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 mathematical insights from them.
My naïve expectation would be that philosophy might take a difficult construction or proof, and clarify it by isolating the key ideas behind it. Having isolated the key ideas, philosophy might then highlight their relevance and thus point the way forward. Beyond this, I would hope that philosophy might elucidate the `true meaning' of axioms and of definitions by examining their ontology in a wider context.
In reality, to the best of my knowledge (please prove me wrong!) both of the above tasks seem to be carried out exclusively by mathematicians, physicists, computer scientists, and other natural scientists, as far as I can see. To play the devil's advocate, philosophy seems to me like it might historically have largely played an opposite role, labeling certain objects as "unreal" and "unnatural" which in fact later turned out to be fruitful to study (negative numbers, irrational numbers, complex numbers...).

Question: Has it ever happened that philosophy has elucidated and clarified a mathematical concept, proof, or construction in a way useful to research mathematicians? 

Philosophers have created much new mathematics (e.g. the work of C.S. Peirce, much of which is bona fide mathematical research), but the question is not about this, but rather about philosophy as practiced by philosophers providing elucidation, explanation, and clarification of existing mathematics.
 A: Logic (and formal systems) is a contribution of philosophy.
The point is that up to recent times (~1800), mathematics was a discipline of philosophy, just as music was a discipline of mathematics at the same level as arithmetics, geometry and astronomy. It was impossible for a mathematician not to study philosophy.
So you question is meaningless up to 1800. Just read Leibniz or Newton to see that the question of how god made the universe was a mandatory base to justify axioms, which we nowadays consider as hypothesis.
As far as modern science is concerned, philosophy is a science of human ideas and beliefs which escape to any mathematical models, and will probably ever be.
During the transition period (1800-1900), lots of work has been done on the common part of mathematics and philosophy, which resulted in a formalisation of logic and a pedestal to base mathematics.
A: Perhaps it has been cited elsewhere in this context, but Hilary Putnam's 1980 short essay "Models and reality" brings together mathematics and philosophy in its discussion of $V = L$ and the Loewenheim-Skolem theorem: http://www.princeton.edu/~hhalvors/teaching/phi520_f2012/putnam1980.pdf
Another source of recent interaction is the field of axiomatic theories of truth (Kripke, Herzberger, Leigh and Rathjen and others). For example: http://www1.maths.leeds.ac.uk/~rathjen/AbsoluteEnd_Truth.pdf
Maybe these are examples of mathematical logic clarifying philosophy...
A: To add to the discussion: the very dichotomy Philosophy x Mathematics didn't make sense to most thinkers of the past. That still holds true for many of today's philosophers - Philosophy is not a discipline, like Mathematics or Chemistry.
That being said, I could list tens of philosophers that satisfy your expectations: Frege, Russell, Tarski, Smullian; the list goes on and on. Now, for those that don't satisfy your expectations, but may be even more interesting to you for that exact reason, I'll mention Husserl, Wittgenstein and classics like Pythagoras and Plato (yes, they had a lot to say about "Mathematics" and ontology).
Based on this sentence alone: "Philosophy might elucidate the 'true meaning' of axioms and of definitions by examining their ontology in a wider context", I would definitely give Wittgenstein a try. You'd be surprised to know that the sun may not rise tomorrow...
A: Would you expect to learn much about how to practice science (or any sub-branch thereof) from the philosophy of science? I hope the answer would be no; that falls within the study of science.
The philosophy of science and mathematics exists to examine the philosophical underpinnings of those disciplines which lie outside the purview of those disciplines themselves (e.g. do numbers have some kind of objective existence separate from the minds of reasoning beings), and philosophical topics related to them (such as the ethics of scientific and mathematical enquiry). 
Now, that is not to say that this stuff is useless - it can guide the actions of scientists and mathematicians, and identify limitations in the discipline and its methods. However, you're not going to find material which has a simple relationship to the practice of the discipline, and certainly not in introductory texts aimed at people who are not mathematicians.
A: I'm surprised no one has yet mentioned Cantor.  His work leading to Transfinite Arithmetic, and from there to general Set Theory, overturned millenia of received opinion on the concept of the infinite by insisting on the mathematical existence of actual, completed infinities.  He defended this work with a body of philosophical arguments arguing that the earlier conclusion - that infinity could of necessity only be potential - was in fact mistaken.  [source: biography of Cantor by (I forget who, sorry)]
A: Bernhard Riemann's famous 1854 Habilitation lecture "On the hypotheses…" owed more to the influence of the anti-Kantian philosopher Johann Friedrich Herbart than to Kant himself.  Here one could mention an illuminating article by Nowak:

G. Nowak, in The history of modern mathematics, Vol. I (Poughkeepsie, NY, 1989), 17–46, Academic Press, Boston, MA, 1989.

Nowak argues that


*

*Herbart's constructive approach to space, already cited, mirrored the content of Riemann's reference to Gauss in that both discussed construction of spaces rather than construction in space. 

*Riemann followed Herbart in rejecting Kant's view of space as an a priori category of thought, instead seeing space as a concept which possessed properties and was capable of change and variation. Riemann copied some passages from Herbart on this subject, and the Fragmente philosophischen Inhalts included in his published works contain a passage in which Riemann cites Herbart as demonstrating the falsity of Kant's view. 

*Riemann took from Herbart the view that the construction of spatial objects was possible in intuition and independent of our perceptions in physical space. Riemann extended this idea to allow for the possibility that these spaces would not obey the axioms of Euclidean geometry. We know from Riemann's notes on Herbart that he read Herbart's Psychologie als Wissenschaft.
It is possible that, had Kant's ideas about infinite space not been challenged by Herbart, the field we know today as Riemannian geometry may have developed much later.
This is meant to complement Thomas Klimpel's answer on Riemann.
For those with access to mathscinet, more information can be found here.
Note. Peter Heinig pointed out that the historian David Rowe similarly holds that Riemann was more of a Herbartian than Kantian; see comments below and also Rowe's review.
A: 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. 
A: From this article in Wikipedia : "La Géométrie was published in 1637 as an appendix to Discours de la méthode (Discourse on Method), written by René Descartes. In the Discourse, he presents his method for obtaining clarity on any subject. La Géométrie and two other appendices also by Descartes, the Optics and the Meteorology, were published with the Discourse to give examples of the kinds of successes he had achieved following his method".
A: Ludwig Wittgenstein invented the truth table, in trying to clarify the structure of propositional logic. Of course, this is now a standard tool in mathematics and computer science. Wittgenstein was trained as an engineer and working as a philosopher at the time, but was never a mathematician.
A: After Newton and Leibniz developed infinitesimal calculus, a number of critics emerged to criticize the new technique.  These included Berkeley, Cantor, and others.  One of the few schools to battle the negative attitude toward infinitesimals at the beginning of the 20th century was Hermann Cohen's school, also known as Marburg neo-Kantianism.  Their efforts did not bear fruit immediately but they did influence one Adolf Fraenkel, who passed on his interest in infinitesimals to one Abraham Robinson. In his autobiographical work in the 1960s Fraenkel noted that "Robinson saved the honor of infinitesimals."
Thus Cohen's attempts to develop a respectable theoretical basis for a philosophy of infinitesimals eventually bore fruit in Robinson's framework for analysis with infinitesimals.
A: There's a great book called the "Synthetic Philosophy of Contemporary Mathematics" by Prof. Fernando Zalamea that came out a few years ago. I've read about half and would highly recommend.
It's intention is not so much to clarify mathematics for mathematicians but to clarify contemporary advances in mathematics to mathematical philosophers. In a sense, it distills and analogizes contemporary mathematics into "intuitive" nuggets that can be used in a philosophical sense.
I would recommend it. There are also some interesting discussions on the n-category café.
https://www.amazon.com/Synthetic-Philosophy-Contemporary-Mathematics-Fernando/dp/0956775012
Furthermore, work done by Ricardo and David Nirenberg on these aspects, e.g.: http://criticalinquiry.uchicago.edu/uploads/pdf/nirenbergs_badiousnumber_complete.pdf
A: Yes, philosophy has clarified mathematics.
In 'Naming Infinity', by Loren Graham and Jean-Michel Kantor, they argue that Russian mathematicians in the early 1900s were able to introduce new concepts to set theory due to their philosophical (and religious) perspective on free will and naming things.

"At the core of this book is the contest between French and Russian
  mathematicians who sought new answers to one of the oldest puzzles in
  math: the nature of infinity. The French school chased rationalist
  solutions. The Russian mathematicians, notably Dmitri Egorov and
  Nikolai Luzin—who founded the famous Moscow School of Mathematics—were
  inspired by mystical insights attained during Name Worshipping. Their
  religious practice appears to have opened to them visions into the
  infinite—and led to the founding of descriptive set theory."

I think this is a clear example of how our approach to mathematics depends on our philosophical beliefs. Both the French and the Russian mathematicians saw mathematics as a language that describes reality. For the French, this prevented them from advancing in set theory, since it ran against their beliefs of a continuous and orderly world. Meanwhile, the Russians were able to solve the contradictions in set theory and pave the way for new mathematical disciplines.  Egorov and Luzin thought free will and the power of names were universal truths that existed in every discipline, not just philosophy or religion. They saw discontinuous functions as a way to describe freedom of choice. Ultimately, their perspective brought about a lot of progress in mathematics.
A: 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. 
A: 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.
A: 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.
A: George Berkeley’s „The Analyst” is an example. 
Berkeley was a philosopher. He had a philosophical motivation. 
He attacked a mathematical concept of “Infinitesimals”. He clarified that an original approach to Infinitesimals was not rigorous.
It contributed to the development of epsilon-delta definition and non-standard analysis.
I can see an essential difference between Pythagoreans being against irrational numbers and Berkeley criticizing infinitesimals. The Pythagoreans were using metaphysical arguments and it was against the development of mathematics. The contribution of Berkeley who spotted gaps was positive. 
A: Per Martin-Löf has done work spanning mathematics, logic, and philosophy.  I am not sure how much his early work on randomness was influenced by philosophical concerns, but his work on type theory in the 70s and 80s certainly was.  This is especially clear in the 1984 lecture series, On the Meanings of the Logical Constants and the Justifications of the Logical Laws, which was highly influential in computer science.
A: 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
A: I'm not sure that some people would agree... but Popper's work on provability has clarified how things should work in "classical" statistics, that we can only prove something is wrong not that it is right - black swans anybody?  We can only say that we can/cannot reject the null hypothesis... whereas most people want to (incorrectly) accept the null/alternative hypothesis.
A: 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.
A: 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. 
A: Just like René Descartes' La Géométrie from Discours de la méthode (1637) might have had a profound influence on mathematics, also Immanuel Kant's Transzendentale Ästhetik from Kritik der reinen Vernunft (1781) might have had a profound influence on mathematics.
In his Habilitationsvortrag (1854), Bernhard Riemann talked about his success in the task given by Carl Friedrich Gauß to "update" Kant's intuition in the face of new facts like non-Euclidean geometry. When this talk (and a sketch of the elaboration of the "proposal") was published in 1867 as
Ueber die Hypothesen, welche der Geometrie zu Grunde liegen
On the Hypotheses which lie at the Bases of Geometry
it lead to responses like the one from Hermann Helmholtz in 1868
Ueber die Thatsachen, die der Geometrie zum Grunde liegen
The Origin and Meaning of Geometrical Axioms
which in the end also influenced Sophus Lie's On a class of geometric transformations (1871), which in turn also had an influence of Felix Klein's Erlanger Programm (1872).
Well, maybe the influence of Kant or Riemann or Helmholtz or Lie, or even Klein wasn't such important, but still these developments were always responses to earlier published philosophical positions.
A: I am very skeptical of  a philosopher  talking about Math, given the  unavoidable interpretative elasticity of the philosophical discourse and the blunders of some of  its practitioners. (Hegel's name comes to mind.) In particular, I don't  find it productive to learn Math from a philosopher, or to turn to philosophy when your research is stuck hoping to find salvation there. It looks to me that  philosophers' discussions and debates come after major advances in sciences, and not the other way around.   Philosophers may claim that Turing is one of their own. However,  he is famous for his mathematical contributions.   
A: There is a book that gained a cult following in the 1970s, and this book lies at the intersection of philosophy and mathematics.  It's Laws of Form by G Spencer Brown.  
Few serious mathematicians have held it in high regard, and little new creative work can be traced to it.  And it's a little on the abstract side, when compared to most works on philosophy.  But it can be an interesting way to look at logic and arithmetic as two different expressions built on top of a single concept. 
Brown calls this the calculus of distinctions.  Making a distinction may be the most fundamental thought process a child first learns.  
A: Lawvere has evidently applied a philosophical outlook to some effect in his mathematics.
This answer by Urs Schreiber on philosophy stack exchange gets at this and more:
https://philosophy.stackexchange.com/a/9814
A: Can the works of Penelope Maddy ("Believing the Axioms", I and II),  Georg Kreisel (his papers on "Informal Rigour"), and Benjamin Rin's paper "Transfinite recursion and the iterative concept of set" (to name just a few), be said to have not elucidated, or clarified mathematical concepts, proofs, or constructions in a way that was useful to research mathematicians?  
