@EnPassant,

I think your argument is right. As far as I can see, there is no possible alternative reality in which mathematics cannot describe the physical characteristics of the universe. Any such alternative would be unstable and beyond description. The effectiveness of mathematics in the natural sciences isn't "unreasonable". It is ineffectiveness of mathematics that would be unreasonable.

Is it true that "Chaos is chaotic beyond imagining"? I think it is though we might wish to rephrase the argument to avoid the criticism that it's only because we lack imagination. What I'd like to say is that the laws of mathematics aren't as separate from the laws of physics as some of us might imagine. To say that one apple plus one apple = two apples isn't just about numbers. It's true of apples, and also oranges, houses, words, and all other discrete objects. Apples are countable. Their masses and velocities are measurable. They don't wink in and out of existence. Their mathematical properties are as much a part of their being as their mass, their extension (i.e. occupation of space), their color, their taste, and everything else about them.

If countability, addition of velocities, and other descriptions of apples did not follow mathematical laws, then the very most basic principles of our physics, and the most fundamental descriptions of our universe, would not obtain - including the law of conservation of energy. We would not exist either since we too are countable and subject to gravity, additions of velocity, and so on. Your statement about this that "... Once you have space you have law/mathematics." seems to me to be right on target.

I think the argument that you and I (if I understand you correctly) are propounding is intuitive but it probably needs a real philosopher/mathematician/physicist beyond my limited knowledge to properly flesh it out.

Alan

I am not knowledgeable about mathematics or the philosophy of mathematics. What I have to say here may be totally wrong but reading the discussion stimulated me to consider some other ideas on the subject and, what the heck, I'll offer them and see what people think.

First, I'd like to call attention to the law of the conservation of energy. I'll argue that the law implies that the physical world is describable mathematically and I'll also argue that violation of the law requires that something can come from nothing and also that something can vanish into nothing - features of the universe that, for thousands of years have been considered illogical, though I understand that some of our quantum mechanics experts now have reason to believe that the ex nihilo fit principle is not exactly what we took it to be.

When an object strikes another object, if chaos results, that is to say, if the two objects can bounce off in any directions at any speeds, then there must be a net gain or loss of energy in the interaction. If that is so, then if we give up the rule that mathematics must be observed in physics, we also must give up the rule that energy is conserved. That seems to me to be a lot to give up. One wonders if there could even be a universe in which that is possible.

I can't prove it but I'm inclined to believe that "our" universe is mathematically describable, not because of the anthropomorphic principle, but because any universe - even any universe in an "expanding multiverse" in which physical constants may be different, must be so describable. It is the nature of existence that that be so. If it's not, then I suspect that logic and reason will go out the window along with mathematics. Or perhaps I should say, there couldn't even be a window for anything to go out.

Alan