It's a very long post with a mix of technical (math) and philosophical sections. Here are the most striking points to reflect upon IMHO.
> It seems to me that training beginning PhD students to do research [...] has just got harder, since one obvious way to help somebody get started is to give them a problem that looks as though it might be a relatively gentle one. If LLMs are at the point where they can solve “gentle problems”, then that is no longer an option. The lower bound for contributing to mathematics will now be to prove something that LLMs can’t prove, rather than simply to prove something that nobody has proved up to now and that at least somebody finds interesting.
Training must start from the basics though. Of course everybody's training in math starts with summing small integers, which calculators have been doing without any mistake since a long time.
The point is perhaps confirmed by another comment further down in the post
> by solving hard problems you get an insight into the problem-solving process itself, at least in your area of expertise, in a way that you simply don’t if all you do is read other people’s solutions. One consequence of this is that people who have themselves solved difficult problems are likely to be significantly better at using solving problems with the help of AI, just as very good coders are better at vibe coding than not such good coders
People pay coders to build stuff that they will use to make money and I can happily use an AI to deliver faster and keep being hired. I'm not sure if there is a similar point with math. Again from the post
> suppose that a mathematician solved a major problem by having a long exchange with an LLM in which the mathematician played a useful guiding role but the LLM did all the technical work and had the main ideas. Would we regard that as a major achievement of the mathematician? I don’t think we would.
I saw Tim Gowers give a talk at the AMS-MAA joint meeting in Seattle about ten years ago where he predicted that in 100 years humans would no longer be doing research mathematics. I wonder if he’s adjusted his timeline.
At the time I thought the key missing tool was a natural language search that acted like mathoverflow, where you could explain your problem or ideas as you understood them and get references to relevant literature (possibly outside your experience or vocabulary).
After reading this post, I have to admit that I could not understand the mathematical parts at all because they are beyond my current knowledge.
But one thing seems clear to me. If I try to describe the situation in mathematics presented here, it sounds like there were already precedents or existing pieces of knowledge, but humans had not thought to connect them. AI seems to have helped make that connection.
If AI can connect different fields in this way, then perhaps something even more significant could emerge from it.
That said, I could not understand most of the article. And if using LLMs properly requires this level of background knowledge, I honestly worry about whether I can really use them well.
>So if your aim in doing mathematics is to achieve some kind of immortality, so to speak, then you should understand that that won’t necessarily be possible for much longer — not just for you, but for anybody.
On complex problems with lengthy proofs, the first step that I would have done is to ask 5.5 pro in a new, unrelated, session, to be very critical, to try to find flaws in the arguments.
And certainly not to send it to a fellow colleague to ask its opinion first.
LLMs are certainly becoming capable to code, find vulnerabilities, solve mathematical problems, but we need to avoid putting their works in production, or in front of other humans, without assessing it by any possible mean.
Otherwise tech leads, maintainers, experts get overwhelmed and this is how the « AI slop » fatigue begins.
To be clear I’m talking about this step:
> That preprint would have been hard for me to read, as that would have meant carefully reading Rajagopal’s paper first, but I sent it to Nathanson, who forwarded it to Rajagopal, who said he thought it looked correct.
Undergraduate? No. We've had calculators able to solve undergraduate problems for decades. AI doesn't change the need to understand how calculus works any more than calculators did. The foundations remain valuable.
It's a very long post with a mix of technical (math) and philosophical sections. Here are the most striking points to reflect upon IMHO.
> It seems to me that training beginning PhD students to do research [...] has just got harder, since one obvious way to help somebody get started is to give them a problem that looks as though it might be a relatively gentle one. If LLMs are at the point where they can solve “gentle problems”, then that is no longer an option. The lower bound for contributing to mathematics will now be to prove something that LLMs can’t prove, rather than simply to prove something that nobody has proved up to now and that at least somebody finds interesting.
Training must start from the basics though. Of course everybody's training in math starts with summing small integers, which calculators have been doing without any mistake since a long time.
The point is perhaps confirmed by another comment further down in the post
> by solving hard problems you get an insight into the problem-solving process itself, at least in your area of expertise, in a way that you simply don’t if all you do is read other people’s solutions. One consequence of this is that people who have themselves solved difficult problems are likely to be significantly better at using solving problems with the help of AI, just as very good coders are better at vibe coding than not such good coders
People pay coders to build stuff that they will use to make money and I can happily use an AI to deliver faster and keep being hired. I'm not sure if there is a similar point with math. Again from the post
> suppose that a mathematician solved a major problem by having a long exchange with an LLM in which the mathematician played a useful guiding role but the LLM did all the technical work and had the main ideas. Would we regard that as a major achievement of the mathematician? I don’t think we would.
I saw Tim Gowers give a talk at the AMS-MAA joint meeting in Seattle about ten years ago where he predicted that in 100 years humans would no longer be doing research mathematics. I wonder if he’s adjusted his timeline.
At the time I thought the key missing tool was a natural language search that acted like mathoverflow, where you could explain your problem or ideas as you understood them and get references to relevant literature (possibly outside your experience or vocabulary).
After reading this post, I have to admit that I could not understand the mathematical parts at all because they are beyond my current knowledge.
But one thing seems clear to me. If I try to describe the situation in mathematics presented here, it sounds like there were already precedents or existing pieces of knowledge, but humans had not thought to connect them. AI seems to have helped make that connection.
If AI can connect different fields in this way, then perhaps something even more significant could emerge from it.
That said, I could not understand most of the article. And if using LLMs properly requires this level of background knowledge, I honestly worry about whether I can really use them well.
>So if your aim in doing mathematics is to achieve some kind of immortality, so to speak, then you should understand that that won’t necessarily be possible for much longer — not just for you, but for anybody.
This made me a little sad
Now repeat that for every sort of human achievement
On complex problems with lengthy proofs, the first step that I would have done is to ask 5.5 pro in a new, unrelated, session, to be very critical, to try to find flaws in the arguments.
And certainly not to send it to a fellow colleague to ask its opinion first.
LLMs are certainly becoming capable to code, find vulnerabilities, solve mathematical problems, but we need to avoid putting their works in production, or in front of other humans, without assessing it by any possible mean.
Otherwise tech leads, maintainers, experts get overwhelmed and this is how the « AI slop » fatigue begins.
To be clear I’m talking about this step:
> That preprint would have been hard for me to read, as that would have meant carefully reading Rajagopal’s paper first, but I sent it to Nathanson, who forwarded it to Rajagopal, who said he thought it looked correct.
Is the assessment system of undergraduate mathematics education no longer effective?
Undergraduate? No. We've had calculators able to solve undergraduate problems for decades. AI doesn't change the need to understand how calculus works any more than calculators did. The foundations remain valuable.
Graduate? Yes.
Today I learnt that there are mathematics papers titled: paper entitled Diversity, Equity and Inclusion for Problems in Additive Number Theory.