> I therefore disagree with claims that mathematics is over because machines can prove theorems. Without humans who understand mathematics, there is, quite literally, no human mathematics—a fundamental part of human culture. If we agree that we should not abandon mathematics because poorly understood machines bruise our anthropocentric vision of it, then society will need more mathematicians, not fewer. Like… a lot more.
Seems contradictory to me, referring to math as a field as “human mathematics” in one sentence, then deriding “anthropocentric visions”.
My read on Terence Tao's comments of late is that professional mathematicians are principally upset about AI-submitted math papers lacking the kind of broad teaching content that high quality human math papers offer. While the answer might be correct, AI math papers don't teach the reader in the way a human author would have. Professional mathematics is not just about solving problems; it's just as much about spreading knowledge so that others can program their human brains with the same understanding that the paper's author had when they wrote the paper (and achieved the result).
In other words, the crisis in mathematics (as I read it) is that AI can solve without teaching. The result may be impressive, but if the only beings that can move math forward in future are machines grinding away on billions of scenarios that no human would have the patience to undertake, this leaves those in the profession wondering about their place in it.
That's definitely true, but on the other hand I can produce with a month of literal non-stop vibe coding (agents running overnight, with checklists of what to qa for when I'm up) what took years for others to do, working in teams of tens or hundreds. At that scale, it's not a question of "doing it faster and not learning" or "doing it slowly"; the alternative is it not happening at all.
There will still be a place for posing problems and noticing how breaking assumptions can create something new. The history of Non-Euclidian geometry is a good example.
A lot of progress comes from taking assumptions and breaking them. There was a time when everyone assumed that locked checkouts were absolutely necessary for version control, for instance. You could argue that AI could learn how to do this well but humans have context that machines won't. This is just an extension of the fact that things with independent existence will have different impressions of the world. If they didn't they would functionally be one thing.
Two of the things I like about the scientific community is the lectures/conference and the sharing of ideas through papers. I’m nowhere near that level and have to wait for those ideas to trickle down into textbooks. So maybe I’m learning stuff that have been discovered decades ago, but it’s still useful on a daily basis.
> The result may be impressive, but if the only beings that can move math forward in future are machines grinding away on billions of scenarios that no human would have the patience to undertake
Seems like the same thing as neural networks. You have something like a few thousand parameters for computer that works well, but no underlying algorithms and formulas, like the shaders.
I’m not bitter about this anymore — it’s been a long time. But the story holds something.
Back in the day, I knew I was talented at mathematics, and I wanted to pursue a research career. For a freshman research class, I selected a problem that was intentionally outside my comfort zone, and came up with some cool ideas. It wasn’t publication quality, though the course wasn’t really expecting publications.
Framing the problem in a way that it could be solved with calculus turned out to be a key insight. I turned to Wolfram, since the integrals were beyond what I as a freshman could have done myself, and I understood that the insight that produced the solution was my original work. I don’t think anyone in the class could have done those integrals. The instructor himself couldn’t tell me what was wrong with them if anything. Yet, he was unimpressed and subtly dismissive or critical.
Today, I might have been in the wave of researchers developing AI for mathematics work. The instructor could have been proud of shaping mathematics through his mentorship. The university might have invited me for talks. Instead, I ended up leaving mathematics as a field altogether. Now, I write software for a bank.
The world is as rich as we make it. Don’t let your opportunities and potential expire. Embrace the materials that are available to work with, or someone else will and they will out compete you and you will have missed the moment.
I get that there’s a romantic idea of what mathematicians do, and the field changing creates a sort of identity crisis. I would argue that the field has entered a new era. Exploring and shaping this era of mathematics is the opportunity of a lifetime. Embrace it, drive it, and make your mark.
I like everything you said, but do you think AI assisted math may primarily give us a more productive age for individuals, with less need for human-human collaboration?
Maybe it doesn’t matter and maybe that’s just the new flavour of golden.
These are all great points but I'd also point out that the 'dark age' scenario is a highly competitive scenario, while the 'golden age' scenario is a highly collaborative one.
I don't think technology will fizzle out in either scenario. In the 'dark age' highly competitive scenario, you will still have those who 'understand what is produced' seeking progress and advancement. They will cohort with whoever/whichever organization is producing the best, most validated results.
I don't think knowledge would be become alien to human understanding, it would just become highly concentrated again as it was in the past. In the past, only the wealthy and noble could afford to spend time on education.
And usually, those who went deep into any field and provided contributions to science were those who had time to devote entire lives to the study or pursuit of knowledge. We still have these issues today, and these issues will always be present.
There is a supply chain of economy, and we provide different rewards based on the value of what is produced. The jobs never go away, the desire for prestige, greed, etc., never goes away. It's why economies exist, and it's why everyone is considering the value of LLMs, and their place in economies, and academies (presumably which exist to advance human understanding/appreciation, but also to prepare younger generations for future work).
Authorship and credit seem like anti-patterns in math. One of the least accessible parts of math to me is that literally everything is named after people. Navier-Stokes? Who? I don't really want to know to be honest, I'd rather the ideas and information be framed with a prioritization towards understanding, not author's reputation. I feel the same way about all of the Greek notation, it's 2026, can we not use something in regular circulation to broaden the audience?
Maybe AI can actually drastically improve this situation by bridging the gap between those "in the club" and those who want to do something useful with mathematical ideas.
To be honest if Greek letters and things being named after the people that discovered them are that big of a hurdle to you the chances of you actually being able to do something useful with these mathematical ideas are probably close to 0 since those are by far the easiest parts of dealing with advanced mathematics.
I bet for a short golden age where AI will reap the low hanging fruit of easy (for a computer) cross-referencing and inference. And then a dark age with none interested in putting new ideas out in the field. Hopefully, I am wrong.
Some great thoughts, many of which extend beyond mathematics.
One in particular, about sharing secrets, that had not crossed my mind (I have found that virtually no one ticks the "do not train on my data" box...):
> Similarly, it seems that mathematicians already concerned and guarded about what they write in research statements and research proposals. After all, an administrator or a reviewer may upload those documents into an LLM to get help in writing a report, thus potentially compromising the confidentiality of unpublished research.
>One in particular, about sharing secrets, that had not crossed my mind (I have found that virtually no one ticks the "do not train on my data" box...):
When said button is in the basement, at the bottom of a broken set of stairs, locked behind a door that says "beware of the leopard", inside a filing cabinet, and implemented as
I also feel like many of the current guest posts have overlapping ideas and arguments and so I get less out of each one of them. Maybe a separate blog for endorsed opinion pieces would be appropriate.
"Logistical component A is constrained by abstract concept B." am I being a dick, or is this not extremely interesting to read? TBF I've probably over-consumed this particular category of argument recently.
By all means, please read the article if it's interesting to you and ignore it if it isn't. But I believe that it's mostly directed to those already within the research math community, with a far more important purpose than to be interesting.
As the post itself explains, we (research mathematicians, as a profession) need to coordinate on deciding what a bright future will look like in a world with powerful AI, and in plotting a course to get there. And we need to do it fast, or circumstances might decide for us.
Historically, we've rather sucked at coordination. We need leadership -- we need someone to step forward, and begin by floating ideas and getting the conversation rolling. Personally I am very grateful for what Tao is doing here.
In the US, we've already seen the decrease in funding for education & research and closing borders down to many foreign researchers. The current admin and mainstream media is science-skeptic. We're already heading for a dark age. AI's contribution is largely because those policies are coupled with the same admin putting all their eggs into the AI basket, and likewise, basically all the major players in AI being on board with the admin's policies. This should be considered a systemic issue. Things like having a "healthy scientific skepticism and the limits of LLMs’ creativity" are important, but won't solve this alone.
> Private control of frontier LLMs is potentially dangerous.
The author spends a lot of time trying to suggest that LLMs are not the end all be all and humans are still needed for understanding etc. Fine. I understand.
If it is so the case that LLMs don't help that much, surely it doesn't matter if LLMs are privately controlled?
How come LLMs are important enough that private control matters but also not enough for actually advancing mathematics significantly on its own?
In other words, you are almost always better off with LLMs even if it is privately controlled. The mathematics community can just ignore LLMs for the base case or use ChatGPT if they want. In either case, it is strange to suggest that things will get worse for them. Its as if they don't have the coordination or agency to adapt.
This is my understanding of that part you quoted. It's not really a statment about open vs closed models (which is another very important topic, personally I think open is the only way forward). What he is suggesting is mathematicians will use llms for their research because llms are transformative for math and research. As they work on a problem, OpenAI will put their strongest model on their biggest cluster and synthesize all the work by the individual mathematicians to arrive at the answer.
One big problem here is that OpenAI would be hoarding this ability by 1) not releasing their top models for public use,as we have already seen them do, and 2) limiting the capability of the public API, which we have also seem them do.
This not only kills incentive for math, it means that OpenAI becomes the arbiter of math and research for the world. The research that will be done is the research OpenAI decides to do.
This seems very dangerous to me.
Another path forward is distributed local models. Professors can do research on their HPC machines, individuals can have full access to their models. It's a much more democratic approach.
> As they work on a problem, OpenAI will put their strongest model on their biggest cluster and synthesize all the work by the individual mathematicians to arrive at the answer.
But coming to final answer is not important and that’s what the author spends time arguing. If they are right, surely whatever slop OpenAI is producing is worthless to human understanding which is the main goal.
The math community has to change prestige markers to promote people who contribute to understanding and not just solving theorems and that’s on mathematicians to do.
If solving problems is the end goal then I don’t mind OpenAI doing it in which case furthering math is already solved by someone.
The only failure mode is if mathematicians can’t get funding for just “understanding”. But it shouldn’t be that hard to do if their claim is right that their main contribution to society has always been understanding.
I’d argue that getting funding purely for understanding may be even easier to justify because the costs go down dramatically per unit understanding.
> Without humans who understand mathematics, there is, quite literally, no human mathematics—a fundamental part of human culture. If we agree that we should not abandon mathematics because poorly understood machines bruise our anthropocentric vision of it, then society will need more mathematicians, not fewer. Like… a lot more.
This is just incoherent. Even if we "agree" (we don't but...) not to "abandon" math, how does it follow that "we need a lot more mathematicians" exactly?
Even if you reject the idea that AI can displace the current top quartile/percentile of human mathematicians, wouldn't it be reasonable to assume that it will at least displace some parts of math so that there'll be less demand for mathematicians overall, not more? The mental gymnastics are astounding here.
Unless AI can do everything involved in math, their greatly accelerated production of certain parts of math should increase the demand for people to do the remaining parts.
> greatly accelerated production of certain parts of math should increase the demand for people to do the remaining parts
So the pie just gets larger? Large enough to outpace the displacement? I am gonna wait for this to generalize then, looking forward to AI giving us more software developers, mathematicians, etc. The economic laws have been broken. It's an infinite money glitch, as they used to call it.
I love how Terrence Tao went from being AI shiller -- being a walking advertisement for openai and anthropic to a Oh nO MaThS iS KilLeD wHaT aBoUT mY JoB in a few months.
terrence tao didn't even write this post and the conclusion it has is thus:
> Universities, professional societies, funding agencies, and scientific foundations should coordinate the development and governance of competitive LLMs. Scientists will need access to frontier capabilities, transparency about their operation and use of research data, strong confidentiality protections, and the ability to test and reproduce research claims independently.
> The massive computing resources, specialized personnel, data infrastructure, and ongoing investments needed to support this kind of initiative are not within the reach of a single institution, and a huge leadership effort will be necessary to conjure the combination of independent model development, guaranteed access to commercial systems, public computing infrastructure, and open research resources that could do the job.
the post is calling for....an enormous investment in LLM infrastructure.
Thanks for a summary. Based on your summary, I think the parent is directionally correct, at least. All of this seems like panic and attempt to salvage a profession, seeking control and money and other people to do stuff, now! It’s not like this is isolated to university labor. It will kill universities yes, but it will kill a lot of bigger and more important things, likely humanity itself will be killed off. That’s the context.
Same phenomenon as all the godfathers of AI that were apparently oblivious to the plot of the Terminator franchise until after their paychecks had cleared.
The irony is that he’s still very very wrong in my opinion. He’s lamenting the end of math, but it’s not a stretch to think that math is just the beginning. Like the destruction of the buoy before the tsunami actually hits land. His current stance seems to be implicating that the blast radius is only till math or science when it’s just the first domino.
> I find it very weird how he didn't foresee the end of math, being so smart. I mean I did and I'm definitely less smart than him.
IMO this is probably not an accurate assessment of Tao in particular at this time.
But in the general case, I think most people have a set of blinders on that will result in a predictable reassessment of AI with the following steps as AI progresses when applied to whatever it is they do:
1) AI is useless and doesn't impact me at all. Look at how bad it is! (Most developers up until late 2025)
2) Ok, now AI is useful but will not replace me. Sucks for the juniors in the field but I am still needed to guide the AI. AI will never be capable of my judgement (or whatever). (Most developers now)
3) Oh shit, I've been replaced in terms of economically valuable utility by AI. (tbd, but likely sooner than is comfortable)
And not everyone will experience #3, there will be a top 1% or so for most fields where #3 remains true for them, but it is likely that for most people #3 comes eventually.
It is unlikely anyone would actually have zero utility, if only because of comparative advantage. (Although I think there will still be absolute advantage)
Seems contradictory to me, referring to math as a field as “human mathematics” in one sentence, then deriding “anthropocentric visions”.
In other words, the crisis in mathematics (as I read it) is that AI can solve without teaching. The result may be impressive, but if the only beings that can move math forward in future are machines grinding away on billions of scenarios that no human would have the patience to undertake, this leaves those in the profession wondering about their place in it.
Frontier models can easily one shot what I ask for, and make adjustments as ask to quickly make an app that meets what I wanted.
But often when I programmed outside of work (and at work too) I learnt something.
Quite often that my favourite pattern is an antipattern in some use cases.
That might never happen again at the rate of development.
A lot of progress comes from taking assumptions and breaking them. There was a time when everyone assumed that locked checkouts were absolutely necessary for version control, for instance. You could argue that AI could learn how to do this well but humans have context that machines won't. This is just an extension of the fact that things with independent existence will have different impressions of the world. If they didn't they would functionally be one thing.
> The result may be impressive, but if the only beings that can move math forward in future are machines grinding away on billions of scenarios that no human would have the patience to undertake
Seems like the same thing as neural networks. You have something like a few thousand parameters for computer that works well, but no underlying algorithms and formulas, like the shaders.
I’m not bitter about this anymore — it’s been a long time. But the story holds something.
Back in the day, I knew I was talented at mathematics, and I wanted to pursue a research career. For a freshman research class, I selected a problem that was intentionally outside my comfort zone, and came up with some cool ideas. It wasn’t publication quality, though the course wasn’t really expecting publications.
Framing the problem in a way that it could be solved with calculus turned out to be a key insight. I turned to Wolfram, since the integrals were beyond what I as a freshman could have done myself, and I understood that the insight that produced the solution was my original work. I don’t think anyone in the class could have done those integrals. The instructor himself couldn’t tell me what was wrong with them if anything. Yet, he was unimpressed and subtly dismissive or critical.
Today, I might have been in the wave of researchers developing AI for mathematics work. The instructor could have been proud of shaping mathematics through his mentorship. The university might have invited me for talks. Instead, I ended up leaving mathematics as a field altogether. Now, I write software for a bank.
The world is as rich as we make it. Don’t let your opportunities and potential expire. Embrace the materials that are available to work with, or someone else will and they will out compete you and you will have missed the moment.
I get that there’s a romantic idea of what mathematicians do, and the field changing creates a sort of identity crisis. I would argue that the field has entered a new era. Exploring and shaping this era of mathematics is the opportunity of a lifetime. Embrace it, drive it, and make your mark.
Maybe it doesn’t matter and maybe that’s just the new flavour of golden.
I don't think technology will fizzle out in either scenario. In the 'dark age' highly competitive scenario, you will still have those who 'understand what is produced' seeking progress and advancement. They will cohort with whoever/whichever organization is producing the best, most validated results.
I don't think knowledge would be become alien to human understanding, it would just become highly concentrated again as it was in the past. In the past, only the wealthy and noble could afford to spend time on education.
And usually, those who went deep into any field and provided contributions to science were those who had time to devote entire lives to the study or pursuit of knowledge. We still have these issues today, and these issues will always be present.
There is a supply chain of economy, and we provide different rewards based on the value of what is produced. The jobs never go away, the desire for prestige, greed, etc., never goes away. It's why economies exist, and it's why everyone is considering the value of LLMs, and their place in economies, and academies (presumably which exist to advance human understanding/appreciation, but also to prepare younger generations for future work).
Maybe AI can actually drastically improve this situation by bridging the gap between those "in the club" and those who want to do something useful with mathematical ideas.
One in particular, about sharing secrets, that had not crossed my mind (I have found that virtually no one ticks the "do not train on my data" box...):
> Similarly, it seems that mathematicians already concerned and guarded about what they write in research statements and research proposals. After all, an administrator or a reviewer may upload those documents into an LLM to get help in writing a report, thus potentially compromising the confidentiality of unpublished research.
When said button is in the basement, at the bottom of a broken set of stairs, locked behind a door that says "beware of the leopard", inside a filing cabinet, and implemented as
I wonder why.https://proofsandprompts.com/
It'll not necessarily be that great for every mathematician or scientist.
By all means, please read the article if it's interesting to you and ignore it if it isn't. But I believe that it's mostly directed to those already within the research math community, with a far more important purpose than to be interesting.
As the post itself explains, we (research mathematicians, as a profession) need to coordinate on deciding what a bright future will look like in a world with powerful AI, and in plotting a course to get there. And we need to do it fast, or circumstances might decide for us.
Historically, we've rather sucked at coordination. We need leadership -- we need someone to step forward, and begin by floating ideas and getting the conversation rolling. Personally I am very grateful for what Tao is doing here.
The author spends a lot of time trying to suggest that LLMs are not the end all be all and humans are still needed for understanding etc. Fine. I understand.
If it is so the case that LLMs don't help that much, surely it doesn't matter if LLMs are privately controlled?
How come LLMs are important enough that private control matters but also not enough for actually advancing mathematics significantly on its own?
In other words, you are almost always better off with LLMs even if it is privately controlled. The mathematics community can just ignore LLMs for the base case or use ChatGPT if they want. In either case, it is strange to suggest that things will get worse for them. Its as if they don't have the coordination or agency to adapt.
One big problem here is that OpenAI would be hoarding this ability by 1) not releasing their top models for public use,as we have already seen them do, and 2) limiting the capability of the public API, which we have also seem them do.
This not only kills incentive for math, it means that OpenAI becomes the arbiter of math and research for the world. The research that will be done is the research OpenAI decides to do.
This seems very dangerous to me.
Another path forward is distributed local models. Professors can do research on their HPC machines, individuals can have full access to their models. It's a much more democratic approach.
But coming to final answer is not important and that’s what the author spends time arguing. If they are right, surely whatever slop OpenAI is producing is worthless to human understanding which is the main goal.
The math community has to change prestige markers to promote people who contribute to understanding and not just solving theorems and that’s on mathematicians to do.
If solving problems is the end goal then I don’t mind OpenAI doing it in which case furthering math is already solved by someone.
The only failure mode is if mathematicians can’t get funding for just “understanding”. But it shouldn’t be that hard to do if their claim is right that their main contribution to society has always been understanding.
I’d argue that getting funding purely for understanding may be even easier to justify because the costs go down dramatically per unit understanding.
This is just incoherent. Even if we "agree" (we don't but...) not to "abandon" math, how does it follow that "we need a lot more mathematicians" exactly?
Even if you reject the idea that AI can displace the current top quartile/percentile of human mathematicians, wouldn't it be reasonable to assume that it will at least displace some parts of math so that there'll be less demand for mathematicians overall, not more? The mental gymnastics are astounding here.
So the pie just gets larger? Large enough to outpace the displacement? I am gonna wait for this to generalize then, looking forward to AI giving us more software developers, mathematicians, etc. The economic laws have been broken. It's an infinite money glitch, as they used to call it.
2) TFA argues golden age, provided LLM infrastructure is independent.
3) Tao just opened a new SAIR AI platform.
4) Tao just opened a journal for papers with Lean proofs.
5) Academics are known to play both sides of the fence.
It is absolutely bewildering how some people miss the finer points.
It would be cool if some prestigious such program revamped itself to be AI-focused, maybe in collaboration with a CS department.
> Universities, professional societies, funding agencies, and scientific foundations should coordinate the development and governance of competitive LLMs. Scientists will need access to frontier capabilities, transparency about their operation and use of research data, strong confidentiality protections, and the ability to test and reproduce research claims independently.
> The massive computing resources, specialized personnel, data infrastructure, and ongoing investments needed to support this kind of initiative are not within the reach of a single institution, and a huge leadership effort will be necessary to conjure the combination of independent model development, guaranteed access to commercial systems, public computing infrastructure, and open research resources that could do the job.
the post is calling for....an enormous investment in LLM infrastructure.
IMO this is probably not an accurate assessment of Tao in particular at this time.
But in the general case, I think most people have a set of blinders on that will result in a predictable reassessment of AI with the following steps as AI progresses when applied to whatever it is they do:
1) AI is useless and doesn't impact me at all. Look at how bad it is! (Most developers up until late 2025)
2) Ok, now AI is useful but will not replace me. Sucks for the juniors in the field but I am still needed to guide the AI. AI will never be capable of my judgement (or whatever). (Most developers now)
3) Oh shit, I've been replaced in terms of economically valuable utility by AI. (tbd, but likely sooner than is comfortable)
And not everyone will experience #3, there will be a top 1% or so for most fields where #3 remains true for them, but it is likely that for most people #3 comes eventually.