Does Quine's argument that there is no real boundary between analytic and synthetic statements include purely mathematical statements such as 1 + 2 = 3? Granted, sentences in everyday languages contain both analytic and synthetic elements, but cannot formal languages support purely analytical statements? Or does mathematics, being a human creation, inextricably model the natural world around us, and thus contain synthetic information? I'm trying to understand the short and (very difficult for me) book "Knowledge and Reality: A Comparative Study of Quine & Some Buddhist Logicians" by Kaisa Puhakka, which seems to represent Quine's thinking faithfully, but my training as a scientist leaves me ill-prepared for much of it. Thank you.

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Quine's views on this matter vary over the years. Early (meaning in "Two Dogmas" and related works of that period), he was prepared to deny that there are any analytic statements. Later, especially in Philosophy of Logic, Quine's view mellows a bit, and he is prepared to recognize a very limited class of such statements, namely, truths of sentential logic, such as "It is raining or it is not raining" and the like. That's still a pretty limited set, as Quine seems unprepared to regard even what one would normally regard as truths of predicate logic as analytic (e.g., "If someone loves everyone, then everyone is loved by someone"). But mostly this is because Quine thinks there's no clear sense in which that sentence is properly analyzed as a truth of predicate logic. This is connected with the doctrine of ontological relativity. In so far as it is properly so analyzed, I think Quine would regard it as analytic.

So mathematics, for Quine, is quite definitely out as analytic. There are going to be several reasons for this. But, Quine seems to think that, in principle, even such simple and beloved statements as "1 + 2 = 3" could be revised, under suitable empirical pressure. I am happy to grant that this is hard to imagine, and if, unlike Quine, one were prepared to grant that predicate logic is analytic, then perhaps this kind of statement could also be regarded as analytic, since it is, in some sense, more or less equivalent to a truth of predicate logic, viz:

(Ex)(Ey)(Ez)[Fy & (w)(Fw --> y=w) & Gy & Gz & (w)(Gw --> w=y v w=z)) -->
(Ex)(Ey)(Ez)[(Fx v Gx) & (Fy v Gy) & (Fz v Gz) & (w)(Fw v Gw --> w=x v w=y v w=z)]

(I write it out just to make clear that it can be written out explicitly.) Or, perhaps the right thing to say would be that it is the truth of predicate logic that is analytic and that we tend to confuse "1 + 2 = 3" with it.

What Quine would not regard as analytic is something like the statement: Every pair of numbers has a sum. And here, most philosophers of mathematics would agree with him. The reason is that there are very small collections of what look like very simple principles that together imply that there are infinitely many numbers, e.g.:

Every (counting) number has a "successor", a number that is one bigger than it.
Zero is not the successor of any number.
Different numbers have different successors.

Note that we're not even into addition here. And now: Those three principles already imply that there are infinitely many numbers, as you will see if you think it through. Most philosophers balk at the idea that the existence of anything, let alone of infinitely many things, might be analytic. Not all philosophers hold that view, but, as I said, by far most do.

Richard's response is helpful and interesting, but perhaps I would put matters a bit differently. He makes it sound as if Quine accepts the distinction between analytic and synthetic truth and goes on to argue that nothing counts as a truth of the first kind (perhaps "mellowing" his view about this later on). But Quine's position (early, middle, and late) is rather that he can make no sense of the distinction at all. His challenge isn't to the analyticity of logical or mathematical truth; it's rather to the intelligibility of sorting truths into these categories – to the very categories themselves – as the traditional philosopher conceives of them.

Your thought that the distinction can be given some sense in the context of an artificial language is a natural one. Quine explicitly turns to this suggestion in section 4 of "Two Dogmas of Empiricism."

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