In the past, new PhD students would ask how they could succeed when they had to compete with the likes of say, Richard Karp or Avi Wigderson. I would say Karp and Wigderson have limited bandwidth and you can work on problems they don't work on, or think deeper about a problem than Karp or Wigderson has time to.
Now we get the same question but with names like Claude and ChatGPT and it's hard to make the same bandwidth argument. What do we tell them as we get closer to Math AGI?
What even is Math AGI? It's not that every math problem gets solved. I don't expect P vs NP to be solved anytime soon. It would require a completely new approach, and AI doesn't (yet) think outside the box, though it has a very large box.
Math AGI means that with rare exceptions, if AI can't solve a math problem then no human could either. If you need a proof, you'd have to pay for more cycles, or wait for the next new and improved model. Like the Turing test, we'll only truly realize we've reached Math AGI once we've gone well past it.
We haven't reached Math AGI yet and we may never fully get there. We have entered the world of Centaur Math. Mathematicians can still prove theorems AI can't, AI can prove some theorems mathematicians haven't yet proven, but the real strength comes with mathematicians and AI working together. Working with AI today is like having a pretty good PhD student, who has a huge broad base knowledge of mathematics, is a whiz at coding, but still needs direction, encouragement and verification.
Chess had a short centaur moment when humans and AI working together could beat the best human players and AI programs. Now, any human would play worse not following what AI says. Nevertheless, we still enjoy watching two sub-AI humans play chess against each other. I doubt the same would hold for sub-AI mathematicians.
So what do we tell the students? If you love math, do math. Embrace AI, use it to go further, not as a crutch. Challenge yourself and remain agile so you can find success whatever the future might hand us. And remember, math is not ultimately about the theorems we prove but how we understand the principles behind them, and that's a human endeavor not a machine one.
"So what do we tell the students?"
ReplyDeleteNow that you have AI tools that make many things easy, learn more and understand more to ask questions beyond the abilities of these AI tools to answer without your help.
All good things come to an end.
ReplyDeleteThe glory days of Theoretical Physics ended on the day when Richard Feynman spent more hours on his bongo drums than on physics. The glory days of Pure Math ended on the day when Terry Tao spent more time on AI models than on searching for structure. The glory of Theory CS started fading on the day when Illinois Tech trumped up claims of financial exigency and unemployed Lance.
What will possibly be crushed brutally by AI is ego. We still get surprised watching some people can do fairly complicated multiplications without the use of pen and paper, although calculator was invented centuries ago. Robots and cars run faster than humans, but I still enjoy watching running races. Once P vs NP gets solved, possibly by Math AGI, I am sure there would be another problem designed by someone which needs math super-AGI to be solved, and this path drives itself forever :), yet we will still enjoy a good Late, possibly a fine peace of stake, and a poker or chess game on sunny day at a snowy mountain.
ReplyDeleteThe ridiculous banality of it all is excruciating, the intelligence sucking generated content are proliferating like mosquitoes after a summer rain, it's all the same font, the same ridiculous two/three column format, the same bulleted list structure, the same swirly graphics; generated videos are all the same voice over, same looking generated faces, same looking scenes.
DeletePeople seem to now be literally scared of THINKING lest they fail (and who knows maybe learn from failures), all they want is to punch in a prompt and get that hit like a drug junkie; when asked a follow-up question they either simply spit out what the AI says or retort that AI could not solve this; so good luck with your problem.
The funny thing is Charlie Chaplin envisioned all this in his Modern Times; even he did not however foresee that mechanization was coming for thinking as well; Fritz Lang's Metropolis is a more serious take from the same period (both movies are also contrasted here but with a more AI friendly slant https://www.shelidon.it/fragmentationai/).
Will skilled theoreticians be needed to sort the valid proofs from the nonsense ones?
ReplyDeleteA few months ago, Lance blogged about how the P vs NP problem is hard, and suggested trying easier problems. I commented that I'd be curious if AIs could get anywhere on something smaller, like detecting triangles in 8-vertex graphs, but didn't know enough Lean to try formalizing it.
ReplyDeleteIt turns out that I was able to use Antigravity to formalize circuits which detect cliques in Lean. The AI assistance was key in, like, getting the array sizes to type-check (as I don't know much Lean).
I'm not sure if this formalization of the problem is correct. (I mostly used tab-completion; it might have been better to purely vibe-code it.)
Lance mentioned computational complexity (CC) as being hard to formulate. For this particular problem, I'd disagree slightly -- I think formalizing the question is not too bad. It seems like one of those things, like the Collatz conjecture, where the definitions are simple, but results are complicated. (I'm sure there are parts of CC which are hard to formulate.)
I completely agree with Lance that actually proving circuit lower bounds sounds really hard. Even for a tiny problem, like detecting triangles in eight-vertex graphs, and even for AIs...
Let's take a deeper look at what really drove these AI's big math breakthroughs.
ReplyDeleteThey generally didn't come from the bots inventing anything new at a conceptual level, or even applying logic in some clever way. They came from the bot cutting across many hitherto isolated information silos, combining pieces that no human had put into 1 common workspace before.
The implication for students and researchers is this: Don't over-specialize. Break down information (and institutional) silos that can hide the fix for a problem in subfield-A from the people still stuck on similar problems in subfield-B. Learning how to efficiently order around the robots is going to be a critical skill for many subfields, just as knowing how to program already is for some others. And the initial gold rush will be in fields like combinatorics where the content is already very user-friendly to machines (and humans), where progress often comes down to changing a specific number- a bound, an algorithmic constant, etc.- or providing an explicit (counter)example.
Longer term, a mastery of critical thinking and abstraction themselves will be the key skill. Even if truly strong AIs are finally built, humans themselves prove that having a conscious brain isn't the same thing as using it to think clearly. Even major AI breakthroughs will require the same sort of fleshing out, feedback, and refinement of the new content into an optimal form (from the models to the notation) as the human ones did. Building a robot Einstein won't matter if it's only given auto-complete jobs for patent applications.
I still found human results more interesting from content perspective.
ReplyDeleteHere is a nice new paper to read, enjoy:
https://arxiv.org/abs/2608.30945