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.
I believe D. A. Jimenez and C. Lin, "Dynamic branch prediction with perceptrons" is the paper which introduced the idea. It's been significantly refined since and I'm not too familiar with modern improvements, but B. Grayson et al., "Evolution of the Samsung Exynos CPU Microarchitecture" has a section on the branch predictor design which would talk about/reference some of those modern improvements.
All you have done is contribute a wikipedia article which is the second google result if you search the title of the video. Another user made a comment referencing a textbook they used to learn this material as well as some extended comments of their own - this actually provides information unlike a bare wikipedia link presented with a dismissive attitude.
> Why do you think that the 2024 Putnam programs that they used to test were in the training data?
Putnam solutions can be found multiple places online: https://kskedlaya.org/putnam-archive/, https://artofproblemsolving.com/community/c3249_putnam. These could have appeared in the training of the base LLM DeepSeek-V3.2-Exp or as problems in the training set - they do not give further detail on what problems they selected from AOPS and as the second link gives they are there.
> By that logic I can slice open a sphere and call it a sheet
You can do this. If you remove a point (or a line, or really any connected component), you get a space which is the same as the plane. What happens if you remove two distinct points? You end up with with a very thick circle. Three points? It starts to get harder to visualize, but you end up with two circles joined at a point. As you remove more points you will get more circles joined together. From a mathematical perspective, these spaces are very different. If we start to allow gluing arbitrary points in the sphere together it gets even worse, and you can get some pretty wild spaces.
The point of surgery is that by requiring this gluing in of these spheres along the boundary of the space we cut out, the resulting spaces are not as wild - or at least are easier to handle than if we do any operation. To give an example, one might have some space and we want to determine if it has property A. The problem is our space has some property B which makes it difficult to determine property A directly. But by performing surgery in a specific way, we can produce a new space which has property A if and only if the original space did, and importantly, no longer has property B.
For property As that mathematicians care about, surgery often does a good job of preserving the property. In contrast things like just cutting and gluing points together without care will typically change property A, so it does not help as much.
> Likewise I wonder why we need to import a sphere rather than just pinch the ends of the tube shut and say it's now a sphere.
I am not an expect on surgery, but I think from a mathematical perspective, pinching the ends of the tube shut and gluing in a new sphere would be equivalent operations. This pinching operation would be formalized as a "quotient space", and you can formalize the sphere as a "quotient" space equivalent to the pinching.
> In guideline 1v1 a lot of very high level games are decided by garbage RNG which I think is even less interesting than determining who is 0.1pps faster.
I have played a lot of (moderately high level) 1v1 tetris and I would have to disagree. In fact I often felt that the reverse is true - if I felt I died to garbage hole RNG, really that meant I was getting out pressured and would have lost eventually anyways. And while my playstyle was more aggressive, try to out speed opponent, I lost my fair share of games to people playing (much) slower but just incredibly efficient.
I agree there is an overall disappointing amount of interaction between players, though. Watching your opponents board and adjusting to it is hard and takes a while to build the skill to do. And a lot of the times you can just get away with it by playing faster and out pressuring and ignoring the other player.
> I have played a lot of (moderately high level) 1v1 tetris and I would have to disagree. In fact I often felt that the reverse is true - if I felt I died to garbage hole RNG, really that meant I was getting out pressured and would have lost eventually anyways.
To be honest, I was never good enough for it to be a big issue, but it does seem apparent to me that it is an issue for the highest level players. I could be wrong, of course, but assuming I'm not, I think this brings up an interesting question: if it's something that you have to be so good at the game to have impact you meaningfully, does it really matter for 99.9% of players including myself who will certainly never get there? I guess the answer is probably not, but it does have a psychological impact of sorts. It definitely can make tournament outcomes feel less interesting.
So really random garbage just irks me because it seems like an unnecessary addition of RNG into an otherwise skilled game. I don't think random garbage is more fun than deterministic garbage schemes. I would suppose some people disagree.
The lack of serious interaction and a deep meta game, though... That's a bigger problem, yeah. I am not sure you can fix that while still producing something that you can really call "Tetris".
(And even when Nintendo called "Panel de Pon" "Tetris Attack" outside of Japan, I don't think it wound up having a terribly interesting interaction between players, either, despite being an entirely different game from the ground up! Still pretty fun though.)
Thanks for the correction. I hadn't considered bulk memory operations to be part of SIMD operation but it makes sense -- they operate on a larger grain than word-size so they can do the same operation with less micro-ops overhead.
> There have been efforts to reprove it with a more easily verified proof, but they've gone nowhere.
My understanding was that the so called "second generation proof" of the classification of finite simple groups led by Gorenstein, Lyons, Solomon has been progressing slowly but steadily, and only the quasithin case had a significant (but now fixed) hole. Are there other significant gaps that aren't as well known?
Still they have a couple more books of proof left, and I have to wonder how carefully it will be reviewed. This will still be a massive improvement, but I'd be a lot happier if the entire proof could be formalized.
Plus there is still a possibility that there proves to be another significant hole.
If any theorem needs to be formalized, this is the one. No other theorem is this big, this hard to prove, and this important to get right.
I am somewhat surprised issues of scripting and trading even exist in the registration system. Staggering enrollment times over a few days, with new waves every 20 minutes or so, mostly solves scripting issues since you are only competing with a fraction of the student body now. Giving courses waitlists once they are full, instead of allowing people to just directly register once a spot frees up, makes trading impossible since if you could trade you could have just registered for the course anyways.
I understand that the registration system is probably old and tied up in tons of just as old management software, but if the university really cared the solutions should be there.
When I went there it was staggered, which causes this desire for spot trading - seniors register first, so if you are an freshman/sophomore/junior, you beg a senior to register a spot in a class you want then coordinate them deregistering just before you register for the class. This automated that and at scale could be a big issue.
The school I attended in 2010 had a system like this as well. Awful backend with a simple, but still awful, web interface to talk to it. There was rumor you could telnet in and use an actual text interface, but I never saw it.
The system was replaced a few years later with an Oracle PeopleSoft implementation. Everybody hated it more.
It's much easier to build a gateway API for a legacy system than to extend it. Not disagreeing with you. Honestly, though, software systems for academic institutions are ridiculously complicated, because they are essentially a student portal, a school, a sales organization, a rules engine, etc. etc. all wrapped up into one and interconnected in ways that aren't obvious on the surface.
You can just drop the course - pretty much every university (in the United States, at least) allows students to drop courses one or two weeks into the semester without any record (on say, a transcript). Otherwise students cannot possibly plan their semesters, since courses may not make material available until after the semester actually starts.
So if you are planning to sell the slots and it does not work out, you just drop the course, no harm to you.
"No harm" except for the several slots that were taken up by people hoping to make a quick buck (selling the trade) who drop a day or two into classes and cause the students who actually NEED the class to be very stressed and annoyed and possibly have to adjust their plans because they don't think they will be able to take a class even though it will actually be free by the third time the class meets.
It also just fucks with the University's ability to gauge class interest. In my university, if a class filled up early in the registration window, the University would try to increase capacity or add another copy of the class, but that's not always an option.
A reminder that this is not done for technical reasons. Plenty of colleges all across the US, big and small, custom-built registration software or purchased on the open market, have fully functional waitlist features. "first come first serve" is a policy choice.
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.