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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

#261

Post by fooker »

bufferoverflow wrote: post #259
bubble12345 wrote: post #254

Earlier quoted context omitted.

AI will not do math for us, but maybe eventually it will lead to another mainstream tool for mathematicians. Along with R, Matlab, Sage, GAP, Magma, ... It would be interesting if in the future mathematicians are just as fluent in some (possibly AI-powered) proof verifying tool, as they are with LaTeX today.

AI can already do a bunch of math. So "AI will not do math for us" is just factually wrong.

Your idea of ‘do math’ is a bit different from this context.

Here it means do math research or better, find new math.

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

#262

Post by elbear »

didibus wrote: post #251
Syzygies wrote: post #247

"Can AI do math for us" is the canonical wrong question. People want self-driving cars so they can drink and watch TV. We should crave tools that enhance our abilities, as tools have done since prehistoric times. I'm a research mathematician. In the 1980's I'd ask everyone I knew a question, and flip through the hard bound library volumes of Mathematical Reviews, hoping to recognize something. If I was lucky, I'd get…

I think I'm missing your point? You still want to enjoy doing math yourself? Is that what you are saying? So you equate "Can AI do math in my place?" with "Can AI drink and watch TV in my place?"

In a way, AI is part of the process, but it's a collaborative process. It doesn't do all the work.

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

#263

Post by red75prime »

voidhorse wrote: post #257
red75prime wrote: post #148

Earlier quoted context omitted.

Function is the most general way of describing relations. Non-deterministic processes can be represented as functions with a probability distribution codomain. Physics seems to require only continuous functions. Sorry, but there's not much evidence that can support human exceptionalism.

I don't understand your point here. A (logical) relation is, by definition, a more general way of describing relations than a function, and it is telling that we still suck at using and developing truly relational models that are not univalent (i.e. functions). Only a few old logicians really took the calculus of relations proper seriously (Pierce, for one). We use functions precisely because they are less general, t…

I'm not a big fan of philosophy. The epistemology you are talking about is another abstraction on top of the physical world. But the evolution of the physical world as far as we know can be described as a function of time (at least, in a weak gravitational field when energies involved are well below the grand unification energy level, that is for the objects like brains).

The brain is a physical system, so whatever it does (including philosophy) can be replicated by modelling (a (vastly) simplified version of) underlying physics.

Anyway, I am not especially interested in discussing possible impossibility of an LLM-based AGI. It might be resolved empirically soon enough.

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

#264

Post by vlovich123 »

bufferoverflow wrote: post #259
bubble12345 wrote: post #254

Earlier quoted context omitted.

AI will not do math for us, but maybe eventually it will lead to another mainstream tool for mathematicians. Along with R, Matlab, Sage, GAP, Magma, ... It would be interesting if in the future mathematicians are just as fluent in some (possibly AI-powered) proof verifying tool, as they are with LaTeX today.

AI can already do a bunch of math. So "AI will not do math for us" is just factually wrong.

Can AI solve “toy” math problems that computers have not been able to do? Yes. Can AI produce novel math research? No, it hasn’t yet. So “AI will not do math for us” is only factually wrong if you take the weaker definition of “doing math for us”. The stronger definition is not factually wrong yet.

More problematic with that statement is that a timeline isn’t specified. 1 year? Probably not. 10 years? Probably. 20 years? Very likely. 100 years? None of us here will be alive to be proven wrong but I’ll venture that that’s a certainty.

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

#265

Post by br0tl1 »

Someone wrote: post #28
est wrote: post #13

At this stage I assume everything having a sequencial pattern can and will be automated by LLM AIs.

I think that’s provably incorrect for the current approach to LLMs. They all have a horizon over which they correlate tokens in the input stream. So, for any LLM, if you intersperse more than that number of ‘X’ tokens between each useful token, they won’t be able to do anything resembling intelligence. The current LLMs are a bit like n-gram databases that do not use letters, but larger units.

I think that's an oversimplification. LLMs have a limited context window of tokens. But that isn't necessarily a limitation: it's been proven that a LLM can simulate any algorithm even with a limited context window (https://arxiv.org/abs/2410.03170), making it computationally universal.

Even if that weren't true, the context windows can be quite large and will get bigger as people figure out how to optimize LLMs. For example Gemini 1.5 has a context of 2 million tokens. A book is typically around 120,000 words, so that's almost 20 books. So one could argue with a context this big they could construct reasoning chains involving far more disparate pieces of information than humans typically work with simultaneously, and arguably that demonstrates intelligence as well.

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

#266

Post by amanda99 »

Syzygies wrote: post #247

"Can AI do math for us" is the canonical wrong question. People want self-driving cars so they can drink and watch TV. We should crave tools that enhance our abilities, as tools have done since prehistoric times. I'm a research mathematician. In the 1980's I'd ask everyone I knew a question, and flip through the hard bound library volumes of Mathematical Reviews, hoping to recognize something. If I was lucky, I'd get…

> AI is often wrong, never knows when it's wrong, but people are like this too.

When talking with various models of ChatGPT about research math, my biggest gripe is that it's either confidently right (10% of my work) or confidently wrong (90%). A human researcher would be right 15% of the time, unsure 50% of the time, and give helpful ideas that are right/helpful (25%) or wrong/a red herring (10%). And only 5% of the time would a good researcher be confidently wrong in a way that ChatGPT is often.

In other words, ChatGPT completely lacks the meta-layer of "having a feeling/knowing how confident it is", which is so useful in research.

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

#267

Post by br0tl1 »

int_19h wrote: post #210
Someone wrote: post #187

Earlier quoted context omitted.

Why is that unfair in reply to the claim “At this stage I assume everything having a sequencial pattern can and will be automated by LLM AIs.” ? I am not claiming LLMs aren’t or cannot be intelligent, not even that they cannot do magical things; I just rebuked a statement about the lack of limits of LLMs. > Naturally, humans couldn’t do it, even though they could edit the input to remove the X’s So, what are you clai…

If you have a million Xs on the end of each line, when a human is looking at that file, he's not looking at the entirety of it, but only at the part that is actually visible on-screen, so the equivalent task for an LLM would be to feed it the same subset as input. In which case they can all answer this question just fine.

I would also add that if I saw a file full of a million X's, I would not track each X as a distinct item in my mind, I would simplify the visual input down to "this is a file containing a lot of Xs" and work with that lightweight abstraction instead.

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

#270

Post by PeeMcGee »

jvanderbot wrote: post #67
dartos wrote: post #58

Earlier quoted context omitted.

What’s worse about it? It never tells you the wrong thing, at the very least.

When you give it a large math problem and the answer is "seven point one three five ... ", and it shows a plot of the result v some randomly selected domain, well there could be more I'd like to know. You can unlock a full derivation of the solution, for cases where you say "Solve" or "Simplify", but what I (and I suspect GP) might want, is to know why a few of the key steps might work. It's a fantastic tool that hel…

> [...], but what I (and I suspect GP) might want, is to know why a few of the key steps might work.

It's been some time since I've used the step-by-step explainer, and it was for calculus or intro physics problems at best, but IIRC the pro subscription will at least mention the method used to solve each step and link to reference materials (e.g., a clickable tag labeled "integration by parts"). Doesn't exactly explain why but does provide useful keywords in a sequence that can be used to derive the why.

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