Comment by jrflo
1 day ago
Why not? If mathematicians in the field cannot understand the papers, but tools at our disposal can, why not use them? What is "unhealthy" about that? Just seems like it's the reality of the world now.
1 day ago
Why not? If mathematicians in the field cannot understand the papers, but tools at our disposal can, why not use them? What is "unhealthy" about that? Just seems like it's the reality of the world now.
If you can't understand what the machine has done, math is not being done. Math is not pushing symbols around to find sequences that follow all the rules. Math is the process of understanding how those sequences of symbols map to meanings. If you can't follow the proof but have validated it's correct, all you know is that it's a proof of something. Is it a proof of what's claimed in the adjacent plaintext? Who knows!
Math is about clear communication between people first and foremost. That's why Shinichi Mochizuki's claimed proof of the ABC conjecture isn't accepted - he's unable to communicate it to enough mathematicians to get a consensus on its validity. LLMs generating Lean proofs are basically following that same schema. Sure, something has been proven. But what? Where's the clear communication explaining it to the satisfaction of other mathematicians?
> Math is about clear communication between people first and foremost.
I can't imagine I'm the first one to point out that this is perhaps the greatest unsolved problem in mathematics.
> Where's the clear communication explaining it to the satisfaction of other mathematicians?
Coming in about a year for $20/month, sure seems like.
I mean it's just patently absurd. It is trendy to use this hubristic, and frankly bureaucratic, definition or requirement on math.
People can do math alone. They don't have to share it with others. Mochizuki was doing math (with errors). The aliens from Arrival could do math and we wouldn't necessarily be entitled to that. It doesn't make these things not valid math.
The only benefit of defining math through this vulgar social reductivism is to benefit the status quo. So you see some experts and expert-adjacent commentators using these math slogans but never critically justifying those words. Case in point, the above commenter asserts "...math is foremost about communication.", but the very next sentence does not explain why it is foremost, it just uses the sentence as a premise for the following sentence. Guess what, that is exactly how ideologues argue, by question begging fallacy. Ironic that STEM people who ought know better fail to catch this simple rational mistake.
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Human understanding is important, but I don’t think it’s the end all be all. There’s no reason to expect humans to be the best mathematical reasoners in the universe. And if math is important, beyond being a human art form, I think we should explore that math beyond our understanding.
What part of "math is the process of humans understanding math" did you not understand or agree with?
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"Mathematics is commonly explained and recorded in symbolic and concrete forms that are easy to communicate, rather than in conceptual forms that are easy to understand once communicated. Translation in the direction conceptual -> concrete and symbolic is much easier than translation in the reverse direction, and symbolic forms often replaces the conceptual forms of understanding."
https://mathoverflow.net/questions/43690/whats-a-mathematici...
Does math exist without humans?
I mean, I could not understand 95% of all the math out there due to skill issue, and it’s still valid. If machines can understand the math and humans can’t, we can’t validate it, but that doesn’t make the math itself invalid.
It could be logically valid but irrelevant? Allowing for the atrophying of all our skills is dangerous? While simultaneously building ever more complicated edifices that control our lives? Not a doomer but you really just want to shrug and say it ain't no thing?
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> If machines can understand the math and humans can’t, we can’t validate it, but that doesn’t make the math itself invalid.
How can humans tell whether machines understand the math if they themselves don't? Math is a human language for quantitative reasoning.
If large sectors of the population decline in their ability to read and write <natural language>, how will they be able to evaluate machine-generated <natural language>?
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Does math already exist and is discovered? Or is it a thing invented by people?
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Why does this not feel like a healthy place to be?
The author of those sentences is also the author of the main result that this generated improvement builds on, and prior to this improvement his work was thought to be unimprovable. He is, in all of the important ways, literally the single most qualified human on the planet to asses this result. And his assessment is 1) this improvement is real, 2) this improvement is exponentially bigger than his work and the work he improved on, 3) it is effectively impossible for him (and by implication, humanity) to understand the result as presented, 4) it is seemingly trivial for “frontier circa 2026”-class models to understand the result as presented, and 5) to the extent that he is able to understand it, it is only with significant frontier model assistance.
So the place we are at, at least for this highly intellectual field, is “intelligence is the property of ~2 corporations, you must pay them a fee and agree to their terms and conditions to access it”.
At risk of stating the obvious, this is not a healthy place to be because intelligence is now structurally subordinate to the profit motives of particular corporations and at the mercy of all the attendant economic/sociopolitical incentives that apply to them. If the landscape is such that it is in these corporations interests to withhold this intelligence from some and make it available to others, that is simply what will happen.
One trivial example, of which there are an arbitrary number: An elderly leader is convinced to issue a decree that all number-math-computation intelligence is to be made available only to the intelligence agencies of his nation, and failure to comply means no IPO and no further investment. From this point on there would be a permanent nation-state-shaped shadow over practically all modern cryptography, with no recourse possible.
If the Church had a Large Astronomy Model in the time of Copernicus, the earth would have stood still - regardless of what it may privately prove about heliocentrism, it would never declare e pur si muove.
Leaving aside the (much more important) issue of monopolizing intelligence, and assuming the claimed result is indeed correct, the question is whether a) the correctness proof is indeed too complex to be understood by humans, or b) the creator (the LLM) is not able to write it down in a form digestible by humans.
Observing the LLM output on problems that have clear solutions invented by humans, I tend to believe that too many people are jumping to conclusions too soon and interpreting b) as a).
If the LLMs are truly so great, we should see (in a short time, i.e. in order of months) a transformation of the math proofs generated by LLMs from the current state of indigestible mess to something that indeed looks like written by human for humans.
Only then would one be able to claim that we got into situation a).
Why should we expect this? Does it help the corporation’s bottom line? How much? Is that worth the R&D spend and compute budget it would need? What advantage do they stand to gain by demystifying their product, that they don’t already get from e.g. Lean verification? Do they gain more if the same research and compute spend on their inscrutable intelligence is directed into making it more intelligent, instead of less inscrutable?
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A more practical approach:
If a paper/proof is impossible to understand by humans, it could be considered useless (irrelevant) for most intents & purposes.
Regardless how interesting or valid looking, might as well be gibberish pulled from /dev/random. Until:
a) Written down in a form that humans can understand. And/or
b) Turned into practical applications that are demonstrated to work.
In other words: safe to ignore, until a) and/or b) occurs.
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> An elderly leader is convinced to issue a decree that all number-math-computation intelligence is to be made available only to the intelligence agencies of his nation, and failure to comply means no IPO and no further investment.
Who would staff those intelligence agencies if humans are no longer able to understand the work product?
…the pre-existing staff, who by virtue of being staff of the intelligence agency, would be exactly the humans who are permitted to use these gated intelligences, and can therefore use them to explain the results, as described in point 5 of my previous comment? I’m not certain I understand your question, is that what you meant? Or did you mean something else?
In any case, it was just one example, I am sure we can think of plentiful others.
Yup, and it always has been, we don’t even understand how our bodies or minds work, yet we use them everyday
The amount of unimaginable complexity contained in a seemingly very tiny amount of matter to us, is incredible, and it’s all around us in nature
We’ve never had the capacity to fully comprehend neither ourselves nor the environment we live in. But we do our best, and we do survive and sometimes even thrive
well math has always been about "understanding things" -- hence proofs and etc is the main tool in the field
(and why math differentiates between conjectures (maybe true) and theorems)
so that "oh some 3rd party person/machine understood it for me" can feel... a bit icky
Even pre-AI, there have been proofs that took years to be verified, understood and some were disputed forever and still are (like the Mochizuki / IUT drama). So, even without AI, maths has reached a point of specialization, where only a few people in the world know the areas used in a proof well enough to check it. When „new math“ is invented for a proof, people may even be more disincentivied to put in the effort. So, I‘d argue, AI makes understanding new proofs easier, not more difficult.
And yet for someone like me who is not a math expert, it has (all) always felt unapproachably foreign.
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IDK why this is relevant to the topic at hand.
Everyone in advanced math research understands the topics they are reading about deeply.
This person is just literally being victimized by the AI incompetence apocalypse.
Information overload + trained laziness.
The mistakes that lurked in radium, in arsenic, in leaded gas, in fallout downwind of nuclear tests ... all had the best tools that the previous generation knew.
None of them were okay. Those were small small tools with few (sometimes only one) uses.
Generations earlier, famines, diseases and lead-leeching plumbing made rulers insane and entire cities of people die without any help from the "best tools".
Scifi is always written about the best tools. The issues arrive in the tools and in the hands that wield them both.
There's nothing that makes this paper the be and and end all opinion pieces of the paper. The community will review his review, and people who wish to understand more will read more and more of the paper. I see nothing wrong with this process of progressive understanding.
What a funny pov?? Mathematics and other pure sciences are about human understanding. You're saying if humans cannot understand the thing that was for human understanding, that's ok? That's like saying all the world's food is now inedible, but that's ok because a blender can consume it still.
One day AI might use all these complex math to build a ftl rocket and you will still be what's the point if we can't understand it?
And one day AI might use this complex math to eradicate humans and you will still be "it's all completely fine"?
The fundamental argument boils down to: "who is all this being made for"? And the answer should be "us, humans". A lot of people here seem to breathlessly fawn over AI as some sort of value in itself, but this is incorrect. There is no absolute value in the universe (as far as we know), the only value to us as humans is subjective.
If AI is not generating value for humans, it isn't generating value, full stop.
Hence the question: "what's the point if we can't understand it?" is perfectly valid and should be discussed. The broader question of "what's the point of AI if not for improving human lives" gets lost in breathless adoration of a technology where nobody knows the outcome.
I'm not saying we need to abandon the pursuit of AI, but we must ask difficult questions at every step because if the technology ends up being as ground-breaking as many here think, it could just as well precipitate a dystopia.
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The "top layer" of maths being this is one thing. Now imagine another layer of this. Then another. Then another. Then another.
At one point intuition helps to build out usage of the math itself. See stuff like "oh I can do rotations through matrix multiplication"-style ideas.
It is helpful for us to be able to reach for those tools quickly. The more tools that require advanced prep to use, the less we'll be able to use the results in easier contexts. Everything will be hard
Of course nothing really prevents people from looking for the simpler tools and proofs. Lots of math stuff starts off complicated and gets simpler.
Just important that this is still a goal in mind
There are a few increasingly powerful companies that own the frontier models that can produces these results. In the future, it may only be these models that understand the results. Should we be beholden to a few companies for basic understanding of foundational principals?
Because the tools don't understand the papers either. We know for a fact that:
1. These tools are wrong a significant % of the time
2. They have an upper complexity bound where they start to fall apart
Relying on AI to check AI papers is a terrible idea
So, just like humans then?
But where they do differ is they do have an infinite stamina, and can work on problems in loops, self-correcting their own mistakes if a formal way of checking the correctness of the solution exists. Which does happen to be the case with math.
Humans can be wrong, but luckily we have an entire process for dealing with this, which is broadly called this new fangled 'science' thing
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The trope of superficially likening every aspect of LLMs to humans is specious and lazy.
Yes you're replying to astroturfers or AI psychosis victims.
Tools don't "understand" anything. They put together a probable string of tokens.
As you did ^
We don’t actually understand what “understanding” means. We’re just uttering plausible strings of words around the topic.
That's empirically false. Breakthrough technologies are novel by definition, their discovery is very improbable conditional on prior thinking.
> Why not? If mathematicians in the field cannot understand the papers, but tools at our disposal can (...)
How can you tell?
Especially for tools built to tell us we’re right and happily pretend to solve problems they didn’t. I think the OpenAI “revert” of their math problems speaks volumes of how volatile this can be.
I'm not a mathematician but from what I've heard from them that's generally not how mathematics works. You can "use" whatever you want, of course, but a mathematician's goal and main activity is finding and understanding proofs. The ones I knew also liked compiling error lists for math textbooks and sending them to the authors. (It opened my eyes because I admit to have been one of those ignorants who implicitly assumed "it's formal stuff in a book, sure it's going to be right".)
The tool may malfunction. Indeed there are theorems that say the tool must malfunction at least sometimes.
If you don't understand the subject and relies on the understanding of the tool, how will you have a frigging chance of catching an error?
Anyway this reminds me that scene in Matrix 2, where the mayor (of sorts, don't remember his exact title) of Zion tells Neo about the machines that keeps everyone alive, but that there is nobody left that understands how they work or could repair them if they broke.
>Indeed there are theorems that say the tool must malfunction at least sometimes.
Gödel's incompleteness theorem only states that there are true statements that can never be proved to be true. That limits what tools can do, but it's not a "malfunction". The exact same limitation applies to humans. A genuine malfunction would be if the tool "proved" a false statement to be true. That actually has happened before because of bugs in proof checkers, but software bugs can be found and fixed, and proof checkers are simple enough that it's possible to write many different implementations. It's extraordinarily unlikely that the same malfunction occurs in all of them.
Sounds like a disassembler.
because the tools can't? Lol. That's not how it works. Am I in the twilight zone?
All this anthropomorphism of literal statistical text generators.
Yes they exhibit generated sentences of text that describe coherent responses about a topic when there's an inordinate amount of research material in the training data from which they are distilling sentences of sound and rational thought. That's the ONLY way it can possibly work. There are so many scenarios where the training data isn't setup like that and low and behold the generated responses are just literal garbage.
Honestly this is starting to feel like a "God in the gaps" argument.
how so?
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