I wonder how we could formalize the notion of „interesting“ problems in a way that would allow us to automatically generate new interesting questions from the existing corpus of mathematics.

"Taste"

Let us know when it is clear that UCLA does not hire the candidate with the most top-tier journal papers.

It’s more likely that instead of spending a 100K/year direct grant on two PhD students, PIs will hire 1 and have the student spend 50K on AI.

He is right if model intelligence stalls. If model intelligence continues to improve soon there's no need for the prompter to understand anything or for any workshop as a mathematician will just be able to ask the model to explain how the proof works and models will do a good job at walking them through it step by step.

There will be no gap in understanding. Now there is because the models are discovering things at the edge of what they can do and so suck at explaining it. There's nothing particularly special about a newly solved problem in terms of learning it.

If we accept AI can explain all of existing math nicely, why shouldn't it be able to explain new proofs?

Related:

AHM Statement on OpenAI's October 6 Release of Mathematical Documents

https://news.ycombinator.com/item?id=50000421 / https://news.ycombinator.com/item?id=49999159

I think we can say by now that we should not listen to the early nay-sayers and just wait a bit. With every trend, not just "AI". They still have some points (the ethics and environment etc), but the we don't hear from the Stochastic Parrot folks anymore.

Of course it's good to have the discussion... So maybe, we listen to the nay-sayers, but defer judgement on the matter... That's wisdom.

Edit, to be clear, I consider Tao to be the wisdom provider, not an early nay-sayer!

Even when a problem got solved, there has always been value in publishing simpler proofs and corollaries that give better intuition into the broader field.

If I understand Tao correctly, he's saying that's going to have to be the focus going forward. I just default to thinking the models are going to be much better than us at that, too.

This is a distilled version of what people say about the tech industry in the past year or so. Replace math with any field, and the statement is still relevant.

A very balanced perspective, and the concerns he raises are reasonable. He acknowledges that AI is going to transform mathematics, but simply dumping proofs on the math community and expecting others to do the grunt work of verifying, refining, and expanding on them is hardly a productive way to advance the field.

There seems to be more interest in hitting some arbitrary benchmark (we proved X unsolved problems) than in genuinely contributing to mathematics. But what else is to be expected? It's become a maniacal race with too much money. Too much effort is being invested in proving that the exponential curve is still holding.

It's been very interesting watching Tao's evolution on his thinking on LLMs. Of course the LLMs have themselves evolved so that shouldn't come as a surprise.

The job of professional mathematician might be the first to be completely eliminated by LLMs, save for those who can make money from a patron. I am hoping they are able to figure something out to save their profession, as other professions could use it as a blueprint as AI comes for them next.

Does this response properly anticipate how math will change further with the next N model generations? Exposition and exploration may fall well within the capabilities of future models.

>the mere knowledge that a solution exists "contaminates" efforts by both humans and AI to find alternate routes to the problem that reveal additional insights

This really expresses the heartburn you see across all fields, not exclusive to careerism. I certainly have friends in decomp and fan translation spaces that have been demotivated by the current rash of efforts happening there.

The rush to be "first" has always been over-celebrated, but it would be nice to believe there's a way to get beyond that thinking.

I imagine doctors will also clutch their pearls when Ai starts curing disease. "But curing disease was never the point! These arbitrary dumps of AI cures for cancers is unsustainable! Who will think of the doctors and who will build their communities further? From now on progress in medicine must be redefined as what makes doctors thrive, not what generates cures!"

The same could be said of Software. Instead of giving up on creating novel projects and instead just taking other peoples ideas and porting them to Rust, we could be embracing AI to push software and computers farther.

Im not sure how that will work, but im convinced the current paradigm of just pushing agents into codebases for not much reason other than you can is going to make building software incredibly boring and push creative people away from the field and stagnate progress.

My prediction is software gets boring and building hardware projects will be the new frontier for creative engineers looking to push computing further. Which is probably a good thing.

I suspect the whole field of mathematics will simply disappear as a career path. It seems obvious that the trajectory is for the machines to be able to provide proof on demand for any solvable problem. Whether or not the proof is understandable by humans is perhaps irrelevant in the larger sense. Doing hard math will simply become another black box tool in the larger AI toolkit for goal optimisation. Is this sad and should we try to prevent it? Is it any less sad than the venerable London cabbie who spent a life time memorising every street to gain "the knowledge" and almost overnight supplanted by machine intelligence.

Most of the problems that have been solved are problems on which a great deal of progress had already been made. Those who work on well known problems posed by famous people are those who suffer the most from this. Those who do their own thing and pose new problems, on the contrary, benefit from it. Suddenly raw technical power and great memory are not so valuable as a broad perspective, structural insight, and wild ideas. Who can be successful in this new ecosystem is different. Some of the elites are (correctly) more threatened by it than some "mid tier" mathematicians. I see lots of opportunities to overcome obstacles in my research program some of which had confounded me for years.

On the other hand, it puts a premium on resources. AI is not cheap for mathematicians. Folks are fancy universities in rich countries with forward thinking ministries of science will have an advantage over the rest.

What is clearly in immediate crisis is the traditional model of doctoral education. Most of the problems that were "given" to ordinary doctoral students are solvable (quickly) even by something like Claude pro. Mathematicians need to adopt training models more like what is done in experimental and laboratory sciences - collaborative and structured.

Where Tao is wrong is in regards to exposition. AI already writes better lecture notes, problems, and exercises for mid level undergrad math classes than do most of my colleagues. It's exposition is generally well structured and clear and it can adjust level on request quite well. It writes research better than most professional mathematicians too.

Working heavily with LLMs for the past year has be nodding strongly with this approach.

AI only take us as far as our imagination thinks to ask it. This can be exhilarating when new models drop every month and we are continually reach a new threshold, basically for free. But it is a one time gain and ultimately short-sighted. Where I get continuous value is using LLMs to help my understanding.

I use LLMs all day long as a SWE and I have tried many approaches, but the most satisfying and consistent approach is to lean heavily into understanding a problem space and a solution space. Yes, it whips up architecture and code, but I spend most of my time peppering it with questions about the design and how it handles certain situations, what about this edge case and that security concern and this future product need. I have it write a report breaking down the feature and how it integrates with existing code and if the report is too confusing I have it simplify either the report or the code until it makes sense to me, sometimes scaling back the work to a more manageable state. I do all of this before I look at any the code.

The difference is I then am reviewing a PR that I fully understand. I can eyeball it quickly for anything that doesn't fit my mental model. Only after I am happy with the bones do I consider the meat and skin of it.

What I find most concerning is how frontier AI companies all seem to have this Math 1.0 perspective that they only want to type "solve Riemann" into the chat box and have the magic to happen. It is the same problem Google ran into, where a simple, no thinking solution serves most of the people best and most profitably, so you fully ignore or remove everything else (boolean operators, exact phrase search, verticals, filters, infinite pages of results, "nothing found" if there isn't, etc.) But that choice leads to the situation Google is in now, scrambling to stay relevant. In a different world, Google would have continuously augmented their search capabilities and eventually built a smooth, guidable AI interface.

But no, we must only have an input box and a Go button.

Everything looks like a nail when you build hammers, sell hammers, have infinite hammers to play with however you like and your company mission is to build a hammer starship to explore the hammerverse, whether or not that is even possible.

Can we please stop reducing human activity to "taste", conferences, talks, "understanding"? I think this is a very unproductive trap.

There is a world where we get to the edge of AI capabilities, and we build on top of that. As humans have always done with every new technology.

There is another more pessimistic view where LLMs just replace every human capability, and our economic overloads dont need us for anything and we just eat the small pieces of bread that are left.

This comes down to the fact of:

is human existence/intelligence just the simbolic representations we make in our brain? Or are they just a tool?

I tend to think of Godels incompleteness theorem as a proof that on the limit LLMs are useless. The real question for me is at what point approaching this limit becomes an issue, and if it has any practical co sequences.

I mean, he still seems to underestimate what future models will be able to do. The exposition, connections, and new research directions he wants to reward are also things future models will do far better than humans.