Plato pondered the relationship between physical reality and mathematics 2500 years ago. He used the allegory of his cave to show that we see traces of structure in the physical world, but we do not see the full picture. I don’t particularly hold to the idea of Platonia, but the idea is food for some thought. It certainly suggests that we may never know the answer to whether reality is ultimately some sort of mathematics which is reified by some means. Penrose has suggested this reification is the existence of mind, where a mind is a way that this Platonia has to become aware of itself and observes this system as “reality.” Again, I don’t know whether it is worth embracing this as Truth, for I see no way it can ever be verified.
Tegmark has proposed some ideas along these lines. He argues that mathematics, or that set of it which is first order and halting and non-halting if it converses to some describable set, in his mathematical universe hypothesis defines the entire set of reality. The ensemble has some statistical weight to its elements which defines the probability the mathematics exists in some “universe” or exists in some reality. Of course the problem is that this set is infinite. An infinite set is one which admits a bijective map from the set to any nontrivial (finite) subset. Such a set of maps will include the Cantor diagonalization and as a consequence the Godel incompleteness issue. So it would appear that Tegmark’s system is computed within the Chaitan halting probability function, which is itself not computable. We may then never know if this system has any bearing on reality.
It seems that a determination of the truth value of these conjectures require that we somehow access information or knowledge which is outside of physical reality. Without doing that this question might amount to chasing one’s own tail endlessly. It is very unlikely this question will be satisfactorily addressed, even as a hypothesis supported by some data set, by ordinary means of science. So in general this question is best something thought about in evening over scotch and cigars.
To conclude one might ask whether reality is fully based on mathematics, or whether on some deep level it is based on magic. By magic we might mean supernaturalism or some theological imposition of an infinite will. If physics is not fully based on mathematics, then there is ultimately some sort of magic deep down in the “rivers of hidden funk.” The question might not be resolvable, but the negative seems far more disturbing.