Mathematics in the age of AI

Tao's Rule of Thumb (which applies very well to software):

> My own suggested rule of thumb: if the authors cannot convincingly demonstrate that they are able to give a clear, expert-level talk on their results, one that is correct and properly attributed, then the result should not be published. A proof that no human can properly explain should be viewed as incomplete, even if it has been formally verified.

Terence Tao's quote about AI's math proofs is relatable outside of pure math: "the writing very often dwells at length on trivialities while passing briefly through — or even actively obscuring — the most interesting and novel portions of the argument."

What is being made is "what are our core values?" argument. One does not need to be a mathematician to know how poorly this worked for large communities when incentives are misaligned...

If a subset of mathematicians, use AI to condense timelines focusing on goal 6.2 exclusively and make rapid progress and reach a proverbial inflection point — one where value proposition of the using this new normal is too enticing to give up — everyone will ask: "This thing is so awesome. Why should I care about your values?"

The chess analogy doesn't quite work for me. In chess, an engine's move is useful because it helps you win. In math, a proof is useful because it helps you understand something - and from that, you can build more. If a proof is incomprehensible, it's like a chess move that only works in that one specific position. Useless. The ABC conjecture is a perfect example - Mochizuki's proof might be correct, but no one can follow it, so it's basically dead. AI proofs are going to be like that, but way more of them. Tao's essay is a great starting point

If you're having dinner like me and prefer to watch: https://www.youtube.com/watch?v=M0--ZH1lOzg

I don't know why anyone should care about understanding the results if the AI is better at math than us. It'd be like demanding that human mathematicians are banned from publishing until their cats understand the theorems.

If Amazon uses AI math to come up with better routing, the cats can benefit from cheaper delivery fees just as much as humans can. No understanding needed.

The human brain is being obsoleted, soon thinking is going to be a recreational activity like weightlifting. If you want to think as a hobby, that's fine, but most people will be free of that toil of unwanted brain labor.

Terence argues that explanation of results ("understanding") will be the new bottleneck in math research but I am not sure this is the real bottleneck for progress.

Understanding was critical for the field to progress when only humans were involved but if humans are not needed to make progress, I wonder if we split into two worlds: an AI math-world where amazing new results continue at a rapid pace bottlenecked only by compute/cost and a human math-world where we understand a subset of the AI math-world as a hobby (similar to Stockfish vs human chess).

The talk contains references to what is clearly a subset of questions and hypotheses. Anyone got a link to somewhere with all of these?

Terence Tao sees a role for AI in science. I'm no genius but he basically described what I've thought all along... We don't need to be "all in" or "all out".

It's the old cliche of "if you only have a hammer every problem looks like a nail". Let's not fall into the trap of thinking that our life needs to be 100% about AI or completely devoid of AI. We can really use this thing to make our lives better.

Instead of wasting time on the question of whether we should use it, let's focus on HOW we'll use it.

And one thing about Tao: it's really refreshing to have an influential genius "around" who isn't a egomaniacal psychopath trying to rule the world through their XYZ corporation but, instead, being a reasonable and well-balanced person. Big fan.

one thought: AI can often help us solve a problem once posed. E.g., try to prove that X is True. But formulating good conjectures is something altogether different. right now we have a backlog of interesting / good conjectures that ai can grind on. but once those are done, will we still need people to sniff out interesting new ones?

Anyone else print their white papers before reading? (At least the short ones)

AI also can replace a lot of expert attention too. Why not? What is useful or what is not useful is based on the expert's narrow opinion. An AI system can do much more and deep value comparison. It looks like if our current technological advancement continues, in the space of what is possible (or even impossible), AI can find the optimal solutions better than any human or human organizations. But I think there is only one think will remain for humans to go for these solutions: what we value. that will be the last resort I believe and hopefully ai systems won't start manipulate us too as we are very fragile on manipulation.

Maybe the Hitchhikers Guide to the Galaxy series was predictive in pointing out the problems of ill defined questions (The Answer to the Ultimate Question of Life, the Universe, and Everything).

Goal 6.4 reminds me always of the numerous times AI generated n PR's for a feature and I revolted and threw my laptop because it was incomprehensible or unworkable when viewed as a process/workflow.

Not using AI puts one at a huge disadvantage in a career setting. Ai can find deep references better than humans now, let alone actually doing the math. The challenge is knowing which problems to tackle given the cost limitations. If you have $10k to spend on tokens, you have to choose problems that can conceivably be solved within this budget.

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If the title have said in the age of "LLMs", I might have given it a try.