You certainly don't need to be able to do original research in maths to be able to work on the philosophy of maths. But you will need to be able to follow whatever maths is particularly relevant to your philosophical interests. How much maths that is, which topics at which levels, will depend on your philosophical projects. For example, compare and contrast the following questions (not exactly a random sample -- they all happen to interest me!):

1. "Is our basic arithmetical knowledge in any sense grounded in intuition?" Evidently, you don't need any special mathematical knowledge to tackle that.
2. "Can a fictionalist about mathematics explain its applicability?" Again, I guess that acquaintance with the sort of high school mathematics that indeed gets applied is probably all you really need to know to discuss this too.
3. "Just what infinitary assumptions are we committed to if we accept applicable mathematics as true?" Here you do need to get more into the maths, and know quite a lot about what can be reconstructed in various weak subsystems of full analysis, and about what infinitary assumptions these subsystems need.
   
4. "Is there a unified justification for all the axioms of the standard set theory ZFC?" You better know a bit of set theory to tackle this -- but you perhaps needn't know much about e.g. large cardinal axioms that take us beyond ZFC!
5. "Do we need axioms that take us beyond ZFC? -- and if so, what?" For this, by contrast, you will need to know more about the fancier reaches of set theory.
6. "Does category theory in any sense provide 'foundations' for mathematics rivalling the set theoretic approach?" Well, plainly, you better know some category theory to tackle this one.
   
7. "Why is the question whether P = NP so hard?" And you'll need to know qutie an amount about complexity theory for this one.
   
8. "How satisfactory is Lakatos's model of the growth of mathematical knowledge?" Here, by contrast, you'd be better equipped if you know a little about a lot of mathematics, rather than a lot about a little, so your discussions aren't to suffer from an impoverished diet of examples.

If the term 'proposition' is used to mean a sentence -- a string of symbols, be they spoken, written, gesticulated or whatever -- then I suppose there could be uncountably many propositions if we allow there to be propositions of infinite length. In that case, one ought to be able to diagonalise on them just as one does with the infinite decimal expansions of the real numbers. But I think it's reasonable to stipulate that we're only going to countenance finitely long sentences. After all, they wouldn't be much use in communication, if you could literally never get a sentence out.

Alternatively, if the set of symbols is itself uncountable, then that will certainly lead to an uncountable infinity of strings of such symbols. But it seems reasonable to stipulate against that case too. Communication would once again be thwarted, because we don't seem to have the perceptual capacity to discriminate between uncountably many different symbols -- indeed, our discriminatory abilities probably only extend to a finite (though no doubt very large) number.

But 'proposition' has been used in other ways over the years. So what if we take it to mean not the string of symbols itself, but rather the non-linguistic thing that such a sentence is expressing: a thought, or something of that kind? Well, then maybe there could be uncountably many propositions: we'd need to have a great deal more spelt out about the details of this notion before we'd be in a position to assess that. But (leaving aside any eccentric conceptions of sentences and symbols like those just discussed) there could no longer be a one-to-one correspondence between propositions and sentences. Which would mean that sentences wouldn't do a very good job of expressing propositions -- which is, after all, what they're there for. For there to be uncountably many propositions, and yet only countably many sentences, a single sentence would need to correspond to more than one proposition. But then your interlocutor would have no way of judging which of these various propositions you were actually intending to communicate.

If I remember correctly, and I may well not, David Lewis explicitly argues that there are uncountably many propositions in Plurality of Worlds and uses this as an argument against any view that would try to reduce propositions to sentences. At the very least, he does consider this issue. So here's an argument that I think I remember from that book that we can consider, anyway. It is based upon the claim that, for any real number x, there ought to be a proposition---a possible content of thought---that I am shorter than x inches tall. Indeed, each such proposition could be expressed by a sentence. All we have to do is give the real number x a name, say, "Fred", and then the proposition will be expressed by the sentence "I am shorter than Fred inches tall". But if so, then there are at least as many propositions as there are reals.

The key to this argument, note, is the observation that the claim "For every proposition p, there could be a sentence S that expressed it" is much weaker than "For every proposition p, there is a sentence S that expresses it". It's the latter that the reductionist needs. But the argument needs only: If there could be a sentence S that meant that p, then there is a proposition that p---assuming that other, uncontroversial conditions required for the proposition to exist are met. E.g., there might have been propositions other than the ones there are, because there might have been people other than the ones there are, and then there would have been propositions about them that there in fact aren't. But that's not what's happening in this case.

That there should be uncountably many propositions for this kind of reason does not, so far as I can see, imply anything bad about the possibility of communicating. Each actual (meaningful) sentence corresponds to---that is, expresses---just one proposition (modulo worries about context). But there are lots of propositions that are not, in fact, expressed by any sentence.

My own preference, for what it is worth, is to refuse to talk of propositions, period. But that's a different matter. If we are going to talk of propositions, then it actually seems quite hard, for the reasons just given, to keep them countable.

Well, there is a logical truth in the vicinity of 1 + 1 = 2. Or perhaps better, a whole family of logical truths. Fix on a pair of properties F and G. Then it is a theorem of first-order logic that if exactly one thing is F and one thing is G and nothing is both F and G, then are exactly two things are either-F-or-G. Here the numerical quantifiers 'exactly one thing is' and 'exactly two things are' can be defined in standard ways from the ordinary first-order quantifiers and identity. And the theorem holds whatever pair of properties we choose. This elementary logical result probably captures what is driving your intuition that in some sense 1 + 1 = 2 is "reducible to formal logic". (For a bit more on this sort of thing, see my Intro to Formal Logic §33.3 -- or any other standard logic text!)

But all that is quite compatible with Gödel's first incompleteness theorem. For Gödel's theorem isn't about some limitation or incompleteness in our ability to prove (or disprove) simple equations relating particular numbers like 1 + 1 =2. Rather it is, roughly speaking, about the incompleteness of any theory rich enough to prove quantified claims about numbers. For any given rich enough axiomatized theory, there will some quantified truth about numbers it can't prove. So that means we can't wrap up all the truths of arithmetic (including all the quantified ones) into a single axiomatized theory . Hence, a fortiori we can't wrap them up into a single axiomatized theory that might be thought of as part of logic. (For a bit more on this sort of thing, see my Intro to Gödel's Theorems -- or any other standard text which covers incompleteness results!)

Well, mathematicians all the time talk about infinitary structures. To start with the very simplest examples, they talk about the set of all natural numbers, {0, 1, 2, 3, ...}: and they also talk about the set of all subsets of the natural numbers. And they introduce "infinite cardinal numbers" which indicate the size of such infinite sets. There's a beautiful theorem by Cantor which shows that, on a very natural understanding of "number of members", the set of all numbers has a smaller number of members than the set of all subsets of the set of all numbers. So there are different infinite cardinal numbers, which we can order in size. Indeed, again on natural assumptions, there's an infinity of them.

Is that little reminder about what mathematicians get up to enough to settle the question whether "infinity exists"? Well, perhaps not. There remain a number of questions here. Here's one (so to speak) about the pure mathematics, and one about applied mathematics.

First, then, we might say "Sure, mathematicians talk about infinite sets, and infinite numbers: but then Conan Doyle talked about Sherlock Holmes. But Holmes doesn't really exist: he's just in a story. And maybe the mathematicians are just telling a story, and there aren't really infinite sets and infinite numbers." But to get to grips with that suggestion, we'd have to tackle the whole question of whether "fictionalism" about mathematics is a viable position -- and that's far too much to take on here!

Second, even if we do allow the existence of infinite structures as entities populating the mathematician's abstract realm (denizens of "Plato's heaven"), we might still wonder whether such structures are exemplified in the physical world. We do use heavily infinitary mathematics in our best going physical theories (they presume, e.g., that space is infinitely divisible); but is that really essential? Could it be that, e.g., the world is ultimately grainy at the small scale, and is finite too at the big scale, and there are no physically existing infinite structures? Again, that's far too big a question to take on now.

So all I've been able to do here is make some preliminary clarifications, and suggest two different ways of sharpening up the original question into something we might get our teeth into. But then a lot of philosophy is like that!

The background issue here is: what's the best way of extending our talk of one set's being "larger" than another from the familiar case of finite sets to the infinite case?

Now, in the finite case, we can say that [L] the set A is larger than the set B if and only if there is a 1-1 mapping between B and some proper subset of A (i.e. there is an injection from B into A, which isn't onto). But equally, in the finite case, we can say that [C] the set A is larger than the set B if and only if there is a 1-1 mapping between B and some proper subset of A, but no 1-1 mapping between B and the whole of A.

In finite cases, [L] and [C] come to just the same: but in infinite cases they peel apart. [L] says the set of naturals is larger than the set of evens; [C] denies that. So isn't [L] the more intuitive principle to adopt?

No! For take the case where A and B are both the very same set, the natural numbers. There is a 1-1 mapping from the natural numbers into the natural numbers which isn't onto: take the mapping S: m --> m + 1. This correlates each number with its successor; and of course zero isn't in the image of the natural numbers under this mapping. So if we adopted that suggestion [L] about what it is for one set to be larger than another, we'd have to say that the set of natural numbers is larger than itself. That is an unwelcome implication to say the least! We've lost a key basic property of the larger than relation that makes it a strict order relation, namely its asymmetry.

Now you might think that we have a straight choice here: do we adopt [L] with its radically counterintuitive implications (e.g. the set of naturals is larger than itself), or [C] with its counterintuitive implications (the set of naturals is no larger than the set of evens)? But the situations aren't symmetric. For [C] preserves at least something of the idea of comparisons of size by keeping larger than an order relation: we get a workable concept of cardinal order. By contrast, [L] destroys too much of the idea of order-by-size to be any use.

Which is why [C] is the definition to adopt.

Here's a simple argument. (1) There are four prime numbers between 10 and 20. (2) But if there are four prime numbers between 10 and 20, then there certainly are prime numbers. (3) And if there are prime numbers, then there must indeed be numbers. Hence, from (1) to (3) we can conclude that (4) there are numbers. Hence (5) numbers exist.

Where could that simple argument be challenged? (2) and (3) look compelling, and the inference from (1), (2) and (3) to (4) is evidently valid.

So that leaves two possibilities. We can challenge the argument at the very end, and try to resist the move from (4) to (5), saying that while it is true that there are such things as numbers, it doesn't follow that numbers actually exist. This response, however, supposes that there is a distinction between there being Xs and Xs existing. But what distinction could this be?

Well, someone could use mis-use "exists" to mean something like e.g. "physically exists": and of course it doesn't follow from there being Xs that Xs physically exist. But it isn't physical existence that is in question when we ask whether numbers exist (or whether God exists, etc.) -- trivially, numbers aren't the kind of thing you can weigh or stub your toe on! It's granted on all sides that numbers are not physical things.

But once it is clarified that "exists" is not being used in a restrictive way to mean, e.g., physically exists it is very difficult to see what the supposed distinction is supposed to be between there being Xs and Xs existing. Certainly, few modern philosophers believe that there is such a distinction to be drawn. (In a slogan, most philosophers think that existence is indeed what is expressed by the so called existential quantifiers, 'there is', 'there are'.) So we'll set aside this challenge.

The other option is to challenge the initial assumption (1). But how can that be done? 11, 13, 17, 19 are the only prime numbers between 10 and 20, and so there are four prime numbers between 10 and 20. That's a simple truth of arithmetic, surely.

But ah, you might say, we need to distinguish: it's certainly true-according-to-arithmetic that there are four prime numbers between 10 and 20, just as it is true-according-to-the-Sherlock-Holmes-stories that Holmes lived in Baker Street. But it isn't really, unqualifiedly, true that there was a man called Holmes living there, and it isn't plain true either that there are four prime numbers between 10 and 20. And from the granted assumption that it's true-according-to-arithmetic that there are four prime numbers between 10 and 20 the most we can reasonably infer is that it should (given arithmetic is a consistent story) be true according to arithmetic that numbers exist. And that doesn't show that numbers really do exist.

This sort of "fictionalist" line that treats arithmetic claims as being claims within a story (the story of arithmetic) has its warm supporters. They will deny that numbers really exist, but at the high price of denying that arithmetical truths are plain true (a reversal, then, of the traditional philosophical ranking of mathematical truths as the most secure truths of all!). If you are not willing to pay that price, and are also not willing to play fast and loose with a supposed distinction between there being numbers and numbers existing, then the simple argument above for the conclusion that numbers exists will look rather compelling.

My guess is that a main source for the whacky interpretations is the claim that has repeatedly been made that the theorem shows that we can't be "machines", and so -- supposedly -- we must be something more than complex biological mechanisms. You can see why that conclusion might in some quarters be found welcome (and other technical results in logic generally don't seem to have such an implication). But as Franzen explains very clearly, it doesn't follow from the theorem.

I'm really glad you enjoyed the Gödel book!

I don't know if anyone has actually tried to prove that establishing Goldbach Conjecture needs additional axioms added to "regular arithmetic". But if someone had succeeded we would certainly have heard a lot about it!

Why so? Well, suppose T is your favourite theory of "regular arithmetic" (and we can assume T contains at least Robinson Arithmetic, Q). And suppose we can show that GC can't be proved by T as it stands. Then, trivially, T + not-GC is consistent. So, since GC is a Pi₁ sentence, it follows from Theorem 9.3 of my book that GC is true! In other words, proving GC can't be proved by T in fact proves Goldbach's Conjecture.

So we can put it this way: if it does need new axioms to establish GC, then proving that is so is at least as hard as proving Goldbach's Conjecture itself. Which, the evidence suggests, is very hard!

As to the "odds": my hunch is that GC is true, and can be proved in PA -- but I wouldn't bet even a decent meal out on it!!