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LLMs stand to be much better at research mathematics than humans. Breakthroughs in mathematics often come from taking a concept from one area of mathematics and

LLMs stand to be much better at research mathematics than humans. Breakthroughs in mathematics often come from taking a concept from one area of mathematics and applying it to another area. Humans find this difficult because it's hard to be a research-level mathematician in more than a few areas, a human life isn't long enough to learn that much. Humans learn depth-first. LLMs learn breadth-first. You can go ask ChatGPT about literally any field of mathematics and it will give you an answer (probably a bad one). It's been trained on the entire Arxiv and (likely) library genesis. When they do become as good at humans at abstract reasoning, they will be as good at humans on all fields at once: an ultra-Tao.

omnicognate 4y ago
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.
saycleearly 4y ago
I hope that intelligent mathematicians don't contribute too much to feed the AI. First develop a method by which all of us get better financial outcome in the long run, and then teach the AI some basic thinks just to please investors.

omnicognate 4y ago

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.

saycleearly 4y ago

I hope that intelligent mathematicians don't contribute too much to feed the AI. First develop a method by which all of us get better financial outcome in the long run, and then teach the AI some basic thinks just to please investors.