However, the claim I was actually thinking of, which is right I think, is that the maths used in the physical revolutions of the turn of the century (SR, QM, GR, and probably QFT, QED, and QCD as well) was invented by physicists or by mathematicians working with physicists for the express purpose of developing this theories, not the other way around.
Also, the basis of mathematics and the first few thousand years were indeed motivated by these kinds of concerns.
In mathematics, transformations equivalent to what was later known as Lorentz transformations in various dimensions were discussed in the 19th century in relation to the theory of quadratic forms, hyperbolic geometry, Möbius geometry, and sphere geometry, which is connected to the fact that the group of motions in hyperbolic space, the Möbius group or projective special linear group, and the Laguerre group are isomorphic to the Lorentz group.
Mathematicians were following up on "what happens when you discard one of Eucilids Axioms" and discovering there was an entire world of consistent hyperbolic geometry and more.
Some time later:
In physics, Lorentz transformations became known at the beginning of the 20th century, when it was discovered that they exhibit the symmetry of Maxwell's equations. Subsequently, they became fundamental to all of physics, because they formed the basis of special relativity in which they exhibit the symmetry of Minkowski spacetime, making the speed of light invariant between different inertial frames.
If you read mathematics histories it's a common complaint that it's nigh on impossible to discover something new and esoteric that doesn't soon end up with a military application; the ongoing search for interesting but useless mathematics is akin to the search for the fountain of youth.
It is the case (IIRC) that quaterions arose directly from Hamilton's search for a better way to describe mechanical motions in three dimension spaces - ie created to be useful from the outset.
I think there's a lot of fascinating mathematical "dualism" in how many of those were developed at the same time together by both "practical" mathematicians (such as physicists) and "theoretical" mathematicians. You feel it is easy to argue that because the practical mathematicians had an easily defined "need" (hypothesis/experiment) they were the "leaders" and the arrow flowed from them to the theoretical mathematicians working with them, but there's just as much evidence in some of those cases that those theoretical mathematicians were already doing the theory building on their own and had a "need" to find practical use cases/outlets. In some cases we know the theoretical mathematician sought out the physicist to try to find ways to test a theory and were really the ones building the hypotheses. In some of the cases we know that though both are generally credited for "deep" collaboration after the fact, because they never really worked together and did all of their work in parallel and it is likely both would have completed just about the same work even if they never crossed paths. Newton and Leibniz famously never corresponded until after both published their own takes on the fundamental principles of The Calculus. Alonso Church had already developed the Lambda Calculus before corresponding with Alan Turing on the fundamentals of Computing and Alan Turing couldn't even share most of his practical work because it was still state secrets (and there was an ocean's distance in their correspondence anyway).
I think as often as not the "arrows" in the diagram point both directions at the same time: the practical needed the theorist to explain the patterns they were seeing and the theorist needed the practical to take the simple beautiful thing they were working on and make it practical and find the edge cases and complications.
That sort of "dualism" seems an interesting pattern in math.
Yes, I agree to some extent with the restricted claim, which only (slowly) started to break down in the 17th century in the west.
A lot of Indian mathematics was rather abstract going back to Vedic times, but since they didn't develop the concept of proof, it sadly had little impact on other mathematics practice (except as inspiration to Persian and Arab scholars) other than the the famous cases of zero and positional notation. The mathematical documents I've seen from that practice have been in the form of essays.
I know little of Chinese or Mesoamerican mathematics and wonder where they were on this axis. It seems pretty likely that maths started in support of astronomy/planting predictions in the cultures I know of so likely also for East Asia and the Americas, but whither thence did it go?
However, the claim I was actually thinking of, which is right I think, is that the maths used in the physical revolutions of the turn of the century (SR, QM, GR, and probably QFT, QED, and QCD as well) was invented by physicists or by mathematicians working with physicists for the express purpose of developing this theories, not the other way around.
Also, the basis of mathematics and the first few thousand years were indeed motivated by these kinds of concerns.