Most importantly, the leap that if a medium can be used to create accurate representations, it must contain inherent, unquestioning truth about that represented phenomena, is just hogwash. Michelangelo could sculpt masterful representations of the human form, but the beauty and accuracy of the representation are the products of a skilled master, not the inherent humanness of marble.
It's not just the medium _can_ be used to create accurate representations, by a skilled master. It's that mathematical concepts created with no regard for the natural world but purely on their formal elegance, turn out, years later, to be an elegant way to describe and predict natural phenomena.
I don't think it "proves" anything, it's just... mysterious. The mystery doesn't go away if we insist that math is just an arbitrary human invented formal system. There's no a priori reason it should work to describe and predict the natural world at all, is there?
We could invent other arbitrary formal systems based on completely different axioms and rules that _weren't_ useful to describe and predict the natural world at all, could't we? I think?
I don't know where this characterization of "proof" comes from. It is merely an essay, and not intended as anything other than one. When I read it years ago, I did not get the sense that Wigner was asserting inherent mathematical truth.
>It focuses on cherry-picked phenomena that are easily represented by simple math, but ignores those where math struggles or appears to be inherently incapable.
Does he assert anywhere that any phenomenon will yield easily to mathematics? I don't think so. I think his point is that there's no good reason that even the phenomena that are well explained by mathematics should yield so easily to it. It's not "All of nature is surprisingly mathematical." It's "I see no good reason that a lot of phenomena is mathematical."
I'm too lazy to reread the essay, so I may be remembering it wrong.
Any particular "arbitrary formalized language of symbols" is validated by experiment, either explicitly or implicitly. Humanity/ culture keeps the string games that works and jettison the ones that don't; it is a slow, collaborative process. We have worked out a pretty good "arbitrary formalized language of symbols", but not because we derived these string games or intuited them, but rather because we test them daily.
To be a formalist doesn't imply that a person under-appreciates the mysterious power of mathematics, just that they don't believe in an underlying psuedo-Kantian "truth".
Also, as I understand it, advanced physics is necessarily done formally only, since our intuition is so ridiculous at the atomic scale and all the equations like Schrodingers are only postulated, then verified empirically.
Also, it is snarky and under-informed, but I think of this debate as Hilbert (Hilbert!) and Some Guy.
"The Unreasonable Effectiveness of Mathematics in the Natural Sciences" by Eugene Wigner https://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.htm...
> The first point is that mathematical concepts turn up in entirely unexpected connections. Moreover, they often permit an unexpectedly close and accurate description of the phenomena in these connections. Secondly, just because of this circumstance, and because we do not understand the reasons of their usefulness, we cannot know whether a theory formulated in terms of mathematical concepts is uniquely appropriate. We are in a position similar to that of a man who was provided with a bunch of keys and who, having to open several doors in succession, always hit on the right key on the first or second trial. He became skeptical concerning the uniqueness of the coordination between keys and doors.
> Most of what will be said on these questions will not be new; it has probably occurred to most scientists in one form or another. My principal aim is to illuminate it from several sides. The first point is that the enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and that there is no rational explanation for it. Second, it is just this uncanny usefulness of mathematical concepts that raises the question of the uniqueness of our physical theories.
(The article happens to mention the Brouwer-Hilbert controversy but only in a footnote).