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Mathematicians are more than just number crunchers. Perhaps the most important part of a mathematicians job is formally defining problems. Even if the next gene

by valine 4y ago

Mathematicians are more than just number crunchers. Perhaps the most important part of a mathematicians job is formally defining problems. Even if the next generation of ChatGPT can solve any math problem you give it satisfactory, you still need a mathematician to ask the question. Presumably this AI won’t be omnipotent either, and there will be some limit to its mathematical prowess. I suspect most human effort will be spent inventing new ways to push the AI further.

thomasahle 4y ago
Why does society have mathematicians right now? It's some mix of "the people find mathematics intellectually worthwhile" (public universities) and "we sometimes need to ask them to solve hard problems". If AI makes it less intellectually interesting, the first reason goes away. If engineers can just ask the AI to solve a hard problem instead of the mathematician, they don't need the mathematician as a middle man. So AI also breaks this reason. > I suspect most human effort will be spent inventing new ways to push the AI further. Assuming the AI can't just do that itself too.
- yohannesk 4y ago
  I am quite interested in this AI that can improve upon itself or come up with genuinely new and novel ideas. What are you referring to exactly? LLMs?
  - thomasahle 4y ago
    I'm replying to a comment about a hypothetical AI that can do math better than the mathematicians.
- wholinator2 4y ago
  You know, this conversation had given me a short story idea. I wonder if anyone's written anything like this. Humanity produces an intelligent mathematical AI. It starts by solving already solved problems, then leaps to relatively simple unsolved problems, then leaps directly to solving any problem that can be defined to it, including problems that humanity has battled for centuries. It even will give fully consistent proofs and the small group of researchers we're following are astounded. The resident proof experts say that the proofs it's giving are strange and sometimes inelegant but always logically sound. They go about proving many different problems until eventually someone disagrees with the request, saying obviously this one isn't true, it's logically impossible, it won't be able to prove it. But then the machine does, and gives a perfectly valid and consistent proof that to the highest of humanities knowledge, appears correct. Then the mathematician requests it to prove the inverse to explore his fault logic, and again the ai spits out a fully consistent and valid proof, for the opposite condition. No one can find any fault in either proof. They go through asking it to prove all the opposite conditions as they had requested earlier and it always outputs intelligent and fully self consistent proofs. No one can find a flaw in anything it outputs, even when it outputs conflicting information. It's as if it's playing on some higher plane, able to just toy with the absolute maximum of human knowledge to make the results look like anything it wants. Our once thought rigorous foundation is easily thrown away by the machine. Everyone claps and looks scared. The end
zarzavat 4y ago
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.

by valine 4y ago

thomasahle 4y ago

Why does society have mathematicians right now? It's some mix of "the people find mathematics intellectually worthwhile" (public universities) and "we sometimes need to ask them to solve hard problems". If AI makes it less intellectually interesting, the first reason goes away. If engineers can just ask the AI to solve a hard problem instead of the mathematician, they don't need the mathematician as a middle man. So AI also breaks this reason. > I suspect most human effort will be spent inventing new ways to push the AI further. Assuming the AI can't just do that itself too.

yohannesk 4y ago
I am quite interested in this AI that can improve upon itself or come up with genuinely new and novel ideas. What are you referring to exactly? LLMs?
- thomasahle 4y ago
  I'm replying to a comment about a hypothetical AI that can do math better than the mathematicians.
wholinator2 4y ago
You know, this conversation had given me a short story idea. I wonder if anyone's written anything like this. Humanity produces an intelligent mathematical AI. It starts by solving already solved problems, then leaps to relatively simple unsolved problems, then leaps directly to solving any problem that can be defined to it, including problems that humanity has battled for centuries. It even will give fully consistent proofs and the small group of researchers we're following are astounded. The resident proof experts say that the proofs it's giving are strange and sometimes inelegant but always logically sound. They go about proving many different problems until eventually someone disagrees with the request, saying obviously this one isn't true, it's logically impossible, it won't be able to prove it. But then the machine does, and gives a perfectly valid and consistent proof that to the highest of humanities knowledge, appears correct. Then the mathematician requests it to prove the inverse to explore his fault logic, and again the ai spits out a fully consistent and valid proof, for the opposite condition. No one can find any fault in either proof. They go through asking it to prove all the opposite conditions as they had requested earlier and it always outputs intelligent and fully self consistent proofs. No one can find a flaw in anything it outputs, even when it outputs conflicting information. It's as if it's playing on some higher plane, able to just toy with the absolute maximum of human knowledge to make the results look like anything it wants. Our once thought rigorous foundation is easily thrown away by the machine. Everyone claps and looks scared. The end

zarzavat 4y ago

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.