Maths are models of reality which may or may not be correct and can only be verified by comparing them to reality. While they serve as the most simple way to explain reality, don't confuse them with reality. The map is not the territory.

What prompts this reply? I'm not saying that math is reality, merely that solid mathematical models have been the foundation of progress in science, not "explanations."

I just don't agree with what you're saying, progress comes from understanding reality via some intuition, understanding and intuition lead to the development of testable models. Testable mathematical models are the result of progress in science, not the foundation of it.

> Testable mathematical models are the result of progress in science, not the foundation of it.

I think that the information can flow either way. For example, the existence of anti-matter, and portions of quantum mechanics (and, let's say, much of Witten's scientific work—though here I am tossing out a bit of folk knowledge, and couldn't point to anything in particular), are examples of letting the math guide the physics.

The math certainly points at places to look for physics, so yes I agree to a point, but the math itself didn't just happen, it came from a mathematician using his intuition and understanding of physics to explore possible ideas that might be true with most of the possible maths not leading to possible physics. Math is just a language to communicate ideas accurately, those ideas however came from someones intuition about what should be. The math it led to is a result of a process, not the creator of it, the model the math lays out could be right or could be wrong and as such simply points out possible places for experimental physicists to go searching. The math didn't come first, the intuition came first, the mathematical model second, and the physical proof last.

> the math … came from a mathematician using his intuition and understanding of physics to explore possible ideas that might be true with most of the possible maths not leading to possible physics.

This is famously not true for the matrix mechanics in Heisenberg's picture of quantum mechanics and the Riemannian geometry underlying relativity, both of which were created without (so far as I, and history, know) any intended or perceived connection to physics.

On the other hand, I guess you could argue that that math was irrelevant to physics until Heisenberg and Einstein came along and found it an appropriate expression of their physical intuition.

> I guess you could argue that that math was irrelevant to physics until Heisenberg and Einstein came along and found it an appropriate expression of their physical intuition.

Exactly.

Just the math is not enough, you need some connection to reality. For instance, your formula does not describe reality for large v. This can be tested.