He's a Field's Medalist so that's automatically one of the most important living. He is good at explaining things; I leaned on his Analysis textbooks when I was taking analysis and functional analysis to great effect; in a research class I was trying to calculate Fourier transforms of algebraic sets and found various almost throw away comments on his blog that were extremely enlightening (alas only to the extent I could follow them). He's a legit great mind of modern mathematics - and also able to communicate well; a historical rarity indeed.

The topic being discussed here has been addressed specifically in his mastodon posts - the pre-LLM math process was understood to be "state a proof, validate a proof" - but the real aim was the not explicit "digest the proof so we can extend to other results" - with LLM/lean making the first two a lot easier and possible to do without accomplishing the digestion into the mathematical community; so we need to add "digesting proofs (from where ever)" into the job description. Thread starts here: https://mathstodon.xyz/@tao/116477351524980995

Not even the most dogmatic of the set theorists ever argued mathematics was possible without reason, however. For mathematics, logic is the world, as the copula makes no distinction between substance and existence. In the same sense that the earth is not matter itself, but it is a material thing.

Putting that aside, to make things more clear: computer science is mathematics. Computer scientists are mathematicians. That was a categorization decided long before you and I ever lived. In the sense that you mean, you're only referring to a very small fraction of what "mathematics" refers to In the true sense of the word. It is just as irreconcilably disjointed as Logic is, not unified and fundamentally non-unifiable.

I too think it would be better if "mathematics" was reserved for the gated suburb of ZFC. But that's not the world we live in, courtesy of the same people who pushed ZFC as a foundation to begin with.

I think that's a good way of putting it. I would addend that most people working in mathematics aren't generalists, their primary interest isn't in a broad picture. Rather, most are hyperfocused into a single domain with a strong backbone of reflexive intuition built up. By virtue of sheer human limitation, there's only so much someone can care about what's happening outside of their world while still making serious contributions within it. This doesn't even just extend all the way to shifting foundations, but number theorists can hardly be expected to keep up with the forefront of graph theory, for example.

For the pragmatists Logic as a field commits the immortal sin: it blasphemes the intuition that mathematicians spend years honing by obliterating it. Not just for a singular domain, but for all domains. Of course, that doesn't really explain the whole picture. Formalism built a holy walled city. Logicians, by nature of their work, leave the safety of the walled city to survey, exploit and die in the tangled jungle outside. Some don't even speak the holy language of the glorious walled city, they talk in absolutely gibberish modalities and hyperstructures. There is a political tension held against logic and logicians as a result.

Mathematicians care about interesting ideas, not whether their theorems are true :-)