Comment by chongli
1 day ago
We don’t trillions of dollars in LLM investment to build things mathematicians don’t understand. We already have plenty of those, even from ancient times.
As to your second point, Terry Tao already has an answer [1]: the proof isn’t the contribution, shared understanding is. This issue was already raised back when the four-colour theorem was proved. Machine proving and machine proof checking are useful tools but they don’t mean anything without the interpretative work and the communication necessary to build shared understanding.
If humans have nothing to contribute then shared understanding is a pointless endeavor. It makes sense now in the "centaur" period where human + AI > AI alone, but when AI mathematicians are both more rigorous and more elegant, then taking the time dumbing down their proofs to a human level of understanding is like requiring that we ensure all our current proofs be understandable by a monkey.
> If humans have nothing to contribute then shared understanding is a pointless endeavor.
I agree, but as a software engineer this gives me pause because I keep trying to insist on coding standards but I’m unable to come up with a compelling reason why it matters. Ostensibly the reason we cared about things like DRY and code quality was so that it would be easy to understand and easy to maintain and easy to make changes to later. But it now seems like a shared understanding of the codebase is less important than ever, and it’s more about shoveling requirements in without breaking any existing functionality.
Is a well tested slopfest better? That seems to be the conclusion for mathematics, so why not software too?
> easy to make changes to later
IMO it's still a problem with LLMs; we still have to build in a way that makes it easier for an LLM to make changes later and arguably it's the same things that made software development easier for humans. IME LLMs tend to not know how to do that for themselves and instead just amplify/copy patterns that already exist.
If an LLM can't pave the way for itself then ultimately shared understanding is required to take advantage of LLMs in the first place.
It used to be the same with assembly. Programmers complained the one generated by compilers was not pretty, but now in 99.999% of the cases, it does not matter because nobody look at it.
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Where is the value in an unintelligible gibberish proof?
We already have countless examples of such filling up the arXiv, written by hacks long before LLMs started writing proofs. No one cares about them. You might as well build a box blasting radio static into the void. You could save a lot of electricity that way.
An unintelligible but correct proof is better than no proof. These first AI proofs may be overly complex and un-elegant, but they are the worst that frontier math proofs will ever be. AI math in 2030 will be leaps and bounds ahead of humans both in rigor and elegance.
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But but but... Does not AI exists (or has to exist) to only serve us.
I suppose it could compete with us.
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Why not? We build cranes to hoist weights construction workers can't lift. We build electron microscopes to measure things physicists can't see.
Why is it so hard to imagine we can build tools to think thoughts we can't comprehend?
If there's commercial value, I think it's inevitable. We don't fund mathematicians because it's cute when they understand a problem, but because their work tends to have applications with commercial value. The value can be captured without understanding the details.
Can you give an example of an incomprehensible piece of writing (any writing, never mind a proof) that has commercial value commensurate with the costs involved here?
> Can you give an example of an incomprehensible piece of writing (any writing, never mind a proof) that has commercial value commensurate with the costs involved here?
It depends on what exactly you mean by "commercial value commensurate with the costs involved" but I'd volunteer the 3G/4G/5G specifications and the other documentation required to implement the mobile network protocols. 5G is currently sitting at over 50,000 pages and it's one of the reasons Qualcomm/Broadcom/Apple are the only ones who can realistically make a mobile radio.
I don't think there is a single human to whom more than a few thousand pages would be comprehensible at a time except for the occasional genius.
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Yeah, how many people do you think understand the Linux kernel in full? What percentage of the people using it to great commercial effect can understand it?
How's your understanding of Schroedingers "An Undulatory Theory of the Mechanics of Atoms and Molecules"? You seem to be using the results of it as applied to semiconductor engineering just fine. And, I promise you, most semiconductor engineers haven't read it in full, they just accepted the results as passed on by several layers of teacher.
I have a paper on routing algorithms, which I have attempted to read to my cat. I don't think my cat retained much, but they seem to be enjoying the cat food that got delivered using the results.
I'd suggest that we're going to be a lot closer to the cat than the author of the paper when AI takes off.
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Can you give an example of a 4-wheeled vehicle that could move at tens of miles per hour before the automobile was invented?
I know right? And there’s only a market for maybe 5 computers in the whole world.
Paul Krugman (1998): predicted the internet’s economic impact would be no greater than the fax machine’s.
The 1876 Western Union memo dismissing the telephone as having too many shortcomings, and the banker telling Horace Rackham not to invest in Ford because the automobile was a novelty.
We are in good company!
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The idea about the goal of mathematics being shared understanding seems to come at a convenient time.
Mathematicians have never been known to communicate their ideas very clearly.
Regardless, even that target llms will likely win - an llm will likely be more efficient at teaching me string theory than a professor in a room with 463 other students.
The llm is the shared understanding.
I don't think "shared knowledge" means "shared knowledge between mathematicians and lay persons" (there is no much point in that, the same way a smartphone technician knows how a smartphone works deep down to the details but there is no big interest for society to have every lay persons being informed about it). I think it means "shared knowledge between mathematicians".
And at this level, while there are anecdotical exceptions, mathematicians have always been pretty decent (with their conferences, workshops, paper publications, international collaborations, ...).
So, it does not mean "teaching the subject", it means "creating a human network of people that share the understanding". LLM can be useful at telling a human, but you still need a human. The point of Tao is not that LLM is not good at providing explanations, it is that "providing explanations" is not the contribution to science, "the human network" is. It's like saying "LLM are great cook, they generate tons of food in space", but the point of having cooks is so that people can eat food and not die. Having LLM generating mathematical proofs is as useless as having LLM generating food that no one can access: the point was never to "generate proofs" or "generate food", the point was "creating a shared human understanding" or "eating the food so human can survive".
I don't get the argument.
I get that there is a cultural benefit to keeping it alive. Just like we ideally want the languages represented at the universities.
But keeping humans in the loop does not appear to be necessary in order to call it science, and certainly not in order to have progress or dessiminate that progress.
I don't have a problem with people doing math. As long that we don't idiomatically hold on to that way of doing things.
I do, however, find it hard to belive that individual humans will play a big role from here and forward, in any scientific desciplines.
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