> We have now almost a century of stagnation around any really new mathematical concepts that would help us push further into "in your face" empirical manifestations, e.g. around complex phenomena.
Could you substantiate this rather outrageous claim with some sources?
No offense but you made a very broad and rather surprising (= anything but well-established) claim, so the burden of proof is on you I'd say.
> Our core explanatory framework (underpining eg quantum or relativity) is essentially still serviced by 19th century mathematics.
Eh what? Quantum mechanics and high-energy physics largely build upon functional analysis and representation theory which were largely developed in the 20th century because of quantum mechanics.
General Relativity did pick up some concepts which were already developed in the 19th century already but today's formulation is quite a bit more modern (and rigorous) I'd say and there have also been lots of advances on the mathematical side since then.
Generally speaking, the issue in fundamental physics is not that we lack the mathematical tools; the issue is that we lack physical insights (from experiments).
We are drifting into specifics (eg Hilbert spaces were already a first decade of the 20th century thing) that are really not relevant for my argument.
The core question is whether the Universe is somehow mathematical (which I would say is the really extraordinary and clickbaity claim of attention seeking physicists with no more serious stuff to work on).
The empirical data point that I am trying to derive (so that we don't talk metaphysically) is focusing on the historical development of mathematical physics. The fact that it had spectacular success up to a point but then it stagnated. Does this suggest something about "maths is in our brains" versus "maths is running the universe"? Maybe it does, maybe it doesnt (keep reading).
Please note that my asserted stagnation does not concern fundamental physics (where indeed experimental barriers have become the limiting factor). More telling imho is that we have no real mathematical tools to understand complex systems, even of fairly benign nature (non-equilibrium thermodynamics), let alone extraordinary phenomena like biology. We have become comfortable that all complexity can "in principle" be explained, but that is not exactly actually grasping the mathematics of it. Pressumably a mathematical universe is mathematical at all scales, so where is the math?
So to recap, my question/challenge is essentially: if the universe is mathematical and we managed to explain quite a bit of it, why not more? There are now more living scientists than ever before. Discoveries are made continuously, but not of the type that counts as a mathematical understanding breakthrough.
The answer might be:
its just a matter of time (it takes cultural evolution, resources and trial and error for brains to "reconnect" with the underlying math universe) and a near century-long hiatus is part of how this works
OR
we had a lucky run in the last two/three centuries (some aspects of brain function can do a good job emulating certain aspects of the underlying "non-math" universe) but there is no reason to expect that this will be an ongoing process of "ever closer union".
I would argue it's both: Yes, there are more scientists than ever but the scientific landscape has also become considerably larger. Everyone is doing research in a different niche and every niche occupies maybe 5-10 people. Also, no one knows all fields or research anymore. So any sort of breakthrough will take time.
And we probably did have a lucky run in the sense that the phenomena in fundamental physics are comparatively simple – at some point you will have solved them. They are not nearly as complex as phenomena in condensed matter, for example, where there might be some overarching patterns & schemes, but ultimately every material is still going to be slightly different and come with unexpected (or at least very material-specific) properties. (The recent discovery of the room-temperature superconductor might be a prime example here.)
The thing with complexity is, well, that it's complex. Why are you expecting any general breakthroughs?
Could you substantiate this rather outrageous claim with some sources?