Good question, and maybe I should have provided an alternative description rather than just criticising.

Unfortunately, it is actually surprisingly hard to pin down, and I think mathematicians (and, as a student, I count myself as one to some degree at least) now have the task of making this a lot clearer. If we want to justify our existence in the face of new machines that can seemingly ‘do our work for us’ (so far in a restricted context), we should give a robust defence of our practice. If we can’t do this, we simply don’t deserve the funding (which, by the way, again contrary to some misguided statements here, isn’t very much anyway!). I think all of this will become clearer to outsiders as time passes, but for now it’s not easy to give a quick answer — though I can try.

Mathematics is about understanding things. Isn’t that what every subject is about? Well, I suppose so, but mathematics more specifically does something like the following:

(1) observe some phenomenon in ‘reality’.

(2) attempt to formalise that phenomenon in such a way that it can be manipulated purely symbolically.

(3) use this (perhaps fairly arbitrary; remember that we can invent as many formal systems as we like) system to deduce from our initial assumptions new facts that would otherwise have been very non-obvious.

It seems like outsiders have a decent grasp of (3) and the application of AI to it, but have very little idea about the other two steps. It seems to be widely assumed among non-mathematicians that problems are essentially god given and that the job of a mathematician is therefore to chug away on these problems, manipulating symbols and trying out tools, in the hope of learning a yes/no answer to each one.

The first two steps are by far the hardest and most important, and they’re also the parts that AI seems currently unable to help with.

NOTE: this is not a deeply insightful description of what the subject is about, and there are many better characterisations out there. I think Tao and various others have written recently about why complicated and inscrutable AI-generated proofs aren’t nearly as valuable as one might imagine. (That’s not to say there’s no value to such proofs; perhaps in time, as technology improves, mathematicians will come to accept AI as part of the process.)

If you want to understand all of this issues better, reading the recent slew of guest posts on Tao’s blog would be a very good start.

My understanding when I was practicing is that the trend in modern mathematics is to focus on spaces with a certain kind of structure, and maps between those spaces that preserve it, and then what are invariants are preserved by those maps. Structure-preserving maps between categories of such spaces - "functors" in the language of category theory - are especially neat.

That's certainly different from Olympiad-style problems.

Pure mathematics reduces to abstract "relationships" and their implications largely.

To know what something is about, a natural way is to do it yourself.

Wondering about the world, using the simplified language of "mathematical logic"

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