Mathematics in the Age of AI

jjonbaer about 4 hours ago 22 commentsRead Article on arxiv.org

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Discussion (22 Comments)Read Original on HackerNews

sonicrocketman•about 2 hours ago

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give a clear, expert-level talk on their results, one that is correct and properly attributed, then the result should not be published.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.

czgov•about 2 hours ago

I wonder what his views on the 4 color problem are. One can explain it as the computer checked a bunch of cases and all maps reduce to one of these cases. It doesn’t take an expert to state this.

Properly explain is an enormous grey area. Soon, I think, there will be proofs of results that are verified in Lean that are so long that no one will be able to “properly explain”. I don’t think they should be discarded.

Resolution of singularities is a famous theorem of Hironaka. Abhyankar claimed that no one truly understood the proof of the theorem. He said that he and Zariski couldn’t get through the paper with a full understanding. But everyone accepts this theorem as being correct.

ChadNauseam•about 1 hour ago

> One can explain it as the computer checked a bunch of cases and all maps reduce to one of these cases. It doesn’t take an expert to state this.

Hmm, doesn't it take an expert to explain why those cases are exhaustive, and why the code that checked them is correct?

Tangentially, I'm not a mathematician but I wonder if one "opaque" proof that is too complicated for anyone to understand, but that we know is correct via formal verification, might end up being built on with "transparent" human-understandable proofs. For example, it's my understanding that there are many conjectures that have been proven true conditional on the riemann hypothesis being true. In that case, an opaque proof of the riemann hypothesis would enable those conjectures to be known and built upon

czgov•40 minutes ago

That will certainly happen. Humans will extend AI generated results. But what will also happen is that AI can “think” much longer than a human can and can have a vastly greater base “knowledge” than humans can have and so there will be a bewildering amount of new results. Humans may not be able to keep up.

To your first point. There a large number of cases that maps can be reduced to. Very few people have checked these reductions themselves. In 50 years there will be no human alive that will have checked the reductions by hand. Do we then discard the theorem? More importantly, do we trust the people that claim to have checked all the reductions? There are hundreds of cases. I trust a computer verification much more than I’d trust human verification. Humans will likely make mistakes due to the tedium. And some will claim understanding of all cases but be wrong in their understanding in some of the cases.

a2ff6eeb0•14 minutes ago

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.

highfrequency•about 2 hours ago

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."

paulpauper•30 minutes ago

Similar to Ai writing. Lots of bloat.

rramach•about 2 hours ago

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).

tocs3•about 1 hour ago

In some sense "understanding" (understanding if it is true, if it is important, how to use it) is about the only bottleneck in math. Any theorem that you can write down or imagine is already true, false, not provable already. In some ways we can already start iterating through all the theorems. We will never get to the end (or really get very far down the line) and most all of them be trivial (I think the Busy Beaver[1] project is a fascinating example, ymmv).

I am wary of AI in all aspects I am seeing it in but in many ways in mathematics seems to me the least troubling. It will change things in and the field will not be the same. Blacksmithing has not really gone away. You can still work as a farrier, if you like that sort of things. The tools that replaced a man working over a forge with a big hammer are part of a giant industry that is still producing works for the modern world.

[1]: https://bbchallenge.org/8226493

GPerson•32 minutes ago

How is either of those situations more or less like a hobby than the other?

plastic-enjoyer•about 1 hour ago

Somehow, I feel that progress, in your understanding of what progress is, loses all meaning.

paulpauper•29 minutes ago

It will be the same as before: some effort will go into checking proofs and the other into creating them. AI speeds up both.

sonicrocketman•about 3 hours ago

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

ivansavz•about 1 hour ago

Absolutely. I feel I gain at least 10 IQ points when reading something on paper.

This is also the strategy I use for editing drafts of my books. I bring a printed draft to someplace nice (e.g. coffee shop or park) and read it all carefully, then I transfer the edits back to the .tex sources. I do several passes of this, until I feel the text + explanations are solid.

Reading on screen just isn't the same...

magneticnorth•about 2 hours ago

When I was in academia and had easy access to a good printer, I always did. I miss it now that it's easier to just read on my screen.

tocs3•about 3 hours ago

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).

glimshe•about 1 hour ago

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.

a2ff6eeb0•1 minute ago

I think it's impossible to be half in. AI will eventually be better at things than people, and people will simply be rocks in the gears of progress.

The only thing to do is to be all in, or get run over.

GPerson•28 minutes ago

I think it’s more that he seeks to preserve and promote human understanding of mathematics, and sees that grappling with this new technology is necessary. One reason is that for human mathematical practices and institutions to retain legitimacy, they need to justify their value. As Tao explains, one obvious answer to that is made less obvious now with AI.

qsera•about 1 hour ago

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

GPerson•27 minutes ago

You should give it a try. Tao is a wonderful person and is trying to help humanity.

paulpauper•about 2 hours ago

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.