Using mathematics to model physical reality cannot be a limitation because mathematics allows arbitrary structures, operations, and theories to be defined.This is an incorrect view of math.
No matter what new phenomena we discover, it will always be possible to describe our observations mathematically. It doesn’t matter how crazy nature is because mathematics is just a language for expressing our intuitions, and we can always add words to the language.
Math has been axiomatized as Zermelo–Fraenkel set theory (ZFC). The content of math consists of the logical theorems of ZFC. It is not just some language or some arbitrarily expandible set of intuitions.
I wonder whether many physicists understand this point. I doubt it.
I thought that everyone understood that math has limits, and the point of my essay was to argue that there might be no ultimate theory of physics within those limits. But if someone does not even accept that math has limits, then the rest of the argument is hopeless.
If you agree that math has limits and that they may not include possible physical theories, then it ought to be obvious that nature may not have a faithful mathematical representation. Saying that there is such a representation is an assumption that may be unwarranted.
So I guess I should have explained the ZFC issue better to the physicists who will be judging my essay.
An example of a limit of math is the unsolvability of quintic polynomials by radicals. That is what keeps us from having something like the quadratic formula for more complicated equations. This fact does not necessarily stop us from solving the equations, but certain kinds of formulas just won't work.
My essay "public rating" is not very high, but my "community rating" must be much higher. If you look at the list of essays sorted by community rating, then you will see that mine is near the top.
Update: Bob Jones argues below that ZFC is not enough because we need mathematical constructs like the Grothendieck universe that are outside ZFC. That is an interesting example, as noted here:
Colin Mclarty says that the big modern cohomological number theory theorems, including Fermat’s Last Theorem, were all proved using Grothendieck’s tools, making use of an axiom stronger than Zermelo–Fraenkel set theory (ZFC). He says that there is a belief that these theorems can be proved in ZFC, but no one has done it.I would really be surprised if cohomological number theory needs something more than ZFC. And it would be even more amazing if theoretical physics found some mathematization of the universe that could be formalized in an extension of ZFC, but not ZFC.