly on its own.
Best Answer
It depends which areas of the philosophy of mathematics you want to study. We can usefully divide the field into (A) very general Big Picture questions about how mathematics fits into our general views about the world and our knowledge of it; and then there are (B) more specific questions that arise from reflecting on some of the details of mathematical practice. You don't need a lot of mathematical knowledge to tackle the first sort of question (though you need a lot of other philosophical knowledge); you need more, perhaps a great deal more, mathematical knowledge (but less general philosophy) to tackle the second sort of question.
Let me spell that out a bit (with the preliminary remark that I wouldn't want to say that there is a really sharp division here: still, I suggest that it is a very useful first approximation to think in terms of there being two different sorts of question here.)
(A) There’s a lovely quote from the great philosopher Wilfrid Sellars that many modern philosophers in the Anglo-American tradition [apologies to those Down Under and in Scandinavia ...] would also take as their motto:
Concerning mathematics, then, we might wonder: how do the abstract entities that maths seems to talk about fit into our predominantly naturalistic world view (in which empirical science, in the end, gets to call the shots about what is real and what is not)? How do we get to know about these supposed abstract entities (gathering knowledge seems normally to involve some sort of causal interactions with the things we are trying to find out about, but we can’t get a causal grip on the abstract entities of mathematics)? Hmmmm: what maths is about and how we get to know about it — or if you prefer than in Greek, the ontology and epistemology of maths — seems very puzzlingly disconnected from the world, and from our cognitive capacities in getting a grip on the world, as revealed by our best going science. And yet, … And yet maths is intrinsically bound up with, seems to be positively indispensable to, our best going science. That’s odd! How is it that enquiry into the abstract realms of mathematics gets to be empirically so damned useful? A puzzle that prompted the physicist Eugene Wigner to write a famous paper called “The Unreasonable Effectiveness of Mathematics in the Natural Sciences”.
Well, perhaps it’s the very idea of mathematics describing an abstract realm sharply marked off from the rest of the universe — roughly, Platonism (for a short-hand label) — that gets us into trouble. But in that case, what else is mathematics about? Structures in some sense (where structures can be exemplified in the non-mathematical world too, which is how maths gets applied)? — so, ahah!, maybe we should go for some kind of Structuralism about maths? But then, on second thoughts, what are structures if not very abstract entities, after all? Hmmmm. Maybe mathematics is really best thought of as not being about anything “out there” at all, and we should go for some kind of sophisticated version of Formalism -- perhaps it is all just symbol shuffling, that doesn't have to hook up to some abstract Platonic reality).
And so we get swept away into esoteric philosophical fights, as the big Isms slug it out. Well, I caricature of course! -- but the key point is that you don't need to know a great deal of advanced maths to follow these fights, they arise from quite elementary reflections on the school-room beginnings of maths and on the applications of elementary mathematics. But to get anywhere, you do need to be able to follow arguments in general metaphysics and epistemology, i.e. follow arguments about what there is and about how we can know about it.
(B) However, philosophers of mathematics talk about much more than this Big Picture stuff. To be sure, the beginning undergraduate curriculum in the philosophy of mathematics tends to concentrate in that region: e.g. for an excellent textbook see Stewart Shapiro’s very readable Thinking about Mathematics (OUP, 2000). But the philosophers also worry about more specific questions like this: Have we any reason to suppose that the Continuum Hypothesis has a determinate truth-value? How do we decide on new axioms for set theory as we beef up ZFC trying to decide the likes of the Continuum Hypothesis? Anyway, what’s so great about ZFC as against other set theories (does it have a privileged motivation)? In what sense if any does set theory serve as a foundation for mathematics? Is there some sense in which topos theory, say, is a rival foundation? What kind of explanations/insights do very abstract theories like category theory give us? What makes for an explanatory proof in mathematics anyway? Is the phenomenon of mathematical depth just in the eye of the beholder, or is there something objective there? What are we to make of the reverse mathematics project (which shows that applicable mathematics can be founded in a very weak system of so-called predicative second-order arithmetic)? Must every genuine proof be formalisable, and if so, using what grade of logical apparatus? Are there irreducibly diagrammatic proofs? …
That's only the beginnings of a list which could go on. And on. But the point is already made. These questions, standing-back-a-bit and reflecting on our mathematical practice, can still reasonably enough be called philosophical questions (even if they don’t quite fit Sellars’s motto). They are more local than the Big Picture questions which arise from looking over our shoulders and comparing mathematics with some other form of enquiry and wondering how they fit together. These are questions are internal to the mathematical enterprise, discussed by mathematically informed philosophers, as well as by philosophically minded mathematicians -- and you do need varying amounts of serious mathematics to tackle them. You can't, for an obvious example, discuss the foundational significance of category theory if you know no category theory.
So, in summary, you don't need a lot of mathematics to follow debates on some central Big Picture ontological and epistemological questions in the philosophy of mathematics. But other areas of the philosophy of mathematics focus in on specific areas of mathematical practice, and then you do have to know quite a bit of maths to follow them.