I think we’re going to find out the hard way that the proofs left to solve are very much not elegant.
AI in mathematics is forcing big questions
41–50 of 193 posts
Re: AI in mathematics is forcing big questions
#42There's yet another major issue of the centralization of power and knowledge:
> Some worry about the accessibility of AI tools. Traditionally, mathematicians have required little more than intuition, training, and a pen and paper to advance their field. If this slow, deliberative process is no longer valued by society, and particularly by research funders, then mathematics could become an elitist activity, only practiced by select organizations that can afford to work with proprietary AI models.
This can be true of anything LLMs do better than existing options. Because the best LLMs require enormous resources to develop, access to them can be very limited. Right now they are priced for market share. What happens when your small law firm attorney, or self-representation, goes up against a large firm with LLM expertise? Can the kid from the poor high school compete with the kid from the rich school with premium LLM access, in mathematics for example?
Re: AI in mathematics is forcing big questions
#43>more recently, a new general-purpose AI system from OpenAI disproved an important conjecture in combinatorial geometry. This result would have been worthy of publication in a major mathematics journal if humans had been the authors The quality of the mathematics is a function of who has authored it?
I suspect it's more that LLMs are not allowed under current journal rules to be authors.
Re: AI in mathematics is forcing big questions
#44The use of computers in mathematics has been somewhat controversial from the very start. There are of course all the computer-assisted proofs (see 4 color theorem), as well as the partially-assisted ones (see Viazovska et al on packing problems in dimensions 8, 24). But even finding a solution numerically, then rigorously verifying its properties can leave a lingering sense of incompleteness, of a gap in understandin…
Reminds me of a quote from Tsoding
> “Programming is understanding. If you don't understand what you are doing, you are not programming. You are generating text.”
Perhaps a proof without understanding is just generating numbers.
Re: AI in mathematics is forcing big questions
#45>more recently, a new general-purpose AI system from OpenAI disproved an important conjecture in combinatorial geometry. This result would have been worthy of publication in a major mathematics journal if humans had been the authors The quality of the mathematics is a function of who has authored it?
Worthiness of publication in a journal.
Re: AI in mathematics is forcing big questions
#46The use of computers in mathematics has been somewhat controversial from the very start. There are of course all the computer-assisted proofs (see 4 color theorem), as well as the partially-assisted ones (see Viazovska et al on packing problems in dimensions 8, 24). But even finding a solution numerically, then rigorously verifying its properties can leave a lingering sense of incompleteness, of a gap in understandin…
Reminded me of this quote: the problem with machine learning is that it's the machine that does the learning
A montage is a fantastic device in a movie.
But a montage about weight lifting does not a body builder make.
Re: AI in mathematics is forcing big questions
#47Turns out you have to be Terence Tao to know when an LLM is right or wrong
You don't have to be as good as the model you are overseeing, but it sure helps, otherwise you will only be able to evaluate partial claims, missing mistakes, and potentially the big picture.
Re: AI in mathematics is forcing big questions
#48It’s a well known problem in higher mathematics that even if you’ve solved a problem, often the proofs are incredibly long and complex and require an extensive amount of time spent by peers to review it. It would be great if someone could explain to me how AI improves this situation. Even if AI thinks it’s solved a problem, unless the proof is incredibly efficient and well explained, it will be difficult to verify th…
> It would be great if someone could explain to me how AI improves this situation. It's main utility is in the search step, not the verification step. The search is the bulk of the work and creativity. Separately, as the sibling commenter pointed out, it will likely get better at the verification step as well, with integrations of tools like Lean. > One hallucination in 300 steps of logic is enough to destroy the ent…
>The situation with human mathematicians is not much different. Eg, Wiles original proof of Fermat's Last Theorem contained errors found by reviewers, which he later repaired.
In fact, it was Wiles himself who realized there was an error.
Re: AI in mathematics is forcing big questions
#49The use of computers in mathematics has been somewhat controversial from the very start. There are of course all the computer-assisted proofs (see 4 color theorem), as well as the partially-assisted ones (see Viazovska et al on packing problems in dimensions 8, 24). But even finding a solution numerically, then rigorously verifying its properties can leave a lingering sense of incompleteness, of a gap in understandin…
Reminds me of a quote from Tsoding > “Programming is understanding. If you don't understand what you are doing, you are not programming. You are generating text.” Perhaps a proof without understanding is just generating numbers.
programming is also solving problems
in medicine they use all kinds of drugs which they don't really understand how they work. anesthetics is a great example
Re: AI in mathematics is forcing big questions
#50Much can be resolved when it is understood math is discovered not created. AI is a tool. if it makes discovery or proof easier that is still mathematics. A proof stands on its own logic regardless how it is derived. The root concern is how ai may provide uplift for mathematical discovery outside of socially expected channels.
You're not concerned about mathematics disappearing as a profession?