(in a hurry now)Well, Einsteins theory of relativity for example relies on Riemannian geometry, which is only a very recent "discovery" considering the long standing tradition of Euclid. Similarly, statistical mechanics incl. Brownian motions is also largely indebted to "recent" developments of probability in early 20th century, just as quantum mechanics relies on "recent" results of functional analysis and operator algebras. My point being, that Newton in his days described adequately a certain range of phenomena, however by only using a restricted subset of mathematical tools. Of course, this does not refute the fact that mathematics can reach its kingdom and provide physics all the tools it needs to describe reality. To me, it seems very unlikely this will happen. And if it does, one perhaps might pull off a Gödel-in-physics (even though I don't have the time to think about this properly).
I'm not sure that I comprehend the source of your doubt re. the eventual (approximate) completion of physics's grand endeavor. I'm not saying that you're necessarily wrong, but what makes you so confident that physicists 1000 or so years from now
won't be plying their theories within the rough confines of a tentative unified field theory of everything?
Similarly, while mathematics proper may be something
like a human invention, a la technology or literature, it's really nothing more than a group of highly imaginative formal extensions of pure logic and commonsensical human intuition regarding the nature of space, shape, measure, number, and so forth. In this sense, the superficial workings of math collectively comprise the one thing a human can ever claim to 'know' to a certainty, since almost all mathematical truth claims are ultimately tautological/deductive in nature. In keeping with its unique epistemological status, math also seems to enjoy a cozy relationship with the aesthetic (empirical) world in which we live - after all, we use math for just about everything, and why shouldn't we? It just works so damn well in so many instances. When one considers the science we call 'physics' as simply another branch of maths (indeed, one for which new kinds of math must often be developed) the issue becomes even more problematic for the epistemological relativist.
If you're saying that Newton didn't have access to the kinds of mental machinery to which Einstein was indebted for his theories, this would not, I'd argue, imply that Newton's world was literally constrained in comparison to Einstein's in any intuitively meaningful way. Newton just didn't (couldn't) fathom any more physics/maths than he did because he didn't have the proper tools for the task at hand. In other words, though it may sound good to proclaim that Riemannian geometry was just 'made up' by humans in the obvious sense, just
how fictional Riemann's postulates really are is exceedingly tricky to specify. The way I see it, there's absolutely no good reason for me to believe that the intricate workings of the universe were significantly different in Newton's time than they are in ours simply because Newton experienced a (perfectly understandable) failure of imagination beyond his integral and multivariable calculi. Obversely, I would be just as skeptical if you told me that it was Newton's
mind, and not the universe, that was somehow fundamentally different on an unfathomable level: From what little I know of the man, if you could magic yourself back to the 17th century with a mercury chelator and a ream of mathematical documents summarizing the developments of the past ~250 years, nothing short of an act of god would stop Newton from poring over the materiel, sharpening his quill, and setting to work. His 'vocabulary' (more appropriately, his semantics, I suppose?) was not the issue. It was just the symbolic
syntax to which he had been exposed, which is really the stuff out of which math is (mostly) made.
In conclusion, there are at least two intuitive perspectives one can take regarding the relationship between deductive truth and physics/maths: Either you can say, without contradiction, that quantum mechanics (or at least the interactions that the mechanics describe) is really nothing new, in the sense that it's been around - almost literally - forever; or you could claim that it's just a twentieth century
idea about the world in which it was originally thought up and experimentally honed. Both are correct in their own little self-contained way. Anyway, I could prattle on about this for hours; the philosophy of mathematics is a favorite subject of mine, because no matter which way one tries to slice it, math just
has to be one way or the other, either a human invention, a useful fiction of sorts OR a kind of low-level programming language describing the undeniable, deductively certain architecture of reality - yet it's somehow both, and upon exhaustive study, so much more.
