Mathematical and Physical Spaces

Question

I think that mathematical spaces that are homeomorphic to the Euclidean space are physically real. By extension, all mathematical spaces that are homeomorphic to such spaces would also part of the extended physical reality. The advantage of such classification is that such a list would provide us with a set of mathematical spaces that would be useful to advance physics. Contrariwise, mathematical spaces that are not homeomorphic to the Euclidean space won't be useful in physics. By extension, a set of mathematical spaces that are homeomorphic to each other form a distinct mathematical universe. I would like to know if this is true based on examples so far and arguments for and against.

Answer 1

I'm afraid: I do not see any reason why Euclidean spaces should stand out as the only candidates for topological spaces relevant for physics.

Euclidean spaces are not compact. While the unit circle, the 2-dimensional sphere, the 3-dimensional closed ball are all compact spaces. It is apparent that these topological spaces are relevant for physics.

I think one cannot decide about your issue on the basis of a-priori reasoning.

Answer 2

Physics makes use of spaces all the time that don't map into the (3-space + time) that we inhabit- for example, the infinite-dimensional Hilbert space used to describe the fundamentals of quantum mechanics. This paradigm is immensely important and useful and has without doubt greatly advanced physics.

Answer 3

Euclidean space consists of an uncountably infinite number of dimensionless points. There is no physical theory of any such thing, and it's unlikely that there ever will be.

For one thing, the Planck length limits the ability of physics to meaningfully talk about anything below a certain size. Dimensionless points of zero size are not discussible in current physics and are not going to be discussible ever. All physical measurement is approximate, and we could never measure something with zero size and dimension.

Secondly. there are no infinite quantities of anything in the physical world according to contemporary theory; and it would be a huge revolution in physics if that were to ever change.

Your proposal implies an uncountably infinite set of dimensionless points, in one-to-one correspondence with the mathematical real numbers. That is very unlikely to ever be the case. For one thing, it would make all the mysteries of set theory, such as the Continuum hypothesis and large cardinal axioms, subject to physical measurement. That is, physics postdocs would be applying for grants to count the number of dimensionless points in a meter of space, to see which Aleph number it is. That's not sensible. It's not going to happen.

Euclidean space is purely an abstract mathematical fiction, which happens to be extremely useful for scientists to model continuous processes, although it is not physically real. The mathematical real numbers are far too strange to be physically meaningful.

In short, Euclidean space is a spherical cow. It doesn't exist. It's an idealized abstraction that's useful to model aspects of reality with. It's not reality itself.

Answer 4

Real, physical space is not Euclidean. Locally, it is bent by gravity, most obviously near black holes, so we can be sure that it is not everywhere locally Euclidean.

Also, we do not know its global topology, so it is impossible to say what mathematical spaces might or might not be homeomorphic to it. Ignoring local distortions, it does appear to be flat within our experimental accuracy. But there are several such spaces, especially toroids; Euclidean space is only one possible solution. And it may be globally curved on a grander scale than we can detect, anyway.

Answer 5

In this animation both sets of orbital predictions are displayed in Euclidean space, although the space of causation is different for each. https://en.wikipedia.org/wiki/Schwarzschild_geodesics#/media/File:Newton_versus_Schwarzschild_trajectories.gif