Has philosophy ever clarified mathematics?
I've recently been reading some standard textbooks on the philosophy of mathematics, and I've become quite frustrated that (surely due to my own limitations) I don't seem to be gleaning any mathema...
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Two points: one, firstly understanding mathematical processes can be of immense pedagogical value. See e.g. Polya's How to Solve It (and he wrote a more academic book with these themes), or Lakatos' Proofs and Refutations. I found this book New Directions in the Philosophy of Mathematics which had interesting essays as well.

Secondly, remember that broadly the point of philosophy is to make things not philosophy. In extremely simplistic historical terms, once natural philosophy becomes rigorous it becomes science, once philosophy of language became rigorous it became linguistics, and today we're seeing philosophy of mind turn to neuroscience.

So philosophy that elucidates mathematics is simply... mathematics. Most obviously Russell and the development of set theory. Modernly I don't know: I think the interesting stuff is happening at computer science/philosophy and physics/philosophy which trickles into mathematics. I'm posting this largely because I think the question is slightly broken because philosophy doesn't really work to clarify a field where it has already been clarified.

I find the case of Alan Turing's development of the concept of computatibility to be an example. Before Turing, the logicians had no clear concept of what it means to say that a function is computable. Even Gödel had despaired to ever have a formal notion of computability, because he had expected that for any such formal notion of computability, we would be able to diagonalize against it and thereby find a function that was computable in an intuitive sense, but not with respect to the formal notion. This was true of the class of primitive recursive functions and other extensions of that idea.

Meanwhile, Turing proceeded on a mainly philosophical level to consider what it was that a human did when undertaking a computation, imagining a person with paper and pencil and plenty of time, following a rote computational procedure, and was thereby led to his notion of Turing machine, which led to the fields of computability theory, complexity theory and so on.

As already remarked by others: If one tries a narrow interpretation of your question, you are asking a lot. You want someone whose specialty is not mathematics to elucidate a mathematical argument in a way useful and interesting for mathematician in their practice of mathematics, without creating new mathematics.

Interpreted in a slightly broader sense, the first men to come to my mind are Bolzano and Frege.

Bolzano was certainly a philosopher (and priest). His work on what we now consider set theory can very well be regarded as a philosophical elucidation of the mathematical treatment of, I would say, infinite sets, only that prior to Bolzano noone has really thought in these terms! Of course, there is other work by Bolzano, which more purely qualifies as mathematics, as his efforts to put analysis on a firm foundation and his example of a continuous and nowhere differentiable function, which first of all required clear concepts of these terms! But it should be clear that these questions had then a strong philosophical bend; only now that we take these modes of thinking for granted, we can claim that they are just part of mathematics and not philosophy.

Frege is considered by many analytic philosophers as one of the foremost philosophers of the 19th century, although one has to remark that he studied mathematics. In his Begriffsschrift he essentially invented formal predicate logic. In his later work he applied logic to the foundations of mathematics. Formal logic has certainly done a lot for mathematics. Of course, you could claim that formal logic is really the creation of new mathematics, but I think it was not primarly so. Coming up with formal logic was foremost an act of thinking hard about what an argument is - a typical philosophical activity.

There also names like Quine, Kripke and Dana Scott, where part of their works blurs the distinction between set theory, logic, philosophy and mathematics.

Many of the mathematicians active in foundations had also a strong philosophical interest. I want just to mention the names Cantor, Hausdorff, Gödel and MacLane. Their philosophical interests had certainly influence on their mathematics, although this is probably hard to prove.

Even one step further: Mathematicians have certainly done interesting philosophical work on mathematics, even if they do not claim so. I just want to mention Tao's What is good mathematics? and Thurston's On proof and progress in mathematis.

Even more ordinary mathematicians use from time to time phrases like "from a philosophical point of view" in their mathematical musings. One might mark this just as a non-rigorous mode of mathematical thinking, but also as a philosophical-bend mode of thinking - both are correct at the same time, I suppose.

In order to address this question, I think it is important to first take a step back and examine with a critical eye something that we normally take for granted, namely the professionalization and compartmentalization of academic departments. It is common to think of "philosophy" as "that which is practiced by professional philosophers" and "mathematics" as "that which is practiced by professional mathematicians." However, in my opinion, the divvying up of academic "turf" is driven more by sociological and economic factors than by any intrinsic divisions in the intellectual subject matter. To put it crassly (and somewhat exaggeratedly), my fellow academicians and I stand to extract more money and prestige from the rest of society if we agree to slice up the pie in a certain way and not fight too much internally over the division, conserving our energy to be directed outwards.

But if we actually want to understand the true relationship between "philosophy" and "mathematics," we shouldn't confuse ourselves by insisting that what mathematicians do is by definition "not philosophy" and that "philosophy" is by definition what is done by people whose paycheck comes from a philosophy department of an academic institution.

If we accept this point of view, then we should have no qualms about using the word "philosophy" to describe the sorts of activities that you described as being "carried out exclusively by mathematicians and scientists." In that sense, philosophy has always clarified mathematics and will continue to clarify mathematics.

Now, you might still have a lingering question, which might be rephrased as follows: "Does the kind of philosophy of mathematics that happens to currently be the bread and butter of those people who happen to be paid by academic philosophy departments stand to offer any clarification of mathematics?" Phrased this way, we can see that it is more of a sociological question (and a historical question, since the political agreements about who owns what turf change over time) rather than a question about philosophy and mathematics per se. But still, we might be interested in the answer.

Generally speaking, the kind of thing that is covered in "textbooks of philosophy of mathematics" interacts with mathematics via the foundations of mathematics in general. The influence that this kind of philosophy has had in clarifying mathematics in general is clearest if we look back at the early 20th century. For example, today we enjoy an immensely clearer notion of what a "proof" is than anyone had in the 19th century, and that is thanks to the clarifying work of those who worked in foundations. We also have a much clearer notion of the distinction between constructive and non-constructive mathematics, thanks to foundational work surrounding the axiom of choice, intuitionism, uncomputability, etc. For a modern example, Voevodsky and his collaborators are now pushing homotopy type theory as their preferred approach to foundations. (Admittedly most of their paychecks don't come from philosophy departments but until recently, I think most authors of books and papers on type theory were paid by philosophy departments.)

In my opinion there is a continuum between foundations of mathematics in general and foundations of specific areas of mathematics (e.g., the foundations of algebraic geometry—think of topoi), but if you insist on defining "philosophy" according to which pot of money the paycheck comes from, then the foundations of algebraic geometry won't count because our society is set up so that such people experience pressure to migrate to the mathematics department. Thus so-called "philosophy of mathematics" is more-or-less forced to limit itself to clarifying the foundations of mathematics in general.

Turing has already been mentioned in previous excellent answers as someone whose ideas sit at the boundary of philosophy and mathematics, conventionally understood. I want to mention Ludwig Wittgenstein in this context as an example of someone grappling with a host of similar ideas but who arguably took a more "philosophical" approach to them, one that explicitly resists formalization.

The book "Wittgenstein's Lectures on the Foundations of Mathematics, Cambridge, 1939" is a transcript of a seminar given by Wittgenstein which was attended by Alan Turing, among a handful of other notable young Cambridge scholars of the day. It contains an interesting exchange between Turing and Wittgenstein about what happens when people disagree as to the result of a calculation. Turing insists that some are right and some are wrong, and Wittgenstein wonders what that might mean. Turing says that if you build a bridge that depends on a wrong calculation, it will fall down. My recollection is that Wittgenstein argues that this doesn't cause the bridge to fall down, but that the bridge falling down might serve as a definition of what it means for the calculation to be wrong.

I gather that many people find Wittgenstein to be obtuse and/or just plain confused, but a more charitable reading suggests that he was wrestling with ideas related to undecidability, albeit from a much broader sociological vantage point. See, for example, "A note on Wittgenstein's notorious paragraph on the Gödel theorem". If a bridge standing or falling down depended on a proposition that was undecidable, what then? I believe, but am not sure, that the issue of "in which formal system" would not necessarily have been in the air at the time.

So, on the one hand we have a nice example of philosophical questions stimulating what went on to become a much more formal (and elaborate) theory in the work of Turing. On the other hand, I feel that the hypothetical scenario about the bridge is a "purely" philosophical question that is both interesting and challenging and that is not addressed by the subsequent formal developments. Naively put, which formal system does mother nature obey? Moreover, how can we make better sense of this question? Philosophy gropes at such questions; once they have been sufficiently sharpened, mathematics constitutes the work of refining and extending our understanding.

Kant has featured in Geog Cantor's argument that time is irrelevant for describing the continuum.

This is a partly a (tangentially) relevant answer to the question, and partly a relevant comment on a very interesting comment at 'Asaf Karagila Oct 7 '14 at 10:47' in this thread, which in effect points out that Cantor considered Kant harmful and a bad mathematician.

I just stumbled over an 1883 passage of Cantor's in which a thirty-years-younger (than the Cantor who wrote the 1911 letter to Russell, that is) has Kant have sort-of a cameo appearance in what arguably is transfinite-set-theory's foundational-document: in [G. Cantor: Grundlagen einer allgemeinen Mannichfaltigkeitslehre. Teubner 1883, p. 29] one finds:

[my translation:]

First of all, I have to declare the following: in my opinion it is not correct to adduce a concept of time or of a (intuitive-)view-on-time [translated intentionally non-idiomatically] if one explains the much more original and general concept of the continuum; I think that time is an idea which, in order to be explained clearly, presupposes the the concept-of-the-continuum, which is independent of time; also, even with the help of the concept-of-the-continuum, [time] can neither be objectively conceived of as a substance, nor subjectively conceived of as a necessary apriori form-of-intuition, rather time is nothing other than an auxiliary and relative concept, by which one determines the relation between distinct movements which occur in nature and are perceived by us. [Some would argue that Cantor is anticipating both Einstein and Barbour here.] Something like objective or absolute time nowhere occurs in nature and therefore one also cannot view time as a measure of movement, rather, to the contrary: movement could be viewed as a measure of time, were it not for the fact that, which contradicts the latter interpretation, that time, even in its modest role of a subjectively necessary apriori form-of-intuition, has failed to reach any fruitful, unchallenged state of prosperity [Cantor gets rather ornate at this point, and, in a sense, weakens his own case by negating an extreme condition ('time has not reached an unchallenged etc etc'; what wonder.)], even though it has had enough time to do so since K a n t.

In short, Cantor uses a reference to Kant's influential philosophy to strengthen his argument that time is irrelevant for explaining the continuum.

This can be seen as an example that "philosophy" "clarified" "mathematics".

Of course (thanks to Asaf Karagila for pointing out that this was not made clear in this post), one can interpret Cantor as writing rather ironically about Kant here, proposing something like the following rhetorical question (I am paraphrasing):

'Now if what Kant wrote were so true, how come that a hundred years later [which is roughly the time span that had passed when Cantor wrote this][which is roughly the time span that had passed when Cantor wrote this] the view on 'time' as a 'subjective necessary apriori form-of-intuition', this point-of-view still has not taken hold?'

Now one can argue that Cantor's argumentation itself is a fallacy, or a rather weak charge against Kant, and by itself does not speak against Kant (let alone contradict him), since of course *it does not necessarily imply the wrongness of a view that a view is not widely held*.

This post is not to 'endorse' either of these interpretations, rather to point out a relevant and historical interesting aspect of Cantor's work.

The present observation can be seen as philosophy having clarified mathematics, if only in the indirect way that (0) Cantor seems to have disliked Kant, whether at age 37 or at age 66, and therefore (1) distanced himself from, and avoided, the use of any notion of 'time' in his mathematics.

I paraphrase with italics from the original post to give my perspective:

My naïve expectation would be that metamathematicians * might take a difficult construction or proof, and clarify it by isolating the key ideas behind it. Having isolated the key ideas, they might then highlight their relevance and thus point the way forward. Beyond this, I would hope that such practitioners might elucidate the `true meaning' of axioms and of definitions by examining their ontology in a wider context.

I first took meta-mathematics hoping to discuss things like the duality between the notions of points and lines in axiomatic geometry, so that one could find ways to discover and prove more results by exploiting the symmetry of the foundations of a given theory. I found something completely different, of course, with only occasional references to duality or n-ality. I put * after metamathematicians, because, after some years seeing how some good mathematicians practiced (they would not stop at a single result: they would poke it, prod it, extend it this way or that, try to find limits by tweaking some aspect of the assumptions or proof and come up with counterexamples, and use other methods to look for wideness of application, or anticipate when such attempts might fail), I realize now that the first term should be "professional mathematicians", or just "mathematicians" perhaps.

(This is not to dismiss philosophy from the question, so much as to ask for an alternate basis for the question. I think philosophical examination might yield guidance as to which logical system to use in certain pursuits, but it would still depend on the naive expectation above being carried out by mathematicians.)

Gerhard "Mathematicians Know What 'Is' Is" Paseman, 2014.10.01