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.
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 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.
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.
"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.