alt Hacker News“Math 2.0” will need to value mathematical progress more holistically
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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.
The authors of the proof are invited to give many talks, and meet with other experts in the area. Workshops are set up to discuss the proof, as well as other recent developments.
problems are being solved autonomously by AI prompters who have no interest in the broader field itself once their initial target is "solved", and do not understand the AI output well enough to answer questions on the result
The valuable part that Tao is feeling the absence of is the insight, and you can only get that from talking to the swarm of agents that developed the original proof with all of their context.
So to me it feels like we don't need Math 2.0, but Authorship 2.0. I want to "meet" the context that generated these proofs. I mean luckily these were not generated by faceless systems like a SAT solver, you can actually talk to it, but I'm not sure if we can step beyond our pride and grant the true authors of these proofs that recognition.
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.
The real question is how do you come up with a credible 3-5 year research program that is unlikely to be scooped or become pointless overnight.
3-5 years is the period of a grant, and grants have to make research progress, or you don’t get the next grant.
Why were we doing math in the first place? We should be happy that math problems were being solved, since presumably they were blockers for other problems in science and the like. But it feels like math was really more about seeking enlightenment, like a form of mental yoga or something. If so, we can just ignore AI proofs and continue on maybe?
>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 think this focus on a "holistic" approach applies to everything AI is touching now, not just math. On Twitter I see people one-shotting games, or reproducing games. If the goal is to just one-shot a game using AI, it's done. But if the goal is to produce immersive medium that people can truly enjoy, admire the story and the craftsmanship, and can find entire new ways of bringing a story to life, that's something else entirely.
I hope Terry has somewhere private where he can safely express his anger at how poorly math has been treated by these AI corps. I understand that as a recently pro-AI public figure he is limited to ambivalence, so I understand him adding caveats like “maybe math 2.0 has a place for AI”, but it can’t feel good to say stuff like that just days after OpenAI so drastically salted the earth.
He is right, I think this would be a good direction for sectors and careers that are at risk of becoming redundant.
I love smart people like this; even when there's a threat, instead of just being in denial or boycotting out of anger, they figure out a new path for their community
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.
OpenAI counterexample solves the millennium problem in a Yes/No manner - we do know in general that singularity could arise, but it still isn't solved in terms of "Why", ideally we could take an arbitrary physical problem and decide if Navier-Stokes equation can be applied to it.
This is similar to general solvability of the quintic equations - Abel provided a proof first but only with the advent of Gallois theory we could basically understand it in full and decide for any quinitic if it's solvable by radicals or no.
I said this when OpenAI announced they had solved a Millennium prize problem: solving open problems for the sake of it will lose its cachet. AI companies will no longer benefit by making these announcements. They've proven the effectiveness of their tool. If people want to use them to advance human knowledge then let them do that. There's no benefit to humanity to turn electricity into proofs just for the sake of it.
It is surprisingly similar to "Catching crumbs from the table" by Ted Chiang, a sci-fi perspective published in Nature in 2000, https://www.nature.com/articles/35014679
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.
> But at the current time, the opposite is often occurring: problems are being solved autonomously by AI prompters who have no interest in the broader field itself once their initial target is "solved", and do not understand the AI output well enough to answer questions on the result, give talks, or otherwise interact with the rest of the field.
Sounds like Terence Tao would have said the same about Ramanujan who basically just "solved" problems without much explanation / reasoning / communication other than it just arrived from god.
In the case of Ramanujan, others took on the responsibility of socializing and community building knowing that he wouldn't do it himself. Why can't the same approach happen here?
There will be people who want to just "solve" math problems now that they have a new tool that lets them express themselves this way. Maybe the don't want to participate in the broader math community, etc. Why discourage them, or add friction / a barrier to them participating in their own way? Why not take on the burden of socializing, making sense of, and community building yourself?
There may be valid reasons here I'm missing, but to me this seems a bit like wanting others to approach a field in a particular way even though the field can support many ways.
Feels very similar to the emotional rollercoaster programmers are going through. And presumably soon everyone else
Stephenson’s Anathem solves some of these problems with cloistered groups (“maths”) that only interact on an epicyclic basis so that old wisdom becomes stabilized with new knowledge has supports for metabolization. I would enter a math.
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.
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.
I feel like there first will be a lot of rationalizations, hamd-wringing, and existential tummy aches, but then the eventual resigned acknowledgment (just like in Chess), that humans do math because they like to do math, not because humans will ever again be as good as computers are at math, then come to make discernments between human math proofs, and the work of an engine, maybe they'll even start start doing proofs on short time controls and stream it on Twitch.
Because the other solutions are to a) quite literally become inhuman, with cyborg integrated TPUs running local models and networked interfaces to propierary models run in data centers, or b) assert dominance of human ignorance by burning civilization down, which doesn't sound pleasant.
What's happening in the math field at the moment represents on a smaller scale the issue we (probably) will face when AI becomes smarter than us in general. Do we slow the AI down so we can understand what it does and if it's correct (and align with our values for bonus points); do we, humans, adjust the way we work to the new speed or do we give in and let the AI advance while we're not completely sure of what it does and the correctness of it. The math field is in the position of showing us the way.
When you mechanise, you take the "meaning" out of the effort and that, in of itself, is death of human pursuit of the venture.
The industrial revolution did that to battles and wars and it inspired Tolkein's lores to a considerable degree. He loathed what mechanisation had done.
I feel something similar is happening to Mathematics. I shudder to think what would come of other human pursuit this mechanisation targets next.
I feel like OpenAI should have secretly contacted a bunch of mathematicians and offered to give them "credit" for these "discoveries" as long as they also credited ChatGPT with helping in research. Mutual interest etc.
This is basically On proof and progress in mathematics by Thurston restated. When Thurston wrote it in 1994, many people didn't understand what he is talking about.
Off topic, but the font is barely legible on my screen? Firefox, Win11 https://imgur.com/a/8LrOojX
I have this feeling that the frontier of maths is going to accelerate faster than humans can keep up with it, even if machines get exceptionally good at math exposition. There’ll be some event horizon of new discoveries which are so complex we’ll never understand their intermediate steps. Beyond that point, humans will revert to “Math 1.0”, where we’ll need to rediscover proofs that have already been solved by machines, and we’ll have a pair of frontiers each for the humans and the machines.
> placed a premium on being the first to solve an open problem
Which is a simple measurable goal requiring little bureaucracy. The mythical "all you need is a pen and paper and a lifetime of dedication"
> "Math 2.0" will need to ... value mathematical progress more holistically
which is directionally the opposite
> community building ... AI can contribute positively
what is this belief based on? Any other communities can illustrate?
A well-known MIT professor gave a presentation about the advancement of mathematics, back in 1965: https://youtu.be/W6OaYPVueW4
It is interesting to see that we are using this term Math 2.0 so quickly, after just a few months/year of seeming progress on previously hard problems, for something that has been around for thousands of years.
I mean, he still seems to underestimate what future models will be able to do. The various directions he wants to reward are also things future models will do far better than humans. I suspect we're better suited to pursuing math like we do pleasure reading... it's enjoyable, can be useful in various situations, but we're clear-eyed that we're not gonna advance the field... and that's okay and doesn't mean it's not still worthwhile.
After seeing the cancerous monster AI "proofs" it became obvious as day that we need better abstractions in several fields. The good part is: there is still a lot left to do for humans as they actually have the abstract thinking ability, as opposed to crazy cancerous token generation.
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.
he put in an elegant way, that its not just about the solution it's about how we would leverage the AI for better good.
Which should improve collaboration, Research and Clarity.
I would really appreciate if we come up with protocols for using ai in STEM field's it might be award at first but we could regulate properly using this method.
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.
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!"
Working heavily with LLMs for the past year has me nodding strongly with Tao's mindset.
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 can continually reach a new threshold, basically for free. But it is only a one time gain and ultimately short-sighted. Where I find continuous value is using LLMs to help my understanding, full stop.
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 of the code it writes.
The difference from this approach is that I am not suffering reading through 3000 lines of AI slop but I am reviewing a PR that I fully understand. I can eyeball it quickly for anything that doesn't fit my mental model and dig deeper or quickly revise it. Only after I am happy with the bones do I consider the meat and skin of the code.
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.
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?
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.
""Math 1.0" placed a premium on being the first to solve an open problem, even if the solution was not initially well understood"
Arbitrary conclusion. This is the corporate take on "mathematics"
Mathematics were meant to further our understanding of nature and solve people's problem. Not to serve corporate delusional CEOs for their psychopathic purposes.
> And our community will need to explicitly re-evaluate its criteria for education, publication, and career advancement, to reflect the "Math 2.0" era.
I think most academic disciplines would benefit from such a re-evaluation. AI is still a scourge on the earth, but I suppose if it spurs such changes that's a modicum of a silver lining.
We go Math 2.0 before GTA 6
Curiously fitting that the Pope is also a Mathematician by training. I'd imagine he agrees with Tao on the fundamental aims of mathematical research and how these proof dumps largely miss the point
Perhaps it's the moment that the likes of Terence Tao hand over the reins to the likes of Grant Sanderson.
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 overlords 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 consequences.
I think it's disappointing [1] for a group of people (however smart they are, whatever titles and fame they have) to think they are the gatekeepers of mathematics, the core of human knowledge. In this respect "mathematics is what mathematicians do" is no longer true, perhaps it was never a useful thing to say. Mathematics is (downstream) consumed in some shape or form by every human on Earth, and no elite, appointed group gets to decide how new mathematics is created. I feel this is a reaction to the sting of the reality check for these people, who are gifted with amazing genetics, and can think faster and deeper than other humans, that their gifts are relative to humans, and machines can still outdo them.. but this is already true for so many other areas, from physical strength to playing chess. Also, I feel this reaction is just non-sensical, it reminds me of when commercial software company CEOs [2] said open source is communism in the early 2000s, because they felt open source is threating their business model.
Lastly, deep down I don't really get what mathematicians are so upset about. All open problems, once solved, are not solved by 99.9999% of mathematicians, because it's solved by one or a handful of others, and the others just learn of the solution/proof. Mathematicians can now still organize conferences about these proofs, discuss them, digest them, think of new avenues of research, etc. They don't even have to invite OpenAI, in 6 months whatever model is available on chatgpt.com will be this smart anyway, and they can use it in the workshops for explanations, etc.
[1] I was going to write "I'm a bit disappointed by the response of the math community..", but then I remembered, whatever T. Tao writes is not the position of the math community, it's his position. Then I was going to write "I'm a bit disappointed by the response of T. Tao..", but then I remembered, I don't know Tao personally, so why am I disappointed?
[2] Steve Ballmer of Microsoft, I believe
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
The future of "math" is not just solving, but communicating what was solved, and why that solving is important. "Math" as a discipline is only ever been half created, they abandoned explaining themselves like it was beneath them. Well, now in "Math 1.0 finally whole" they will be explaining what they did to the rest of us. If you can't explain you are not really there.