Here is an example of philosophy helping a breakthrough in mathematics (in differential topology). The breakthrough happened last year, the philosophy that helped it come into existence happened 200 years ago, via a formalization suggested in the last two decades.

A long-standing open problem in differential topology and in mathematical physics was the definition of differential cohomology theories that are "twisted". This plays a role notably in quantum anomaly cancellation in quantum field theories, such as in the Freed-Witten-Kapustin anomaly in the worldvolume theory of the type II superstring, where it is twisted differential K-theory that is relevant. As this example shows, the question is of profound relevance for the foundations of our most advanced theories of fundamental physics. It was long known how to do the twist and the differential refinement separately, but their combination used to be elusive. The right framework was as much missing as it was known to be necessary.

It should be clear at least from the sound of the technical terms here that this is a question that involves messing with the very foundations of modern geometry. Topology, differential structure, homotopy theory, generalized cohomology, fundamental physics (string physics, hence perturbative quantum gravity if you wish) all intimately interact in differential cohomology theory. That should make it plausible that if you get stuck here with your formal mathematics, it might help after a while to step back, put on a philosopher's hat, and try to see if for a moment you might be helped by adopting more of a "natural philosophy" perspective to regauge your formal tools.

Now it turns out that just this is what William Lawvere had been doing throughout his life. Lawvere is famous in pure mathematics as being the founder of categorical logic, of structural foundations of mathematics, and of intuitionistic logic embodied in topos theory. But what less people know is that in all these developments he was to a large extent motivated by finding foundations for a geometry of physics (see here, for him it was specifically classical continuum physics, but we'll see how the insights he gained there inform also modern quantum field theory).

Lawvere discovered that in order to lay foundations for geometry of physics in the foundations of mathematics, it was surprisingly useful to read Hegel's metaphysics, the "Science of Logic" from 1813, if only one translated the notorious "unities of opposites" that structure this text into the formal concept of pairs of adjoint modalities (Lawvere called them: "adjoint cyclinders"). Indeed, in his famous and at the same time (I think it is fair to say) widely underappreciated "Some thoughts on the future of category theory" he follows Hegel, formally defines "categories of being" (mathematically, in category theory!) in which "nothing" and "being" combine to "becoming" in a genuine formalized precise mathematical sense, and suggests that these categories of being are where the foundations of the geometry of physics is to be looked for. Later he speaks instead of categories of "cohesion" to amplify the differential geometrical aspect more. Lawvere uses Hegelian terminology in much of his mathematics, and it seems clear -- preposterous as that may seem in the eyes of the anayltic philosopher -- that reading Hegel helped Lawvere develop the intuitionistic mathematics and the application of cohesive toposes. Indeed, once you follow Lawvere and accept that whenever Hegel speaks of his infamous dualities he is secretly (intuitively) describing an adjoint modality in intuitionistic type theory, then it feels a bit as if one can suddenly see the Matrix behind the mysterious string of greenish symbols, and Hegel's seemingly gnostic metaphysics suddenly reads much more like axioms for a practical axiomatic metaphics.

Nobody picked this up for years, because I think nobody recognized it. Then homotopy toposes (infinity-toposes) appeared on the scene (there is another story to be told here about the philosophy of constructivism causing a fantastic breakthrough in the foundations of mathematics via homotopy type theory, but this should wait for another post) and founding fundamental physics (in particular gauge theory) in (higher) topos theory became ever more compelling.

In any case, at some point it became clear that equipping an infinity-topos with the structure of a Hegelian "category of being" in the formal translation via Lawvere, hence making it a "cohesive infinity-topos", is the step necessary to obtain a working formal foundation for differential cohomology.

Indeed, last year Ulrich Bunke, Thomas Nikolaus and Michael Völkl realized (see here ) that the famous "differential cohomology diagram", which is a diagonally interlocking pair of two excact sequences of cohomology groups that has been postulated to be the very characteristic of differential cohomology, universally follows for every stable object in any "homotopy topos of Hegelian being and becoming", whence in every cohesive infinity-topos. And based on that more profound understanding of the foundations of differential cohomology, Uli Bunke and Thomas Nikolaus could now solve the problem of twisted differential cohomology. (This followup article should be out soon.)

To sum this up, I think one lesson is the following. Sure, once you have a formal system that formalizes what previously was "just" natural philosophy -- such as when Newton finally had his laws of motion nailed down -- then reasoning with that formal system will be far superior to what any philosphical mind un-armed with such tools may possibly achieve. But these formal systems -- our modern theories of mathematics and physics -- don't just come to us, they need to be found, and finding them is in general a hard and nontrivial step. Often once we have them they appear beautifully elegant and of an eternal character that makes us feel as if they had always been around in our minds. But they have not. And this is the point where philosophical thinking may have deep impact on the development of science, at that edge of science where the very formal mathematical methods that feel so superior to bare philosophical reasoning -- end.

In fundamental physics it is (or at least was in the 1990s) common to declare with a certain awe and also pride that quantum gravity, non-perturbative string theory and such like will force us to do things like "radically rethink the foundations of reality" or similar. Unfortunately, that rethinking has mostly been what I think is fair to call a bit naive. One cannot just talk about it. It needs both, a technical understanding of the core formal mathematics up to that very edge up to which we do understand the formal laws of nature, and a trained profound philosophical mind who can stand at that cliff, stare into the misty clouds beyond and suggest directions along which further solid ground of formalism might be found. Once it is found, true, then the philosopher should probably better step back and watch those mathematicians and physicist built a tar road over it and then run heavy truck load back and forth through what had been uncharted territory. But before that is possible, the new stable ground has to be found first.

I conclude with a personal note. Back as a kid I was thrilled by philosophy, but then got appalled by the philosophy that I was fed in school, turned to science instead and held views much like those exppressed by Aaronson above. Then the philosopher who profoundly changed my view of philosophy was David Corfield, philosopher of science and mathematics from University of Kent. His philosophical commentary and prodding as he watched me develop maths as in my recent "Homotopy-type semantics for quantization" have considerably helped and propelled some of these developments. I am thankful for that.