Toby on Something Special About Three Dimensions

This actually has explanatory power!

See Toby's Video About Madame Wu and Parity and her video about Roger Penrose’s impossible 4D object

Actually watching that again I realise that she didn’t say this was the 4D object Roger Penrose said he had invented. Someone ought to ask Roger Penrose. I recall him saying in an interview that he once invented (I suppose he would say he discovered) something that, if a 4D person stumbled across it they would say “My God, what’s that?!”

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Henry Segerman and co did a nice video about cohomology that I didn’t really appreciate the first time I saw it:

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What does that his have to do with knots? I don’t know, but I think it’s probably a lot:

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It’s all to do with Stokes’ theorem, apparently:

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You could have invented homology:

Part 2:

 

Part 3:

See the full set of three videos in one playlist.

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Gabriele Carcassi on why space is three dimensional.

I have serious trouble following that argument, which is probably why I am not allowed in their Discord group. I don't understand why a physical degree of freedom implies two independent variables, it's all confusing to me (1:51). Maybe Matroids would help. According to Wikipedia, in classical mecanics "conjugate variables are related by Noether's theorem, which states that if the laws of physics are invariant with respect to a change in one of the conjugate variables, then the other conjugate variable will not change with time (i.e. it will be conserved)." Ah wait, he explains why two independent variables at 8:45!

He also did a talk about The equivalence between geometrical structures and entropy.

See the book: https://assumptionsofphysics.org/ 

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I like arguments like this one. This is particularly interesting because it is the theorem, the proof of which led L.E.J. Brouwer to the idea of intuitionistic mathematics when he found his own proof of the fixed-point theorem unconvincing:

The fundamental distinguishing characteristic of intuitionism is its interpretation of what it means for a mathematical statement to be true. In Brouwer's original intuitionism, the truth of a mathematical statement is a subjective claim: a mathematical statement corresponds to a mental construction, and a mathematician can [justifiably?] assert the truth of a statement only by verifying the validity of that construction by intuition.  

So in Brouwer's view mathematics is an activity of a community of people who develop a shared intuition. See Jennifer Nagel on Epistemic Collaboration and Common Knowledge. 

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