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First, I link the following video: https://www.youtube.com/watch?v=Vfh21o-JW9Q

It demonstrates the 'Dirac belt trick', which was created by (I believe, Dirac) to demonstrate the calculus of spinors. Particularly, it demonstrates the double-covering of the 3-dimensional rotation group SO(3) (essentially a set containing all the possible 3D rotations) by the group SU(2) (a set containing 2D complex matrices). A 'covering map' essentially describes the way that one space can 'wind' around another.

The demonstration of this trick is absolutely independent of science - we live in a (locally!) flat 3D plane which we can assume to be normal 3D space and demonstrate the belt trick. This belt trick is mathematical truth - not science. Science does not actually have anything to do with this. It's just a mathematical fact about the topology of rotation group SO(3).

We can derive the mathematical fact and then see it in our real 3D world.

Does this prove that mathematics is real? Does this prove that 'abstract' mathematics is indeed real and right in front of us? Can I therefore justify that my mathematics is not 'made up', and rather is indeed real, if you look in the right places?

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    When some philosophers say "mathematics is real" they don't simply mean it's useful at describing reality.
    – Fizz
    Mar 30 at 10:23
  • I'm gonna try to find a more philosophical take on this, but in the meantime, you could look at "does infinity exist?" merely from a mathematical standpoint. The answer is actually not as obvious as you might think.
    – Fizz
    Mar 30 at 10:40
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    What you talk about here is called the "Indispensability Argument" iep.utm.edu/indimath
    – Fizz
    Mar 30 at 10:56
  • "Another line of criticism of the indispensability argument is that the argument is insufficient to generate a satisfying mathematical ontology. For example, no scientific theory requires any more than א1 sets; we don’t need anything nearly as powerful as the full ZFC hierarchy. But, standard set theory entails the existence of much larger cardinalities. Quine calls such un-applied mathematics “mathematical recreation” (Quine 1986b: 400)."
    – Fizz
    Mar 30 at 11:13
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    How about the trick of untying any knot in 4D, which we never get to see no matter how hard we look? Does it prove that abstract mathematics is "unreal"? Or does it prove that "real mathematics" isn't mathematics at all, but rather physics of our space mathematically described, which happens to be, approximately, flat 3D.
    – Conifold
    Mar 30 at 11:19
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The Dirac belt trick is simply experimental evidence that our universe has three macroscopic spatial dimensions. If our universe had more than three macroscopic spatial dimension then the belt trick would not work (although there are more direct ways of demonstrating this e.g. seeing how the volume of a sphere is related to its radius). And the belt trick does not rule out additional microscopic dimensions, which the belt is simply too large to “notice”.

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There're several major schools of realism views towards math. Platonism long ago already believed math objects as real forms existing independently of our physical world. Modern Structuralism instead regard internal structures as real. Formalism and Constructivism (including Intuitionism) usually regard math as merely symbolic rules or subjective mental constructs without independent math reality. From here u even have Fictionalism which looses math to mere metaphors, seems possible from their common etymology. And most importantly, as always in metaphysics, a single trick most likely won't sufficiently prove any logically sound ontological view since it's not a strictly closed deductive system. As @gandalf61 pointed above, the trick only shows our physical space dimensions.

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