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.
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.
Mathematics serves as a bridge to what Neoplatonists call the intelligible world. Currently, mathematicians navigate that world on foot. It's exciting to see that soon we will have cars and trains in that world so we no longer have to walk everywhere painstakingly on foot.
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.
I think when your work is no longer required, you will probably come to regret this sentiment, not that it matters.
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That’s definitely not your actual principle. I can conjure several reductios that demonstrate various scenarios you’d prefer to not happen that involve a massive “advancement of knowledge”.
It’s completely trivial. 20% more knowledge except an orphan is killed per 0.001%.
Let’s try to think posts through.
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.
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.
Does it terrify you to look at children?
Not so many years from now, some of them will surpass you. A few years after that all (that survive to that point) will surpass you.
Does that terrify you just as much?
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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?
People who enjoy thinking. Ya know, the "intellectual" part.
This is the beginning of thinking, not the end...
The so called "progressives" prove that they were the same ones crying after the printing press, automobile, calculator, washing machine, etc
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.
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?
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.
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.
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.
Seems rather depressing to me but maybe I am a Luddite.
Exactly. I would rather we let these discoveries stay hidden for a while longer such that human ingenuity may untangle them from the coils of reality. A machine? A mechanical man? Deins to produce something as pure as mathematics without the divine fervor of the ineffable spirit of Man?! It's just not what God wants.
Human ingenuity is untangling perhaps the deepest question of all- what is the essence of Reason and the intellect that so privileges man? I don't know if it's what God wants, but it's certainly getting close to some existentially fundamental questions.
While many seem to be anxious or pessimistic about the future of intellectual/artistic pursuits (understandable although I disagree), I do find the utter lack of curiosity or interest in the incredible machinery that is causing all the fuss to be striking.
I can't tell if this is satire.
I had the exact same thought. It depends entirely on what voice you read it in, I suppose.
Haha, now I do think it is more likely satire. I don't think HN has many people who would post like this sincerely.
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