Comment by samuelknight
8 hours ago
Slide 12 and 13 sum up my conclusions from the Oct 8 100+ reactions to 100+ solutions. Several reported OpenAI drop included answers put a wrap on problems they had been working for years. Several said that they have to completely rewrite grant requests that they had just submitted. Others described learning of the solutions problems they had been working on like losing an old friend or a lover. The general sentiment was to bemoan the loss of a field, as if it would have been better be born in the early 20th century and conclude their career arc before reaching this point. My take away is that the field needs to get it through their heads that their old problems are no longer ambitious, and that their job in the short term is to find the new frontier.
One idea Terrance Tao conjectures, which is highly doubtful, is that spamming the AI button will solve open problems without producing insightful new methods. But the OpenAI drop would seem to disprove this. The sub O(nlogn) proof for DFT for example violated very old human assumptions. Decades of work in the field was incremental progress on sub optimal method that nobody questioned hard enough. More generally, we should always be able to go back to a super-human AI and say, "Attack this problem, but don't use a method tried before."
I don’t think your second paragraph is a fair representation of what Tao is saying or contradicts his argument. He is arguing that AI can be useful in the service of human understanding in math and the examples you gave are exactly that. Whilst OpenAI just spammed the AI button the mathematicians that engaged with the output were able to progress their understanding about the problems in some ways (some of which they don’t really like). You need both parts for this to be useful to the field. What’s not clear is whether the juice is worth the squeeze: will all the money spent on spamming the AI and human mathematician time spent studying the outputs progress the field “better” than without the AI?
> What’s not clear is whether the juice is worth the squeeze: will all the money spent on spamming the AI and human mathematician time spent studying the outputs progress the field “better” than without the AI?
The AI proofs are a side-product of benchmarking current and in-development models on especially hard problems. They're clearly cost effective for frontier AI firms, and free for the taking as far as human mathematicians are concerned. The real issue with them is that they look like bizarre nonsense as written, so they need mathematicians familiar with those specific areas of math to "decode" and digest them.
> They're clearly cost effective for frontier AI firms
It's not obvious that this is a given. "Cost-effective" implies a comparison between cost and output. OAI spent millions to race human researchers on Navier Stokes, and that doesn't even account for the training cost. And how does one value the output? OAI is for some reason still hiring armies of humans instead of automating roles like "AI support engineer" or "Product Designer" (https://openai.com/careers/search/).
Calling the proofs a "side-product" is also rather dubious when OAI employs a team of mathematicians specifically to train its theorem proving capabilities.
Yeah, right now the juice isn’t only about the progress of math - it’s also the advancement of the LLM tech and the marketing benefit to the AI companies which feeds back into developing the tech. Those each have a different juice to squeeze ROI.
Software engineers are primarily outcome-oriented.
Mathematicians are primarily understanding-oriented.
Leveraging AI to tackle new frontiers without true understanding converts mathematicians to engineers.
I usually hate analogies because they're imprecise.
But in this case, isn't use of llm / generative ai to tackle new frontiers just another instrument to be used to understand the universe?
An engineer picks up a telescope, and uses it to build new applied technologies, a scientist uses the data it produces to answer new mysteries.
No one's saying, "we have to be careful with the information the telescope produces".
Well at the end of the day, the point of trying to understand the universe is to achieve specific goals leveraging that understanding.
It seems to me that all of these hundreds of proofs we've seen recently are glorified academic exercises, whose purpose is curiosity for its own sake without any practical application, or we'd already hear about at least one of them being implemented to some gain somewhere. It's all woefully unimpressive. It's not like anything stops mathematicians from trying to find more elegant solutions to their machine solved pet problems, since that's what they were going to try and do anyway despite it being completely pointless in practice.
> Well at the end of the day, the point of trying to understand the universe is to achieve specific goals leveraging that understanding.
Nope.
1 reply →
Actually both are outcome oriented and both can use AI to compress decades of progress. One camp accepts this naturally. Other camp is making their profession to be mysterious and spiritual to run away from the implications of AI
Since when are code poets no longer a thing? /s
> The sub O(nlogn) proof for DFT for example violated very old human assumptions.
btw people has massively improved the lower bound (from 1-2^-182 to about 1-2^-10) in the past couple of days: https://github.com/CrocSwap/integer-mult-bounds
Looking at this I'm reminded of https://en.wikipedia.org/wiki/Polymath_Project, in particular "Yitang Zhang's 2013 breakthrough on bounded prime gaps, eventually lowering the upper bound on the gap between consecutive primes from 70,000,000 down to 246".
I don't think we should be using the term "spam the AI button".
We should name the explicit mechanism that was employed - telling the model to "believe in yourself".
There is something quite humorous but also poetic about how the manipulation of this term worked. Doubtless in the model's weights lies the echoes of generations upon generations of humans telling each other to believe in themselves.
In pursuing the "new frontier" as you rightly put it, mathematicians would do well to remember the same. It's ok, don't be afraid of the future. Believe in yourself.
Was that phrase specifically used in the prompt?
Yes
> solve open problems without producing insightful new methods
> sub-O(nlogn) proof disproves that
How? Re-iterating, creating and understanding new proof techniques is the point of most of modern mathematics. Your statement is that proving a particular result is evidence of AI creating and understanding new proof techniques. I don't see how that follows, and I'm inclined to believe Tao is right for now.
And yes, results matter too, but if we stop at our current body of techniques and strip-mine results then we'll kneecap our future selves.
The DFT paper is an example against AI 'strip-mining'. The paper introduces a new method, and researchers are already trying to improve on it. If anything, the OpenAI dump re-vitalized that branch of study.
Agreed. Basically, if you don't make any effort to understand the proofs and you just look at the final answer, it will look like this proof dump is "strip mining" entire branches of math. But this is a pretty short-sighted way of looking at the issue: there will be plenty of novel approaches to be uncovered here.
Should we feel sorry that a mathematician had to write a grant request they submitted because an automated tool did what they wanted to do?
I don't really know the answer to that. I am happy when my own work is replaced by automated tools ("script yourself out of a job every six months!").
I think it makes sense to feel sympthy towards people facing disruption in their plans, even if you believe society is better off overall.
It's like getting scooped. If you just founded a startup based on tech XYZ, should you be happy when someone releases an open source XYZ? Should a news reporter be happy when another network breaks the story they were working on? On the one hand, society got the value of the thing you wanted to do. On the other hand, now you need to find something else to do, which might be really annoying.
I am a bit disappointed in this Math 2.0 concept. It is poorly proposed and weakly argued. Although I agree with his main point that AI should focus on helping human understanding, that doesn't preclude AI from finding answers first then figuring out how to explain it afterwards.
This is exactly what happened with that counterproof chat he posted a month or so ago - AI gave us an answer, he used AI to back into insights about the answer.
You don't address seismic shifts with a sweeping new approach, they are too multifaceted and present complexities and conflicts. He can say AI should help human understanding, which is a good end goal, but that doesn't mean AI dumping solutions isn't progress. That doesn't mean if AI builds 5,000 proofs in Lean and no human ever looks at them that they aren't useful, especially if other LLMs can access and build on those results.
This is exactly, exactly the same as when computers took over. "Oh, we don't need accountants any more" - not true, we just need acountants to deal more with human concerns than adding columns of numbers. That is called human progress, not a threat to humanity.
The sub O(n log n) significance is not in its practical "optimality". But rather that the previous lower bound was assumed to be true "by symmetry" and that result challenges some of our strongest intuitions and expectations about mathematical results. I still find this result very unsettling and a part of me remotely expects/wishes that there is some mistake somewhere.