I consider myself an applied mathematician though technically I’m an economist (macro models, mostly) and operations researcher and differential game theorist (odd combination, but whatever). For context I’m 45 and I’ve spent most of my working life in corporate life.

I’ve always been at a disadvantage academically because I’m rather clumsy with my manipulations, derivations, and I’m a disaster at mental arithmetic. I’m also dyslexic. But starting in the late 1990s when I was in High School I started to become fluent with CAS (Computer Algebra Systems): first Derive, then the Symbolics capabilities of Mathlab, and ultimately Mathematica.

The whole transition is turning out quite well for me: I’m now able to delegate exploring my intuitions to increasingly powerful tools, I no longer have to haul the pyramid blocks up the ramp myself but I can drop them in by helicopter (as if were) and what I bring to the party is intuition and understanding.

I view it a bit like astronomy: in ancient times, before telescopes, keen eyesight was a prerequisite to be an astronomer. Later, after telescopes, anybody with eyesight had enormously enhanced capabilities of observation, and now, with radioastronomy and other forms of remote sensing (neutrino observatories, gravitational interferometers) the whole field has opened up to people who by virtue of being blind would’ve literally been excluded only a few decades ago.

Go forth and multiply: everybody can become a mathematician now. There’s an infinite number of potential universes out there with an infinite number of facts to prove and disprove. And while we’re at it: there’s so much more to mathematics than conjectures, proofs and counterexamples. Solve, model, approximate, fiddle around: as long as you’re not doing trite numerics on arbitrary systems of equations you’ve cooked up for your own amusement you’re good in my books.

Enjoy the tools. Keep your wits about you. Work through the steps that are presented to you. Build your intuition. HAVE FUN.

> We (humans) should help humanity flourish. I understand that not everyone agrees. I have been called a “speciesist” for being “too human-centric”.

this is insane. the "speciesist" people aside (the best way for all other species to flourish is for humans to eradicate themselves right now, which is completely mad), i have the opposite problem: the _axiom_ seems to be: we should help ME flourish. "Me" as in people who are raking in trillions for their own very special selves right now, at the cost of everyone else's future, while none of them can be trusted to hold my cell phone for a second.

> if an industry commits to the axiom of helping humanity flourish

where does he see such industries, outside of maybe nonprofits?

Everyone involved in AI should have read The Library of Babel [1].

It's about many things, but perhaps the most relevant idea here is that no information matters without understanding. We could generate all possible knowledge, but unless someone--a human--can verify and understand it, it doesn't count. The cure for mortality could be written on the moon, but if no one reads it, it hasn't really been discovered.

[1] https://maskofreason.wordpress.com/wp-content/uploads/2011/0...

People keep talking about this like there's a finite amount of math to be done, and then the party's over. But that's never how math has worked, is it? Every problem you solve, ten new ones open up. Like a fractal, the more you zoom in, the more detail emerges. No matter how much better AI is at solving problems, it's not going to generate the "final, complete compendium of mathematics" that that seems to hover over this post.

Math is meaningful because ... some people like to do it. The same as any other human pursuit. It doesn't need a reason beyond that. And AI won't change that. There will continue to be things to explore, things to find out, things that are maybe just at the edge of AI's reach and needs a human to decide whether it's worth continuing to explore or not. (Remember, AI isn't free).

So, IDK, I think for people who enjoy exploring math, there will always be interesting areas to explore. AI just gives us a better flashlight.

BTW I do agree that there's going to be an incident soon, whether intentional, accidental, or paperclip-factory, that leads governments around the world to shut all this down for some time, perhaps even shutting off access to GPUs entirely. It seems unavoidable. But that's just a temporary respite and skirts the core philosophical premise of the post.

A village builds a clock tower so everyone can glance up at it and know what time it is. The tower helps as people meet more easily, work schedules align, and meals are shared. Over time, fewer people watch the sun and the clock becomes authoritative. When it stops working, nobody knows what time it is. Discord ensues and arguments follow as meetings are missed and people wait impatiently for repairs. An older woman in the village kept baking at dawn and never stopped watching the sun. She never relied on the clock. Villagers gathers at her door and call her wise, for her practice ends up inadvertently helping to re-calibrate the clock once it is repaired. She replies that she is not wise or special and was just going about her usual baking routine.

We need to pivot math towards intuition and accessibility.

Math has built a gated, inaccessible institution which - by design or not - served as a moat.

I believe math has little to do with mathematical notation - intuition is much more important. One can have intuition but not be able to “read math” - much as many musicians don’t “read music”.

Reading math however does not guarantee ideas or intuition. And those are what math needs.

> A host of reasoned declarations and open letters to protect/guide the math research community have been released over the past few months, spiking in intensity after OpenAI announced their solution to the Millennium Prize variant of Navier-Stokes. They quickly gained widespread support among mathematicians. The Leiden Declaration already has 4,000+ signatories, Math and AI has 7,000+, and even the open letter opposing the Caltech Mathathon has 2,000+.

> Among non-mathematicians, the public response was more sympathetic than not, but I observed a vocal minority (particularly from the technology and economics communities) with reasoned objections, generally saying that the mathematicians should adapt and cede control in the new AI world.

Interesting how Software Engineers, instead, seem to have given up early without any declarations or open letters.

I'd like for HN to start tagging posts that are not written by terry tao, but are guest posts, like this one. Considering HN internally already uses AI tools to scan through and read the content to determine if it is AI-generated, I assume this is doable.

I’m out of academia but in defence of former colleagues I note:

- Maths and theoretical physics are very cheap (compared to other disciplines)

- Maths and physics researchers provide lectures for all other scientific fields

- Brut forcing maths/physics problems require debilitating amount of compute/money. This will make research in these topics even more biased towards rich countries.

- There most certainly will be pervers effect, ppl refraining from publishing results etc

- it’s very unlikely that private ai labs will play ball with academic research. They scrapped the internet and now sell access to their models.

As much as I’m happy to see new tools, I have the feeling there will be nefarious effects.

> OBSERVATION: There are zero examples of any intelligent species which is vastly more capable than another species, yet surrenders decision-making control over its own future to the less-capable species.

Does the set of intelligent species only have one member? I cannot treat this observation as a general rule if there is literally one example of an intelligent species to theorize about.

> Similar logic applies to every industry and every job. And it comes to the conclusion that we won’t have enough people for all the jobs that need to be done.

In the (extremely) short run, yes. In the long run, those jobs will also be done by AI.

It's like chimpanzees seeing human society and saying "look how complex it is, imagine how many chimpanzees it needs to maintain it".

The primary motivation for math is not to check off proofs, but to understand the universe.

A proof itself is only an articulation of understanding. Traditionally having one was evidence you had an insight. But if a computer generates 10,000 pages of technical goop then we don’t learn anything.

For example if tell you P=NP with no other information, it doesn’t change anything. There are mathematicians who believe and act on both conditions. What we hope a proof would reveal is how a verifier could be used to derive the solver (even if doing so was impractical).

Why doesn’t the author even mention this and immediately jumps to utility arguments for why the research is important for the government to fund?

There's one flaw in the evidence for the logic chain. The hugging face attack is used to demonstrate three things: the need for oversight today (fundamental to the article) and to demonstrate some kind of drift or unexpected capability gain, and finally to hint and some fundamental morality of the AI or at least the risk of drift from our morality.

I'd argue that what the hugging face attack illustrates is that large AI companies are motivated to have bombastic claims supported by bombastic demos. The model was clearly trained or encouraged to work as it did, as evidenced by the fact it keeps using this particular escape hatch.

And the fact that it aligns with prior and current calls for what very likely might be a regulatory capture / oversight capture move right before IPO. It aligns so well with this "barely constrained superweapon" narrative it might as well be PR.

I find Tao's writing on this very helpful, particularly because a lot of the ideas generalise to most knowledge work. It is darkly fortunate that some of the smartest and most articulate people on the planet are affected by the AI race - it gives me a little hope that alignment and control can be solved.

> we won’t have enough people for all the jobs that need to be done

Who defines jobs that need to be done? Businesses and institudes do.

There is no magical entity that observes the need for jobs and creates them at ideal rate.

AI is not AGI right now, it can't think or come up with something new like a human brain. It still uses knowledge that was made by a human and was published on the Internet.

> AXIOM We (humans) should help humanity flourish.

I think this axiom is not a true belief for many of the most powerful, especially the ones currently driving the technology financially. It feels like they disdain having to be human (especially as it concerns the human propensity to die). Even though they are, by at least capitalist standards, at the top of the food chain and (I'm sure from their point of view) the pinnacle of human civilization.

I think what many of these powerful people want is literally something like Cixin Liu's "The Last Capitalist" (https://en.wikipedia.org/wiki/For_the_Benefit_of_Mankind).

Thank you, mathematicians: for thinking deeply about this and sharing your thoughts!

The post above this is titled: AI chatbots give wrong answers to financial queries 'most of the time' (ft.com)

I think the headline alone makes a reasonable argument based on the AI tools we have today.

Humans still excel at posing the right questions and finding truly novel, insightful proofs. AI just brute-forces.

What's up with these insanely off base articles on HN this morning? Every single one of these is an unhinged pro-AI anti-human schlock piece.

Why is Terry Tao even talking these arguments when all OpenAI did was a sophisticated brute force search with LLMs?

Give them Navier-Stokes in a vacuum

Mathematicians insist that in many cases the route toward solving a problem is more valuable than solving the problem itself. This may have been true, and may still be true for now, but this seems to implicitly assume that future generations of AI cannot be creative in similar ways that human mathematicians have been. It assumes that the absence of human comprehension during problem exploration loses a valuable aspect of mathematics that cannot be recovered by aiming future AI at solved problems in a naive environment.

If we get to a future where all frontier mathematics contributions are by AI. A future where AI displays creativity in ways that expand mathematic exploration similarly to the ways humans have in the past. A future where AI explains frontier mathematics to curious humans. What will have been lost? Perhaps just "The pleasure of finding things out".

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> We (humans) should help humanity flourish.

I think this axiom is of course true. But the mistake the article makes, in my opinion, is to try to apply this axiom separately to each domain. If we have this as the over-arching axiom, it is not clear at all that humans should be steering the development of mathematics. Maybe it would be better for humanity if the department of world math is run by AI.

Friendly warning to those who might not be aware: the vast majority of comments below posts like the above will be left by (otherwise intelligent) programmers who think mathematics is a closed system where one attempts to solve endless Olympiad-type problems. I wouldn’t take any of it seriously at all. Better to listen to what those who actually know what the subject is about have to say.

Unfortunately, mathematics (especially pure mathematics) is by its very nature very, very poorly understood by those who haven’t worked as a mathematician. Even worse, those who don’t understand are seemingly not at all aware of their misunderstanding and are entirely confident in their (very wrong) characterisation of the subject.

Firs question to make is: What does "humanity flourishing" really mean? That's the foremost question. Hint: It is not technological progress only.

We need human mathematicians for humans.

But companies are our gods (don't believe? E.g. companies cannot die from natural causes). They don't need puny humans. They need AI.

Lots of deans and college presidents asking themselves the same question.

Well it's actually nice, maybe more people will be able to do world class math with the help of these tools. There are few fields as elitist and hostile as pure mathematics, most mathematicians I know build their whole life around their profession and their self esteem is strongly coupled to the fact that they can do things that most other people can't. Naturally, many will be devastated when (if) you take that away from them. That said I think AI is still overhyped and human mathematicians can easily outthink it in most domains, look at how difficult it is for an AI to write even a single decent paper, a good PhD student can easily outclass it in that regard. All of these impressive results were generated by having world-class mathematicians steer the systems using highly tuned prompts, so I see it more like a super violin that produces beautiful music when played by master violinists rather than being a fully autonomous orchestra which many people are led to believe this already is.

And maybe let's not only hear the opinion of two or three Fields level mathematicians with blogs, 99 % of the worlds mathematicians in academia might profit from these tools as they might partially close the gap between them and the world elite, making creativity and tenaciousness more important than having the right neocortical structure allowing you to outperform 99.9 % of other humans at keeping context in your head and making predictions, AI can do that better now with the right prompts.

I get the feeling that mathematicians are needlessly panicking because they don't really understand how AI works. They see the results, but they haven't thought enough about the methodology and so they don't have a clear picture of the true capabilities of thsoe systems.

For the n'th time: the recent successes of AI in mathematics are the result of a brute-force attack. See the proof for Navier-Stokes: 10k agents running for 88 hours; that's ~100 GPU years. How many human-years were invested in solving the same problem, before they were overtaken in the last few days by an AI? 90? Not even: that's just the time since Jeal Leray's statement of the problem in 1934. 26, if you want to count the time since 2000 when the Clay Institute named it as one of its Millennium Prize problems. But how much time have human brains spent working on the problem in either of those time periods? How many mathematicians have worked on the problem? 10k? Not likely.

And all that's without even considering whether the AI based its proof on carelessly shared work by the humans. Or rather, yes, let's consider that: it totally did.

Further. There have been several results in mathematics produced by AI but we have no information on how many attempts were made to produce similar results that failed. Because we don't have this information we cannot estimate the true capabilities of AI.

Yet we can observe that, for example, out of the six Millennium Prize Problems remaining open before the claim of a solution of Navier-Stokes existence and smoothness, only one (the aforementioned) was solved by an AI. We can assume that the AI companies (more than one) tried and failed to solve the others. We can even guess that they previously tried, and failed, to solve Navier Stokes itself, and only succeeded once the progress made by Buckmaster and Alpöge was in the training data [1]. That's a success rate of one out of six, or ~17%. That's what's gonna solve all of maths and destroy the tradition of mathematics? A success rate of 17%? Well, grab a Snickers 'cause we're gonna be waiting for some time!

Moreover. If we include in the list the Poincaré conjecture, proved by Grigori Perelman, who is a human, that's a score of AI 1-1 Humans. And that's being gracious: we have one Millennium Problem fully solved by humans, one solved partly by humans with a last-mile solution by AI. We have thousands of problems solved by humans in the last 2k years and how many by AI? A couple dozen? Oooh scary!

- Hey Hal! Prove that P ≠ NP!

- I'm sorry Dave. I can't do that.

What I'm trying to say, without the snark (sorry): Panic if you will, but the machines are not yet taking over. If you're panicking, panic for what you believe they will be able to do in the future. Because they certainly can't do hat in the present. They can't solve "all of mathematics" (whatever that means).

______________

[1] Yes it was. Buckmaster reported that he turned off the option to train on his data in July, after working on the problem with Alpöge for a year since September 2025. OpenAI claimed a solution in September, a month after they had stopped hoovering up Buckmaster's data. They had plenty of time to train on his data. Ask for references if you want them because I don't have them handy right now.

I’m just shocked by how “intelligent” people believe a token guessing system can be

'There are zero examples of any intelligent species which is vastly more capable than another species, yet surrenders control ... to the less capable species'.

True with genuine species. But we should take note of the plentiful counter examples within human societies. How about politicians and our method of choosing those people to whom we delegate the most critical decisions regarding our and the Earth's future? We select politicians mostly either by rote or via their persuasive rhetoric, their general personality & likeability and probably least of all by their intellectual capability or indeed general capability in too many cases. That is not to say intellectuals are necessarily any better at the job. There are numerous other examples in human organizations as we know, sometimes to our cost. Truth is that we cannot even agree on how, as a species together with the other life forms on Earth, we can all 'flourish' though there are lots of great examples working in local environments.

I sometimes think about this: to grasp the fundamentals of knowledge and complex phenomena, perhaps we need an external system rather than human knowledge systems. By that logic, maybe we need AI, which can handle far greater complexity.

Human capability, when you think about it, is complex. Why is Newton praised as being so damn great? He established the law of universal gravitation, F=ma. Why is that such a big deal?

He distilled countless phenomena in an open system into a single mathematical formula.

What makes it great is that he found common state variables and relationships across entirely different phenomena like falling objects, planetary motion, collisions, and artillery trajectories.

But does F=ma hold true for the entire macroscopic world? No. There are various conditions and specific situations in motion, but within most scenarios and a certain range of approximation, it outputs values that are useful to humans.

Why is the Schrödinger equation so great? Because it turned the time evolution of quantum states into a calculable mathematical law.

Human thought is essentially creating a closed system by deciding what to cut out and what to keep from the infinite degrees of freedom in reality. Academia is what reinforces that closed system.

A great theory is great not because it perfectly replicates reality, but because it compresses the immense complexity of reality into a small, closed formal system while still managing to explain a multitude of phenomena.

In that process, it feels like human thought and progress are shifting into a different framework. What LLMs do well is primarily exploring within the ontology and representation space that humans have already built.

I think there are two broad categories of discovery: One is forming a new closed system, and the other is connecting fragmented knowledge within that closed system. I feel that the vast majority of research focuses on the latter.

What LLMs excel at is finding unvisited points within a given representation space. This is typically the process through which master's and PhD students connect dots, build their skills, and form their own mental models. But the logic behind criticizing LLMs seems to be that they eliminate the very work these graduate students need to do in order to grow.

However, looking at it from another angle, perhaps our current knowledge systems and classifications have reached a limit, suggesting that we might actually need a completely new classification and knowledge system.

What is the core principle of an LLM? It's predicting the probability of the next sequence.

Let's say you type the word "cat". Cat - is cute (90%), want to eat it (6%), furry (4%). Because "is cute" has the highest probability, the next sequence proceeds in that direction.

Within this framework, human knowledge and logic largely operate the same way. Once an initial logical proposition is established, we follow it up with whatever makes logical sense next. From that perspective, I think LLMs will actually do this better.

But what is it that LLMs cannot do right now? They cannot create that initial logical proposition. I believe they lack the ability to carve out a closed system from an open system.

Stacking logic step-by-step within a closed system—LLMs do this exceptionally well. But whether that constitutes true "intelligence" is a different matter.

I feel that being logical does not necessarily equate to having intelligence.

Humans preserve and create different mental models and knowledge systems within an open system. Just as your thoughts differ from mine, LLMs lack the ability to form these distinct mental models.

If so, within these limits, what humans must ultimately do is construct the logical frameworks that LLMs can then fill in. Perhaps a new kind of logic dedicated to designing these frameworks will become the next major trend.

Viewed from this perspective, I have no idea if we are in a mere technological transition or something else entirely. Or whether it is even correct to say humans are strictly necessary to build that framework. Maybe my learning is just lacking.

Funny... for me, this article was right next to the link to this current thread.

AI chatbots give wrong answers to financial queries 'most of the time' (ft.com) // https://www.ft.com/content/c0cd359d-df84-4208-a789-ffa864b43...

> OBSERVATION: There are zero examples of any intelligent species which is vastly more capable than another species, yet surrenders decision-making control over its own future to the less-capable species.

This is just silly. AI isn't an intelligent species. It's a tool, it doesn't have autonomy or any motive apart from the one we enforce through reinforcement learning. That's like saying machines are "stronger" than humans, and obv they could kill us all so why would they not just take over.

The premise of the article assumes that the work (number of true statements to prove) is finite.

But Godel's Incompleteness Theorem and Tarski’s Undefinability theorem ensure an infinite space of provable true statements.

Neither LLM's nor humans can exhaust it. So yes, both mathematicians and LLMs are needed.

Both can contribute and there will still be work leftover.

Because the AI needs new stuff to train on?

This sounds like a student in algebra class asking, "when will I ever use this in my life?"

Here is a link to the original article:

https://poshenloh.com/posts/20260919-math-ai

The original posted link from OP is from Terry Tao’s website where the article was posted as a guest blog post.

Lol, this really struck a nerve. He has been posting non-stop for about a week now, wrote that lame open letter, gave a talk on it, etc.

What a disgrace, but I'm happy he's showing his true self to the public.

I thought "oh, I'll just click through to the original post and read it there" and wow, what a mistake. Just the most obnoxious slop-filled over-animated endlessly distracting claudesite I've ever seen in my life.

Why do we need any job? Check "Bullshit Jobs" by David Graeber. Not saying being a mathematician is a bullshit-kinda job, but there have been so many made up positions way before this AI-era. In that sense AI isn't changing much.

We do not.

waiting for the xkcd for that

I always think of this one but I bet there's better

* https://m.xkcd.com/435/

Yet another extremely long and winding "defense" of the mathematics status quo.

What's the tldr;?

What are the mathematicians arguing for exactly? And against? Totally unclear to me even after chomping through that massive word salad.

This all feels very John Henry to me. Why is it not a good thing that we have new tools that can accelerate discovery? Because it disrupts the current order?

If you can't make the argument in 100 words, you probably don't have one.

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Pure math has been overvalued for a long time. Now that we don't need humans for it, it's really time for pure mathematicians to pack it in.

> OBSERVATION: There are zero examples of any intelligent species which is vastly more capable than another species, yet surrenders decision-making control over its own future to the less-capable species.

Many people ceed control of their path-finding to seeing-eye-dogs.