(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...
(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...