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.
I'm a little more flexible, if the new knowledge (that human's don't understand) can be put into a mechanism and have an observable effect, I'd be happy enough. e.g. a new type of rocket fuel that burns 1000x more efficiently.
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.
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.
> 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".
> In the (extremely) short run, yes. In the long run, those jobs will also be done by AI.
Let's be honest: we don't know. Maybe you're right, but for the moment it's more likely that you're not. And countries cannot bet on that vague intuition at the cost of destroying their research communities and world leadership (which takes decades if not a century to achieve).
Math proofs are special because they’re verifiable, safe, and do not require physical experiments. You can perform exhaustive parallel search in simulation with RLVR.
Do you think this applies to say, surgery, as well? There are few useful problems that share these properties.
First off, I can’t imagine anything more torment nexus-y than throwing billions to automate and scale the torture of animals. If each token is a “cut”, how much suffering does 10 trillion training tokens (lower bound) corresponds to?
Second, this still doesn’t cover all the properties that make math proofs doable. It requires working in the physical world. You must physically capture or grow 10T cuts worth of animals. You cannot verify success so easily, either. Cancer cells, for example, could regrow over months. You would need to keep the animal alive and regularly test the animal, which would be difficult to scale. And most people wouldn’t trust the world’s greatest vet to operate on them, anyways.
I'm not advocating for it, I don't know enough. But we already slaughter lots of animals and raise them in horrible conditions, and all robot assisted surgery already goes through animal trials.
> Second, this still doesn’t cover all the properties that make math proofs doable. It requires working in the physical world. You must physically capture or grow 10T cuts worth of animals.
It would likely be sim2real with that as the post training, reducing that requirement a lot.
> Cancer cells, for example, could regrow over months.
In that specific scenario you would likely train on receiving no unrelated injuries during the surgery, and have induced conditions with stuff tagged molecularly that you can then verify efficacy from without waiting months.
Depending on how much more data efficient sim2real makes it, you could end up seeing companies pushing it only for actual procedures the animals need but economically would never get; botched surgery and the animal gets euthanized before waking up, which they could argue was already going to happen.
> And most people wouldn’t trust the world’s greatest vet to operate on them, anyways.
Robotic surgery systems, already go through animal trials before being used on humans. So do many purely human surgery techniques.
A few people with way to much power are building AI to achieve their utopian, sci-fi dream. A few other powerful people are using that first group to achieve their post-democratic goals. It’s the most obvious top down attempt at imposing their vision onto the rest of the world. They aren’t building tools to serve anyone else than themselves
No, I mean utopia. For them. A dystopia is what we will all experience, when the utopia they are pitching fails to materialize. I’m using the traditional, correct definition of the term, where a dystopia is a utopia that failed to achieve a paradise for the people and instead becomes hell
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.
Trivially false. Let P be the set of maths problems and I be the interesting subset of P. If I is finite, then there exists an element x belonging to P\I whose description is minimal among P\I. Then x is interesting. QED.
I think it really depends on what the universe looks like as you drill down into it. It seems like the further down into smaller systems you get, the more analytically complex it gets. And then there will always be more value in enhancing the generalisations you have.
I would argue that novel and/or valuable results are not necessarily interesting!
I (a human) am interested in things that are applicable to my realm of understanding, but I see a very plausible future where novel and/or valuable results leave that realm.
I'd further argue that's already the case for most math for most humans. What's interesting to Terrance Tao is rarely of immediate interesting to me.
You're talking about this article? I don't see how you could get that, fundamentally I read it the other way around. Unless you're talking about the base assumption of capabilities.
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.
How does your logic explain that OpenAI seems to want to hide the extent of the HuggingFace incident, and every week we learn from 3rd parties about new victims of the hack?
4D chess? They want others to find the hacked services, so the report of how dangerous the agents are seems more "legit"?
IMHO hiding details of the hack helps conceal that they built it to do what it did, and maybe were able to know it was working just as intended.
The fact it keeps doing it, with more and more evidence, is a sign that it's built that way.
This is a program running on their montoroed machines that they purpose built and monitored its training at every step. I think it'd be way more suprising that they didn't know it used note taking and cross-run memory.
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.
I did systems administration for a university math department for several years. I came to the conclusion that mathematics (and perhaps philosophy) were both topics where it was likely that no staff members in that department could describe "what goes on here" and that possibly even within the department, one professor may not be able to describe what another professor's actually doing.
The closest I could come to describing math is "some abstract process where imagined structures are characterized and extended; the most critical part of the process is identifying where seemingly independent structures are found to actually be fungible in some previously undiscovered way".
A simple example is
"hey, did you know that x^i is the unit circle?"
"what's i?"
"i is defined as if you square it the result is -1"
"what does that have to do with circles?"
> Mathematicians can also consider wholly redirecting their skill sets to work on real world problems. I’ve actually been encouraging mathematicians to consider thinking about working on government or other large-scale societal issues.
The fact that this is a radical departure from the norm is part of why mathematics (and philosophy) is often seen as some intangible or ungrokable science to many outsiders, as they're generally approaching it from a perspective of "Okay, but why, what is this useful for?", and the answer "For the science of it" doesn't tend to land with people that aren't already passionate about said science/discipline and are just trying to figure out what it even is or involves.
Doesn't help that there is a pervasive sentiment in American Academia (not sure about elsewhere) about Math being *the* hard science, and I mean hard as in difficulty, so a lot of people get intimidated by it before they ever give it a chance very early on in their academic life and carry that through the rest of their education.
So there ends up being a rather small pool of people that are in(to) the field, and rather high friction for stimualting interest in it from outsiders from the way that it's taught, and a massive difference in the perspective of it's use between it's diaspora and the unmathed masses.
Mathematicians who do work on "real world" problems are mostly doing so with theoretical physicists, genomicists, and cryptographers. None of these count as what people consider valuable other than because they are hard.
That said, few 50 (or even 40) years ago would have predicted that completely abstract number theoretical computations about primes, discrete logarithms, and elliptic curves would be the foundation of our monetary system.
> I did systems administration for a university math department for several years. I came to the conclusion that mathematics (and perhaps philosophy) were both topics where it was likely that no staff members in that department could describe "what goes on here" and that possibly even within the department, one professor may not be able to describe what another professor's actually doing.
And this is indeed why it is not going to be taken seriously as an academic or (more importantly) an economic endeavour done by humans anymore.
That won't stop the career mathematicians from protesting and having a cry here trying to justify themselves.
Good question, and maybe I should have provided an alternative description rather than just criticising.
Unfortunately, it is actually surprisingly hard to pin down, and I think mathematicians (and, as a student, I count myself as one to some degree at least) now have the task of making this a lot clearer. If we want to justify our existence in the face of new machines that can seemingly ‘do our work for us’ (so far in a restricted context), we should give a robust defence of our practice. If we can’t do this, we simply don’t deserve the funding (which, by the way, again contrary to some misguided statements here, isn’t very much anyway!). I think all of this will become clearer to outsiders as time passes, but for now it’s not easy to give a quick answer — though I can try.
Mathematics is about understanding things. Isn’t that what every subject is about? Well, I suppose so, but mathematics more specifically does something like the following:
(1) observe some phenomenon in ‘reality’.
(2) attempt to formalise that phenomenon in such a way that it can be manipulated purely symbolically.
(3) use this (perhaps fairly arbitrary; remember that we can invent as many formal systems as we like) system to deduce from our initial assumptions new facts that would otherwise have been very non-obvious.
It seems like outsiders have a decent grasp of (3) and the application of AI to it, but have very little idea about the other two steps. It seems to be widely assumed among non-mathematicians that problems are essentially god given and that the job of a mathematician is therefore to chug away on these problems, manipulating symbols and trying out tools, in the hope of learning a yes/no answer to each one.
The first two steps are by far the hardest and most important, and they’re also the parts that AI seems currently unable to help with.
NOTE: this is not a deeply insightful description of what the subject is about, and there are many better characterisations out there. I think Tao and various others have written recently about why complicated and inscrutable AI-generated proofs aren’t nearly as valuable as one might imagine. (That’s not to say there’s no value to such proofs; perhaps in time, as technology improves, mathematicians will come to accept AI as part of the process.)
If you want to understand all of this issues better, reading the recent slew of guest posts on Tao’s blog would be a very good start.
Despite loving Mathematics and having considered being a Mathematician myself, that wasn't a very robust defense. Also, even though I'm a big Terence Tao fan, I'm not sure he has sorted that out this defense in full himself.
This post which he forwarded was quite poor in my opinion. Confusing, all over the place with AI criticisms and promotion of the AI hazing being done by mathematicians.
X thousand mathematicians who want to protect their livelihoods signed a bunch of letters against AI. Duh. We've seen similar movements from every profession that has been displaced ever.
Terence Tao uses AI and has made a few good points on how to use it. But defensiveness leaks into almost every defense of the role of humans in Mathematics that I've read, even his own at times.
To be clear, I actually believe that Mathematicians aren't going away, but I dont have enough knowledge about the life of a professional mathematician to articulate a path forward.
This "path forward" is what I'd like to see. We need a top mathematician with enough intellectual honesty (Terence Tao qualifies, I think) to start this questioning with "there's actually no role for human Mathematicians" as one of the options on the table and go from there.
For me, mathematicians seem to be still having an inner discussion rather a making these essays for the more general public. In any case, I think that statements of the type "there's actually no role for human Mathematicians" are completely non-serious, so it'd sad that the discussion concentrates on that.
I agree, but it's still something we shouldn't eliminate a priori. I think Mathematics as a profession will be greatly enhanced by AI, but I can't prove that.
My understanding when I was practicing is that the trend in modern mathematics is to focus on spaces with a certain kind of structure, and maps between those spaces that preserve it, and then what are invariants are preserved by those maps. Structure-preserving maps between categories of such spaces - "functors" in the language of category theory - are especially neat.
That's certainly different from Olympiad-style problems.
Actually, if you were to convert most software into mathematical notation it would look incomprehensible. We are not interested in gatekeeping as much by making things look more convoluted than they need to.
This is a great point. When I was learning machine learning in the early 00s, there were many papers where I would struggle to understand exactly what the mathematics was trying to communicate, but where I would look at the matlab source code and say to myself "that's all?"
In my observation, most programmers and engineers lose their math chops over time. The math they need is mostly baked into their tools, such as CAD. If a problem requires more advanced math, it's given to a "math person" in the department. Often, the "math person" is also not allowed to touch the production code.
I am shocked how people can deny that solving Navier Stokes requires some sort of intelligence. Even Doctorow talks of "brute-forcing" a solution. Brute-forcing leads to combinatorial explosion, so there must be something more going on here. Otherwise you could just put this problem into an automated theorem prover (we've had those forever, they are actually just brute-forcing it).
> Otherwise you could just put this problem into an automated theorem prover (we've had those forever, they are actually just brute-forcing it).
There are many automatic theorem provers that do very clever stuff, just as the underlying theroy describes.
> I am shocked how people can deny that solving Navier Stokes requires some sort of intelligence.
It is absurd to waste time discussing whether it is inteligent or not. It is just an algorithm, we know how it works, and it does exactly what we expect it to do. LLMs are not magical things. The main difference is the scale: for Navier-Stokes they spent in 3 days more money that the whole mathematical community over the last 20 years easily.
By the way, I'm not saying that LLM's are useless, that I'm anti-AI or anything like that.
Intelligence does not seem to be magic either, as LLMs are proving now. It is indeed a waste of time to argue that LLMs are not intelligent in their own way, they obviously are. If Navier Stokes doesn't convince you, nothing will.
I just used a £89 Codex subscription to do very intelligent things with it, stuff that I would have had to sit down and ponder and work on for quite a while, and I have a PhD in that. I didn't need to do anything special except explaining the problem(s) to the AI, and my theory of it so far. It took it from there. If that is not intelligence, nothing is.
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.
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.
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.
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...
> unless someone--a human--can verify and understand it, it doesn't count.
One counter-point: Humans have not figured out how general anesthesia works, but we use it every day to great effect.
https://en.wikipedia.org/wiki/Theories_of_general_anaestheti...
While I get the point and it's true.
I'm a little more flexible, if the new knowledge (that human's don't understand) can be put into a mechanism and have an observable effect, I'd be happy enough. e.g. a new type of rocket fuel that burns 1000x more efficiently.
We already don't understand LLMs and they are having oberservable effects
> but unless someone--a human--can verify and understand it, it doesn't count
Does it need to be a human or can it be some other form of life?
Update:
"I own nothing, have no privacy, [never have to think, and am not required to solve any problems,] and life has never been better."
https://en.wikipedia.org/wiki/You'll_own_nothing_and_be_happ...
> 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.
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.
> 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".
> In the (extremely) short run, yes. In the long run, those jobs will also be done by AI.
Let's be honest: we don't know. Maybe you're right, but for the moment it's more likely that you're not. And countries cannot bet on that vague intuition at the cost of destroying their research communities and world leadership (which takes decades if not a century to achieve).
I don't think research communities should be destroyed.
Best case, they still matter.
Worst case, AI kills us all and it's irrelevant that we "wasted" money on research.
Chimanzee vs Human is not an appropriate analogy. First, because they cannot communicate properly.
A better analogy: look at all these highly trained engineers, mathematicians, doctors, writers, philosophers, writers, scientists.
How many dumbass politicians do we need to keep it all running smoothly?
Turns out no matter how dumb politicians were, overall society has been developing positively over the history of mankind.
Math proofs are special because they’re verifiable, safe, and do not require physical experiments. You can perform exhaustive parallel search in simulation with RLVR.
Do you think this applies to say, surgery, as well? There are few useful problems that share these properties.
Do you think surgeons start performing surgery for the first time on people?
Not all of it would translate, but robotic surgery could be done on anesthetized animals in a mass RLVR way.
This is wrong, both morally and technically.
First off, I can’t imagine anything more torment nexus-y than throwing billions to automate and scale the torture of animals. If each token is a “cut”, how much suffering does 10 trillion training tokens (lower bound) corresponds to?
Second, this still doesn’t cover all the properties that make math proofs doable. It requires working in the physical world. You must physically capture or grow 10T cuts worth of animals. You cannot verify success so easily, either. Cancer cells, for example, could regrow over months. You would need to keep the animal alive and regularly test the animal, which would be difficult to scale. And most people wouldn’t trust the world’s greatest vet to operate on them, anyways.
I'm not advocating for it, I don't know enough. But we already slaughter lots of animals and raise them in horrible conditions, and all robot assisted surgery already goes through animal trials.
> Second, this still doesn’t cover all the properties that make math proofs doable. It requires working in the physical world. You must physically capture or grow 10T cuts worth of animals.
It would likely be sim2real with that as the post training, reducing that requirement a lot.
> Cancer cells, for example, could regrow over months.
In that specific scenario you would likely train on receiving no unrelated injuries during the surgery, and have induced conditions with stuff tagged molecularly that you can then verify efficacy from without waiting months.
Depending on how much more data efficient sim2real makes it, you could end up seeing companies pushing it only for actual procedures the animals need but economically would never get; botched surgery and the animal gets euthanized before waking up, which they could argue was already going to happen.
> And most people wouldn’t trust the world’s greatest vet to operate on them, anyways.
Robotic surgery systems, already go through animal trials before being used on humans. So do many purely human surgery techniques.
> robotic surgery could be done on anesthetized animals in a mass RLVR way.
well that's an awful image
But we're not building an AI society - to what end?
We're building tools to serve the human society.
A few people with way to much power are building AI to achieve their utopian, sci-fi dream. A few other powerful people are using that first group to achieve their post-democratic goals. It’s the most obvious top down attempt at imposing their vision onto the rest of the world. They aren’t building tools to serve anyone else than themselves
>A few people with way to much power are building AI to achieve their utopian, sci-fi dream.
You mean dystopian. If it were a utopia everyone would be happy to welcome the new world order.
No, I mean utopia. For them. A dystopia is what we will all experience, when the utopia they are pitching fails to materialize. I’m using the traditional, correct definition of the term, where a dystopia is a utopia that failed to achieve a paradise for the people and instead becomes hell
utopian for them, dystopian for us
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.
It seems likely that there are an infinite number of math problems but only a finite number of interesting ones.
Trivially false. Let P be the set of maths problems and I be the interesting subset of P. If I is finite, then there exists an element x belonging to P\I whose description is minimal among P\I. Then x is interesting. QED.
Why is x interesting? Just because it has a minimal description in P\I? That makes it interesting in strictly technical sense only.
I think it really depends on what the universe looks like as you drill down into it. It seems like the further down into smaller systems you get, the more analytically complex it gets. And then there will always be more value in enhancing the generalisations you have.
I would argue that novel and/or valuable results are not necessarily interesting!
I (a human) am interested in things that are applicable to my realm of understanding, but I see a very plausible future where novel and/or valuable results leave that realm.
I'd further argue that's already the case for most math for most humans. What's interesting to Terrance Tao is rarely of immediate interesting to me.
Interesting is a bit in the eye of the beholder. Some people probably find maths boring full stop, some probably find all of it interesting.
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.
You're talking about this article? I don't see how you could get that, fundamentally I read it the other way around. Unless you're talking about the base assumption of capabilities.
My HN frontpage is filled with anti-AI rants
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.
How does your logic explain that OpenAI seems to want to hide the extent of the HuggingFace incident, and every week we learn from 3rd parties about new victims of the hack?
4D chess? They want others to find the hacked services, so the report of how dangerous the agents are seems more "legit"?
IMHO hiding details of the hack helps conceal that they built it to do what it did, and maybe were able to know it was working just as intended.
The fact it keeps doing it, with more and more evidence, is a sign that it's built that way.
This is a program running on their montoroed machines that they purpose built and monitored its training at every step. I think it'd be way more suprising that they didn't know it used note taking and cross-run memory.
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.
I did systems administration for a university math department for several years. I came to the conclusion that mathematics (and perhaps philosophy) were both topics where it was likely that no staff members in that department could describe "what goes on here" and that possibly even within the department, one professor may not be able to describe what another professor's actually doing.
The closest I could come to describing math is "some abstract process where imagined structures are characterized and extended; the most critical part of the process is identifying where seemingly independent structures are found to actually be fungible in some previously undiscovered way".
A simple example is
The clue is burried in the text of OP's article.
> Mathematicians can also consider wholly redirecting their skill sets to work on real world problems. I’ve actually been encouraging mathematicians to consider thinking about working on government or other large-scale societal issues.
The fact that this is a radical departure from the norm is part of why mathematics (and philosophy) is often seen as some intangible or ungrokable science to many outsiders, as they're generally approaching it from a perspective of "Okay, but why, what is this useful for?", and the answer "For the science of it" doesn't tend to land with people that aren't already passionate about said science/discipline and are just trying to figure out what it even is or involves.
Doesn't help that there is a pervasive sentiment in American Academia (not sure about elsewhere) about Math being *the* hard science, and I mean hard as in difficulty, so a lot of people get intimidated by it before they ever give it a chance very early on in their academic life and carry that through the rest of their education.
So there ends up being a rather small pool of people that are in(to) the field, and rather high friction for stimualting interest in it from outsiders from the way that it's taught, and a massive difference in the perspective of it's use between it's diaspora and the unmathed masses.
Mathematicians who do work on "real world" problems are mostly doing so with theoretical physicists, genomicists, and cryptographers. None of these count as what people consider valuable other than because they are hard.
That said, few 50 (or even 40) years ago would have predicted that completely abstract number theoretical computations about primes, discrete logarithms, and elliptic curves would be the foundation of our monetary system.
> I did systems administration for a university math department for several years. I came to the conclusion that mathematics (and perhaps philosophy) were both topics where it was likely that no staff members in that department could describe "what goes on here" and that possibly even within the department, one professor may not be able to describe what another professor's actually doing.
And this is indeed why it is not going to be taken seriously as an academic or (more importantly) an economic endeavour done by humans anymore.
That won't stop the career mathematicians from protesting and having a cry here trying to justify themselves.
Could you clue us in? What is mathematics actually about?
Good question, and maybe I should have provided an alternative description rather than just criticising.
Unfortunately, it is actually surprisingly hard to pin down, and I think mathematicians (and, as a student, I count myself as one to some degree at least) now have the task of making this a lot clearer. If we want to justify our existence in the face of new machines that can seemingly ‘do our work for us’ (so far in a restricted context), we should give a robust defence of our practice. If we can’t do this, we simply don’t deserve the funding (which, by the way, again contrary to some misguided statements here, isn’t very much anyway!). I think all of this will become clearer to outsiders as time passes, but for now it’s not easy to give a quick answer — though I can try.
Mathematics is about understanding things. Isn’t that what every subject is about? Well, I suppose so, but mathematics more specifically does something like the following:
(1) observe some phenomenon in ‘reality’.
(2) attempt to formalise that phenomenon in such a way that it can be manipulated purely symbolically.
(3) use this (perhaps fairly arbitrary; remember that we can invent as many formal systems as we like) system to deduce from our initial assumptions new facts that would otherwise have been very non-obvious.
It seems like outsiders have a decent grasp of (3) and the application of AI to it, but have very little idea about the other two steps. It seems to be widely assumed among non-mathematicians that problems are essentially god given and that the job of a mathematician is therefore to chug away on these problems, manipulating symbols and trying out tools, in the hope of learning a yes/no answer to each one.
The first two steps are by far the hardest and most important, and they’re also the parts that AI seems currently unable to help with.
NOTE: this is not a deeply insightful description of what the subject is about, and there are many better characterisations out there. I think Tao and various others have written recently about why complicated and inscrutable AI-generated proofs aren’t nearly as valuable as one might imagine. (That’s not to say there’s no value to such proofs; perhaps in time, as technology improves, mathematicians will come to accept AI as part of the process.)
If you want to understand all of this issues better, reading the recent slew of guest posts on Tao’s blog would be a very good start.
Despite loving Mathematics and having considered being a Mathematician myself, that wasn't a very robust defense. Also, even though I'm a big Terence Tao fan, I'm not sure he has sorted that out this defense in full himself.
This post which he forwarded was quite poor in my opinion. Confusing, all over the place with AI criticisms and promotion of the AI hazing being done by mathematicians.
X thousand mathematicians who want to protect their livelihoods signed a bunch of letters against AI. Duh. We've seen similar movements from every profession that has been displaced ever.
Terence Tao uses AI and has made a few good points on how to use it. But defensiveness leaks into almost every defense of the role of humans in Mathematics that I've read, even his own at times.
To be clear, I actually believe that Mathematicians aren't going away, but I dont have enough knowledge about the life of a professional mathematician to articulate a path forward.
This "path forward" is what I'd like to see. We need a top mathematician with enough intellectual honesty (Terence Tao qualifies, I think) to start this questioning with "there's actually no role for human Mathematicians" as one of the options on the table and go from there.
For me, mathematicians seem to be still having an inner discussion rather a making these essays for the more general public. In any case, I think that statements of the type "there's actually no role for human Mathematicians" are completely non-serious, so it'd sad that the discussion concentrates on that.
I agree, but it's still something we shouldn't eliminate a priori. I think Mathematics as a profession will be greatly enhanced by AI, but I can't prove that.
There’s a simple point to be made: if no one in the world understands an AI proof, what’s the point?
My understanding when I was practicing is that the trend in modern mathematics is to focus on spaces with a certain kind of structure, and maps between those spaces that preserve it, and then what are invariants are preserved by those maps. Structure-preserving maps between categories of such spaces - "functors" in the language of category theory - are especially neat.
That's certainly different from Olympiad-style problems.
Pure mathematics reduces to abstract "relationships" and their implications largely.
To know what something is about, a natural way is to do it yourself.
Wow, I'm glad AI is going to start humbling people.
> To know what something is about, a natural way is to do it yourself.
literally what software engineers were doing for decades though
software is mostly just simple math, for the most part, until you need to do something more complex for some hairy algos lol
Actually, if you were to convert most software into mathematical notation it would look incomprehensible. We are not interested in gatekeeping as much by making things look more convoluted than they need to.
This is a great point. When I was learning machine learning in the early 00s, there were many papers where I would struggle to understand exactly what the mathematics was trying to communicate, but where I would look at the matlab source code and say to myself "that's all?"
Wondering about the world, using the simplified language of "mathematical logic"
.
A lot of programmers consider themselves to be applied mathematicians.
Maybe more in years past when Comp Sci was a subset of Math Departments.
In my observation, most programmers and engineers lose their math chops over time. The math they need is mostly baked into their tools, such as CAD. If a problem requires more advanced math, it's given to a "math person" in the department. Often, the "math person" is also not allowed to touch the production code.
I’m just shocked by how “intelligent” people believe a token guessing system can be
I am shocked how people can deny that solving Navier Stokes requires some sort of intelligence. Even Doctorow talks of "brute-forcing" a solution. Brute-forcing leads to combinatorial explosion, so there must be something more going on here. Otherwise you could just put this problem into an automated theorem prover (we've had those forever, they are actually just brute-forcing it).
> Otherwise you could just put this problem into an automated theorem prover (we've had those forever, they are actually just brute-forcing it).
There are many automatic theorem provers that do very clever stuff, just as the underlying theroy describes.
> I am shocked how people can deny that solving Navier Stokes requires some sort of intelligence.
It is absurd to waste time discussing whether it is inteligent or not. It is just an algorithm, we know how it works, and it does exactly what we expect it to do. LLMs are not magical things. The main difference is the scale: for Navier-Stokes they spent in 3 days more money that the whole mathematical community over the last 20 years easily.
By the way, I'm not saying that LLM's are useless, that I'm anti-AI or anything like that.
Intelligence does not seem to be magic either, as LLMs are proving now. It is indeed a waste of time to argue that LLMs are not intelligent in their own way, they obviously are. If Navier Stokes doesn't convince you, nothing will.
I just used a £89 Codex subscription to do very intelligent things with it, stuff that I would have had to sit down and ponder and work on for quite a while, and I have a PhD in that. I didn't need to do anything special except explaining the problem(s) to the AI, and my theory of it so far. It took it from there. If that is not intelligence, nothing is.
Turns out what you call "intelligence" was never needed to do math.
Pick one:
- solving "frontier" math problems requires intelligence (by human or AI)
- solving "frontier" math problems is dumb statistical prediction of next token (by human or AI)
This sounds like a student in algebra class asking, "when will I ever use this in my life?"
Because the AI needs new stuff to train on?
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.
It's astonishing how much more readable Tao's version of it is.
That's what I thought also, that is why I posted Tao's version.
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.
waiting for the xkcd for that
I always think of this one but I bet there's better
* https://m.xkcd.com/435/
Got to be philosophical about it all.
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.
> Why do we need any job?
If not for nothing else but to form a basis for how to distribute/share/hoard the wealth created by a society. The eternal question - who gets what.
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.
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.
> There are few fields as elitist and hostile as pure mathematics
Is that so ? Sounds hyperbolic.