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?

Software engineers are primarily outcome-oriented.

Mathematicians are primarily understanding-oriented.

Leveraging AI to tackle new frontiers without true understanding converts mathematicians to engineers.

> 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

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.

> 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.

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.

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!").

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.