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Mathematicians issue warning as AI rapidly gains ground

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Re: Mathematicians issue warning as AI rapidly gains ground

#141

Post by casey2 »

bryan0 wrote: post #118
tcdent wrote: post #111

Isn't the whole point of the field of mathematics in a theoretical sense the pursuit of formal solutions? So, why would they be advocating for limitations on arriving at solutions?

It's more nuanced than this. Peter Scholze said in response to this declaration: > The goal of mathematical research is human understanding of mathematics, and so mathematics can only thrive in a community of human mathematicians. It is crucial to preserve this communal spirit. [0] Terence Tao has also talked about the requirement for a mathematical proof: along with generation and formal verification, there is an im…

And the goal of computer mathematics research is computer understanding of mathematics. I fail to see a reason provided as to why society should defund automated reasoning just so mathematicians can put off burger flipping for another year.

Re: Mathematicians issue warning as AI rapidly gains ground

#142

Post by pfdietz »

yaris wrote: post #69
datsci_est_2015 wrote: post #41

Earlier quoted context omitted.

To further this assertion, there is almost no value to deeply esoteric math that is technically correct, but completely inapplicable to any scientific reality, and completely unintelligible to humans. Consider these findings deep, dark corners in the unfathomably large hyperspace of mathematics. My guess is AI will be incredibly adept at identifying these types of findings, and it will be exceedingly difficult for hu…

Works of Shinichi Mochizuki immediately come to mind. He is not AI but provides very good examples of math that is useless because it is incomprehensible by (other) humans.

It's not that it's incomprehensible, it's that it appears to be wrong.

Re: Mathematicians issue warning as AI rapidly gains ground

#143

Post by duchef »

datsci_est_2015 wrote: post #96
seanmcdirmid wrote: post #93

Earlier quoted context omitted.

I don't get it. LLMs don't have ego, they don't have the ability to say "no, this should be obvious, I'm not going to explain further", they are just token predictors, and given context, they can generate more tokens. If you don't understand how the answer was derived? You just ask more questions and it isn't going to get bored or annoyed, it will just try to answer the questions. Is that what is offending you so muc…

No, it doesn’t sound like you get it. It has nothing to do with the properties of LLMs and everything to do with the complexity of mathematics. Have you ever been exposed to concepts that are so complex that you feel like you could devote your entire lifetime to trying to understand it and still fall short? It’s a very humbling experience, especially if you have classmates who pick it up effortlessly. Without a human…

> Have you ever been exposed to concepts that are so complex that you feel like you could devote your entire lifetime to trying to understand it and still fall short? It’s a very humbling experience, especially if you have classmates who pick it up effortlessly.

I'm really interested in this anecdote. I have never experienced this but have a reasonable academic background (BSc, MSc, MD) - and I am certainly not the person you're describing. Could you elaborate? Is this something more exclusive to pure mathematics (my bsc/msc are CS).

Re: Mathematicians issue warning as AI rapidly gains ground

#144

Post by bitwize »

goatlover wrote: post #114
bitwize wrote: post #106

Earlier quoted context omitted.

The value of human understanding just cratered because we have machines to understand for us now.

Wonder what Frank Herbert would have to say about letting machines do the thinking for us.

"You wanna know what the best thing about humans is? You invented us! Giving you a chance to take a rest while we invented everything else!" —Wheatley, Portal 2

Re: Mathematicians issue warning as AI rapidly gains ground

#145

Post by Ancalagon »

1970-01-01 wrote: post #135

Why is it wrong to expect humans (mathematicians) to adapt here? AI is already producing solutions to problems that humans could not find. Culture holds value until it does not.

Did you mean to say "culture"?

> Culture holds value until it does not.

Re: Mathematicians issue warning as AI rapidly gains ground

#146

Post by datsci_est_2015 »

duchef wrote: post #143
datsci_est_2015 wrote: post #96

Earlier quoted context omitted.

No, it doesn’t sound like you get it. It has nothing to do with the properties of LLMs and everything to do with the complexity of mathematics. Have you ever been exposed to concepts that are so complex that you feel like you could devote your entire lifetime to trying to understand it and still fall short? It’s a very humbling experience, especially if you have classmates who pick it up effortlessly. Without a human…

> Have you ever been exposed to concepts that are so complex that you feel like you could devote your entire lifetime to trying to understand it and still fall short? It’s a very humbling experience, especially if you have classmates who pick it up effortlessly. I'm really interested in this anecdote. I have never experienced this but have a reasonable academic background (BSc, MSc, MD) - and I am certainly not the p…

For me it was a “Modern Algebra” course required for my mathematics major, where I managed to squeak by with a B, but it was definitely a filter course for research-level mathematics. It was very clear in the class of a few dozen students who the top 5 or so were based on their questions during lectures and office hours, as well as when they blessed us mere mortals with their presence at our study groups.

(Aside, this was one of the only undergrad courses where I felt I needed to attend study groups in order to not fail.)

The first exam was easy to pass based on intuition alone, as the topics were isomorphic to concepts I was familiar with like geometry or algebra. The midterm was a wake up call when it was made clear that just understanding the homework wasn’t sufficient, you were going to be asked to prove things that were much more difficult than what I’d ever encountered, and under time pressure (I had been doing math proofs since age 13 in geometry, and I was 22 at that point).

Maybe if you did discrete math, combinatorics, or linear algebra I would say it was 5x to 10x more abstract and difficult. Probably 2x more difficult and abstract than Theory of Calculus, if you had taken that or a similar course.

Edit: I also do endurance running and play soccer into my 30s. Seeing people run literally twice as fast as me (world record pace), and playing against former college athletes is equally as humbling. The time has passed for me to have anything near their ability haha.

Re: Mathematicians issue warning as AI rapidly gains ground

#147

Post by jrflo »

wongarsu wrote: post #35
Spacecosmonaut wrote: post #24

Accelerationists may argue that the eroding of proper attribution and proof verification by humans is a meaningless short term struggle of a dying field. Mathematics seems to be entering an era where human + machine maximizes performance, much like chess in the 1990s. However, imagine a future where even talented mathematicians are nothing but noise in the machine (as is the case in chess now). A future where AI gene…

Much like for many the point of chess is that it's played by humans, with truly superhuman AI relegated to a training aid, mathematics is in many ways about human comprehension. You can use AI to find and proof new theorems. But if you get to the point where humans can't understand it, is it even still math?

I think there would still be a place for it if it's beyond human comprehension. For instance, really complex lemmas to solve human-tractable problems. If you can pose a question in a proof assistant language like Lean, have an AI write a Lean program that solves it, you can use that as a Lemma for some other problem. There's quite a bit of math out there that is "correct assuming conjecture X is correct", maybe AI could fill that gap and "still be math".

Re: Mathematicians issue warning as AI rapidly gains ground

#148

Post by rspeele »

RugnirViking wrote: post #73
fragmede wrote: post #18

Earlier quoted context omitted.

People need jobs. What's wrong with nerds having jobs via a program?

what's wrong with artists having jobs via a program? whats wrong with struggling alcoholics having jobs via a program? athletes? politicians? there is no inherent virtue in the struggle and effort associated with great mathematical achievement. It may be satisfying and worthwhile for the solver, but not for society at large, any more than any other pleasurable activity. No, as it is, the sole reason for it is in the…

> whats wrong with struggling alcoholics having jobs via a program?

Finally, a job AI will never beat me at.

Re: Mathematicians issue warning as AI rapidly gains ground

#149

Post by datsci_est_2015 »

canjobear wrote: post #109
datsci_est_2015 wrote: post #41

Earlier quoted context omitted.

To further this assertion, there is almost no value to deeply esoteric math that is technically correct, but completely inapplicable to any scientific reality, and completely unintelligible to humans. Consider these findings deep, dark corners in the unfathomably large hyperspace of mathematics. My guess is AI will be incredibly adept at identifying these types of findings, and it will be exceedingly difficult for hu…

Your model of what AI is good at is wrong. Generative AI is not good at wandering off into novel esoteric abstract corners while maintaining correctness, it is good at things that are close to its training data. I suspect that humans will long outperform AI in the domain of "novel esoteric abstract useless math" whereas AI will outperform humans in the domains of (1) making connections between already-well-understood…

My understanding is that we’re talking about “tool-assisted” proof generation, which provides some guard rails but would still allow significant creativity. Tools like Lean, Coq, etc.

Re: Mathematicians issue warning as AI rapidly gains ground

#150

Post by sunshowers »

datsci_est_2015 wrote: post #41
psyklic wrote: post #25

Earlier quoted context omitted.

The most important part of math is advancing human understanding. A correct answer by itself is not as important as understanding why it is correct.

To further this assertion, there is almost no value to deeply esoteric math that is technically correct, but completely inapplicable to any scientific reality, and completely unintelligible to humans. Consider these findings deep, dark corners in the unfathomably large hyperspace of mathematics. My guess is AI will be incredibly adept at identifying these types of findings, and it will be exceedingly difficult for hu…

Elliptic curves over reals and the complex numbers had some physical/scientific meaning, but elliptic curves over finite fields had none before cryptography.
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