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The counterpoint to this comes from chess. High level engines "prove" certain lines correct (not in the mathematical sense) but those "engine lines" are really hard to explain to humans, even by GMs. They can sort of explain that something is a good line but not why. Engines crush GMs and are considered ground truth even if noone really understands what is happening. Would it be a nightmare if math was the same, not sure. Especially for counterexamples LLM solutions seem fine. They stop humans from wasting time on pointless things. For proofs it gets more hairy but I think if it is formally verified a proof is a proof. Attribution is a problem (should the person who wrangled the answer out of an LLM get the credit, I guess so).

I think these are non-trivial epistemology and science theory problems.

I believe this rule of thumb will come to fail. The combination of superhuman mathematical reasoning and synthesis in upcoming AI models plus the rapid build-out of scalable formal verification infrastructure means this exponential in math is going to take off quite explosively, and we've barely seen anything yet. Mathematics is going to decisively move beyond human ability fairly soon (within our lifetimes, if not much more abruptly). It seems abundantly clear to me that much of the work will only be immediately accessible to AI, and rather than trying to explain all of it back to humans we will rather focus on explaining the portions that humans would benefit disproportionately from understanding.

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.

I don't know, it sounds analogous to how the early Amish would have started their doctrine: "if the craftsman cannot do the task by hand, then they shall not use a machine ..."

I was thinking about that too.

I am probably being too optimistic, but wouldn't it solve the problem if peer-review had a pre-screening phase where you give a presentation about your work? Similarly to how a PhD presentation is given. It could give back the publishing power to the expert, rather than the journals.

Once you have validated that the knowledge you want to publish is yours and that you actually understand and own the work, then it doesn't matter if the paper is written by a LLM or if the LLM assisted you in doing the work.

This is a statement about what Tao values in the proofs that he consumes, as a world-class, human mathematician.

For many of the rest of us, mere consumers of mathematical results, it’s sufficient to know that a^2 + b^2 = c^2 was proven by somebody or some machine at some point.

The problem is that there will be far more formally verified proofs than that human mathematicians around the world can read, much less explain. What then? Would the role of mathematicians just become explainers of AI generated proofs?

I don't think the mathematicians are going to be able to make that work, because journals are already struggling to keep up with their review load, and AI seems like it will make that harder. So a solution that involves "journals will do a lot more effort to review each paper" doesn't seem practical.

It would work better as a bar for hiring, rather than as a bar for publishing.

What if the result is a counter-example? A fact that disproves the conjecture? Is that not publication-worthy?

The problem with that rule of thumb is that unless there's some status/reward for completing the result, it won't happen. People will just put up the formally verified result and call it a day, and there's no incentive for them or anyone else to clean things up.

We'll end up with incomprehensible math because comprehensibility isn't rewarded. No one is going to get a Fields Medal, or tenure, for digesting someone else's results.

Tao has yet to produce work that outshines those whose work he studied and memorized. Not worth the reverence merely being a VHS copy of history.

He's a typical person otherwise, politically aware of how he barters for food; until proven otherwise this can be seen as little more than social moat defense.

To paraphrase a quote attributed to Upton Sinclair; hard to get a worker to understand something when their paycheck relies on them not understanding it.

The only interesting thing here is the frogs high up admitting they feel the heat.

I just don't see that to be true. If tommorow someone pulls a proof that n = np out of their ass but is not able to explain it, it will still have immense value.

This artificially limits mathematics to the limit of human ability.

It should be ignored and refused.

I think any idea that is contingent on a human being in the loop, solely to the property of being a human is most practically doomed to fail, but is inherently anti scientific.

Science,at its core, does not care about the credentials or institutions. It cares about the results and to what extend they can be falsified.

This feel a bit like "we know all about physics, we can only get more precise" - moment

I saw an analogous argument posted on LinkedIn the other day from one of the opencode guys: the job of a programmer is still to be able to answer questions - from memory - about how the system works and why.

Ive wondered whether a possible outcome of LLM slop is a retvrn to oral wisdom traditions. Ironically that's the most anthropological form of understanding and pedagogy.

Someone just brought up this point to me a few days ago on here, I'm definitely increasingly convinced that it's one of the main reasons (maybe even the main reason?) AI prose is so annoying to read through, and so rarely seems able to convey true understanding. It assigns the same narrative importance and dramatic tone to everything (the load bearing whatever, the crucial insight, the smoking gun) even when it's trivial.

That's also true for regular math proofs.

No one talks about why the proof works, but they will happily spend thousands of pages explaining how it works.

>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."

I noticed a long time ago, that the more people focus on trivialities like typos when arguing against someone online, the more compelling the original argument is. Basically, bikeshedding.

The most compelling evidence of the compelling nature of the original argument is when the most-upvoted reply is a joke or a meme. That's when you really know that those responding have nothing else to say. It's a white flag being run up, or the dog turning over and exposing its belly.

Similar to Ai writing. Lots of bloat.

The essay On Proof and Progress in Mathematics by a Field's medalist is worth reading:

https://arxiv.org/abs/math/9404236

He wrote it in 1994.

He writes about how he almost "destroyed" a subdiscipline in mathematics by becoming so good at it that he outclassed everyone. PhD students were advised to stay away from the whole field.

When he discovered this, he realized his error was that he was focusing on producing results, and not focusing on explaining his thought process. It's that thought process that is valuable in advancing the frontier - results alone won't do it. It didn't matter how many theorems he proved, if he was the only one who had the mental framework in mind on how to think about the whole field.

I'm sure we've come across abstruse books where every theorem has a rabbit being pulled out of a hat, whereas other readers find it intuitive. It's because the latter has developed a mental model for the discipline, and you haven't.

So he set about slowing down, and focusing on holding lots of seminars where he worked with other mathematicians to explain the thought process. Eventually others started publishing proofs of key theorems.

When people publish in a journal, they are not merely doing it to show the result. They are having a conversation with other mathematicians. If they cannot explain their own proof, they're not having a conversation.

This is why even decades after the Four Color Theorem was proved, plenty of mathematicians don't consider it "mathematics".

If you are free from physical and mental labor, you are in fact, not supplying labor, and are therefore surplus to requirements.

> I don't know why anyone should care about understanding the results if the AI is better at math than us

This is a big if, right? AI can still generate subtle or even silly mistakes that any normal human, let alone a mathematician, wouldn't make. Besides, math is more than just getting a conclusion but to understand and to generalize new ways of solving problems. After all, mathematicians are a curious bunch. To quote Hilbert's epitaph: We must know. We shall know.

Your analogy is great.

What happens when it's the cats who get to decide what's published?

Not an ideal scenario, but that's exactly the situation here. Mathematicians decide what gets reviewed and published in a a top journal.

In the long run, this can and should make journals obsolete.

If a result has a real-world application, then it can easily be published in an engineering or applied scientific journal in which it is already the norm to present methods that work empirically with little to no understanding of how.

I’m not anti AI but thinking the human brain is obsolete and using it will become a hobby is a dystopian view of the future where no one has any agency anymore. By your logic since our brains provide no value why not just shoot ourselves in the head while we’re at?

> If Amazon uses AI math to come up with better routin

Most research mathematics is pure mathematics which is completely useless. No routing algorithms. It's only relevant because we (or at least mathematicians) are interested in it. So an AI producing incomprehensible proofs would be completely pointless. That's why Tao insists on the importance of human understanding.

What if the better routing leads to an outage that the AI can't explain or fix and all the humans who might have understood it were laid off or otherwise unavailable?

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

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

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

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

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.

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.

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...

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.

But how would „what we value“ still be relevant?

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

Out-of-hand dismissal of Terence Tao is certainly a take.

And the term "artificial intelligence (AI)" has been the name of the field for 70 years and counting. If anything, "LLM" is a misnomer that's been lingering around since 2018-19. When the term was coined, these systems were relatively small, experimental, and could only produce impractical facsimiles of the English language. This is obviously no longer the case today.