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(commenting on part I because there's no comment box on part II)

The idea that virtual reality will lead to significant new results, or new proofs of old results, seems dubious to me. I am tempted to promise that if there is a VR proof of the Poincare conjecture I will quit mathematics, both because of how unlikely that is, and because I would be disappointed with the state of mathematics if literal hand-waving becomes an accepted measure of proof. (I know this is a bit at variance with my other comment.)

We can look at a series of different visual tools roughly ordered in increasing depth or fidelity:

Diagrams < Drawings or computer graphics with the illusion of 3d < actual 3d objects < videos < video games < VR experiences.

Certainly at least the first four have already been useful tools in understanding mathematics, though I think only diagrams and drawings have really been used in mathematical proofs.

The difference between VR experiences and videogames in terms of their applicability to mathematics seems less to me than the other differences on the list. A VR experience is basically a videogame that you control with your hands or other body movements, where the screen is pressed up against your face. Neither of these seems like a significant change compared to the ability to manipulation unphysical three-dimensional objects in real time that videogames offer.

But have videogames been used in mathematics in a significant way? As far as I know they have not. So I don't think VR will be useful either.

But writing this I fear that comment may have been more a joke than I understood at first...

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I actually envisaged envisaging a VR system that would produce a standard mathematical proof upon manipulation. So when my virtual avatar squeezes a 3-manifold the VR works out the inequalities. As long as we're speculating, it's no less plausible than automated translation of a standard human proof into a formal language.

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Oct 19, 2022·edited Oct 20, 2022

But wouldn't this require just as much, if not more, AI work, to translate from the 3d movements into inequalities? The point is that the process of translation cannot be done purely mechanically, by following formal rules, and must require some creativity in interpretation, which, if it is coming from a computer program, will likely require machine learning techniques.

As I gestured towards in the other post, I do think it would be reasonable to speculate about how such a system could work and phrase it as a challenge to AI math people! When you have a group of people, at least some of whom are prone to making very bold claims, trying to get them to claim to be able to do something you actually want is a win-win: either you get what you want or you prove them wrong.

Maybe you could interest Meta's AI team...

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I'm just putting out hypotheses, and Meta is only a placeholder for VR research; I would hope less toxic alternatives will emerge.

No doubt such a system would be very hard to design. Even a system that fell far short of providing the basis for a rigorous proof could help to develop intuition and conjectures.

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