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Can AI do maths yet? Thoughts from a mathematician

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Re: Can AI do maths yet? Thoughts from a mathematician

#101

Post by newpavlov »

bondarchuk wrote: post #84
newpavlov wrote: post #38

Earlier quoted context omitted.

>in the quantum research to demonstrate any valuable forward progress you must compute something that is impossible to do with a traditional computer This is factually wrong. The most interesting problems motivating the quantum computing research are hard to solve, but easy to verify on classical computers. The factorization problem is the most classical example. The problem is that existing quantum computers are not…

No, it is factually right, at least if Scott Aaronson is to be believed: > Having said that, the biggest caveat to the “10^25 years” result is one to which I fear Google drew insufficient attention. Namely, for the exact same reason why (as far as anyone knows) this quantum computation would take ~10^25 years for a classical computer to simulate, it would also take ~10^25 years for a classical computer to directly ve…

It's a property of the "semi-artificial" problem chosen by Google. If anything, it means that we should heavily discount this claim of "quantum advantage", especially in the light of inherent probabilistic nature of quantum computations.

Note that the OP wrote "you MUST compute something that is impossible to do with a traditional computer". I demonstrated a simple counter-example to this statement: you CAN demonstrate forward progress by factorizing big numbers, but the problem is that no one can do it despite billions of investments.

Re: Can AI do maths yet? Thoughts from a mathematician

#102

Post by wbl »

Xcelerate wrote: post #49

So here's what I'm perplexed about. There are statements in Presburger arithmetic that take time doubly exponential (or worse) in the size of the statement to reach via any path of the formal system whatsoever. These are arithmetic truths about the natural numbers. Can these statements be reached faster in ZFC? Possibly—it's well-known that there exist shorter proofs of true statements in more powerful consistent sys…

2 is definitely true. 3 is much more interesting and likely true but even saying it takes us into deep philosophical waters.

If every true theorem had a proof in a computationally bounded length the halting problem would be solvable. So the AI can't find some of those proofs.

The reason I say 3 is deep is that ultimately our foundational reasons to assume ZFC+the bits we need for logic come from philosohical groundings and not everyone accepts the same ones. Ultrafinitists and large cardinal theorists are both kinds of people I've met.

Re: Can AI do maths yet? Thoughts from a mathematician

#103

Post by mensetmanusman »

swalsh wrote: post #87

Every profession seems to have a pessimistic view of AI as soon as it starts to make progress in their domain. Denial, Anger, Bargaining, Depression, and Acceptance. Artists seem to be in the depression state, many programmers are still in the denial phase. Pretty solid denial here from a mathematician. o3 was a proof of concept, like every other domain AI enters, it's going to keep getting better. Society is CLEARLY…

The reason why this is so disruptive is because it will effect hundreds of fields simultaneously.

Previously workers in a field disrupted by automation would retrain to a different part of the economy.

If AI pans out to the point that there are mass layoffs in hundreds of sectors of the economy at once, then i’m not sure the process we have haphazardly set up now will work. People will have no idea where to go beyond manual labor. (But this will be difficult due to the obesity crisis - but maybe it will save lives in a weird way).

Re: Can AI do maths yet? Thoughts from a mathematician

#104

Post by bondarchuk »

newpavlov wrote: post #101
bondarchuk wrote: post #84

Earlier quoted context omitted.

No, it is factually right, at least if Scott Aaronson is to be believed: > Having said that, the biggest caveat to the “10^25 years” result is one to which I fear Google drew insufficient attention. Namely, for the exact same reason why (as far as anyone knows) this quantum computation would take ~10^25 years for a classical computer to simulate, it would also take ~10^25 years for a classical computer to directly ve…

It's a property of the "semi-artificial" problem chosen by Google. If anything, it means that we should heavily discount this claim of "quantum advantage", especially in the light of inherent probabilistic nature of quantum computations. Note that the OP wrote "you MUST compute something that is impossible to do with a traditional computer". I demonstrated a simple counter-example to this statement: you CAN demonstra…

Apparently they can't, right now, as you admit. Anyway this is turning into a stupid semantic argument, have a nice day.

Re: Can AI do maths yet? Thoughts from a mathematician

#105

Post by esafak »

casenmgreen wrote: post #4

I may be wrong, but I think it a silly question. AI is basically auto-complete. It can do math to the extent you can find a solution via auto-complete based on an existing corpus of text.

Humans can autocomplete sentences too because we understand what's going on. Prediction is a necessary criterion for intelligence, not an irrelevant one.

Re: Can AI do maths yet? Thoughts from a mathematician

#106

Post by derangedHorse »

newpavlov wrote: post #38
LittleTimothy wrote: post #21

It's fascinating that this has run into the exact same problem as the Quantum research. Ie, in the quantum research to demonstrate any valuable forward progress you must compute something that is impossible to do with a traditional computer. If you can't do it with a traditional computer, it suddenly becomes difficult to verify correctness (ie, you can't just check it was matching the traditional computer's answer. I…

>in the quantum research to demonstrate any valuable forward progress you must compute something that is impossible to do with a traditional computer This is factually wrong. The most interesting problems motivating the quantum computing research are hard to solve, but easy to verify on classical computers. The factorization problem is the most classical example. The problem is that existing quantum computers are not…

> This is factually wrong.

What's factually wrong about it? OP said "you must compute something that is impossible to do with a traditional computer" which is true, regardless of the output produced. Verifying an output is very different from verifying the proper execution of a program. The difference between testing a program and seeing its code.

What is being computed is fundamentally different from classical computers, therefore the verification methods of proper adherence to instructions becomes increasingly complex.

Re: Can AI do maths yet? Thoughts from a mathematician

#107

Post by asadotzler »

josh-sematic wrote: post #52
QuadmasterXLII wrote: post #45

Earlier quoted context omitted.

To give a sense if scale: It’s not that o3 failed to solve that red blue rectangle problem once: o3 spent thousands of gpu hours putting out text about that problem, creating by my math about a million pages of text, and did not find the answer anywhere in those pages. For other problems it did find the answer around the million page mark, as at the ~$3000 per problem spend setting the score was still slowly creeping…

If the trajectory of the past two years is any guide, things that can be done at great compute expense now will rapidly become possible for a fraction of the cost.

The trajectory is not a guide, unless you count the recent plateauing.

Re: Can AI do maths yet? Thoughts from a mathematician

#108

Post by hash872 »

mensetmanusman wrote: post #103
swalsh wrote: post #87

Every profession seems to have a pessimistic view of AI as soon as it starts to make progress in their domain. Denial, Anger, Bargaining, Depression, and Acceptance. Artists seem to be in the depression state, many programmers are still in the denial phase. Pretty solid denial here from a mathematician. o3 was a proof of concept, like every other domain AI enters, it's going to keep getting better. Society is CLEARLY…

The reason why this is so disruptive is because it will effect hundreds of fields simultaneously. Previously workers in a field disrupted by automation would retrain to a different part of the economy. If AI pans out to the point that there are mass layoffs in hundreds of sectors of the economy at once, then i’m not sure the process we have haphazardly set up now will work. People will have no idea where to go beyond…

If there are 'mass layoffs in hundreds of sectors of the economy at once', then the economy immediately goes into Great Depression 2.0 or worse. Consumer spending is two-thirds of the US economy, when everyone loses their jobs and stops having disposable income that's literally what a depression is

Re: Can AI do maths yet? Thoughts from a mathematician

#109

Post by LittleTimothy »

0xfffafaCrash wrote: post #41
LittleTimothy wrote: post #21

It's fascinating that this has run into the exact same problem as the Quantum research. Ie, in the quantum research to demonstrate any valuable forward progress you must compute something that is impossible to do with a traditional computer. If you can't do it with a traditional computer, it suddenly becomes difficult to verify correctness (ie, you can't just check it was matching the traditional computer's answer. I…

I agree with the issue of ”is the test dataset leaking into the training dataset” being an issue with interpreting LLM capabilities in novel contexts, but not sure I follow what you mean on the quantum computing front. My understanding is that many problems have solutions that are easier to verify than to solve using classical computing. e.g. prime factorization

Oh it's a totally different issue on the quantum side that leads to the same issue with difficulty verifying. There, the algorithms that Google for example is using today, aren't like prime factorization, they're not easy to directly verify with traditional computers, so as far as I'm aware they kind of check the result for a suitably small run, and then do the performance metrics on a large run that they hope gave a correct answer but aren't able to directly verify.

Re: Can AI do maths yet? Thoughts from a mathematician

#110

Post by thomashop »

lordnacho wrote: post #93
jampekka wrote: post #19

I just spent a few days trying to figure out some linear algebra with the help of ChatGPT. It's very useful for finding conceptual information from literature (which for a not-professional-mathematician at least can be really hard to find and decipher). But in the actual math it constantly makes very silly errors. E.g. indexing a vector beyond its dimension, trying to do matrix decomposition for scalars and insisting…

The only thing I've consistently had issues with while using AI is graphs. If I ask it to put some simple function, it produces a really weird image that has nothing to do with the graph I want. It will be a weird swirl of lines and words, and it never corrects itself no matter what I say to it. Has anyone had any luck with this? It seems like the only thing that it just can't do.

Ask it to plot the graph with python plotting utilities. Not using its image generator. I think you need a ChatGPT subscription though for it to be able to run python code.
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