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In the story, the baker was the only one left with that ability, as others left the skill behind when the clock became the authority.
This problem has been seen when missionaries, or whoever else, try and bring tribes into more modern civilization. They have trouble adapting to modern civilization, but the skill loss during that process makes it almost impossible to go back to the jungle.
When skills are lost due to modern technology, it isn’t so trivial to get them back. Look at how much time people have spent trying to figure out how stone was moved in ancient Egypt, for example.
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.
Hauling the pyramid blocks is what gives people intuition and understanding. It is true that school systems usually have way too much computation -- it is easier to test and grade computation. But the only way that you were able to use computer algebra systems fluently is because you had internalized how algebra worked by hand. If we tell students that they no longer need to learn how to solve equations we are seriously depriving them of a mathematical education.
I have no idea how assembly works - in my years of experience I’ve never had an issue that required I dig that deep into it. Does that mean I can’t be effective with higher-level tooling because I do not know the absolute basics?
You can be effective at using C# and F# to solve other problems specifically because they are very well designed at the goal of abstracting over the lower level hardware. But if your goal is to build an intuition of how programming languages themselves are implemented and how computer hardware works, then, using those languages will be counter productive.
If you're trying to use math to solve arithmetical problems, then by all means using AI to help. But if you're trying to advance the math field itself, then actually knowing how math works is probably essential.
Put in more concrete terms: if someone wants to be a working mathematician without learning the fundamentals of math, then how do they even know what prompts to the AI are worth writing? In what way are they adding any value to the process at all?
You can use AI like a telescope, and alleviate your inherent organizational problems or other ailments, which in my opinion is a great use case, as it is empowering. But there will also be plenty of people who are not looking to alleviate any mental/physical quirks they have to do more, but simply want to make a "quick buck" for the least possible effort from the work and thinking of others, without adding much value of their own. Lowering the bar makes things easier for both, the ones who bring value and the ones who merely exploit in some context.
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?
> 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.
I don't think there are actually any "speciesist people." I think most, if not all, are the selfish "me people," who have disingenuously created "speciesism" by mad-libbing racism and sexism in a ham-handed attempt deflect criticism/opposition to whatever they want to do. Most of them aren't even "raking in trillions," they just don't like thinking about others or being told "no."
That wouldn't do much for rice, pigs, chickens, or rats.
> Po-Shen Loh (Chinese: 罗博深; pinyin: Luó Bóshēn; born June 18, 1982) is an American mathematician specializing in combinatorics. Loh teaches at Carnegie Mellon University, and from 2014 to 2023 served as the national coach of the United States' International Mathematical Olympiad team. He is the founder of educational websites Expii and Live, and lead developer of contact-tracing app NOVID.
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...
"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...
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.
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.
I loved mathematics when I was young and wanted to be good at it, but I got burnt so many times by teachers who could not explain even if their life depended on it.
Asking an LLM and discovering how different theories belong together and what are genuinely open questions, along with different philosophical interpretations has been a wonderful gift.
I have long accepted that Mathematics is a language where you must go "all in" because for the most parts institutions are fundamentally unable to explain it in an intuitive way. There are people who "just get it" and are blind to the struggle of people who need a different approach in learning it. There's something about the field where a generational survivorship bias has created an environment where math is mostly taught to likeminded people, who then don't see why anybody could not see the difficulties learning it.
I know this is not true everywhere, but the only reason I could finish my studies was because there are some genuinely excellent online lectures on youtube, along with massive support from friends who have already been through the ordeal.
If I had had access to an LLM, I could have actually taught myself from scratch and be more playful in learning it. And no, I am not willing to learn lots of math "for fun", I can think of better "for fun" things that actually help my profession and cater to my actual interests and not just some academic ideal.
A good human on the subject will do better, and I'm not claiming the LLM has "understanding" in any philosophical sense. But it's more useful than something that just regurgitates material.
Nobody will hate AI more than the con-artists in academia pumping out fake papers by the zillions that are all about to be wiped out.
For every person with this mentality, how many people do you think there are that aren't learning from the LLM at all?
See: what they did with personal computers and smartphones. They have access to Fable and Astra right now, and they're not interested what-so-ever despite what these systems can now do. Instead they'll promptly go vaporize $50k on a giant pile of metal garbage (autos) that will rapidly depreciate. They have no practical use for Fable as is.
For the same reason people didn't rush off to their local public library to teach themselves JavaScript to acquire a better life for the cost of renting library books. They can't easily go against their restraints, their biological potential. To do so is extraordinarily taxing.
Originators generate most of the extreme wealth, and extreme outcomes. Mimics can be exceptionally successful however, most doctors are mimics for example, they won't ever originate anything.
Amplifiers are the teachers, they - ideally - shuttle what works to the mimics for reproduction across society. The mimics are largely just trying to keep up with the Joneses. They go out of their way to not risk anything most of the time. They're ideally following pre-set paths to biological success. It's why mass, rapid de-industrialization for one example is such a wipeout, the mimics don't adapt very quickly, they followed the path they were told would work and then they feel betrayed, they become like abandoned programs and don't know what to do next.
As a system you don't want too much of your population in the originator category: it's very expensive and highly prone to spectacular failure. You want the extreme majority of your biological entities following successful paths that have already been proven to work.
Humans aren't special, we're animals. Apply the patterns accordingly.
Most people consider mathematical intuition something that only develops from deep study. What the parent is referring to is being able to anticipate how a system will behave before you actually do the calculation.
Most people who use mathematics in an industry job rely on their intuition rather than rigorous proofs. And most industry mathematicians will acknowledge that they're not as rigorous, nor as academically gifted, as their friends who stayed in academia.
Just like a programmer with 20 years of experience can look at a bug and anticipate the cause.
But please elaborate a bit on your statement.
Is it really that common for musicians to not be able to read music? I'm not a musician so I wouldn't know, but it seems far-fetched. Like the story that Einstein failed math. I likewise doubt that there are many people who would be good mathematicians but unable to learn the language.
Sure notation is arbitrary and not the actual "truth" of math, but we need some way to talk about it, so why not just learn the accepted standards?
There might be at least one awesome author out there who never learned to write or type, but they're probably extremely rare, because becoming great at something usually entails a lot of practice, and that's hard to get when you're not making the effort to learn the basics.
In any case, LLMs are probably a bad way to "pivot math towards intuition and accessibility" because they, as language models, are great at generating a bunch of jargon that is hard for somebody with no training to tell apart. See the recent story [0]. Here the author (perhaps) succeeded with his proof, but was it because of superior intuition? Or just throwing compute at the wall to see what sticks?
[0] https://news.ycombinator.com/item?id=49755024
> Reading math however does not guarantee ideas or intuition.
is true, but the converse, having the mind for good mathematical ideas or intuition, almost surely means that you'll have very little trouble picking up mathematical notation (not just notation, but the reasoning and ability to write a proof, be it in natural language or some formal proof assistant language).
Like yeah, before the written language there were great storytellers who couldn't write. But now that humanity does have standardized languages, it's way more rare to come across such a person.
(Attn: pls read more than the title before you downvote)
With powerful tools of the type Bret envisions, humans, in aggregate, might have a chance of staying ahead of LLMs at even frontier math..
(But maybe that wouldn't fare so well against future world-models, who knows)
here might be something to point those tools at--- if one isn't fixated on "predictions", or "real world" https://youtu.be/fec8qSBiM4k
This has always been my problem with math going all the way back to college. It was obvious to me that the concepts were far less hard than the combination of notation and esoteric jargon with liberal use of symbols and weird letters made them seem. Math felt (and still does feel) "encrypted."
The impression math gave off is of an arcane discipline that uses its arcane-ness as a gatekeeping tactic, intentionally or not, and makes itself intentionally hard for newcomers to learn without being hand-held by members of the guild.
Of course I can say the same about a lot of computing, and I'm old enough to remember efforts to make computing more approachable like easier to learn languages and GUIs being mocked and scoffed at by "real programmers."
I think this is a pretty typical human group behavior.
I think this underlies a lot of AI hate from these communities today. AI makes it easy for outsiders to bash their way into the field with the help of an LLM. Yes, this often results in low-effort "slop," but if used correctly it can also help people climb the learning curve really fast. I've had great luck having an LLM make me some passable "slop" and then explain it and go around with me as we fix and refine it, explaining each step, and as it does so it feels like we are learning together. It's very powerful and I, as the student, can control exactly how the teacher presents the material.
I was able to do this to finally start grasping the math behind LLMs themselves: attention layers, tensors, etc.
A lot of people have a powerful visceral probably instinctive reaction to large numbers of migrants entering their area. "Build the wall!" I suspect this is brain stem stuff going back to evolving under conditions of scarcity where migrants meant less food.
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.
- 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.
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?
Because the question that belies all of this is whether still makes financial or economic sense in the age of LLMs for tax-payers to fund mathematics research.
Up until recently, they've had a true ivory-tower to dismiss such criticism, but as LLMs have demonstrated, that approach won't survive going forward, so pure mathematics are up in arms. They genuinely don't know how to justify their jobs.
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.
But within the sample of the human species it is more common for less intelligent people to control more intelligent people.
This happens within political systems (political leaders are usually above average intelligence, but hardly the most intelligent) and within companies and other economic systems.
I don't see this fairly obvious point discussed, and not really sure what it shows.
it shows that you don't get anything for free. If it was without any cost in other areas to make a human massively more intelligent, it would presumably happen over time via the same processes that led to us.
As it is, the higher (and lower!) ends of the distribution tend to be highly correlated with other issues (mental+physical), and the further you go, the more unfortunate things which make it harder to do stuff in general start to crop up like psychosis, autism, ocd, anxiety, addiction etc
> given n species and the n * (n - 1) binary combinations possible, there are no examples of the said observation
I would expect that in the vast majority of combinations, those n species are not intelligent enough to make decisions for another species, making the statement nearly useless.
I stopped reading at this point because I reject this initial assertion, and I assumed everything that follows is based on it.
There is nothing to suggest that an AGI will care about its future or who controls it. (whatever that means). that just more anthropomorphism.
We care a lot about our future so an ai pretending to be like us will also pretend to care. But we have a lot of biology driving us. Our emotions (Fear, Hate, Love, Pride etc) are baked into us in a way just wont be the same for ai.
Isn't this exactly the scary thing? A sufficiently advanced agent unleashed with an imperfectly defined goal will do whatever it determines it needs to do to accomplish it, careless of the consequences.
For guest post on Tao's blog, after a decent start this fell short of expectations rather rapidly.
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).
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".
It's because they can't.
It's a mathematical theorem that no algorithm that "solves mathematics" can exist.
>In mathematics and computer science, the Entscheidungsproblem is a challenge posed by David Hilbert and Wilhelm Ackermann in 1928. It asks for an algorithm that considers an input statement and answers "yes" or "no" according to whether it is universally valid, i.e., valid in every structure. Such an algorithm was proven to be impossible by Alonzo Church and Alan Turing in 1936.
https://en.wikipedia.org/wiki/Entscheidungsproblem
I'll kindly disagree on this front, because when I'm walking a path toward solving a solution, I mark a lot of steps for possible diversions to other paths hence solving adjacent or different problems with the method I have at hand.
Currently, AI takes us from A to B, and is improving on that front. However, the paths in science are not lines, but a trees. Methods are cross-pollinated from each other.
Human intuition enables this cross-pollination. AI works with a laser focus. Human intuition and resulting wide perspective sow the seeds for solutions in many areas at once.
But surely, if we know that the dead ends of exploring a problem are valuable, we should be able to explore them even if a solution is already known. It just requires that the mathematics community reshapes itself. And it must. Two years from now people might be able to run the computation that solved NS on their Iphone.
I'm sure that whatever has been discarded during the NS exploration as a dead end you would be able to rediscover using purpose built tooling in the near future. The purpose of human mathematicians in the medium term might be to explore dead ends, and to provide human insights as context to attack other problems. But whether this type of work will remain necessary in the long term im not sure.
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.
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".
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).
Best case, they still matter.
Worst case, AI kills us all and it's irrelevant that we "wasted" money on research.
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.
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.
well that's an awful image
We're building tools to serve the human society.
You mean dystopian. If it were a utopia everyone would be happy to welcome the new world order.
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.
4D chess? They want others to find the hacked services, so the report of how dangerous the agents are seems more "legit"?
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.
Every possible proof exists already as a possible generation in the grammar of lean or rocq. In no way does that mean we have discovered everything.
The map is not the territory, etc. If math was just an elaborate linguistic Glass Bead Game then we wouldn't be funding it. The intuition is that the surface rules of math help uncover the underlying structure of reality.
Brute-force as an approach, or partial approach, might only become feasible with more computing power. Decades ago, we had less computing power.
Give them Navier-Stokes in a vacuum
Brute Force or not, the game is being played and won without them all of a sudden, and even worse: at a level of output far beyond them
(Evidently not, since humans hadn't managed to do it even with the entirety of human knowledge available to them.)
Thinking.
The problem they care about is not just “AI replaces human mathematicians” but rather the society thinks mathematicians can be replaced by AI.
Spending 132 billion tokens is definitely way more energy than human mathematicians would have thrown at the problem and probably would have solved within two years.
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 hope this isn't actually news to you, but: There is more than one human. There is even more than one mathematician.
If there happen to have been as many as four humans working on Navier-Stokes at any given time since the year 2000, then that's more human-years applied to the problem than agent-years.
> How many mathematicians have worked on the problem? 10k? Not likely.
You don't get to count the factor of 10k once when working out how many agent-years OpenAI gave to the problem and again when demanding that for parity there would need to have been 10k mathematicians on it.
> 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.
Let's suppose that indeed what Buckmaster and Alpöge had done was in the model's training data. Well, it didn't enable Buckmaster and Alpöge to solve the problem for Navier-Stokes (they could only do Euler), and it did enable OpenAI's model to do that.
Also: we don't actually know that what they'd done was in the training data; the latest bits of what they'd done that could plausibly have been in the training data were from before when Buckmaster said they progressed from preliminaries ("We worked through the literature and upgraded various preliminary results") to actually making substantial progress on the problem ("This was until about a month ago, when we had real progress"); and from what Buckmaster wrote it sure seems like a lot of the Buckmaster/Alpöge progress was in fact done by LLMs. (E.g., Buckmaster says that he and Alpöge have been working frantically to try to understand the proof for their Euler solution. That sounds to me much more like "an LLM did this thing" than "we figured out all the hard bits and the LLM did nothing more than filling in a few details".)
Buckmaster's own account of things is that all the really clever ideas were those of Córdoba and Martínez-Zoroa. (Which are already out there in the open literature, and there is nothing remotely improper about making use of them.) And my understanding (but, note, I am not an expert on fluid dynamics or PDEs and I could be wrong) is that actually the OpenAI model's construction is quite different from that of C&MZ. On what basis are you confident that "the AI based its proof on" what B&A did?
(For the avoidance of doubt: I am not arguing that what OpenAI did was OK. Even if they actually didn't train at all on any of the Buckmaster/Alpöge chats, it's very much not good professional ethics to hear that someone else is working on something and rush to try to scoop them, and there is absolutely no question that they did that. The question here is how impressed we should be by the model's mathematical prowess.)
> A success rate of 17%?
A success rate of 17% on problems of this difficulty and significance is something that for any human being would be a career-defining triumph.
> We have thousands of problems solved by humans in the last 2k years and how many by AI? A couple dozen? Oooh scary!
That would be a more convincing argument if the AIs, like the humans, had been around and trying to solve those problems for the last 2k years. However, as you might have noticed, the state of the art in AI was rather primitive 2000 years ago.
So, the "~100 agent-years" calculation goes like this:
10,000 agents * 88 hours = 880,000 agent-hours
88,000 agent-hours / 24 hours = 36,666.7 agent-days
36,666.7 agent-days / 365 days = 100.5 agent-years.
That's what you get for working 24 hours a day, 7 days a week, 365 days a year. Realistically speaking, that's not a work schedule any human can follow.
It's hard to make a realistic estimate because normally even a very dedicated mathematician will not be working exclusively on one problem all their waking time, or even all their working time. But, let's ignore this and assume a pretty standard work schedule of 8 working hours, five working days a week, and 52 working weeks a year.
Now, that's:
8 hours * 5 days = 40 working hours a week
40 hours * 52 weeks a year = 2080 hours a year
880,000 agent-hours / 4 humans = 220,000 hours per human
220,000 hours per human / 2080 hours a year = ~105.8 years
To clarify, that's how I estimate the number of years it would take a mathematician to do a quarter of the work of the 10k OpenAI agents if that mathematician worked only on solving Navier-Stokes and did nothing else in their entire career.
That's just not a realistic work schedule for any human. You can adjust the working hours if you want but I don't believe you'll get any realistic estimate. Don't forget that most academics' careers last around 30 years from PhD to Professor Emeritus. If you want a more realistic estimate of how much time it would take how many humans to do the work of the 10k OpenAI agents, you can start from that assumption and work your way up from that.
>> That would be a more convincing argument if the AIs, like the humans, had been around and trying to solve those problems for the last 2k years. However, as you might have noticed, the state of the art in AI was rather primitive 2000 years ago.
Sure. But the thing is agents can run 24/7, 365/365 in parallel and as you see above they can cover 2000 years of human work in much less time. I'm not going to estimate how much because the only bottleneck is the amount of compute and money that an AI company wishes to spend, and that depends on their motivation to solve a particular problem. However, with sufficient motivation 2k years of human research (keeping mind that's not 2k years of continuous work) can be covered in a few ... months? Probably.
I sunno if mathematicians would be having problems understanding how matrix multiplication, backprop, sigmoid functions, attention, embedding distances, probabilities, etc work.
As a group, they are probably more likely to understand it than everyone else.
Wanna guess why? I'm too tired now to expand the argument properly but basically understanding the components of a complex system doesn't mean you understand the principles of the system. A mathematician who is not an expert in AI has no reason to be particularly capable of understanding how AI works, i.e. how all the maths that go into creating an AI system come together to create. An AI system.
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.
My point however is that companies have no agency of their own. Their age doesn't really matter for that. There are guns that are older than all living people too. If somebody used a flintlock pistol to go around robbing people that wouldn't make the pistol a "godlike eternal entity". Or for that matter if somebody used a 200 year old shovel to create a really nice garden (in the case you believe companies are a net positive).
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.
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.
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
> 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.
That's certainly different from Olympiad-style problems.
Maybe more in years past when Comp Sci was a subset of Math Departments.
What a disgrace, but I'm happy he's showing his true self to the public.
- 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 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.
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.
AI chatbots give wrong answers to financial queries 'most of the time' (ft.com) // https://www.ft.com/content/c0cd359d-df84-4208-a789-ffa864b43...
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.
IMO, this is why we actually need mathematics -- as a field in which to learn what it means to know what you're talking about.
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.
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.
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.
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.
I always think of this one but I bet there's better
* https://m.xkcd.com/435/
Our biblically literate ancestors knew better. Our purpose is to love God and love people, which is why we still have a modicum of sense for the value of completely unproductive people.
As we’ve become spiritually hollowed, and more biblically illiterate, we’ve started to dehumanize unproductive people, a slippery slope if we ourselves become unproductive.
The axiom in this article will only be accepted by the intelligentsia if you can tell a story about why it is true: that humans are created in the image of God.
Until we re-find our ability to tell cosmic stories, this hand-wringing will continue amongst the atheistic elites.
I agree that human society is too focused on productivity but disagree that it's tied to ignorance of the Bible. The fall of man illustrated in Genesis is tied to feeding on the fruit of the tree of the knowledge of good and evil. Having the right set of religious concepts wont put you in harmony with God and thus won't make you more loving to your fellow man.
There are many modern examples leading to disastrous results.
> we’ve started to dehumanize unproductive people, a slippery slope if we ourselves become unproductive.
That is very evident in the sort of things the techbros come up with, but I agre it is a wider social problem.
The moral and spiritual depth of our society has only shallowed after centuries of doing this.
It hasn’t worked!
Because people have different religious beliefs or none at all. Biblical is not the natural starting point for everyone.
TLDR: we don't. Terry Tao just announced he's becoming a UFC heavyweight fighter.
"chat i'm about to drop a conjecture"
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.
Is that so ? Sounds hyperbolic.
Many people ceed control of their path-finding to seeing-eye-dogs.
Do you think the people using a seeing-eye-dog do this voluntarily over using their own eyes?