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I'd like to start this comment with a question:
When you talk about a "Platonic world" you mean a distinct, clearly defined world, completely different from the physical world, or is it a short-hand term used to describe the underlying order and internal coherence of physical phenomena?

If the former, I don't think a discussion is possible, because: where do you draw the line? As you very well put it in your article, a hypothesis such as that would be, for all practical purposes, virtually indistinguishable from a religious one, because it entails assuming all order and coherence are derived from a world to which we cannot have access empirically, and is thus impossible to prove scientifically.
As Albert Szent-Györgyi once put it "Thus Aristotle laid it down that a heavy object falls faster than a light one does. The important thing about this idea is not that he was wrong, but that it never occurred to Aristotle to check it."
My point being that the empirical aspect of scientific inquiry is just as important as the logical/mathematical one, as Galileo himself so aptly proved.

If the latter, then I don't see why the formal justification of mathematics would be wrong, provided you assume that the symbols (and their possible relations) are adjusted to coincide with physical phenomena, and, to a great extent, are able to predict them.
According to this definition, mathematics would ultimately be a scientific enterprise concerned with the coherence and consistency of sets of symbols and relations we invent as we come across new phenomena, and, since these relations can be studied without empirical analysis, mathematics can go further than other, more heavily empirical sciences, which allow it to predict the behavior of phenomena we have yet to come across (assuming these phenomena follow the same underlying principles as those encountered before).

I hope the above made sense, and that I did not misunderstand any of the concepts you used.



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