Some mathematicians claim they can visualize R^4, or even higher-dimensional spaces. I think this is probably possible for the human mind to do, in principle; the human mind often proves surprisingly flexible. But I think it probably takes a lot of concerted practice, and because you are stretching your mind beyond its intended limits in a purely-internal, imaginative way with no external reference points to check against, I think it probably has some things in common with certain types of meditative practice.
It's also common to hear mathematicians say that if humans had visual intuition for dimensions higher than 3, higher-dimensional geometry may have advanced much farther, since visual intuition is so often crucial in thinking of proofs.
So, ok, I suspect that AI will obviate the utility of this before it could ever have the chance to get off the ground, but this all makes me imagine a world where techniques for learning to visualize higher dimensions are well-known and practiced, and have become a functional necessity for being a working geometer in higher-than-3-dimensions. And these techniques require a lot of persistent practice and training, which (by the nature of the thing) is hard to precisely communicate to students. So part of becoming a geometer involves training in what is basically a meditative practice, where, à la Zen, much of the process involves not directly teaching the student but giving them prompts and mental exercises that are meant to trigger internal, incommunicable revelation. But instead of enlightenment it's geometric intuition. And so if we're getting a math PhD in, say, differential equations or something, it's mostly like it is today. But if your PhD is in low-dimensional topology there's like a whole monastic apprenticeship style thing that just comes along with it. People sometimes drive themselves insane. People sometimes drive themselves insane.
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