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.
Right, easy comparison to make the the open source community for software.
And it's not like this is something where we're loaned some top math genius for a limited amount of time and we have to make the most of it. Rather, this is a new high water mark. The accessibility of the results is no longer scarce. The scarcity has shifted, and that's where the focus of the math ecosystem should shift as well. And it doesn't help for a frontier community to saturate and take over messaging pipelines that were typically managed by the math ecosystem. It's not about "stay in your lane" but rather "we need coherence and be careful not to break the system."
Just two cents from someone who could screw up basic cashier math on any given day.
That seems short sighted though. A few years ago models couldn't do this at all, I'm not sure there's any evidence to suggest exploring and refining results is outside their capabilities or will remain so.
OAI obviously have a fiscal incentive here, but to presume a year from now we won't see improvements and more succinct work on the results coming from models?
The problem is that one a person writes a 60 page proof in theory that person has spent an inordinate amount of time on the proof and can answer questions, describe some insight, etc etc.
If a random person is given a 60 page proof to digest and not the author, those hidden insights that _aren't_ in the paper might be completely inaccessible. Maybe the AI will "just" be able to provide the insights. Maybe. But pedagogy is tricky work, and despite these AIs being able to do all this fancy math we can't get them to write good cover letters yet, so....
Ultimately we might be left with just a bunch of intellectually unsatisfying proofs. This means way less drive to simplify the proofs or rework them.
End result: we generate a layer of "less efficient" mathematics, that won't get built upon. We will not actually have any shoulders upon which to stand.
> I'm not sure there's any evidence to suggest exploring and refining results is outside their capabilities
OP didn’t suggest that.
The bar has been raised. Everyone has to meet it now. An inelegant solution squatted onto the internet doesn’t count as discovery per se, even if it’s impressive.
Your phrasing is illuminating that perhaps they aren't engaged in the creation, understanding, or integration of these proofs by humanity; they just have them. For them, this is a slidedeck they can pass to investors, creditors, the marketing department. Something they can add to the employee onboarding pamphlet.
What should they do? Hyperbolic maybe, but perhaps engage with humanity.
It’s fine that OpenAI posted their findings. It’s not fair to claim these problems have been solved. Not until someone can understand and verify the proof and then communicate the core, novel methodological element to someone else.
No need. In the end the mathematicians that don't like this can just not look at the proofs or use them. They have that choice. Just like they didn't "ask" for them, they don't have to even acknowledge they exist.
Those few thousand mathematicians are now getting a taste of their own medicine. After all, it was people with extraordinary mathematical talent who developed machine learning and large language models, leaving hundreds of millions of people who earn their living through speaking, writing, or teaching worried about their future job prospects.
Still, I believe almost everyone will be fine. Perhaps AI will also prove good at coming up with new conjectures, and some mathematicians may shift towards applied mathematics or other sciences.
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.
People wrote many books about software engineering, all from valuable experience from buildng expensive software systems. But in the age of AI, is there still anything learnable from generated code?
Personally I always ask AI to summarize its findings and lessons in a .md file. And I always learn something from it.
But could AI utilize some new patterns and paradigms I wasn't aware of? Very likely. Because we only learn from our personal grave mistakes, a summary from others gets neglected and forgotten
Sort of. An elegant proof is useful beyond what it shows. It hints at new mathematics, and can prompt discovery in applied fields. I don’t think I’ve heard of elegant code leading to discovery on its own.
> I don’t think I’ve heard of elegant code leading to discovery on its own.
I’ve not heard of it either, but code is an abstraction of math, so I don’t see why this couldn’t theoretically happen.
Anecdotally, I’ve started spending time advancing my math skills beyond the early college level I stopped at and I’ve frequently found I already know concepts of more advanced math - I just didn’t know what they were called or how to apply them to an equation on paper, but I’ve been using them for years and intrinsically grasped the underlying academics.
Models have limits too, I don't think software will become boring. I think it'll become more interesting, in the not so distant future one software engineer will be able to do so much more than today.
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.
>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.
This has already been the case in AI/ML and computer vision papers via flag planting papers. Have an idea, super quickly publish a hasty work based on it that doest actually work, methodological and eval issues, engineering terrible, slow, bad results etc. But it was the first so now your concurrent work that was much better evaluated, better implemented, etc is suddenly worthless and unpublishable.
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.
> 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.
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.
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 don't think it's that simple. I divide AI-impacted fields into three buckets:
1. Some present a unified line that the whole point of their craft is the human experience, and that automation is the antithesis of that. Marathon runners don't care that a car can get there faster, poets don't care that Poem Bot 2000 can write poems too. I think this is smart if you can credibly take this position. The difficulty is mostly convincing the buy side, which requires being very outspoken about your views.
2. Some appear to be undecided, with one faction taking the pro-human stance and another rushing to accelerate things with AI. A good example of this is mathematics, and I really wonder where they end up in the long haul. They have a very good claim on #1, because mathematics is pretty close to an art form and is robustly insulated from the pressures of the marketplace. But they can also choose option #3, below.
3. Some crafts prioritize results above all else, practitioners either rushing to extract as much money as possible before it all collapses, or believing that they can out-prompt everyone else forever and that their prompting skills are indispensable to their employers in the long haul. That's software engineering. I think this is going to be interesting to watch.
Seems like neat buckets but you may want to find one example for the first which can actually be done by an llm as running is not its strong suit afaik
I gave two examples for #1 and one of them is something done by LLMs. Other examples include painting / drawing, songwriting & composing, etc. In all of these, gen AI output is pretty widely stigmatized.
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.
> job of professional mathematician might be the first to be completely eliminated by LLMs
Strong disagree.
Do you work in a math adjacent field? I do and I find having a mathematician around invaluable.
It's like a non-software person writing software. Yes, using a LLM will get you to a solution that works. But just talking with a software engineer will make the quality of that solution enormously better.
I find the same with math - I can get something to work using an LLM, but if I speak to a mathematician they'll say some magic words to try and I put that in the LLM and it is "oh yes this is a much better solution".
This is very different work to generating proofs though. Its things like "I'm trying to get my confidence intervals to properly deal with census like sampling but at small sample sizes" (yes, I know stats not pure math but still..)
That's an uphill battle. People will hand wave that the s curve has flattened, and think it will stay at current levels. They claimed this confidently year after year over the last few years.
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.
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.
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.
This is wildly shortsighted view of mathematics. Historically many of the subfields of math which are presently most valuable were considered useless for decades or centuries. Number theory, non-Euclidean geometry, group theory, and Boolean algebra, to name a few.
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.
The interviewee is worried about the future of math research. He is not strictly worried about being replaced, instead he is worried that he will no longer be able to launder math-as-a-hobby through math-as-something-useful as is the case today. He lays out very clearly that grant proposals claim to have useful outcomes while the proposers know those claims are nonsense.
Business as usual in math, and frankly in all the other sciences, is to do research that furthers the researchers careers or personal interests and pretend that it’s somehow useful. This would be absolutely fine if it were privately funded, but it’s not, this is public money.
In every other endeavour, lying in order to get money is considered fraud.
We have collectively wasted a huge amount of taxpayer money and human time, entire careers, on things not likely to ever matter to anyone.
I look forward to science becoming automated so that we can have real progress instead of the current broken system.
People thought, back in the 17 century, that imaginary were useless (except as a trick for some calculations). Turns out the research into these numbers back then is amazingly useful today, 300 years later, in electronics and such.
Publicly funded maths research should continue, even if some taxpayers feel it's a waste of money.
Nobody determined the usefulness of the internal combustion engine or the aeroplane after the fact. They were goals specifically worked towards. For every useful discovery that comes from this hobbyist approach there are far more that aren’t, and those would have been discovered during a goal orientated research program.
However, with AI it actually may become so cheap that the scattergun random approach becomes more viable rather than less. It’s when human time and resources are scarce that you need to optimise. The hobbyist approach may therefore ironically continue, but without the hobbyists.
The whole point of public funding for science is that seemingly pointless research yields useful but hard to monetize discoveries. Otherwise VCs would be doing it.
Actually yes, I think that AI will end up in a place where additional human effort, even at the highest level of suggesting what directions to look in, will become a rounding error compared to what the AI will achieve by itself.
So as measured by utility, I absolutely believe we don’t need humans doing science into the future. I’m sure people will continue doing it, but not for utility, for enjoyment - as a hobby. Probably we’ll all end up as dedicated hobbyists.
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?
That doesn't make sense to me. Terence Tao is great at explaining things, but he could spend months explaining some of his proofs to me without me understanding it. Even if the AI had a superhuman ability to explain things it's no guarantee that it could make a human understand.
I am not so sure that's true. Even if AI intelligence were to plateau at today's levels, there are still many gains to be had in speeding up today's intelligences. ASICs with burned in weights could become economical to invest in as they would retain usefulness longer than 18 months, and I feel we have only scratched the surface on possible usecases of local AI and what it means for almost any technology product or interface.
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!"
That's a strong contender for the worst analogy I've ever seen on HN, and it's a crowded field.
Edit: If you'd like a better medicine based one, look to radiology, where AI is an omnipresent tool but claims that radiologists are no longer needed, based on an ignorant view that a radiologist's job is "classify images according to what diseases they indicate" have only contributed to a crippling worldwide shortage of radiologists.
I think the difference is that with cancer cures we mostly care that it works as proved by trials, and understanding it is a bonus.
Up until now the prize in pure (as opposed to applied) mathematics was the _understanding_ and the machine can't do that for you. What does it mean if we get "super powered alien maths" but humans can't do it? It's like inter univeral teichmuller theory all over again but imagine if Mochizuki was right and it came with a lean proof?
It was the understanding for some, and the result for others. Math is going to bifurcate along those lines. Many are the builder type who use math for a purpose. To accelerate an algorithm, to improve the numerics or convergence of a computation, to verify statements about real things, to use it in angineering applications, etc etc.
In my view, over the last century, math has turned into an intellectual analogue of extreme bodybuilding competitions. A navel gazing runaway optimization in making useless stuff just to demonstrate cleverness. That's fine, why not. But society has no obligation to fund that, just as it doesn't fund other extreme hobbies. Ideally if we ever get something like UBI, math can be still their hobby.
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 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!
Yann LeCun's criticisms are at least balanced with an alternative approach he has teams actively working on and showing good progress in areas LLMs are weak.
Oh, sorry, misinterpreted you! Although I am not sure I agree with your dismissal of nay-sayers. We can equally dismiss the opinions of early proponents. Perhaps maybe we should be hedging on all early opinions regardless.
Yeah agree with you, and perhaps both are even important in our public opinion forming... Recently I've been hearing people like Grant Sanderson on AI, and they are all very "wise" and informed, neither dismissive nor mindlessly (p/b)ro, but really thinking implications of this technology through. Like Tao does here. I love it, these people provide real direction.
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.
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.
Right, easy comparison to make the the open source community for software.
And it's not like this is something where we're loaned some top math genius for a limited amount of time and we have to make the most of it. Rather, this is a new high water mark. The accessibility of the results is no longer scarce. The scarcity has shifted, and that's where the focus of the math ecosystem should shift as well. And it doesn't help for a frontier community to saturate and take over messaging pipelines that were typically managed by the math ecosystem. It's not about "stay in your lane" but rather "we need coherence and be careful not to break the system."
Just two cents from someone who could screw up basic cashier math on any given day.
That seems short sighted though. A few years ago models couldn't do this at all, I'm not sure there's any evidence to suggest exploring and refining results is outside their capabilities or will remain so.
OAI obviously have a fiscal incentive here, but to presume a year from now we won't see improvements and more succinct work on the results coming from models?
The problem is that one a person writes a 60 page proof in theory that person has spent an inordinate amount of time on the proof and can answer questions, describe some insight, etc etc.
If a random person is given a 60 page proof to digest and not the author, those hidden insights that _aren't_ in the paper might be completely inaccessible. Maybe the AI will "just" be able to provide the insights. Maybe. But pedagogy is tricky work, and despite these AIs being able to do all this fancy math we can't get them to write good cover letters yet, so....
Ultimately we might be left with just a bunch of intellectually unsatisfying proofs. This means way less drive to simplify the proofs or rework them.
End result: we generate a layer of "less efficient" mathematics, that won't get built upon. We will not actually have any shoulders upon which to stand.
> I'm not sure there's any evidence to suggest exploring and refining results is outside their capabilities
OP didn’t suggest that.
The bar has been raised. Everyone has to meet it now. An inelegant solution squatted onto the internet doesn’t count as discovery per se, even if it’s impressive.
But what should they do? They got all these proofs, should they just have sat on them?
>They got all these proofs...
Your phrasing is illuminating that perhaps they aren't engaged in the creation, understanding, or integration of these proofs by humanity; they just have them. For them, this is a slidedeck they can pass to investors, creditors, the marketing department. Something they can add to the employee onboarding pamphlet.
What should they do? Hyperbolic maybe, but perhaps engage with humanity.
> should they just have sat on them?
It’s fine that OpenAI posted their findings. It’s not fair to claim these problems have been solved. Not until someone can understand and verify the proof and then communicate the core, novel methodological element to someone else.
No need. In the end the mathematicians that don't like this can just not look at the proofs or use them. They have that choice. Just like they didn't "ask" for them, they don't have to even acknowledge they exist.
Why not asking another chatbot to verify, like what we are doing with coding ?
That's not the hard part. The hard part still requires humans (and/or _maybe_ a bunch of tokens) and a lot of work.
Yes, it is indeed a balanced view.
Those few thousand mathematicians are now getting a taste of their own medicine. After all, it was people with extraordinary mathematical talent who developed machine learning and large language models, leaving hundreds of millions of people who earn their living through speaking, writing, or teaching worried about their future job prospects.
Still, I believe almost everyone will be fine. Perhaps AI will also prove good at coming up with new conjectures, and some mathematicians may shift towards applied mathematics or other sciences.
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 was about to comment the same.
People wrote many books about software engineering, all from valuable experience from buildng expensive software systems. But in the age of AI, is there still anything learnable from generated code?
Personally I always ask AI to summarize its findings and lessons in a .md file. And I always learn something from it.
But could AI utilize some new patterns and paradigms I wasn't aware of? Very likely. Because we only learn from our personal grave mistakes, a summary from others gets neglected and forgotten
> same could be said of Software
Sort of. An elegant proof is useful beyond what it shows. It hints at new mathematics, and can prompt discovery in applied fields. I don’t think I’ve heard of elegant code leading to discovery on its own.
> elegant code
Usually it's the opposite. "That's in prod? And it works? It shouldn't work and I thought it was doing something else. Why does it work?"
> I don’t think I’ve heard of elegant code leading to discovery on its own.
I’ve not heard of it either, but code is an abstraction of math, so I don’t see why this couldn’t theoretically happen.
Anecdotally, I’ve started spending time advancing my math skills beyond the early college level I stopped at and I’ve frequently found I already know concepts of more advanced math - I just didn’t know what they were called or how to apply them to an equation on paper, but I’ve been using them for years and intrinsically grasped the underlying academics.
you must not be familiar with haskell
Models have limits too, I don't think software will become boring. I think it'll become more interesting, in the not so distant future one software engineer will be able to do so much more than today.
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.
>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.
This has already been the case in AI/ML and computer vision papers via flag planting papers. Have an idea, super quickly publish a hasty work based on it that doest actually work, methodological and eval issues, engineering terrible, slow, bad results etc. But it was the first so now your concurrent work that was much better evaluated, better implemented, etc is suddenly worthless and unpublishable.
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.
There’s value but it is not rewarded commensurately with the contribution. Same for doing peer review.
> 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.
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.
https://arxiv.org/abs/math/9404236
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 don't think it's that simple. I divide AI-impacted fields into three buckets:
1. Some present a unified line that the whole point of their craft is the human experience, and that automation is the antithesis of that. Marathon runners don't care that a car can get there faster, poets don't care that Poem Bot 2000 can write poems too. I think this is smart if you can credibly take this position. The difficulty is mostly convincing the buy side, which requires being very outspoken about your views.
2. Some appear to be undecided, with one faction taking the pro-human stance and another rushing to accelerate things with AI. A good example of this is mathematics, and I really wonder where they end up in the long haul. They have a very good claim on #1, because mathematics is pretty close to an art form and is robustly insulated from the pressures of the marketplace. But they can also choose option #3, below.
3. Some crafts prioritize results above all else, practitioners either rushing to extract as much money as possible before it all collapses, or believing that they can out-prompt everyone else forever and that their prompting skills are indispensable to their employers in the long haul. That's software engineering. I think this is going to be interesting to watch.
Seems like neat buckets but you may want to find one example for the first which can actually be done by an llm as running is not its strong suit afaik
I gave two examples for #1 and one of them is something done by LLMs. Other examples include painting / drawing, songwriting & composing, etc. In all of these, gen AI output is pretty widely stigmatized.
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.
> job of professional mathematician might be the first to be completely eliminated by LLMs
Strong disagree.
Do you work in a math adjacent field? I do and I find having a mathematician around invaluable.
It's like a non-software person writing software. Yes, using a LLM will get you to a solution that works. But just talking with a software engineer will make the quality of that solution enormously better.
I find the same with math - I can get something to work using an LLM, but if I speak to a mathematician they'll say some magic words to try and I put that in the LLM and it is "oh yes this is a much better solution".
This is very different work to generating proofs though. Its things like "I'm trying to get my confidence intervals to properly deal with census like sampling but at small sample sizes" (yes, I know stats not pure math but still..)
Yes, I am projecting out into the future, not talking about the current state. I said "might be the first".
That's an uphill battle. People will hand wave that the s curve has flattened, and think it will stay at current levels. They claimed this confidently year after year over the last few years.
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.
More likely is that frontier labs give huge discounts of EDUs if they consent to allowing training.
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.
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.
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.
For me this path means that our role as humans is just the understanding of intelligence?
Everything else is secondary (or the last of our priorities) and would be better automated?
This is a hard pill to swallow
We should think about rejecting this AI future.
Or maybe we should not let pure mathematicians decide which problems are interesting, but reward the practical applications instead.
This is wildly shortsighted view of mathematics. Historically many of the subfields of math which are presently most valuable were considered useless for decades or centuries. Number theory, non-Euclidean geometry, group theory, and Boolean algebra, to name a few.
If you only judge by practical applications you eliminate vast swaths of human endeavor.
Human endeavor and achievement does not make number go up for next quarter’s earnings
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.
“doing math responsibly” is just a euphemism for “ensuring we can continue doing our paid hobby”.
Take a look at this interview from two days ago: https://m.youtube.com/watch?v=oQypVVv1u1o
The interviewee is worried about the future of math research. He is not strictly worried about being replaced, instead he is worried that he will no longer be able to launder math-as-a-hobby through math-as-something-useful as is the case today. He lays out very clearly that grant proposals claim to have useful outcomes while the proposers know those claims are nonsense.
Business as usual in math, and frankly in all the other sciences, is to do research that furthers the researchers careers or personal interests and pretend that it’s somehow useful. This would be absolutely fine if it were privately funded, but it’s not, this is public money.
In every other endeavour, lying in order to get money is considered fraud.
We have collectively wasted a huge amount of taxpayer money and human time, entire careers, on things not likely to ever matter to anyone.
I look forward to science becoming automated so that we can have real progress instead of the current broken system.
usefulness can only be determined after the fact.
People thought, back in the 17 century, that imaginary were useless (except as a trick for some calculations). Turns out the research into these numbers back then is amazingly useful today, 300 years later, in electronics and such.
Publicly funded maths research should continue, even if some taxpayers feel it's a waste of money.
Nobody determined the usefulness of the internal combustion engine or the aeroplane after the fact. They were goals specifically worked towards. For every useful discovery that comes from this hobbyist approach there are far more that aren’t, and those would have been discovered during a goal orientated research program.
However, with AI it actually may become so cheap that the scattergun random approach becomes more viable rather than less. It’s when human time and resources are scarce that you need to optimise. The hobbyist approach may therefore ironically continue, but without the hobbyists.
>We have collectively wasted a huge amount of taxpayer money and human time, entire careers, on things not likely to ever matter to anyone.
How much..?
The whole point of public funding for science is that seemingly pointless research yields useful but hard to monetize discoveries. Otherwise VCs would be doing it.
> look forward to science becoming automated so that we can have real progress instead of the current broken system
Do you really think a society with zero human mathematicians or scientists will outperform one with both human and AI ones?
Actually yes, I think that AI will end up in a place where additional human effort, even at the highest level of suggesting what directions to look in, will become a rounding error compared to what the AI will achieve by itself.
So as measured by utility, I absolutely believe we don’t need humans doing science into the future. I’m sure people will continue doing it, but not for utility, for enjoyment - as a hobby. Probably we’ll all end up as dedicated hobbyists.
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?
That doesn't make sense to me. Terence Tao is great at explaining things, but he could spend months explaining some of his proofs to me without me understanding it. Even if the AI had a superhuman ability to explain things it's no guarantee that it could make a human understand.
>if model intelligence stalls. If model intelligence continues to improve
So much in AI is dependent on which of these two outcomes occur.
I am not so sure that's true. Even if AI intelligence were to plateau at today's levels, there are still many gains to be had in speeding up today's intelligences. ASICs with burned in weights could become economical to invest in as they would retain usefulness longer than 18 months, and I feel we have only scratched the surface on possible usecases of local AI and what it means for almost any technology product or interface.
If the mathematician doesn't understand anything, are they even a mathematician?
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!"
That's a strong contender for the worst analogy I've ever seen on HN, and it's a crowded field.
Edit: If you'd like a better medicine based one, look to radiology, where AI is an omnipresent tool but claims that radiologists are no longer needed, based on an ignorant view that a radiologist's job is "classify images according to what diseases they indicate" have only contributed to a crippling worldwide shortage of radiologists.
Yours is the worst comment I've seen today. But this one is good because I tell you why: because you didn't tell why.
I think the difference is that with cancer cures we mostly care that it works as proved by trials, and understanding it is a bonus.
Up until now the prize in pure (as opposed to applied) mathematics was the _understanding_ and the machine can't do that for you. What does it mean if we get "super powered alien maths" but humans can't do it? It's like inter univeral teichmuller theory all over again but imagine if Mochizuki was right and it came with a lean proof?
It was the understanding for some, and the result for others. Math is going to bifurcate along those lines. Many are the builder type who use math for a purpose. To accelerate an algorithm, to improve the numerics or convergence of a computation, to verify statements about real things, to use it in angineering applications, etc etc.
In my view, over the last century, math has turned into an intellectual analogue of extreme bodybuilding competitions. A navel gazing runaway optimization in making useless stuff just to demonstrate cleverness. That's fine, why not. But society has no obligation to fund that, just as it doesn't fund other extreme hobbies. Ideally if we ever get something like UBI, math can be still their hobby.
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.
>There is a world where we get to the edge of AI capabilities, and we build on top of that.
We don't build on top of that. No need for us to. AI does. That's sort of the whole point of this endeavor is it not? Humans need not apply.
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!
Yann LeCun and Gary Marcus are still at it - the latter having shifted goalposts.
Gary Marcus has has a long career now of shifting goalposts
Does he have time to do anything but shift goalposts, given how fast his are moving?
I believe he now just leaves them in his truck and drives it a bit further each day.
Yann LeCun's criticisms are at least balanced with an alternative approach he has teams actively working on and showing good progress in areas LLMs are weak.
The less said about Gary Marcus the better.
Tao was not an early nay-sayer.
Certainly not implying that! He's the one with the deferred judgement and the wisdom.
Oh, sorry, misinterpreted you! Although I am not sure I agree with your dismissal of nay-sayers. We can equally dismiss the opinions of early proponents. Perhaps maybe we should be hedging on all early opinions regardless.
Yeah agree with you, and perhaps both are even important in our public opinion forming... Recently I've been hearing people like Grant Sanderson on AI, and they are all very "wise" and informed, neither dismissive nor mindlessly (p/b)ro, but really thinking implications of this technology through. Like Tao does here. I love it, these people provide real direction.
"Taste"
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.
> the whole field of mathematics will simply disappear as a career path
Almost certainly not. It's just going to jump to a higher level of abstraction.
It's already unbelievably abstract as it is...
Did the London cabbie explore unknown territories to find new streets to explore?