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Comment by TheMayorOfDunce | Hacker News Reader

this is immediately disprovable and embarrassingly naive in the year of AI generating cancer vaccines and solving Navier-Stokes. You can argue "the vaccine is just interpolating chemicals together" and "the solution uncovered is just interpolating mathematical operations together", but by that standard there is literally nothing new under the sun.

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The Navier-Stokes solution was an interpolation of existing data.

admittedly I am not a mathematician, so the Navier-Stokes solution is just an example I am using. But how is it not something "new" if it did not exist before? At what point could something ever possibly be new, if "new" means "this uses absolutely zero existing elements"? Nothing in math would ever be "new". Nothing in physics or chemistry would ever be "new", by this standard.

It seems to me that the only reason to declare this solution "not new" is specifically to dismiss AI. If a human had deduced the Navier-Stokes solution, who would bother to scoff "that's not new! the numbers already existed!"?

The claim is OpenAI stole the work of mathematicians who had the same proof that they had developed using chatgpt conversations. Since by default, 'sharing' is turned on, and it often 'turns itself on' -- it is plausible gpt6 had been trained on the work of mathematicians who had effectively solved this problem in private.

If inrember correctly, this mathematicians worked on a simpler version of the problem and called transforming this in the solution to the wider problem a "remarkable" thing to do. Based in this, solving this problem is very impressive

Maybe. Or maybe its evidence that the frontier of mathematics is knowledge-bound rather than understanding-bound or even just manpower-bound. In many cases where proofs have emerged, that I have read, the LLM has retrieved some antique lemma unknown to the mathematician.

OpenAI spent 10-20m USD in energy costs to produce that proof with likely substantially similar prior work in the training data. What does this say? Who knows.

It continues in the tradition of using measurements of intelligence in humans, applied to LLMs with the hopes the "stolen valour" transfers. Here, the NS problem was a useful framing problem for mathematics to progress because of how it interacted with the development of mathematics broadly -- ie., how it progressed techniques, ideas, understandings, etc.

When we apply these issues to LLMs (whether IQ tests or mathematical proofs) we always discover something substantial lacking beneath the interesting facade of useful answers. The process isnt useful. And it is precesiely the process which these tests, in humans, are supposed to help with. The tests themselves (IQ or otherwise) arent the point. No one cares about their answers.

LLMs represent an alternative understanding-free approach to solving problems, with variable success rates depending on how similar the problem is to the training data and its rewarded reasoning traces.

That mathematics is making substantial progress, "10 million USD / problem" at a time, in using understanding-free methods -- says something sociologically interesting about the state of the field. Something which was already know: mathematics has long been full of a vast amount of papers, proofs, theorems and lemmas that few have ever read, or investigated. Mathematics has long been in a crisis of "overproduction of unvisited knowledge", LLMs are exploiting that otherwise unmined gold.

It’s all about what “new” means. It is possible to prove something in a very tedious way using preexisting techniques. There have many times in history been new ideas which are not just very impressive applications of old techniques. I’m still not aware of any famous problem in mathematics being solved by an unambiguous introduction of a genuinely new idea or semantic concept in this sense, as the candidates I previously had in mind have fallen into question by new findings of non-cited work; i.e. many if not all it seems have been impressive applications using ideas from known frameworks. I don’t know if this will continue into the future or not, but I think it’s important to try to make an honest assessment of reality at all times.

Of course in isolation it is a strict positive to have a verified truth value to any particular statement. Mathematicians currently are advocating for the idea that human understanding greater than this also be prioritized. There are in fact utilitarian arguments for this but I won’t go into everything here.

All mathematic breakthroughs could be characterized as an interpolation of existing data

Not really. There’s no honest sense in which quantum mechanics is interpolated from the text of Euclid’s elements, to demonstrate the point with a very extreme example. Many mathematical breakthroughs of the past seem to have involved the observation of semantically interesting concepts, beyond the syntax of known theories.

(To onlookers this particular post makes no claim about AI’s capacity to make the same observations.)

It depends on the subspace in which the interpolation is framed.

If they can "interpolate" existing data to that level then saying "just don't hack" doesn't make any sense, hacking is derived from knowledge of software systems. It's far harder to solve navier-stokes than creating a program that replicates and abuses computer resources.

You could as models improve continue to remove more and more training data, what happens when there is no more data left to remove but a running system still outperforms humans? I think you grossly overvalue data.

Ok, then let's go with that viewpoint and apply it to the original question: How much do you need to remove from the training data such that from what remains, nothing harmful can be "merely interpolated" (which you acknowledge the LLM is in principle able to do) and anything harmful would require something "new" that is not in the training data and that the LLM is incapable of coming up with. The argument is that there won't be much left in the training data if that's your approach.

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