[throwaway67678]

Arguments about beauty don't lead anywhere constructive because they are too observer- and context-dependent. Poincaré himself was decrying continuous non-differentiable functions as abominations. The monster group is, well, just like that. What feels intellectually ugly for one generation is natural for the next, and the field moves on

[potbelly83]

That's not what op is arguing. To use your example, coming up with singular examples of continuous non-differentiable functions is an example of "ugly" mathematics, whereas putting them into a nice framework where they can be analyzed as a whole (i.e. functional analysis, density of such functions, etc...) is an example "elegant and insightful" mathematics. The same with the monster group, on its own maybe nothing special, but then you have the connections with other branches of math. Tao seems so focused on the individual problems and not their connections/generalizations.

[throwaway67678]

Well one does have to come up with continuous non-differentiable functions to begin with, right? Weierstrass had to shock the community with his weird series that's almost everywhere nondifferentiable before people could conceive of a nice framework that includes them. People do not invent whole encompassing abstractions out of nowhere

[potbelly83]

Great point, I think the argument you could make about Tao (fairly or unfairly) is he never tries to build that framework.

[Ygg2]

According to legends Pythagoreans tried to surpress existence of irrational numbers because they couldn't be expressed as ratio of natural numbers

Supposedly even drowned their member that divulged their existence.

[zerobees]

> Arguments about beauty don't lead anywhere constructive because they are too observer- and context-dependent.

Meh. You can successfully argue that there is no objective anything. It's all just our perception and the emotions we associate with it. We built entire civilizations on subjective notions of good, evil, beauty, and so on. So where do you draw the line between "acceptably subjective" and "too subjective"? And are you sure it's not just a subjective code name for "the thing I don't like"?

Ultimately, people practice mathematics mostly for abstract reasons. It's not a field where you routinely ship products and get rich by meeting market demand. If 99% of contemporary mathematicians don't want to become prompt engineers, there's nothing that makes the transition to AI math inevitable. If not mathematicians, the only party with vested interest in that would be the PR departments of frontier labs.

[threethirtytwo]

Agreed, mathematics is ugly without ai. I feel beauty is in massive complexity and intricacy. Every time I see a small proof it feels too easy and trivial. Triviality and simplicity is ugly to me.