Terry Tao is a next level vibe coder: he inspires people to do his vibe coding for him. As someone with a background in advanced math, though never even close to Tao's level, I find myself skeptical about this type of mathematics. I don't personally find it beautiful and it feels like the line between the profound and the trivial (as in of minimal importance not difficulty) is blurry. One could argue for pure mathematics that is of no practical utility but is aesthetically beautiful, but I struggle to see the beauty in a gargantuan lean proof constructed by 100 different people. Perhaps this work will lead to deeper insight about the universe and the human condition, but I catch a whiff of problem solving for the sake of problem solving untethered from a deeper sense of purpose and meaning.
> I struggle to see the beauty in a gargantuan lean proof constructed by 100 different people
Why does it need to be beautiful? Once you proved it it's true and you can use its consequences in math, sciences and engineerings.
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.
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
Great point, I think the argument you could make about Tao (fairly or unfairly) is he never tries to build that framework.
> 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.
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.
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.
I think what people find beautiful in math is largely something that enables the mathematics (or physics) to be translated to something that they can think about intuitively, and what people can handle in an intuitive way is largely an artifact of what the brain evolved to be able to think about "naturally". But it's quite possible that most things that are true about the universe or math are just ugly and unintuitive, and the pursuit of truth shouldn't necessarily be limited by what people can easily reason about and hold in their heads.
Beautiful explanations are lovely when they exist, but we shouldn't wait for them if we can also find the truth through an ugly method.
the analogy with experimental physics is a good one - being sure something is true is a good first step to developing an elegant proof of its truth.
And I thought it was cocaine.
Quantamagazine is essentially Renaissance Fund, which is heavily invested in AI.
This is a clever piece reminding people of Tao's pre-AI Lean efforts. Now, however, Tao and especially Gowers are receiving AI money and have AI positions so they are far from unbiased.
Or maybe they have caught Feynman's "computer disease"? Either way, this is a hype piece.
mathlib and lean are currently too cumbersome for many researchers to use in say algebraic geometry, but maybe more suitable for combinatorics where it has been applied recently.
More accurate title would be "Terry Tao Became an Evangelist for Lean"
the smartest people see AI as an incredible tool that enhances their productivity.
Just like me! I like AI because of how smart I am.
Terry Tao is a next level vibe coder: he inspires people to do his vibe coding for him. As someone with a background in advanced math, though never even close to Tao's level, I find myself skeptical about this type of mathematics. I don't personally find it beautiful and it feels like the line between the profound and the trivial (as in of minimal importance not difficulty) is blurry. One could argue for pure mathematics that is of no practical utility but is aesthetically beautiful, but I struggle to see the beauty in a gargantuan lean proof constructed by 100 different people. Perhaps this work will lead to deeper insight about the universe and the human condition, but I catch a whiff of problem solving for the sake of problem solving untethered from a deeper sense of purpose and meaning.
> I struggle to see the beauty in a gargantuan lean proof constructed by 100 different people
Why does it need to be beautiful? Once you proved it it's true and you can use its consequences in math, sciences and engineerings.
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
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.
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
Great point, I think the argument you could make about Tao (fairly or unfairly) is he never tries to build that framework.
> 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.
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.
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.
I think what people find beautiful in math is largely something that enables the mathematics (or physics) to be translated to something that they can think about intuitively, and what people can handle in an intuitive way is largely an artifact of what the brain evolved to be able to think about "naturally". But it's quite possible that most things that are true about the universe or math are just ugly and unintuitive, and the pursuit of truth shouldn't necessarily be limited by what people can easily reason about and hold in their heads.
Beautiful explanations are lovely when they exist, but we shouldn't wait for them if we can also find the truth through an ugly method.
the analogy with experimental physics is a good one - being sure something is true is a good first step to developing an elegant proof of its truth.
And I thought it was cocaine.
Quantamagazine is essentially Renaissance Fund, which is heavily invested in AI.
This is a clever piece reminding people of Tao's pre-AI Lean efforts. Now, however, Tao and especially Gowers are receiving AI money and have AI positions so they are far from unbiased.
Or maybe they have caught Feynman's "computer disease"? Either way, this is a hype piece.
mathlib and lean are currently too cumbersome for many researchers to use in say algebraic geometry, but maybe more suitable for combinatorics where it has been applied recently.
More accurate title would be "Terry Tao Became an Evangelist for Lean"
the smartest people see AI as an incredible tool that enhances their productivity.
Just like me! I like AI because of how smart I am.