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LLMs aren't new and there has been no sign of them being able to automate maths research so far. If they do, I will acknowledge them as AGI as it has been my go

LLMs aren't new and there has been no sign of them being able to automate maths research so far. If they do, I will acknowledge them as AGI as it has been my goalpost for over 30 years. I've never moved it once, and I won't. Computers were invented in large part for this purpose. The hope was to have a machine that could spit out interesting theorems. It turned out that generating theorems was trivial (you can just enumerate them) but generating interesting theorems was a task of an entirely different nature. To this day, computers are frequently used as tools by maths researchers but cannot do their own independent maths research, i.e. invent and prove their own interesting theorems. This is surprising if you think of research maths as ultimately a game of symbol manipulation, but I think what it tells us is that that is not what research maths is. The discovery that in trying to make a maths machine we made a computation machine instead, and it wasn't inherently any good at maths, made the difference between computation and maths very clear. What we have now is algorithms that are able to learn patterns in symbol use from large corpuses of text, and generate new text by recombining those patterns. Personally, I'm pretty sure this will be a demonstration that that isn't what research maths is either. If LLMs do turn out to be capable of independently inventing and proving interesting new theorems (every word of that is important) I will have to throw away most of what I have come to think about what intelligence is, and I will do so without complaint because it will have been shown to have been wrong. Indeed, I'll do it with joy, because I've always believed AGI is possible and the biggest thing I've always wanted out of its invention is a better understanding of what mathematics is. This will have been provided. However, I don't expect them to. Usually when I mention this people bring up various examples of attempts that have been made to make AIs "do maths" in some way but I'm aware of what has been done so far and none of it comes close to AI "independently inventing and proving interesting new theorems" or looks like its heading towards that any time soon.

lordnacho 4y ago
If it generates a bunch of theorems, won't some of those be interesting? Is there no easy way for a mathematician to sift through what it makes?
- thom 4y ago
  Yes, and that was basically the controversy around Doug Lenat’s automated mathematician: https://en.wikipedia.org/wiki/Automated_Mathematician https://en.wikipedia.org/wiki/Automated_Mathematician
- mkl 4y ago
  There are an astronomical number of true theorems, but almost all of them are uninteresting and not useful. We've had automatic theorem-generating software for decades, and there are so many useless true theorems that it can keep busy with all the CPU cycles you throw at it without necessarily generating anything useful at all. Even if you let it run with huge resources and generated loads of theorems, searching through them seems pretty hopeless. Where machine learning systems like LLMs may be able to help is to spot analogies and ways to translate theorems to different areas to do something useful (but the tendency to hallucinate seems a formidable obstacle for things that must be proven). For real automated headway/assistance, I think we need something like Alpha Go's "intuition" at judging the value of a board state somehow applied to mathematical arguments. That's how mathematicians usually work; we have an intuition that we can use a particular approach to prove something, and then we fill in the precise particulars.
dogcomplex 4y ago
If "interesting" is the keyword here, I'd have hope that language models based around mathematics research would at least be able to narrow the scope of theorems being iterated through to those closest to what mathematicians are already talking about. We'll see
pharmakom 4y ago
Interesting meaning interesting to humans? If an AI is doing all of the work then maybe it doesn't matter what we consider interesting? You gave a prompt asking for 2023 nostalgia pop music in the care home, and the AI found novel theorems to generate it for you, but it didn't even notice or care! Maybe there will be some hobbyist human Mathematians and the AI can throw them a clue once in a while, like a good murder mystery host.
- TimPC 4y ago
  Interesting generally means having some sort of sufficiently broad applicability to be useful and sufficiently narrow applicability to have enough explanatory power. If you are too narrow you don’t get applied in many situations and if you are too broad you’re generally too watered down to prove anything significant about the situations in which you apply. Some exceptions which makes everything harder. 1+1=2 is a theorem but a very uninteresting one because it applies exactly once. Every number n has a double successor S(S(n)) is also uninteresting because while it applies more broadly it’s too watered down. There are no numbers n with n greater than two such that x^n + y^n = z^n is extremely interesting.
  - pharmakom 4y ago
    You are defining useful relative to human understanding. My point is this may not matter in an AI future.
    - TimPC 4y ago
      The underlying concept I’m getting at is that there is more than just human understanding here. Our understanding of mathematics is highly subjective but it seems to me theorems do have some sort of true value with some of them being more useful for doing mathematics and others being things you prove once and never use again. A robot could prove infinite theorems about every number having a double successor, triple successor and so on. If it does so it’s not just human subjectivity saying it hasn’t done any meaningful mathematics.
optimalsolver 4y ago
That's kind of what PowerPlay does: https://arxiv.org/abs/1112.5309 https://arxiv.org/abs/1112.5309
IanCal 4y ago
It would be interesting to try these things out. I've been experimenting with making art with ChatGPT - instead of telling it how to do it or even what to do specifically I'm asking it for ideas, to then pick its favourite and expand on why etc. It's worked surprisingly well. I've moved on to asking it how you'd make an art show of multiple images, then just asking it to "follow step one in your instructions", "now follow step two". Results were good enough feeding that into a SD model that we're commissioning the first one it made to get it in real life. I'm not arguing chatgpt can research maths, but I do think the tools available are now fundamentally different than they were perhaps even last year. I don't think we're even at a point of knowing how best to use what we have, and the models should continue to get better. I'd encourage you to experiment and see how far away you think things are. I'd do it, I probably will, but I lack the mathematical background to understand whether it's responding with garbage or not.
- IanCal 4y ago
  I can't evaluate the results but here's my first chat, may be interesting: https://gist.github.com/IanCal/4426fdf81797c2800e8202a3b4f6c10d https://gist.github.com/IanCal/4426fdf81797c2800e8202a3b4f6c...

lordnacho 4y ago

If it generates a bunch of theorems, won't some of those be interesting? Is there no easy way for a mathematician to sift through what it makes?

thom 4y ago
Yes, and that was basically the controversy around Doug Lenat’s automated mathematician: https://en.wikipedia.org/wiki/Automated_Mathematician https://en.wikipedia.org/wiki/Automated_Mathematician
mkl 4y ago
There are an astronomical number of true theorems, but almost all of them are uninteresting and not useful. We've had automatic theorem-generating software for decades, and there are so many useless true theorems that it can keep busy with all the CPU cycles you throw at it without necessarily generating anything useful at all. Even if you let it run with huge resources and generated loads of theorems, searching through them seems pretty hopeless. Where machine learning systems like LLMs may be able to help is to spot analogies and ways to translate theorems to different areas to do something useful (but the tendency to hallucinate seems a formidable obstacle for things that must be proven). For real automated headway/assistance, I think we need something like Alpha Go's "intuition" at judging the value of a board state somehow applied to mathematical arguments. That's how mathematicians usually work; we have an intuition that we can use a particular approach to prove something, and then we fill in the precise particulars.

dogcomplex 4y ago

If "interesting" is the keyword here, I'd have hope that language models based around mathematics research would at least be able to narrow the scope of theorems being iterated through to those closest to what mathematicians are already talking about. We'll see

pharmakom 4y ago

Interesting meaning interesting to humans? If an AI is doing all of the work then maybe it doesn't matter what we consider interesting? You gave a prompt asking for 2023 nostalgia pop music in the care home, and the AI found novel theorems to generate it for you, but it didn't even notice or care! Maybe there will be some hobbyist human Mathematians and the AI can throw them a clue once in a while, like a good murder mystery host.

TimPC 4y ago
Interesting generally means having some sort of sufficiently broad applicability to be useful and sufficiently narrow applicability to have enough explanatory power. If you are too narrow you don’t get applied in many situations and if you are too broad you’re generally too watered down to prove anything significant about the situations in which you apply. Some exceptions which makes everything harder. 1+1=2 is a theorem but a very uninteresting one because it applies exactly once. Every number n has a double successor S(S(n)) is also uninteresting because while it applies more broadly it’s too watered down. There are no numbers n with n greater than two such that x^n + y^n = z^n is extremely interesting.
- pharmakom 4y ago
  You are defining useful relative to human understanding. My point is this may not matter in an AI future.
  - TimPC 4y ago
    The underlying concept I’m getting at is that there is more than just human understanding here. Our understanding of mathematics is highly subjective but it seems to me theorems do have some sort of true value with some of them being more useful for doing mathematics and others being things you prove once and never use again. A robot could prove infinite theorems about every number having a double successor, triple successor and so on. If it does so it’s not just human subjectivity saying it hasn’t done any meaningful mathematics.

optimalsolver 4y ago

That's kind of what PowerPlay does: https://arxiv.org/abs/1112.5309 https://arxiv.org/abs/1112.5309

IanCal 4y ago

It would be interesting to try these things out. I've been experimenting with making art with ChatGPT - instead of telling it how to do it or even what to do specifically I'm asking it for ideas, to then pick its favourite and expand on why etc. It's worked surprisingly well. I've moved on to asking it how you'd make an art show of multiple images, then just asking it to "follow step one in your instructions", "now follow step two". Results were good enough feeding that into a SD model that we're commissioning the first one it made to get it in real life. I'm not arguing chatgpt can research maths, but I do think the tools available are now fundamentally different than they were perhaps even last year. I don't think we're even at a point of knowing how best to use what we have, and the models should continue to get better. I'd encourage you to experiment and see how far away you think things are. I'd do it, I probably will, but I lack the mathematical background to understand whether it's responding with garbage or not.

IanCal 4y ago
I can't evaluate the results but here's my first chat, may be interesting: https://gist.github.com/IanCal/4426fdf81797c2800e8202a3b4f6c10d https://gist.github.com/IanCal/4426fdf81797c2800e8202a3b4f6c...