or of our mental representation of it?
Edit: Michael Dorfman raises the point, "why would it be amazing that "extremely complicated mathematics" would model the physical world? ... In fact, it would seem that given mathematics of sufficient complexity, one should be able to model anything; if one finds an object or system that resists modelling, one could easily add new axioms or arbitrary complexities until one finds a suitable result."
In response, I claim that it is exactly the fact that this latter does NOT happen that draws some people to prefer pure mathematics above other fields, especially in the humanities and social sciences. The major mathematical investigations in history have always rested on just a few collections of primitives and first principles that change slowly or not at all. And often (as in the removal of the parallel postulate), major reconsiderations of these primitives and principles lead to their simplification, not an introduction of "arbitrary complexity".
When it was first realized that our universe seems not to be Euclidean space, but rather a pseudo-Riemannian manifold, arbitrary complexities were not introduced to the mathematics to better explain the physical world. Indeed, this part of the deductive structure of mathematics was developed well BEFORE physicists had use for it. The definitions needed to defined these manifolds were very old, just needing sets and the usual notions of analytic geometry (to describe a topological space, and then say what it means for that space to be a manifold with structure). All that had been added to the theory to produce the needed manifolds were the right definitions and deductions made from those definitions.
The symbiotic relationship of pure mathematics and physical theory includes contributions the other way, with physicists realizing that mathematics should be able to deduce new truths that explain phenomena they observed. I have never heard it suggested that "arbitrary complexity" be introduced to explain these phenomena --- rather, the physicists are somewhat content to wait for mathematics to produce the requisite addition to the deductive structure through the usual processes.
I hope this better explains my question. Now as far as how "well" mathematics seems to explain physical phenomena, in terms of "approximating", I have a less well-formed notion. But the vast amount of deductive structure built on both the primitives of mathematics and those of physical theory could lead in very different directions. If we had poor ability to ascertain the primitives of the physical world, we might at best only be able to construct mentally a poor facsimile of the physical primitives. Then perhaps, near the base of the deductive structure, there might continue to be some resemblance between mathematical theory and physical theory. But I would expect that if one moved far enough up the deductive tree, building more complicated definitions and theorems about them, we would eventually see something analogous to sensitive dependence on initial conditions and see larger and larger divergences of the two structures. The fact that even some of the deepest (highest? my tree is upside-down!) mathematics we have developed continues to be connected to physics (i.e. connection between the Langland's correspondence and string theory or the no-ghost theorem from string theory leading to mathematical construction of the monster Lie algebra) seems to controvert this. The relationship is so symbiotic that at this level, it is hard to distinguish between mathematicians and physicists (also due to the (present) dearth of falsifiable predictions by versions of string theory).
The last three sections of Wigner's essay (in Joseph Weissman's comment below) pose what I believe to be the same question. He concludes with, "The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning."