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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

#231

Post by bawolff »

busyant wrote: post #138

As someone who has a 18 yo son who wants to study math, this has me (and him) ... worried ... about becoming obsolete? But I'm wondering what other people think of this analogy. I used to be a bench scientist (molecular genetics). There were world class researchers who were more creative than I was. I even had a Nobel Laureate once tell me that my research was simply "dotting 'i's and crossing 't's". Nevertheless, I…

I doubt it.

Most likely AI will be good at some things and not others, and mathematicians will just move to whatever AI isn't good at.

Alternatively, if AI is able to do all math at a level above PhDs, then its going to be a brave new world and basically the singularity. Everything will change so much that speculating about it will probably be useless.

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

#232

Post by kortilla »

CamperBob2 wrote: post #184
bongodongobob wrote: post #158

Earlier quoted context omitted.

You're using it wrong, you can use natural language in your equation, but afaik it's not supposed to be able to do what you're asking of it.

You know, "You're using it wrong" is usually meant to carry an ironic or sarcastic tone, right? It dates back to Steve Jobs blaming an iPhone 4 user for "holding it wrong" rather than acknowledging a flawed antenna design that was causing dropped calls. The closest Apple ever came to admitting that it was their problem was when they subsequently ran an employment ad to hire a new antenna engineering lead. Maybe it's…

No, “holding it wrong” is the sarcastic version. “You’re using it wrong” is a super common way to tell people they are literally using something wrong.

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

#233

Post by daxfohl »

vouaobrasil wrote: post #202
zmgsabst wrote: post #151

Earlier quoted context omitted.

Presumably people who get into math going forward will feel differently. For myself, chasing lemmas was always boring — and there’s little interest in doing the busywork of fleshing out a theory. For me, LLMs are a great way to do the fun parts (conceptual architecture) without the boring parts. And I expect we’ll such much the same change as with physics: computers increase the complexity of the objects we study, wh…

Just a counterpoint, but I wonder how much you'll really understand if you can't even prove the whole thing yourself. Personally, I learn by proving but I guess everyone is different.

My hunch is it won't be much different, even when we can simply ask a machine that doesn't have a cached proof, "prove riemann hypothesis" and it thinks for ten seconds and spits out a fully correct proof.

As Erdos(I think?) said, great math is not about the answers, it's about the questions. Or maybe it was someone else, and maybe "great mathematicians" rather than "great math". But, gist is the same.

"What happens when you invent a thing that makes a function continuous (aka limit point)"? "What happens when you split the area under a curve into infinitesimal pieces and sum them up"? "What happens when you take the middle third out of an interval recursively"? "Can we define a set of axioms that underlie all mathematics"? "Is the graph of how many repetitions it takes for a complex number to diverge interesting"? I have a hard time imagining computers would ever have a strong enough understanding of the human experience with mathematics to even begin pondering such questions unprompted, let alone answer them and grok the implications.

Ultimately the truths of mathematics, the answers, soon to be proved primarily by computers, already exist. Proving a truth does not create the truth; the truth exists independent of whether it has been proved or not. So fundamentally math is closer to archeology than it may appear. As such, AI is just a tool to help us dig with greater efficiency. But it should not be considered or feared as a replacement for mathematicians. AI can never take away the enlightenment of discovering something new, even if it does all the hard work itself.

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

#234

Post by lmm »

joshuaissac wrote: post #147
bondarchuk wrote: post #104

Earlier quoted context omitted.

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

If they can't, then is it really quantum supremacy? They claimed it last time in 2019 with Sycamore, which could perform in 200 seconds a calculation that Google claimed would take a classical supercomputer 10,000 years. That was debunked when a team of scientists replicated the same thing on an ordinary computer in 15 hours with a large number of GPUs. Scott Aaronson said that on a supercomputer, the same technique…

> If they can't, then is it really quantum supremacy?

Yes, quantum supremacy on an artificial problem is quantum supremacy (even if it's "this quantum computer can simulate itself faster than a classical computer"). Quantum supremacy on problems that are easy to verify would of course be nicer, but unfortunately not all problems happen to have an easy verification.

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

#235

Post by lmm »

ludwik wrote: post #81
bambax wrote: post #50

> As an academic mathematician who spent their entire life collaborating openly on research problems and sharing my ideas with other people, it frustrates me [that] I am not even to give you a coherent description of some basic facts about this dataset, for example, its size. However there is a good reason for the secrecy. Language models train on large databases of knowledge, so you moment you make a database of mat…

> But if all models were truly open, then we could simply verify what they had been trained on How do you verify what a particular open model was trained on if you haven’t trained it yourself? Typically, for open models, you only get the architecture and the trained weights. How can you reliably verify what the model was trained on from this? Even if they provide the training set (which is not typically the case), yo…

> Even if they provide the training set (which is not typically the case), you still have to take their word for it—that’s not really "verification."

If they've done it right, you can re-run the training and get the same weights. And maybe you could spot-check parts of it without running the full training (e.g. if there are glitch tokens in the weights, you'd look for where they came from in the training data, and if they weren't there at all that would be a red flag). Is it possible to release the wrong training set (or the wrong instructions) and hope you don't get caught? Sure, but demanding that it be published and available to check raises the bar and makes it much more risky to cheat.

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

#236

Post by lmm »

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…

> and for arithmetical statements, the system must necessarily be sound

Why do you say this? The AI doesn't know or care about soundness. Probably it has mathematical intuition that makes unsound assumptions, like human mathematicians do.

> How likely are each of these outcomes?

I think they'll all be true to a certain extent, just as they are for human mathematicians. There will probably be certain classes of extremely long proofs that the AI has no trouble discovering (because they have some kind of structure, just not structure that can be expressed in ZFC), certain truths that the AI makes an intuitive leap to despite not being able to prove them in ZFC (just as human mathematicians do), and certain short statements that the AI cannot prove one way or another (like Goldbach or twin primes or what have you, again, just as human mathematicians can't).

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

#237

Post by asddubs »

watt wrote: post #207
asddubs wrote: post #14

Earlier quoted context omitted.

it can take my math and point out a step I missed and then show me the correct procedure but still get the wrong result because it can't reliably multiply 2-digit numbers

it's a "language" model (LLM), not a "math" model. when it is generating your answer, predicting and outputing a word after word it is _not_ multiplying your numbers internally.

Yes, I know. It's just kind of interesting how it can make inferences about complicated things but not get multiplications correct that would almost definitely have been in its training set many times (two digit by two digit)

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

#238

Post by _jayhack_ »

vouaobrasil wrote: post #146
busyant wrote: post #138

As someone who has a 18 yo son who wants to study math, this has me (and him) ... worried ... about becoming obsolete? But I'm wondering what other people think of this analogy. I used to be a bench scientist (molecular genetics). There were world class researchers who were more creative than I was. I even had a Nobel Laureate once tell me that my research was simply "dotting 'i's and crossing 't's". Nevertheless, I…

I was just thinking about this. I already posted a comment here, but I will say that as a mathematician (PhD in number theory), that for me, AI signficantly takes away the beauty of doing mathematics within a realm in which AI is used. The best part of math (again, just for me) is that it was a journey that was done by hand with only the human intellect that computers didn't understand. The beauty of the subject was…

If you think the purpose of pure math is to provide employment and entertainment to mathematicians, this is a dark day.

If you believe the purpose of pure math is to shed light on patterns in nature, pave the way for the sciences, etc., this is fantastic news.

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

#239

Post by kens »

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…

What book was it that you found helpful?

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

#240

Post by threeseed »

dns_snek wrote: post #137
fn-mote wrote: post #72

Earlier quoted context omitted.

This level of conspiracy thinking requires evidence to be useful. Edit: I do see from your profile that you are a real person though, so I say this with more respect.

What evidence do we need that AI companies are exploiting every bit of information they can use to get ahead in the benchmarks to generate more hype? Ignoring terms/agreements, violating copyright, and otherwise exploiting information for personal gain is the foundation of that entire industry for crying out loud.

Some people are also forgetting who is the CEO of OpenAI.

Sam Altman has long talked about believing in the "move fast and break things" way of doing business. Which is just a nicer way of saying do whatever dodgy things you can get away with.

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