Heh, I like this. but it should be pointed out that from the other point of view, software developers were the supernatural beings (dare I say demons), which the sorcery of a good project manager could tame (with more or less success) to perform the desired magical effect

Differently than wizards that lock their knowledge in towers and in sects, software development has a tradition of being open for the most part, so the sorcery and wizardry analogy works more like a spectrum. It just depends on how close to the metal the apprentice would like their consciousness.

The sorcerers have always outnumbered the wizards. Before AI, we called them code monkeys.

OP has reportedly been in contact with Prof. Mantova, who actually worked (jointly with S. L'Innocente) on the key human-authored results behind this AI proof and is arguably in the best position to understand exactly what the AI added that wasn't known before. (See the OP's thread on the Lean Zulip.) So this is happening, and we might see an actual paper publication of this result down the line (possibly encompassing multiple roughly self-contained papers, building up to the final result). The current AI-written version is way too obscure for that, and the AI-written human-targeted "summaries" are not really helpful. Again the OP is quite aware of this.

It's impossible for finite number of LLMs to solve all theorems. This would imply that the busy beaver sequence is computable which implies the halting problem is decidable.

For any finite program (eg some LLMs), there is a true math theorem which they cannot prove or disprove (given fixed input of the statement with no other information sources). If that weren’t true, BB would be computable.

Math is beyond computation. Since AI is just bits in bits out, it has this fundamental limitation.

Any magic of AI systems comes from the transformed meaning of its input data. With fixed weights any LLM is just an artifact. For example a human prompting an LLM constitutes an extra information source, which removes the above limitations. In theory any input from the natural world would remove the limitations too. The natural world is a black box and we don't know what kind of meaning or intelligence could underly it.

lol same with people.

"Given infinite thinking time a finite number of humans will solve all theorems"

I also love the angle that this was not intelligence just brute force. As if the mathematicians didn't reeaaally want to solve this they were just too lazy to give it a good try.

What does AI have to actually do before you realize these things are actually smart?

https://www.youtube.com/watch?v=ZYj4NkeGPdM

Yeah it's a pretty tough question! I've resisted doing that until I had a relatively high certainty that their published results contained minor mistakes, which I assumed they would want to know about. I've also been explicitly apologetic and tried to keep it super brief.

A counterpoint - I was told a story by one of my professors in the late 1990s, about one of his professors -- he'd written a thesis, gotten hired somewhere like Princeton, and taught there for a few years as Dr. <Somebody>. One day he received a letter pointing out a construction flaw in his thesis. He brought it to the department head who read the letter, and said "Well, Mr. Somebody, ..." Ultimately he fixed the proof.

Upshot, if there are real errors in published work, I think most mathematicians want to know about them.

I don't think this is your fault; the description isn't very explicit. Let me try to do a bit better. (I'll also try to go somewhat further, and you should not be discouraged if at some point it stops making sense.)

You can think of the "surreal numbers" as being built up step by step. We start out with no numbers at all, and then we repeatedly do a construction that makes some new numbers.

A surreal number is made from two sets of (pre-existing) surreal numbers. We typically call them L and R, for "left" and "right", and sometimes write it as L|R or {L|R} or something like that. The "left" numbers have to be smaller than the "right" numbers. The resulting number will turn out to be, in a certain sense, the "simplest" number in between all the left numbers and all the right numbers.

Now, as I said, we start out with no numbers at all. It might seem like that gives us no way to proceed, but it does: even given no numbers at all, we can still make a set of numbers, namely the empty set! So we can use that for both L and R, getting ∅|∅. Empty sets on both sides. We call this 0, and it will turn out to behave in the way you'd expect the number 0 to behave.

Now we suddenly have another set available, namely {0}, the set containing only zero. Which means that instead of being able to make one number, maybe we can make four: ∅|∅, ∅|{0}, {0}|∅, {0}|{0}. The first of these we already knew about. The last isn't actually admissible -- remember that the "left" numbers have to be smaller than the "right" numbers, which is "vacuously" true when one of those sets is empty (it means "if you have a number x in the left set, and a number y in the right set, then x<y", and if there are no numbers in the left set or no numbers in the right set then that's trivially true) but isn't true when both sets contain 0 because 0<0 is false.

So actually we get two new numbers: ∅|{0} and {0}|∅. The first fits into what OP calls the gap "between nothing and zero". The second first into what OP calls "the gap between zero and nothing". In both cases, "zero" means a number and "nothing" means a space where we don't yet have any numbers.

The number ∅|{0} is called -1 (it has to lie to the left of 0, and there's no constraint on its left, and -1 is "the simplest number less than 0") and the number {0}|∅ is called +1 (it has to lie to the right of 0, and there's no constraint on its right, and +1 is "the simplest number greater than 0").

I should explicitly acknowledge that I haven't defined what "less than" and "greater than" actually mean for these numbers, nor anything else about how they relate to one another that could possibly justify giving these things the specific names 0, -1, and +1. But there are definitions for "less than" and "greater than" and "plus" and "minus" and so forth, and the whole thing does turn out to work very nicely.

Anyway, once we've got these numbers we have eight possible sets that can go on the left or on the right. The requirement for left-things to be smaller than right-things reduces the possibilities somewhat, and the actual new numbers we get next time around are: ∅|{-1}, which turns out to be -2; {-1}|{0} which turns out to be -1/2; {0}|{+1} which turns out to be +1/2; {+1}|∅ which turns out to be +2. We also get some already-existing numbers in new ways; for instance, {-1}|{+1} is actually equal to 0 ("0 is the simplest number between -1 and +1"). Again, I should explicitly acknowlege that I haven't said anything about how you determine when two of these things are actually equal; again, it does all turn out to work properly.

If you keep going with this construction, you produce all the integers, two at a time, and also all the "dyadic rationals", meaning fractions where the denominator is a power of 2. And then, once you've got all those, at the next stage of construction you abruptly get all the real numbers -- e.g., the square root of 2 is L|R where L = {dyadic rational numbers that are negative or have a square smaller than 2} and R = {dyadic rational numbers that are positive and have a square larger than 2} -- and you also get {0,1,2,3,4,...}|∅, conventionally written as a lower-case Greek letter omega, which is an infinite number, larger than all the integers. (And its negation.) And {0}|{1,1/2,1/3,1/4,...} which is an infinitesimal number, positive but smaller than any ratio of positive integers. And you can then proceed further and construct a vast infinitude of numbers, including all the real numbers (which we've already made) and all of the so-called infinite ordinals (which you can kinda think of as being a sort of "infinite positive integer", though there's more to them than that) and much more, all in a system that lets you do arithmetic and suchlike. It's very elegant, if your brain has been twisted into the mathematician-y shape that finds such things elegant.

I made a picture, hope this helps: https://excalidraw.com/#json=zfKWWn1h7GzdFca6RDdXl,plr_WeaCt...

Sorry it was confusing.

Edit: the picture is now edited into the article.

I think it's just we don't have a way to say/write these numbers. So you make up a way to write them (-1 and 1) and continue.

The numbers don't matter and you could replace -1 and 1 with anything. It's just easier to begin your new fake number at - 1 and 1. Because position does matter.

I think he just wrote it in a confusing way. The quote before says:

> (crucially, “to the left of all” and “to the right of all” also count as “gaps”)

So there are two "nothings" here, left of "all" - i.e. the zero - and right of it.

Though I'm not quite sure how you'd get infinite or irrational numbers by this procedure. Wouldn't you simply get the rational numbers by this?

(unless the "put a number" step is doing more work here than it seems. He doesn't really say which number to put there. In the examples, he mostly did "new number = (left number + right number) / 2", with special cases if any number is "nothing" - but he never actually wrote what the rules are here.)

Yeah I'm curious about the difference between "nothing" and "0", but I just decided to roll with it. Until the Greek letters made my head start to spin as they usually do.

The image helped for me at least.

(Apparently I was extremely unclear with this text. For clarity: if you want to actually understand surreal numbers, go and read On Numbers and Games, by Conway, which is a delightful book; or get an LLM to talk you through Wikipedia. Original text follows.)

It’s a terrible explanation. A surreal number is defined as a pair of sets of surreal numbers (where you fiddle around the recursion in that definition by defining them in waves, so strictly speaking you’re defining “the surreal numbers born at time T” for each individual T given access to the surreal numbers born at all earlier times, and then you “take the union across all times”, scare quotes because there are too many times for this to result in a set). Zero is a surreal number but the LLM is using the word “zero” to mean “the set containing just the surreal number 0”; “nothing” here is the LLM’s obtuse word for the empty set. Wikipedia may actually be easier to follow.

> I think I'm officially too dumb for math.

I'm hoping someone develops an interactive tutor that can teach any subject to any depth.

The tutor should optimize its pedagogy. It should use online RL to adapt to a learner's ideal learning style, model what the student understands and to what degree, and understand what the gaps and next steps are.

I'd subscribe in a heartbeat.

I feel the same way. Want to become a dumb and dumber duo? I believe you could be my friend.

Hah, very interesting question!

Let me first clarify my relationship with mathematics. I think of myself as "an awestruck observer from a distance". I find some parts that I understand beautiful, and I have also tried to understand some of the basics rigorously. However, I generally just can't make my way through any serious paper, as I both lack the prerequisites and struggle with the amount of inference mathematics tends to place on the reader. That's the "from a distance" part.

Now, about picking the problem. I am genuinely "pulled by" surreal numbers themselves. I find them irresistibly beautiful. There is also a bit of bitterness around how they haven't fulfilled their promise (yet?) as Conway hoped they would be able to become a better foundation for some mathematics. But they are a bit too difficult to prove things about so far, and we know too little about them. So what "pulls me" also is a possibility of making enough dents in this that we would be able to use them more broadly, and learn even more things about them.

However, I do not know the details of the latest research. I don't know which problems have actually been solved, which pursue Conway's original vision vs narrower approaches, and which are elegant enough to feel "awestruck" enough about. So this is an invitation from me to LLM to share what it "feels pulled by" (for whatever definition; I think of it as just navigating the languagespace) , and then sifting through that list to see if something it lists makes me feel something. I would assume that with the field currently being so small (serious mathematicians mostly don't care about surreals), it's easy to get the LLM "excited" (again, just a vector in the languagespace) enough that it would give me genuinely interesting candidates. Then it's up to me to sift through them and see if they "speak" to me.

It's like asking a mathrock nerd to share their favorite mathrock albums. Niche enough that you'd likely get good results. Then you can listen and form an opinion.

In this particular example, the "ONAG birthday" and "maybe last Conway's unsolved conjecture about surreals" part spoke to me emotionally, the statement itself amazed me with its simplicity, and I felt "blood in the water" related to the recent results bringing the conjecture closer. So I felt the pull myself and went with it.

I use you, me, us interchangably because I don't think it really matters for anything and I don't need to reaffirm my individuality with such words.

My god, that proof map has so many parts O.O

Absolute bad-metaphor-laden slop, in a dramatic writerly voice. These LLMs are trained on too much pretentious writing.

Discovering that poetry was cognitive compression is alone one of the latent findings LLMs unlocked.

You can so easily imagine this shit being read in a 90s slam poetry coffeehouse. Trust me I was there

> I genuinely invite a refutation.

> So, assuming my proof doesn’t rely on a Lean kernel bug, it’s likely to be legit too.

He lacks the understanding to verify his solution properly, and has to lean on those who do have the understanding to verify it, only being able to say himself that it's "likely" to be correct. (And what do those mathematicians get for laboriously checking the generated proof? 40 grand?)

Seems to me problem solving is as dependent on understanding as ever.

Perhaps mathematicians will undergo the same split as what happened to philosophy and natural sciences.

Yes, the timing of this post just after Gowers' post is fascinating. It is almost as if the marketing machine is well oiled.

I've confirmed with the mathematicians working in that field that this is a new result.

On the other hand here we have an amateur that could accelerate their learning and experimentation faster than ever possible before.

People will just limit publishing valid works to avoid becoming a hapless plagiarism victim class. Same thing happened to tech bloggers ripped off by low-effort you-tube content makers.

Isomorphic plagiarism makes people feel 23% smarter, but it also provably degrades core skills by 17%.

LLM are great at context search, but are also trivially proven degenerative under recursive self improvement scenarios. We look forwards to stripping their assets at a heavy discount.

Also, we shouldn't kink shame peoples cognitive dildo choices. =3

Author here! My impression is that it's customary in the mathematical community to take responsibility for the result with your name, regardless of whether it came from LLM etc (as long as you disclose LLM usage). I am perfectly fine calling it "LLM's proof" or somehow else, but it's "my" in the sense that "if there is a mistake in it, it is my mistake".

I don't understand your comment. At first it seems like you think the LLM should get credit for the work. But then you mock the author for asking about the LLM's mood, which makes me think you believe the LLM is just a tool and not capable of receiving credit (FWIW I would agree.)

Someone had to choose the problem, steer the model, and check the output - it's clearly taken a lot of time. That's authorship with a powerful tool, same as it's always been.

They still needed to invest time and other resources into this. It's listed clearly in the 2nd paragraph. Mathematicians, or anyone for that matter, don't figure out everything from scratch. They lean on the work that others have done before them to save time. How is this any different?

When you use a drill to put a hole in the wall, do you take credit for it, or do you credit the drill? Without intent, a tool, whether it be a drill or an LLM, is just an inert object.

LLMs need a lot of help to get to any kind of complicated proof, really. And yes, they get in moods. I spent 3 weeks trying to prove something and frequently had to try and convince Claude that it wasn't impossible and that it could really do it.

Several of my coworkers — it’s not that unusual that if you can max out a couple accounts, the companies will obviously notice you (as a high cost customer), and sometimes offer more

$200 a month is what people pay for morning coffee in the Bay Area.

I don't see why that's weird

ah, if it's anyone it'd be dan abramov