this paradox is particularly useful in demonstrating my argument that physics is essentially impossible to do until we understand space.

put simply, we need the following work flow:

1. understand the space we exist in (space in a kantian or descartian sense, not outer space).
2. start math over from scratch, with a proper understanding of space (in fact, we can find most of the work already done).
3. reconstruct the physics using the math that now exists, which properly understands space.

what we have right now is something more like:

1. do physics that needs complicated math to understand space.
2. ask the mathematicians for it.
3. rely on their expertise that it's "right".

meanwhile, the mathematicians are being perfectly open about the fact that the math they're doing makes no attempt to verify whether it's valid in the universe we inhabit or not. nor do the mathematicians care if the math they're doing is true in this universe or not, either. the circle completes: that's a physics problem.

there was this argument advanced by the likes of kant that math is the perfect representation of perfect knowledge, and philosophers actually ran pretty far with it, but, while kant was writing, gauss (a very competent and very famous mathematician) was actually in the process of disproving exactly what kant thought was perfect knowledge. oops. regardless, this kantian delusion has really set hold in the minds of physicists, for some reason. you'd think physicists would listen to gauss instead of kant! not so, though.

(of course, physicists listen to gauss instead of kant every time they do relativity. but, as they're doing relativity, they repeat the kantian lie that mathematics is some kind of language of nature. the problem is that nobody makes physicists study their own history, or take a credit worth of philosophy classes.)

the math itself is a model. that's what mathematicians will tell you: math is not a perfect description of space, and nature doesn't adhere to it as a script, but is merely a model to better understand it. but, it's a non-empirical model. and, we know from experience that non-empirical models always fail.

maybe you've heard of this, maybe you haven't. if the system of mathematics allows for this, that system is obviously not modeling our universe very well. we should consequently expect that any physics that relies on a flawed system of mathematics will also be flawed; that a system of physics based on a mathematical model that is full of contradictions and paradoxes will also be full of contradictions and paradoxes.

https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox