Speaking as a postdoc in math, I must say that this is rather exciting. This is outside of my field, but the companion remarks document is quite digestible. It appears as though the proof here fairly inspired by results in literature, but the tweaks are non-trivial. Or, at least to me, they appear to be substantial to where I would consider the entire publication novel and exciting.
Many of my colleagues and I have been experimenting with LLMs in our research process. I've had pretty great success, though fairly rarely do they solve my entire research question outright like this. Usually, I end up with a back and forth process of refinements and questions on my end until eventually the idea comes apparent. Not unlike my traditional research refinement process, just better. Of course, I don't have access to the model they're using =) .
Nevertheless, one thing that struck me in this writeup, was the lack of attribution in the quoted final response from the model. In a field like math, where most research is posted publicly and is available, attribution of prior results is both social credit and how we find/build abstractions and concentrate attention. The human-edited paper naturally contains this. I dug through the chain-of-thought publication and did actually find (a few of) them. If people working on these LLMs are reading, it's very important to me that these are contained in the actual model output.
One more note: the comments on articles like these on HN and otherwise are usually pretty negative / downcast. There's great reason for that, what with how these companies market themselves and how proponents of the technology conduct themselves on social media. Moreover, I personally cannot feel anything other than disgust seeing these models displace talented creatives whose work they're trained on (often to the detriment of quality). But, for scientists, I find that these tools address the problem of the exploding complexity barrier in the frontier. Every day, it grows harder and harder to contain a mental map of recent relevant progress by simple virtue of the amount being produced. I cannot help but be very optimistic about the ambition mathematicians of this era will be able to scale to. There still remain lots of problems in current era tools and their usage though.
This is the main thing that I keep harping about that human knowledge is too vast today for a person or even a group of people and llm will change that many discoveries that require serendipity in the past will be more likely than ever
I want to hear James Burke of Connections [1] and his writer team to have a free wheeling discussion for a few hours on what they see will happen with LLM’s making these connections with more conscious intent a lot easier. The awesome compression of knowledge aspect of LLM’s is a far undersold aspect of the technology.
I totally disagree. AI is overhyped, but the demos that have impressed me most are things/elements that very few people are touching and that no one seems to be talking about.
the hype is where the money is, as is always -- marketing & porn. Both touched heavily by AI already.
I am also an AI skeptic, but I would rather have used the 1000 monkeys with a 1000 typewriters will eventually write the whole works of Shakespeare analogy.
When you consider the amount of computation which went into this discovery it is less impressive. Like if you spend a lot of fuel you can travel really fast, much faster then a bicyclist. Similarly Go-engines can beat the best humans at go, but they spend several orders of magnitude more energy to do so.
Mathematicians prove or disprove conjectures all the time and use orders of less energy to do so. Using LLMs is kind of just throwing money at the problem and hoping it works. In this case it did. But this is not the most efficient way to do this, and it won‘t scale.
People are marveling at what AI can discover purely out of time and chance. AI will undoubtedly find awesome things because there's very few things we've thrown this much money at. For every awesome thing AI finds, there's a million mistakes, fake leads, hallucinations, etc. Amaze away but let's not forget this is an exception much more than the norm.
Presumably with automated proofs, hallucinations (aka, creativity or confabulation, if we hadn't botched the naming) is a good thing. It's in the empirical world where trusting an LLM is a stupid thing because there's no automated form of fact checking.
That we never see all the mistakes and dead ends a complex system using AI hits?
Does it even matter if its accuracy rate across all its experiments is < 100% if it can run trillions of experiments in the same time a human could run 1?
We don't see many of the failed attempts of Human Researchers. Why? Because it doesn't matter.
What amazing here is that it shows our society can make discoveries faster in the post LLM world. Thats incredible.
As a human, I have many stupid and wrong ideas all day long - most of those don't bubble up to my conscious awareness. If LLMs hallucinate and come up with crazy things, maybe that's ok given that we can filter out the sensible and novel ones.
And in doing so you spend what, a 100 watt hours per bad idea? Compared to how many megawatt hours of AIs failed attempts at proving math capabilities to investors only to prolong the AI bubble another month?
I bet your stupid ideas also taught you a valuable lesson and you learned at least something from the experience, maybe your next idea won’t be so dumb, and those 100 watt hours weren’t actually wasted (though it may feel like they were). Compered to a failed LLM experiment, where all those billions of billions of computations are completely wasted. the model knows exactly as much after a failed experiment as it did going into it. Those Megawatt hours were simply wasted, turned into heat energy, paid for by raising the power bills of the of the datacenter’s neighbors.
It depends upon what you mean. A dream is a stream of unconscious ideas in a way. One could also look at Jungian psychology and point to the ideas your conscious mind rejects by projecting onto others as unconscious ones, with the idea being that we refuse to face them in ourselves and so must put them outward.
Kind of seems to me, the heart of the critique is that 1. unthinkable amounts of financial and social and political credit have been thrown at this which necessarily has deducted it from other fields we could have invested in, instead. 2. Thus, with such wealth you would expect at least a couple of discoveries.
Not my post, but I think point 1 is stronger than 2.
> 1. unthinkable amounts of financial and social and political credit have been thrown at this which necessarily has deducted it from other fields we could have invested in, instead.
That's not necessarily true. If our only counterfactual to investing resources in project A were to invest them in some other project B, then, yes, the conclusion above follows. But often people just consume the resources.
(In the end, the goal of all economic activity is consumption. We invest resources so that we can consume more later. If there's no good enough project around, might as well consume more now.)
Terence Tao gave a recent talk about this issue (lack of attribution). He called it the decoupling of implicit and explicit goals. AI is only good at solving the explicit goals for now, and humans don't have the bandwidth or the institutions to know how to integrate AI into the field.
That is an odd summary of the talk. He was talking about how the explicit goal of solving a problem is kind of becoming trivialized, but the abundance of 100-page AI generated proofs will not help the implicit goal of furthering human understanding, because we lack the bandwidth to really digest them. Adhering to things like (human-focused) academic etiquette is a different problem and can probably easily be solved by just giving the model the right context. But having humanity keep up with AI insights into math and science is something we might have to give up eventually. Or at least whoever does will be far ahead of us as a society, because most people's lives will only be affected by the explicit results.
I feel like that’s already becoming true. I sometimes work on problems/projects where the AI agent is definitely more qualified than me to call the shots.
For example, this library here for deep learning is 100% ai generated and far beyond my technical capabilities.
The real question is: Do you need to understand it fully for it to improve your life?
For example, if you're in fundamental science (or generally a fan of reductionism), it for sure would be nice to understand the universe instead of just having access to an AI that can comprehend it. But to the majority of the population it only matters that someone (or something) understands it enough to make it useful to others.
Understanding everything fully is futile. But there are many many many things that by understanding you improve your life. So, I feel the question is... not useful, I would say. Yes, you need to search for things that if you knew them you would improve your life. No, you can not know them all beforehand. Yes, there are such things. There always are.
They only improve your life if you actually work on something that you yourself are trying to improve. Most people are fine with the status quo, so if something like LLMs can take over the understanding of complex tasks, they won't even notice, except for the fact that more of these tasks will get done.
Do you really think these models lack the intelligence or language capabilities to handle human etiquette? They can't "read the room" yet because they lack modalities and people don't give them the right context. That's the issue. But I have no doubt that what you two describe here will be solved very soon. And yet the actual implicit goal of all this will need humanity to rethink its priorities.
I am also using these models to accelerate scientific discovery. Yes, they are making all the difference at the frontier. At least, they feel they are. The messy thing is that we still need to communicate with each other and that's not getting dramatically faster or better. As you note the models need to be built so they do more work to participate in our communication economy. Or we will do so much, alone, to get nowhere fast because so much of our behavior is still bound up in old (good, tested, but clunky) ways of building shared knowledge.
> I dug through the chain-of-thought publication and did actually find (a few of) them. If people working on these LLMs are reading, it's very important to me that these are contained in the actual model output.
This is a very important point, especially when the output is from a non-deterministic random walk with some unknown probability distribution.
Yes, I share your optimism overall, although I think it is raising a question of what the future role of the researcher is (much like the current debate on developer roles).
I attended a conference on AI for maths and open science a few weeks ago, and was struck by just how many examples of AI-supported solutions there already are. Virtually every speaker had an example of either their own use of (often the frontier) AI models in solving a problem that was previously too hard (for various definitions of hard).
I wrote up a few notes [1], and most of the speaker videos are available via the conference website [2].
> But, for scientists, I find that these tools address the problem of the exploding complexity barrier in the frontier. Every day, it grows harder and harder to contain a mental map of recent relevant progress by simple virtue of the amount being produced.
AI is going to both help and hinder this process though. At the end of the day, mathematics is mostly a social process at this point. The goal is not raw number of theorems proven, it’s how proving theorems affects the working operational models of mathematicians. Only a rare few new theorems in mathematics nowadays have direct real world applicability.
If AI produced legitimate theoretical breakthroughs at a pace mathematicians are unable to absorb, then the impact will be neutral to negative.
Weird question, do you think AIs might prove a lot of theorems that are mainly useful to other AIs (i.e, make nearly no impact on the human culture of working mathematicians), which then get used to prove results that humans do actually care about?
It seems like if AIs can prove and index a huge number of (largely uninteresting to humans) things there might be sort of "parallel cultures"? Big results are most valuable to humans and AIs both (most context efficient!), but a very large number of less general but still non-obvious results might be an effective approach to solving problems?
> Only a rare few new theorems in mathematics nowadays have direct real world applicability.
I am no mathematician and very naïve about this, but in a world that is rapidly becoming extremely calculation and network dependent that sounds hard to believe.
> If AI produced legitimate theoretical breakthroughs at a pace mathematicians are unable to absorb, then the impact will be neutral to negative.
I think the idea here is that all mathematicians will just be using AI for their future work so they don’t really have to absorb it as long as it’s in the training data.
> > Only a rare few new theorems in mathematics nowadays have direct real world applicability.
> I am no mathematician and very naïve about this, but in a world that is rapidly becoming extremely calculation and network dependent that sounds hard to believe.
I am a mathematician. It is true. The key is we're talking about new theorems, and direct, current real world applicability. Some theorems that have no applicability now may in the future, as theory often precedes applications by a long way and the usefulness is likely to come from other things built on top of the new maths, and a lot of pure maths will never have direct real world applications but contributes to our overall understanding.
The key word in that sentence is “new.” New math is typically explored without expectation of practical use. There are exceptions, but it is generally true.
On the other hand, there are many applied mathematicians and theorists from other fields that mine new maths for applications to their fields. But they are almost always not the ones that come up with the new math.
Historically, of course, mathematics was always driven by the need to explain things. Many of the mathematicians from the 17th and 18th centuries were physicists (or, less commonly, engineers). But for the last hundred years or so that really hasn’t been the case.
Out of interest, what would you estimate the proportion of new maths that is used by other fields to be? Do you think much of this new maths is potentially underutilised as it were?
> Only a rare few new theorems in mathematics nowadays have direct real world applicability.
Has this ever been different?
Math is abstract, rightfully so. It does not have to have direct applicability. Understanding builds over time and applications eventually follow. Number theory used to be a fringe "pure" theory field without applications for the longest time. If we'd only be interested in (and thus fund) what has direct applicability then society would be much worse off.
Side note: I recall my high school class mates rolling their eyes in every math class with "when will I ever need this in my life?" never asking the same question about PE or history or art classes. Now they struggle with their tax return and are routinely getting screwed over by loan sharks. But make no mistake, they can be proud of their A for hitting the goal 5 out of 5 times during soccer in PE class.
I cannot quite share your enthusiasm. The clearest analogy that I can think of to try to explain why I feel this way is that it seems there will eventually be a phantom textbook of all of mathematics contained in the weights of an LLM; every definition, every proof, etc; and the role of a mathematician is going to be reduced towards reading certain parts of this phantom textbook (read: prompting an LLM to generate a proof or explore some problem) and sharing the resulting text with others, which of course anybody else could have found if they simply also knew the right point of the textbook.
To be blunt, this seems incredibly uninteresting to me. I enjoy learning mathematics, sure, but I just don't find much inherent meaning in reading a textbook or a paper. The meaning comes from the taking those ideas and applying them to my own problems, be it a direct proof of a conjecture or coming up with the right framework or tools for those conjectures. But, of course, in this future, those proofs and frameworks are already in the textbook. So what's the point? If someone cared about these answers in the first place, they probably could have found the right prompt to extract it from this phantom textbook anyways.
You could argue for there being work still like marginal improvements and applying the returned proof to other scenarios as happened in this case, but as above, what is really there to do if this is already in the phantom textbook somewhere and you just need to prompt better? The mathematicians in this case added to the exposition of the proof, but why wouldn't the phantom textbook already have good enough exposition in the first place?
I think my complete dismissal of the value of things like extending the proofs from an LLM or improving exposition is too strong -- there is value in both of them, and likely will always be -- but it would still represent a sharp change in what a mathematician does that I don't think I am excited for. I also don't think this phantom textbook is contained even in the weights of whatever internal model was used here just yet (especially since as some of the mathematicians in the article pointed out, a disproof here did not need to build any new grand theories), but it really does seem to me it eventually will be, and I can't help but find the crawl towards that point somewhat discouraging.
In Erdös idiosyncratic nomenclature, all the best proofs are "in the book" and it was always a joyful thing to not only find a proof, but to find the proof that is in the book.
Who cares if it is God's book or the machine's Xeroxed copy?
Long before Erdös, we had Plato and Socrates develop the theory of anamnesis, that there is no such thing as learning, but rather, whatever we supposedly learn, we actually remember (we knew it already and had forgotten it). Presumably this should be understood only of universal facts (like mathematics), not contingent facts (like who was the president of the U.S. in 1950).
Before birth. ...Hey, don't point that pitchfork at me, point it at Socrates. In his defense, that kind of does describe when LLMs acquired their knowledge (if we consider "birth" to be the moment when the already-trained weights are sent to the GPU) https://en.wikipedia.org/wiki/Anamnesis_(philosophy)
Pre-conception is unlikely to be really possible outside some esoteric circles. While in the womb there could be some limited experiences that get ingrained in the mind as memories, but I don't think that's the topic here.
I mean, my reaction to God coming down and saying they were bored of being God and instead they would just sit around and answer all of the mathematician's questions would largely be the same, so yes, who cares if its God's book or the machines Xeroxed copy?
"The Book" is more interesting to me if I am the one coming up with the ideas to fill it in. Maybe this is a bit egotistical, but I'd like to think it is allowed to have a desire that you, personally, are contributing to something in a meaningful way. Like, if you are on a sports team, it'd be more fun to win a game if you were on the field than if you were benched, and I think that's okay. And ultimately I don't find dredging for proofs from an LLM particularly meaningful, nor do I see it as a particularly personal contribution, as anybody else could have done the exact same thing with the same prompt.
This isn't to say I wouldn't love to read the proofs in "The Book" for problems I care about, I just think I'd eventually get bored of only reading. And so its hard to be enthusiastic when this book is being built through an LLM.
If ASI does create an abundant future I think many are going to have that familiar listless feeling of enabling cheats on a computer game and all the mystery and fun is gone.
Technology in general (smartphones, social media, search) even without AI is creating this feeling, as it shrinks the world and makes it less mysterious.
It's worse than boredom it's more like nihilism.
Then when you strip purpose and meaning from a human you get something very bad, despondency being the best case outcome.
Aye, but it’s also possible for people to find their own purpose and meaning. Some find it in religion, some in art, some in love or nature.
It will be a transition, for sure - there would no longer be meaning in “winning the game” in a capitalistic or scientific sense. Anything you want to produce or learn, the AI could already produce or has already learned. Now you have to do it just for the love of the process.
I have a musician friend who likes to say that good artists overwhelmingly make art for their own benefit. Not to advance the world or blow people’s minds, but because something inside of them needs to come out, and art is how they express it. And that part of us isn’t going to go away.
And you just expressed the thoughts of every engineer that writes code for a living who is either left behind, or embracing the technology to hit KPIs and QVRs.
I want to push back against the notion that the math already exists in the weights, both in the practical and the philosophical sense. The LLM had to do an enormous amount of computation to find the counterexample. We know it wasn't looking up the answer from its internal representation, because the conjecture was unproven. The proof came into being when the model output it, and if they'd run it for less time or asked it something else then the conjecture would still be unsolved.
I'm also afraid of a world where AI completely replaces human mathematicians, but if we remain collaborators, then that's a world I can still feel excited about.
It’s funny because the shift from handmade goods to automated factories didn’t seem so bad. Same for mechanized farming instead of mules and people.
Shifting from “human calculators” to machines for arithmetic is also hard to argue against.
I think what makes the AI transition difficult is it impacts a wide range of high-value activities that would have been implicitly assumed to always remain human.
I do have great trouble seeing how a pile of matrices is ever going to be capable of innovation. Maybe with sufficient entropy and scale, it will… The day that becomes practical will be a turning point in history.
Economically, goods and services are often priced based on labor/“value added” aspects. Lawyers’ fees aren’t driven by paper costs! If AI takes a huge bite out of intellectual labor, the future could become very different…
BTW, your book description reminds me of the 2025 movie “A.I”. I thought it was quite good.
There isn't anything functionally special about the human brain - why is there some reason to expect the human brain is capable of innovation but no program, even one far more powerful than the brain, is not?
You admit this possibility so I'm not arguing with you, but it seems far more plausible to me that we can build something better than the brain.
In the limit we can just grow brains and put them in computers anyway, then the debate is moot. That's a really hard problem but of course not physically impossible.
The cool thing about LLMs is not only might they be a database of all mathematical theorems, but they can also apply those ideas to the problems you're trying to solve, which is exactly what you said you're interested in. Not sure why you lack enthusiasm.
LLMs applying the ideas to problems I'm trying to solve is exactly what I said I wasn't interested in, actually. Because the LLM doing this for me reduces back to me simply reading from the textbook, only now I have no problems I'd be interested in applying things to since, again, they're already in the textbook.
Can you describe what the reaction to these results has been like in your department? Obviously many people are excited, but what else? How do grad students feel about this? Are any professors getting worried about becoming obsolete?
I am a PhD student in mathematical statistics. The people I have spoken too think this is very exciting and cool. There is also a sense of unease about what this will mean in terms of being a mathematician, and what effects it will have on our future employment.
I am also a little worried about what it means for your training as a junior PhD. Often you would try and solve a problem your advisor thinks is doable that they assign to you as a learning exercise. It may be more and more difficult to find problems that a junior PhD can solve but that AI can not. Tim Gowers has written about that here: https://gowers.wordpress.com/2026/05/08/a-recent-experience-...
One concern is that it will become more and more challenging to conduct cutting edge maths research without substantial resources only available at very rich institutions (to pay for state of the art AI assistants).
Maybe I'm misunderstanding how these models work, but isn't it more the responsibility of the harness and its prompts rather than the model itself to make sure that a result is generated with explicit sources?
"All" a model is doing is predicting the next words, based on the statistical distribution of words it has seen similar to the ones read/produced so far.
We push a model towards a particular set of distributions through context. If I ask a model "What is the capital of France?", there is a non-zero chance it goes down the dad joke answer of "The letter F". The far more likely option is "Paris", because the joke appears much less often in training material, but if I wanted to be absolutely sure of getting a consistent geography answer I'd address that with additional context. We can add context via prompts, RAG, agents, skills and so on.
However, when training a model, we select the material. We could show it a lot more geography information (or dad jokes!), and skew the statistical distribution in the direction we wanted. We could also decide to design the system prompt towards the direction we prefer - which the user would interpret as "the model" - and so nudge the context model-wide. We can also construct the interaction to iterate on context with a specific framing and call it "reasoning".
In this specific example, you could therefore solve the problem by a) training skewed towards mathematical papers, which likely degrades performance in general and likely for the specific case too, b) train the user to provide better context/prompts for mathematical work, shifting the workload to them which feels very "a la 2024", c) publish agents and skills that are tailored to mathematics work (very "a la 2026"), d) tweak the system prompt for when the model is doing mathematics work, which the user would see as "the model" doing the change, but you and I might look under the hood and say that is in the harness or a specific type of prompt, or e) add "reasoning" execution that is set to focus on mathematical formatting, or f) a mixture of the above.
Right now we're probably looking at agents and skills. I think over time we're going to see smaller models targets towards domains with a mixture of all of it, where some of this sits at user configurable levels, and some is "baked in" via training, system prompts and execution modes, but from a user perspective it's all just "the model".
I don't think you are misunderstanding how models work, but I think the parent comment meant that the training of the models should push them to include attributions in their native output so they will more likely do so without reinforcement through the harness.
> Every day, it grows harder and harder to contain a mental map of recent relevant progress by simple virtue of the amount being produced.
And by opening the door to LLM-generated results, you'll see greater and greater amounts without any hope of ever navigating this field again without machine help.
It's a little like a software project which more and more gets extended by a AI agents with less and less review by human software engineers and in the end the complexity and spaghetti design are so incomprehensible by humans that the maintenance requires an AI agent. The risk is that math as a whole (the field itself) will experience that effect.
I'm no mathematician but it seems like if this happens, we get to a quite intriguing place as a species.
Say we achieve interstellar travel, but nobody actually knows how it works.
Or we cure cancer, but the "cure" requires a microrobotic implant, and it runs as a blackbox AI, and only the other AIs can make one, and there's no guarantee they will know how to make one tomorrow.
Or we solve global warming but it requires giant cooling machines running 24/7 and again, nobody knows how it works, but with the added bonus that the planet is cooked if they ever stop working.
That's already how civilization works. There's no one person that knows everything about (say) modern food production, from top to bottom. If it ever stopped working (because too much knowledge was lost somehow), most people would die. And yet the system seems fairly resilient. Mostly, only local knowledge ever seems to be necessary to keep the whole thing running. Super-intelligence (or even just super-normal-intelligence) might expand the scope of what constitutes local knowledge but it will still run into limits somewhere.
> There's no one person that knows everything about (say) modern food production
True, but it is possible to assemble a team of people that does, with backup for each person. There's also teachers and written knowledge to educate new team members. That's what makes it resilient.
I think that's a very different situation from what's decribed.
Agreed, the food production analogy doesn't really work because the issue is the scale of the problem. On the one end there's the realm where you need a few specialists and a small group could potentially figure the entire thing out from scratch given a bit of time and effort. And then at the opposite extreme there's the realm where everything is built on a giant pyramid of artifacts that currently work, just keeping each individual piece running day to day requires a dedicated expert, and the combined stack took hundreds or thousands of lifetime equivalents to develop.
The idea being that once a toolchain becomes sufficiently complex if you ever have to bootstrap it again for whatever reason you won't be able to speedrun the process the way you might naively expect. I think modern chip production likely already reached this point several decades ago. As evidence I'll point out that China only recently achieved EUV and remains several nodes behind despite directing an obscene amount of resources towards the initiative.
Speaking of pyramid (shapes), this reminds me of an idea in Robert Silverberg's Majipoor series - there's a 30-mile high mountain with populated cities all the way up to the top whose weather and temperature is controlled by 8,000-year-old-tech established by the original colonizers. My memory is that nobody at the time of the series' events knows how to operate the tech - it just works.
People lump them together because of an anti-technology reputation, but I don't think most Amish would have trucked with Luddites. Amish tend to avoid actively participating in popular social movements, and oppose violence and property destruction.
However, you can assemble a team of humans who knows the whole pipeline. This trajectory lands us squarely in "The Machine Stops" of "Pump Six" territory, where assembling such a team or going back to a simpler system is impossible
Sorta. Take a look at a brick in a house. You'd need everyone from geologists to miners to kiln specialists to construction workers and engineers -- not to mention all the people required to make the tools required to make the tools. The team would likely involve well over 1000 people. So, "just assemble a team" is not quite as simple as you make it sound.
I don't think that's true in the sense meant. Sure, to reproduce a near replica of a specific brick from first principles. But not to produce something broadly functionally equivalent. You can (rather inefficiently) manufacture approximately equivalent bricks in your backyard on your own, possibly even from locally harvested material depending on where you live.
Well. Sure. If we move the goal post to “something passable and good enough” you only need a small number of people. In that sense, we are lucky that “black smithing” (as a proper trade) only ended in the last hundred years and many people continue it as a hobby. In that case, “small team of hobbyists” can likely reproduce a few bricks. But bootstrapping mass production of bricks? Unlikely.
Doesn't matter. At the end of the day, the knowledge is embodied by humans, or can be learnt again. Let it be 100, 1000 or 10000 people. At the end of the day, they are made of meat.
When you let the machines do it, and don't care about moving it towards human domain (i.e. meatspace), you're done.
Noble "save the humankind using the tech nobody fucking understands" textbook goals like curing cancer, solving global warming and achieving interstellar travel would always turn up when owners of trillions of dollars place orders on positive AI narratives, but in reality all of that will wither down to "It's what plants crave! It's got electrolytes."
>ve AI narratives, but in reality all of that WILL wither down to...
Looks like you're pretty sure of that. Every time I see argument like this delivered with confidence I wonder how is it different from, say, digital calculators. Or better yet, books - Greek philosophers moaned that young people will stop understanding anything and just check books when they want to know anything.
Knowing the history of the humankind is what makes me pretty sure of that.
The extent of misery and destruction is directly proportional to the level of technological advancements, and I don't like the idea of sacrificing millions of lives in the name of the figurative HVAC, smartphone and other benefits of civilization. Or billions in the name of whatever benefits the next VC money stake should bring.
> I wonder how is it different from, say, digital calculators.
Did a single digital calculator ever stop any war, or liquidate a psychopath who orders people to go kill and die?
> The extent of misery and destruction is directly proportional to the level of technological advancements.
By statistics of war, poverty rates etc this is trivially false. I think you are really, really underestimating how hard life was pre-industrial revolution.
Ten thousand years ago (around 8000 BCE), the global human population was estimated to have been roughly 5 million people. This is significantly smaller than the current population of just Poland (about 36 million).
In absolute numbers there might be more now, even if the percent is smaller. It is difficult to compare this things without having a specific place in mind.
The fact that humankind grew from 5M to 8.3B, while dramatically improving longevity and quality of life speaks volumes. Multiply life quality × population × life duration, not only "misery and destruction" is not the case, but you could rather see powers of positive technology influence.
Are we just going to totally ignore the Polio vaccine? Modern medicine? Modern agriculture?
If you had a magic button that turned off all those "benefits of civilization", millions would die. If you managed to drag agriculture down with the rest, the death toll would be in the billions.
I don't understand how you can possibly think you "know history" without recognizing that technological progress has taken us from constant warfare to such a state of abundance that war is actually rare and noteworthy in much of the world.
> technological progress has taken us from constant warfare to such a state of abundance that war is actually rare and noteworthy in much of the world.
Let's try to have an actual argument. How many people, in absolute numbers, were affected by that constant warfare of past, which past exactly do you mean, and how many people were/are affected by "rare" wars of modern history?
80 million people killed or maimed with arrows, swords and catapults over centuries and 80 million killed or maimed with fruits of industrial revolution over 6 years of WWII are very different figures.
If only 4 people die violent deaths out of a total population of 5, that’s an extremely violent population to be a part of.
If 8 people die violent deaths out of a total population of 100,000. That’s a much more peaceful population to be a part of, despite the greater number of absolute deaths.
To me at least, the intriguing part has nothing to do with the textbook goals of whichever foulmouthed trillionaires you have in mind.
The intriguing part is that we could get objectively good outcomes, but at a cost of being dependent on the machines. So it's not that you couldn't actually unplug Skynet, it's that if you did civilization would collapse (or whatever) because Skynet stops doing its thing.
I'm not sure that gets us to a better place overall, but I doubt we could resist the temptation.
That is fascinating how the more knowledge and reasoning we can get our hands on and actually produce, the higher the risk of us, as a species, to become actually much dumber.
It's hard to describe the feeling of seeing intelligence being delegated increasingly to AI. If that's not a pivotal moment, a revolution, I don't know what is.
> That is fascinating how the more knowledge and reasoning we can get our hands on and actually produce, the higher the risk of us, as a species, to become actually much dumber.
This has always been true. There was a time where someone had to teach farming to others and that information had to spread and be passed down. Eventually, farmers became better than hunter-gatherers and they became known as hunters. The information on what was safe to gather for civilisation got passed down as 'safe to eat on the hunt' because the farmers were farming. The civilisation collectively "forgets" foraged foods as that knowledge becomes niche.
So knowledge becomes meta stable. There was an AIDS drug in the 90’s that we stopped being able to make. IIRC Apparently there were two different crystal/folding structures for the compound and the desired one was not the lowest energy. After years of production, the wrong version was produced and they could no longer make the correct version do to contamination. And every facility that tried to study it, wound up no longer being able to make the correct version. It was like a real life ice-IX situation. Scary that changing weights or model parameters could lead to the same thing happening with knowledge.
I think anything is and will be explainable. Like in the OpenAI proof, I’m sure they were able to understand the solution 100% and could even drill down and ask more clarifying questions to the model. After all, the point of science is so that knowledge can be made logically transparent. If something can’t be explained, it isn’t really understood yet — and the same applies to model outputs. The only question is how much effort it takes to surface the explanation.
I think explanation is itself a rather complex concept. At what point do we consider something as explained? Usually it has to do with identifying some causal factors and their relationships so that we can intervene and explore counterfactuals. But in many cases we are forced to act on the basis of incomplete explanations (e.g. in medicine).
I think there will be regulation that requires some users of AI to provide an explanation upon request. For instance, banks could be required to "explain" why you didn't get that loan. What if the decision is based on a credit score that includes some AI prediction that ultimately relies on the entire training corpus?
The bank can give you a list of factors that play into the decision but they may not be able to explain deterministically why a very similar customer did get that loan. At that point I think we're going to resort to statistics that prove a lack of bias against certain protected characteristics, but that's not really an explanation, is it?
I think we will never get useful and complete explanations for everything that AI does. Society will just accept some explanation-like thing or proxy and move on.
And more intelligence should give an opportunity to increase explain-ability rather than just complexity. It can potentially explain the proof at the level of the listener. Make visualizations. Etc.
The explanation isn't the problem, it's the comprehension.
Your dog will never understand calculus or why Fourier transforms are interesting. There's almost certainly topics that are beyond human comprehension that an advanced artificial or alien intelligence can easily handle.
So I guess sci-fi movies were right all along. Nobody in Star Wars knows how hyperspace travel works, it just works. The little robots know everything but almost no human bothers to care. People just carry on with their bickering lives while the bots whiz in the background, and these robots are astonished at human inefficiency every single time, but rarely do anything about it. And people are still people.
Why can't we (or AI) invent ways to explain information that makes it much more digestible? And the solutions simpler?
Why is it necessary to continue to increase complexity when we get better intelligence? Can't we find more simple solutions? Or at least more explainable.
Is particle physics digestible even if it is explainable? Some things are not simple, cannot be not abstract, and will not be understood by most, or all, people.
The first two are open enough that they may be as you say*, but we already know how to solve global warming, it's more of how much do we want to.
Green energy and transport technology is now at the point where people save the world and get rich trying, just as fast as they can build the factories.
Food's climate impact is harder, because the problem isn't technical, it's convincing people to give up beef (and other things, but mostly beef).
* quantum mechanics and general relativity are famously difficult to get to grips with
I've been thinking about this and I believe the best place to be is a scientist who keeps looking at an AI's output, prods it in the right directions, verifies the proofs, fixes and fills gaps, takes the proof to production with safety, risks etc mitigated and then distribution with a company wrapped around the discovery. I think it wouldn't be black-boxed as much and will require a lot more understanding and reviewing to trust and productize it.
> Or we solve global warming but it requires giant cooling machines running 24/7
That’s not “solving it”, that’s putting a bandaid on it. Solving it would mean correcting the underlying issue to the point it’s no longer a problem which requires maintenance.
I doubt we’d build anything IRL without understanding how it works first. And we’re pretty good at putting 2+2 together once we have the pieces, for a lot of these things we don’t even have those. After all AI can just explain it to us atp
This is pretty much what underlies AI doomer argument of people like EY. Humans will gradually hand over civilization to black box AI they can't understand. As AI becomes more complex and powerful it will be harder and harder to control.
That assumes everyone will do so. Some people won't, and it's not clear you need a large number of such, a priesthood if you like, to survive as a species without AI.
There's a lot of un-contacted (or voluntarily self-isolating, it's hard to tell which) tribes around the world; things go poorly for them when businesses with power tools decide they want the resources of the land they live in, even when they're theoretically protected by law of the nation whose borders both the tribes and the businesses both find themselves within.
This is like in Hyperion where AIs invented the farcaster wormhole portals and no human or team of experts ever came close to understanding how they worked.
But I would say that processor like the Z80 are so simple and well-known and documented in the wild that they would have some chance to be bootstrapped again with what would realistically survive, one way or another, in a doomsday scenario. But yeah 20 year is too short, it's more like 40 years.
The amount of papers produced passed the point of being digestible by humans a long time ago.
I do think we will need to find a way to get away from publishing papers. But I thought that before the AI came along and made mediocre papers something you can produce in a day. The academic system seems utterly incapable of self-correcting on this point though. We haven't even managed to get rid of for-profit publishers. So how this all will go down is anybodies guess right now.
I am curious if LLMs are better at some kinds of problems than others. IIRC this and another big recent one were cases of the LLM producing a counterexample to a conjecture.
IMO, it's due to some problems being better documented, with more well-documented, previous research available. LLMs don't really create novel mathematics, they mostly "connect the dots". LLMs by design are not coming up with anything new, unless by statistical probability, aka "brute forcing".
I don't want to minimize LLMs capabilities, it's pretty cool they are doing this, and it's useful from a research point of view. But it's important to set expectations.
> LLMs don't really create novel mathematics, they mostly "connect the dots".
That is not what the mathematicians are saying. I don't have the knowledge to evaluate this myself, but a number of mathematicians - for example, in the SP - are saying it goes further than that - they really do introduce novel ideas. Of course everything is based on and inspired by some previous work, but that is true of all human mathematics as well.
LLMs that have been trained through reinforcement learning on mathematics are NOT simply token predictors. Only base models can be accurately described that way. They have learned how to do mathematics. They have learned to do coding. Its really amazing we're three years into instruct models and such a large part of Hacker News still does not understand the most basic facts about this field.
> Every day, it grows harder and harder to contain a mental map of recent relevant progress by simple virtue of the amount being produced. I cannot help but be very optimistic about the ambition mathematicians of this era will be able to scale to. There still remain lots of problems in current era tools and their usage though.
Always, always always, the problem with research and development is leadership, not insufficient supportive technology. It is a political problem, there is absolutely, positively no shortage of technologies to support research. Your optimism is totally misplaced. The NSF funding cuts have negatively impacted math more than AI has benefitted it. And guess who supports the administration that cut NSF funding? The people who ousted the PhDs from OpenAI.
I think we’re looking at a new class of wonderful machines that can potentially make meaningful contributions to the sciences and maybe even humanity as a whole, in addition to far more insidious and destructive capabilities.
You are right to point out that the ones who fully own and pilot the machines all belong to the “fuck science and humanity as a whole” group. So the likely outcomes don’t look good.
Echoes the early promise of the internet vs the eventual state and consequences of it, although seemingly primed for far more dire and deeply penetrating consequences.
Not in academia, but the amount of crying over rapid technological and intellectual progress because you're not getting credit validates everything critics say about you.
No interest in human advancement, just attribution.
I’m not in academia myself, and I think AI solving all our problems ASAP is ideal, even if it means no humans get attribution.
What I’m saying is that the ultimate goal of those in power are not these sorts of altruistic or even scientific pursuits, and that the massive labor disruption and hyper concentration of power in the hands of those who are proving time and again that advancement of science and benefiting the whole of humanity are actually antithetical to their goals is likely a bad thing.
Does your average poor person in America have happiness after 100s of years of relentless technological progress for the benefit of human advancement, and especially today, in the age of becoming spaghettied into the AI event horizon?
Much of Europe is close to the ocean, high in latitude, or mountainous, and climates there are more temperate. You don’t need AC there; AC is a luxury.
Southeast or central US has considerably higher wet bulb temperatures than Europe does in summer. Without HVAC, there’s a good chunk of the year where it’s too hot to get much done.
Eh, don't be silly. In the places where the summer is hot enough (or, more precise, where it used to be hot enough), I have seen plenty of AC units on shabby buildings, even on old Commie apartment blocs in Romania.
> I think we’re looking at a wonderful machine that can potentially make meaningful contributions to the sciences and maybe even humanity as a whole.
That's true. But. Maybe you've seen the Oppenheimer movie, there is a moment where Oppenheimer shakes Teller's hand, basically after the guy ruins Oppenheimer's life in a completely immature betrayal. That's what people are angry about, the academy community is Oppenheimer's wife asking, why the fuck did you shake his hand?
At least regarding leadership and funding, I don't know if it's a matter of likely or unlikely outcomes. It's just facts: these guys are collaborators. The commenter might very well have zero graduate students starting next year. What pisses me off is the utter obliviousness that STEM people have about how deeply political their work is.
And perhaps this is the real reckoning for the mathematics community. Not the possibility that AI is going to replace their jobs, it's not going to do that. But that having these intensely myopic and disagreeable personalities mean that basically zero leadership skills have been nurtured in the mathematics community. You cannot name a single politician who is a mathematician. You have to be elected to have power in this country, it's that simple, there are way more billionaires than there are presidents! Leadership is far more scarce. So that's why these disputes matter, and while it's great that people engage on Hacker News about it, it's intensely disappointing that "reduced science funding is really bad" gets downvoted.
That is a result of Hacker News's emphasis on this very 2010s view that it wants to be a place where the math nerds gather (in @dang's words) - he doesn't get that the quality of the discourse was caused by great leadership at many political and academic levels. Nobody credits how much better leaders were during Y Combinator's biggest success stories, or how much we overvalue the intellectual powers of math because it makes money as opposed to enlightening our view of the world.
Why would it excite you, rather than terrifying you? The better LLMs get at math, the closer the expertise you spent your whole life building is to being worthless.
Along with all the rest of what humans find meaningful and fulfilling.
I spent years grinding to learn mathematics because it was the language I needed to solve problems that excite me. If the tools I need to do so change, I can change too. Research training is not so rigid that it can only applied to the single set of skills I developed it in the context of. I can learn this too.
Moreover, truth be told, I don't really see myself doing any less math and requiring less from my skills. At least from the moment I've begun incorporating LLMs into my research workflow to now, the demand I've had from my own skills has only grown. At least in an era prior to Lean formalization.
Because for many people who pursue these fundamental truths, the reward is not necessarily personal fame, fortune, or even personal understanding. Advancing humanity's total knowledge (even if that knowledge is by proxy through AI) is reward enough.
At that point, if AI can do 75-99% of what you do... Why should anyone pay you to live/survive?
Humanity is having those discussions, heck you are in one RIGHT NOW not some Hollywood future.
What is coming of those discussions is the ownership class balks at the idea of raising their taxes (see recent interview with bezos), and therefore balks at the idea that you or I should have any value beyond what we produce... And if AI can replace you or I, well how do we survive if we can't produce in a technological society?
Go ahead and have that conversation with the billionaires running a worldwide satellite grid of data centers to power their AI surveillance dragnet and autonomous robot soldiers. See how far it’ll get you.
If they don't have millions/billions of customers to spend koney on whatever they are selling, their riches become irrelevant too.
Money is valuable only as it changes hands for goods/services, and if you want to get rich, on top of having/producing/controlling something everybody desires, you also need as many people as possible to have money to give you in exchange for a piece of that something.
With AI + robots all you really need is the starting capital + land (minerals, energy, etc). The value of land will not decrease when human labor becomes obsolete.
There's an unstated assumption there, which is that you'll have some reason to continue to want your work to be required.
In the (probably unlikely) event that AI use results in a post-scarcity economy in which there's no need to work to survive, a lot of people wouldn't regret sentiments like the ones in question.
On the contrary, it would mean they could work on whatever they please, including potentially standing on the shoulders of giants - the AIs - and seeing even further.
If we actually worked to create a society that work for the benefit of all its members, there would be a lot less reason to worry about developments like these. Much of the worry arises because for various reasons - none of them really good ones - we've ceded control of these developments to the people least suited to manage it.
And how do you see us getting from what we currently have: a working class and capital/ownership class, where a vast majority of society is required to work 40+ hrs/week to sustain their ability to live.
To a society that provides a livelihood to all humans, equally?
For, I would love to hear how we get from here to there during an era with the largest wealth disparity ever seen in human history. (Yes, it's worse than the robber Baron era of US history). For I have yet to see any signs that the capital/ownership class has any intentions other than vacuuming up even more wealth and power for themselves. And that anathema to your desired outcome.
Part of my point is that this helplessness about the expected outcome is a choice. If everyone is sitting back waiting for "signs", nothing will change for the better.
History is full of examples of situations like this being corrected, at least to an extent. If we learn from those, we can do even better next time around.
If you can reach it... The universe is expanding, and matter is being dispersed by both that and other forces.
Mind you, there are places in the universe that we have no way of knowing ever existed... The non-obserable universe if you will. For when physicists talk of the observable universe, it is only the fraction we have any chance of receiving data/light/radiation of/from
The parent poster isn't saying "advancement of knowledge" is some kind of universal goal for humanity at the cost of all else - and I would agree that it shouldn't be. They're suggesting that as an individual studying pure mathematics, the discovery of new truth is a self-consistent good.
Even taking a purely Kantian interpretation that would scale this beyond mathematicians - and that itself is a logical leap! - making a universal law out of "a discovery can be beautiful regardless of whether created by humans or AI" is is much more specific than the straw extrapolation you've created.
Have you considered that utilitarians actually exist?
If 20% more medical knowledge would save more lives long term, there are actually people, probably some browsing this website right now, maybe the person you're responding to, that actually think killing people up to the expected number of lives saved is justified.
I would personally call that evil, but it is thought through.
At least from my perspective, these sorts of tools could have the possibility of allowing us to reach post-scarcity (I guess a skynet future is another possible outcome, as is just grimdark industrial hell). If we reach that point, then anybody could (in principle- in reality utopias don't exist) pursue anything they wanted.
This is just an application of the philosophy "automate yourself out of a job every 6 months"- I've been doing that for a long time, and the outcome is generally a more interesting job.
If one only found meaning in life through external factors like work (no matter how "intellectually rewarding") then it seems like a life destined for eventual disappointment.
The answer is that we simply need to decouple the "right to exist" from "worth."
You should have the right to exist and explore the world simply because you're human, not because you can use your skills to provide some sort of transactional value to someone else. Deprogramming so many people is going to be hard...
All sane and noble in theory, but in practice, how do you see that happening?
Let's start with the first practical step: how do you dethrone the psychopaths in charge of the world who own about everything on Earth and have all the world's lethal force in their pockets?
That’s kind of a strange comparison. It’s the natural order for a population to thrive, reproduce, age, repeat. I’m not taking a side on the original comment, but the idea of human skill being completely supplanted by AI is not the same thing as having children and getting old.
AI is not a living or conscience entity, no matter what the hype men are selling society.
A child is a living, breathing, growing, and changing conscious entity. It is the natural order for the young to supplant the old, no matter what the politicians and billionaires desire.
"AI" - terrifies anyone who understands the pact our society rests upon: that labor is valued and can be exchanged for goods and services to survive. Thereby enabling a person to support their families without having to do everything themselves.
If AI replaced a noticeable fraction of society, destroying their capacity for work. That threatens and ultimately blows up this compact between working class and capital class... With it, the foundations of a modern technological society.... It may sound like hyperbole, or some fantastical prediction. But really it is basic economics, like econ 101... And personally the last few years have terrified me, not because of AI directly, but because how ignorantly blind many smart and tech savvy people are... You are marching us to collapse with a smile on your face...
Mathematics is a bridge to what Neoplatonists call the intelligible world. Currently, mathematicians navigate that world on foot. It's exciting to think that soon we might have cars and trains in that world so we don't have to painstakingly walk everywhere.
Many of us don't do what we do for our expertise to be recognized or valued by others, rather that is a pleasant side effect. Many of us do what we do for intrinsic reasons related to the nature of the work, and would likely do it for free, or indeed, would pay for the opportunity. Many STEM-types are in this category, and as such, are compelled to continue to tinker as we fancy, and are glad for more tools to help us expand the breadth of our tinkering capabilities.
A dedicated engineer is always looking to automate themselves out of existence, so that they can move on to the next thing to automate. Ongoing repetitive work is less engineering and more akin to toiling on a line.
What's happening is the verbal/linguistic equivalent of the invention of calculus. No intellectual field will ever be the same again. Who wouldn't find that exciting, and want to experience it?
It depends. If you are in a disadvantaged class it is very likely going to err towards a dismal result long term. However if you are a privileged intellectual these models can accelerate and expand your horizon. It isn't the end, surely. It is, however, both impressive and depressive simultaneously and that perspective only depends on your point of view.
But when the bar to entry is beyond expertise in a field or subfield, how does an individual ever hope to attain an unexplored space to explore?
It may be the beginning of thinking, but to many who view things on a longer timeline. It starts to look like it will breakdown the frameworks of which are required to get to that position. Otherwise, you just end up retreading explored ground. This removing the joy of discovery from any humans hand/mind.
Recently I've found my mind reawaken. It's about asking good questions now. The models can find the answers, but you have to know what to ask. Sometimes the model is wrong and you have to challenge it to find an alternative. Being able to explore problem spaces quickly is interesting.
You made up a group in the past and you made up things they say and then draw the inference that a different group in the present is somehow morally disadvantaged by obvious inference.
Perhaps your name-calling is not actually as logically grounded as you think. It definitely seems to depend on unfounded leaps.
I'm not sure I grasp the analogy to the invention of calculus. Calculus helped us solve new and interesting math/physics problems. Repeated for emphasis: helped *us* solve.
This technology is solving interesting math/physics problems for us, which is completely different.
Before the discovery of the fundamental theorem of calculus, enormous ingenuity and whole careers were spent doing calculations which the fundamental theorem trivialized. To be clear, I'm not just saying that the people involved were doing lots of mechanical arithmetic (though they did that, too). I'm saying they did creative, inspired, nontrivial mathematics to calculate certain things, all of which was then trivialized and made obsolete by the fundamental theorem of calculus.
In a way, young people have an advantage over middle aged people. I've spent countless hours as a middle aged person learning skills that are now useless. Better to be a young person than a skilled artisan during the Industrial Revolution even if there's uncertainty.
> the closer the expertise you spent your whole life building is to being worthless.
Perhaps it is time for life to be considered intrinsically valuable, instead of being "worthy" only based on output or capability. Disability, animal and environmental advocates have been fighting for this for a long time. Not too long ago women and minorities were in the same boat. Even now, there are many advocating and fighting for a return to the dark old days.
> Along with all the rest of what humans find meaningful and fulfilling.
Some humans. Many are content to enjoy simply existing, and the beauty of life and the universe around us. Just like many non-scientists today enjoy and benefit from the work of scientists, tomorrow too many will enjoy learning from, and applying the coming advancements and leaps in many fields.
And those of a scientist or other research-type mindset? No doubt they will contribute meaningfully by studying the frontier, noting what remains unanswered, and then advancing the frontier, just like researchers do today; just because scientists in the past solved many questions doesn't mean that there aren't any questions to answer today.
IMHO, AI means that the frontier expands faster, not that it is obliterated. Even AI cannot overcome the laws and limitations of physics/universe: even Dyson spheres only capture the energy of one star, thus setting a limit on the amount of compute, and thereby a limit on intelligence. And we are a loooong way from a Dyson sphere.
PS: I think you're being unfairly downvoted. Your question is not invalid and deserves responses, not downvotes.
> These all are valid, noble points I also used to brood about while being young and financially supported by my parents.
Ah, the proverbial silver spoon. Sadly, I never had that luxury. If you look through my comments, you'll notice I'm more at the get-off-my-lawn point.
Also, what happened? Real world wear you down and turn you cynical? It is possible to be hopeful and cynical at the same time. This tech is something new we're seeing: the future is as yet unwritten. r/LocalLLaMa works well, so there's hope even if corporate ai goes kaput.
My generation has been lucky to see a few new things, though we certainly live in interesting times. Moon Landings. Berlin Wall fall. Moore's Law. EU (I have the old coinage to serve as a reminder). Space Shuttles. China and India integrating with the world. Cellphones. The Internet. Digital Photos. Linux. Solar. 3D-printing. Smartphones. Tablets. Bitcoin. EVs. Mars rovers. Asteroid visits. Internet from space. FTTH. MRNA. Gene Therapy. MRI. Ultrasound. Wi-Fi. Mesh Wi-Fi. Reusable Rockets. Cubesats. Selfies from space. Drones. LoRa/LoraWAN. Maglev HSR. And now AI, real AI. Chinese-like Whale Language.
There's hope for the future yet. You can help make it happen right. But only if you leave the cynicism at the door. Can't give up - it's our kids' futures at stake.
> wish I could share your optimism, but no recent event in the world affairs can help with that for me.
What about Ukraine holding Russia back, and now looks like it might actually win? What about the most recent additions to NATO. Hungary's regime change? Canada's save? EU's pivot to arming itself, and quickly?
Many of my colleagues and I have been experimenting with LLMs in our research process. I've had pretty great success, though fairly rarely do they solve my entire research question outright like this. Usually, I end up with a back and forth process of refinements and questions on my end until eventually the idea comes apparent. Not unlike my traditional research refinement process, just better. Of course, I don't have access to the model they're using =) .
Nevertheless, one thing that struck me in this writeup, was the lack of attribution in the quoted final response from the model. In a field like math, where most research is posted publicly and is available, attribution of prior results is both social credit and how we find/build abstractions and concentrate attention. The human-edited paper naturally contains this. I dug through the chain-of-thought publication and did actually find (a few of) them. If people working on these LLMs are reading, it's very important to me that these are contained in the actual model output.
One more note: the comments on articles like these on HN and otherwise are usually pretty negative / downcast. There's great reason for that, what with how these companies market themselves and how proponents of the technology conduct themselves on social media. Moreover, I personally cannot feel anything other than disgust seeing these models displace talented creatives whose work they're trained on (often to the detriment of quality). But, for scientists, I find that these tools address the problem of the exploding complexity barrier in the frontier. Every day, it grows harder and harder to contain a mental map of recent relevant progress by simple virtue of the amount being produced. I cannot help but be very optimistic about the ambition mathematicians of this era will be able to scale to. There still remain lots of problems in current era tools and their usage though.