W4U doesn't explain his underlying intuition very well.
I think that he's claiming that reality displays structure. He's also claiming that there's something "mathematical" about it. It seems to me that physics agrees. Otherwise, what's up with all those RPenner-style chalkboard hieroglyphs that theoretical physicists love so much? What relationship do the mathematical squiggles have to... physics? To physical reality?
Confusion starts to seep in when the word "mathematics" appears, because it's seemingly being used to refer to very different things.
On one hand, 'mathematics' refers to a system of symbols invented by human beings (number marks like Roman numerals etc.) that correspond to something (what?) that's more abstract: numbers themselves, geometrical shapes, abstract algebraic structures or whatever), and to the logical relationships that human beings intuit as holding true between these symbols compounded in various ways. I'm not sure that I agree that it's all a human construct.
There's that peculiar objectivity to it that I mentioned above. We assume that valid mathematics isn't just true for the person that invented it, it's true for everyone. I think that we can say that number systems (decimal, Roman...) are constructed, but numbers themselves seem (to me, anyway) to be something more objective. Logical systems are much the same. We invent a bunch of arbitrary symbols for variables and constants and logical connectives. We define our logical connectives various different ways. We construct different logical systems employing different sets of these things. Yet the technical properties that logicians try to prove regarding these various logical systems seem to me to be objective. If you construct a logical/mathematical system like this (your choice) then it's going to have those specific properties (not your choice).
And then there's W4U's structure of reality (that he insists on calling "mathematics" as well, confusing all the rest of us, and probably himself as well).
Probably the single thing that most contributed to the "Scientific Revolution" in the 17th century was the application of human-created mathematics in the formal-system sense, to simple physical processes like simple falling objects, pendulums, motion on inclined planes, and interestingly, the motion of the planets in the sky. These just happened to be physical phenomena whose underlying structure, the physical principles that govern them and that they exemplify, was simple enough that they could be effectively modeled using the mathematics (in the logical system sense) that the early moderns were familiar with. So Galileo, Newton and company produced successes that caught the imagination of their contemporaries. And physics was off and running.
So the underlying truth of physics as a science seems to me to be that human-created mathematics (or the abstract structure that it exemplifies) is isomorphic somehow (shares the same form) with the structure of how physical reality behaves. I don't know how to explain it, though I suspect that it has something to do with whatever makes abstract mathematical proofs objective rather than subjective. This allows us to use our human-created/discovered mathematics to model whatever is happening in physical reality.
Those models may be more or less accurate. They will almost always involve simplifications. There will likely be more than one model for a particular physical event, with different models having different advantages.
I get the feeling (sometimes) that this is all that W4U is really saying. There's something, some objective structure that our mathematical constructs imperfectly capture, that's also exemplified in physical reality. That's why physics works.
I don't want to let that rather uncontroversial idea push me off on weird metaphysical tangents. I certainly don't want to say that reality is nothing but "mathematics" while smearing the different senses of that word mentioned above together. But I'm happy saying that there's something about physical reality that our human mathematical ideas capture, and that's what makes our physics successful.
Regarding grand metaphysical conclusions, the best position to take is probably agnostic. We need to admit that reality is mysterious and we don't really begin to understand it. We find ourselves in a thick fog of the unknown with our tiny little intellectual flashlights like science, that don't penetrate nearly as far as we'd like. It's just foolishness to assume that we are seeing everything.