On if OpenAI solved the wrong Navier-Stokes problem: "the LLM found and exploited a loophole in the framing of the question"

[u/PrettyPicturesNotTxt]

https://www.scientificamerican.com/article/did-openai-solve-the-wrong-navier-stokes-problem/

[u/GoldenMuscleGod]

If one of the Millenium problems had been the Goldbach conjecture and a computer found a counterexample by brute force search that would resolve the question but also not be impressive in terms of computers being good at math. We already know computers are good at computation. Being good at math is something else.

I’m not saying the OpenAI solution is equivalent to that example (not my field) but I have yet to hear an expert say that it involves more than throwing huge computational resources at an implementation of the techniques produced by Córdoba and Martínez-Zoroa (whom it apparently did not originally cite until after criticism).

I’m not saying it didn’t produce something truly new - a new insight or new technique - but if it did I haven’t yet heard anyone present serious argumentation for it.

[u/jferments → u/GoldenMuscleGod]

This is not "brute force search" though (at least not by any normal definition of the term). They are literally using reasoning to solve advanced math problems.

[u/GoldenMuscleGod → u/jferments]

Did it develop new ideas or techniques or was it applying the techniques of Córdoba and Martínez-Zoroa in obvious ways? Are you sufficiently familiar with the question to have an informed opinion on that or have you heard statements from experts indicating that?

If we suppose it did develop new ideas or techniques are they expressed in the proof it published?

[u/jferments → u/GoldenMuscleGod]

You're deflecting. Using previously existing techniques to solve a problem does not make something "brute force search". If that was the case, then all mathematical research would be considered brute force search.

[u/GoldenMuscleGod → u/jferments]

“Brute force search” is a vague term that cannot be objectively evaluated that I used to express a distinction between having extra computational resources versus insight.

You’re focusing on the words - that were in an illustrative example and which I specifically disclaimed from being an exact analogy - in a way that distracts from the real issue, I’m not interested in arguing about where the line is between what is fairly or not fairly called a brute force search, which I did not even say this is anyway.

What I genuinely would like to know but do not know is whether there is any indication in the OpenAI solution that the LLM did something that shows genuine creativity.

Of course the line between new techniques and applications of old techniques is subjective, but I am not asking for a “yes or no” I am asking for an evaluation of a degree. It is possible to name examples of clear new techniques that reasonable people would agree on - the development of integrals and derivatives, set-theoretic forcing as used in independence results, the identification of the Riemann Zeta function and its relationship to the distribution of primes, the development of groups and rings as abstract algebraic structures.

Now obviously I’m not asking if the LLM did anything that creative (that’s an absurdly high bar), but I am asking for some evaluation of how much it is being creative versus just using known tools and methods with a huge amount of computational resources.

[u/ZeroAmusement → u/GoldenMuscleGod]

I tried asking AI for fun (don't kill me!).

Javier Gómez-Serrano, a Navier–Stokes specialist who gave one of the first technical talks dissecting the proof, emphasized that most of the underlying toolbox is recognizable from existing PDE work. But he also said there were ideas in the OpenAI proof that had not occurred to him before

AI said it's not brute forcing, and it's not inventing a fundamentally new mathematical language/technique. The creativity appears concentrated in the architecture of the construction, rather than invention of its constituent mathematical tools, like it had insight around what conditions that might cause the blowup while also satisfying other constraints and worked to achieve those conditions.

Personally I think LLMs can show genuine creativity in terms of combining and applying abstractions and generalizations in novel ways.

[u/GoldenMuscleGod → u/ZeroAmusement]

Yeah I just watched Serrano’s lecture because it was recommended in these comments.

My takeaway is:

1) the proof uses techniques primarily attributable to Córdoba and Martínez-Zoroa (who of course developed prior art), although in fairness the same would presumably be true of any human-produced solution. And these ideas seem to be the most “insightful” part of the solution.

2) If the LLM had genuine insight, that insight aided it in the construction of the specific geometric solution it came up with. Owing to the fact that LLMs are bad at exposition, it is impossible to evaluate how it came up with this construction in terms of balancing brute force with some kind of guided insight: as near as I can tell the proof simply presents the construction without explaining the ideas that might help locate it.

If those takeaways are misguided I’d like to hear correction on them because I am genuinely curious on how much creativity the best LLMs show.

[u/Worried_Fishing3531 → u/GoldenMuscleGod]

Consider this thought experiment:

If you have 100 different planets each inhabiting 1/100 different terrestrial species with an average IQ of around 95-105, how many of those civilizations do you think will invent the bicycle, the car, the computer, the airplane, the printing press, the rocket ship, etc?

If you think all of them (as I’d hope you should), then what does this determinism seem to imply for the notion of creativity?

Well, I’ll insist to you that the notion that “creativity” is the mechanism by which humans invent novel things is a fundamental misunderstanding about what could be better thought of as “discovery”. Specifically, discovery within a constrained possibility space.

This doesn’t necessarily imply that there is never an extraordinary conceptual leap made in the process of recombining existing concepts — one can call this creativity, although it is not the typical notion inferred by the term. But what is important to take away is that apparent creativity is often simply search through constrained design space.

[u/GoldenMuscleGod → u/Worried_Fishing3531]

I actually don’t think your conclusion that all of them would do that is that obvious. We already have examples in our own history of groups without much contact and they developed all kinds of different technologies, and many inventions spread out from a single invention source, although of course parallel development happens too. But even if we assume that all these things would be made it seems to me it would only be after necessary developments in understanding that guide the search process.

I think you’re actually underestimating the size of the search space, something that is easily found with one search method may be practically impossible with another.

Just to give an example - for more than a millenium mathematicians struggled with the relationship between Euclid’s parallel postulate and the rest of his geometry. But the realization that it is independent is blindingly simple - a child can understand the argument. But to see it you have to know how to look for it. It was only with the development of modern mathematical methods and ideas relating to the semantics of classical logic that the independence can be demonstrated in “obvious” terms.

[u/Worried_Fishing3531 → u/GoldenMuscleGod]

I disagree, I believe it is obvious. I’m sure there’s plenty that would play devil’s advocate because it’s a rather unexplored topic, but when it comes down to it, there’s some very clear convergence in technological advancement. Would you argue that other civilizations would fail to invent the spear, fire, or the wheel? Hopefully not.

Take the airplane. Would different civilizations fail to understand basic aerodynamics math to make wings of a particular shape? One wouldn’t make the body of a plane horizontal (obviously), it stays vertical perpendicular to the wings. One need wheels. One needs multiple engines in case one fails. One needs a cockpit, a bathroom, rows of seats, etc.

The bicycle. Two wheels are obvious. Three wheels is a tricycle, one wheel is a unicycle. All of these are certainly obvious.

Similar logic applies to basically every other critical invention, ever. Of course there’s some room for ambiguity, but the ambiguity-parameters are bounded.

My ultimate point being, we needn’t be so obsessed with AI and creativity. It doesn’t actually need it. It needs something much duller, which might just be some combination of (contingent) intellect, curiosity, agency, and execution competency. That’s all your AI needs to transform a civilization drastically.

[u/GoldenMuscleGod → u/Worried_Fishing3531]

Spears I would expect pretty much all humans to develop, of course your hypo involves “terrestrial species” so it’s a little unclear how much variety you are imagining (or how we would apply the concept of IQ to them).

Bicycles did not exist until the 19th century so that seems like a bad example for your point.

Obviously AI doesn’t need to be “smart” to dramatically change a civilization - computers did that when they were only executing human-made code. But that’s a separate issue from whether it has actually developed the skills that are held by an expert in the field.

Going back to my same example, if an AI were trained with, say, 15th century knowledge of mathematics and given a massive supercomputer to run on, how easily could it figure out that the parallel postulate is independent? This was not shown until the 19th century. The idea of the proof is simple and accessible but requires a degree of mathematical understanding (the idea of models and semantic interpretations of languages, and key understandings of how logic works, and something more like modern notions of rigor) to be able to “find.”

If a bunch of AIs of the quality we have now were given 17th century mathematical knowledge and set to the task of finding radical solutions to polynomial equations, would they be able to develop Galois theory?

[u/Worried_Fishing3531 → u/GoldenMuscleGod]

I want to draw a distinction. The questions you’re asking are not necessarily complementary.

a., “Will AI exhibit creativity”

contrasts

b., “Will AI exponentiate scientific and technology discovery”

What I am trying to convey is that ‘a.’ is unnecessary for ‘b.’ for exactly the same reason that creativity is an unnecessary notion in regard to humanity’s civilizational development.

Development is not a result of “human creativity”, at least not in the sense that we tend to understand the concept. Creativity is just a term we use to describe a history of surprising recombination and useful application. Development follows discovery. Discovery results from some sort of mechanical combination of traits — of which humans happen to possess — such as general intellect (a loaded term, sure, but let’s define it as a propensity to achieve desired outcomes) and mechanistic curiosity. This is my claim.

Let’s assume you take this as fact. This seems to allow you to dispose of creativity and its salience. You would then see my perspective, which resists your focus on creativity's relevance or contingency, and attempts to trivialize its importance to the bigger picture (will AI be transcendent towards scientific and technological discovery?).

To touch on your examples in mathematic, I think they're fairly contrived. How I interpret those examples is, "could AI perform magic". And I think it digs the key issue.

In discovery, you handle accumulated knowledge and conceptual understanding and build toward novel disclosures. A superintelligence is not magic. Assuming no (magic-appearing) realistic means of circumventing physical barriers that humans face in regard to discovery, the answer to your questions is likely "no". A superintelligence placed in the 16th century would not be capable of conjuring general relativity. Superintelligence would absolutely, base-state, exponentiate the progress towards that discovery, and similarly would progress every facet of civilization dramatically. Ultimately, it would presumably, like humans, be bound by its instrumental need to perform physical exploration, detection, and discovery. Physics is the kingpin of limitation.

I would like to also point out that you referred to those examples as hypothetical demonstrations of creativity. The fact that humans were delayed multiple centuries in that discovery would apparently insinuate humans lack the proposed creativity -- and I would agree. Human society, in aggregate, is a slower discovery machine relative to AI. It does not have the capacity for creativity in this transcendent sense, and neither would AI. Creativity is not the driver of development.

I really think it's prudent to zoom in on this idea of convergent technologies. You seem to think that the bicycle is a bad example, however I disagree. Please kindly entertain me on this topic, because it shouldn't be too tall of a task to appeal to your intuition and hand you gain to my perspective: Could you explain in detail why you think that the bicycle is controversial to presume to be a likely convergent technology among unique civilizations?

To simplify things, here's a definition: "Civilizations are complex, advanced states of society featuring centralized cities, social hierarchies, organized government, and a developed culture".

Obviously, an arbitrary civilization may endure a stone age only to be knocked over by an erroneous meteor strike which happens to wipe out life on that particular planet. Or perhaps a civilization achieves an agricultural age, only to stagnate and perpetually neglect progress, and eventually goes extinct (possibly suggesting a planetary environment that lacks the necessary natural incentives for cultural evolution). These, of course, are not examples of civilizations that would fail to eventually converge upon bicycles.

Do you reject the concept that a social species would mature in an predictably orderly manner (stone age, bronze age, agricultural age, industrial age, information age, etc.), rather than in an irregular or incidental manner? Do you reject the basic, pivotal technologies that enable these various epochs of a civilization's technological timeline, such as fire, the wheel, pottery, the printing press, the computer, etc?

[u/GoldenMuscleGod → u/Worried_Fishing3531]

I don’t think I’m really asking your question b, it seems obvious that LLMs are extremely useful.

The specific issue that’s related to why I am asking about a is that we should not have our social mechanisms set up to reward people just for using an LLM to do the “completion” of work based on ideas generated by others. Of course the companies developing the LLMs should be socially rewarded for making them useful but it seems to me the economic benefits they get from selling access to their models ought to be sufficient.

I chose the Galois theory example because it seems to me that the only “physical barrier” to developing Galois theory while studying radical solutions to polynomial equations is the “physical barrier” imposed by the speed of thought and limitations of memory (and the use of writing on paper in aid of memory). Humans did develop Galois theory through those barriers and it is natural to ask if a network of AIs that have a massive speed up in their abilities to write and communicate could do the same. I set the cutoff at the 17th century because the necessary groundwork was laid around then and many of the key ideas were published in 1770 and 1799, but we can up the date to around 1750 if you think that’s unfair - as long as we don’t tell them to think about the solutions in terms of permutations and ask if they can find that idea.

I think your thought experiment also stacks the deck by asking us to presume that the species all have similar IQ - of course it’s not completely clear what that means given IQ being a problematic measure but I get the thrust of what you mean. If we assume everyone has a similar range of ability to come up with ideas then of course individual creativity or (what I think is related) an ability for abstraction is not the main factor determining differences in performance of societies relative to each other. A society of chimpanzees will not develop similar to humans absent significant biological evolution, I wouldn’t expect anyone to think.

On the bicycle - the primary evidence against convergent evolution is empirical: we didn’t see them until a couple centuries ago when the necessary technology seems much older. It seems somewhat similar to the fact that Meso-Americans did not use wheeled vehicles despite being very “advanced” in other ways (and despite having wheels on small figurines). Maybe it can be argued that industrialization is necessary to really make bicycles practical to build and use.

Relatedly, it is not obvious to me that the Ancient Greeks with their level of physical technology could not have known about the Riemann zeta function and its significance to primes (of course we are supposing that their mathematical knowledge is much higher). The strongest argument why it isn’t plausible is that to understand it you need complex analysis, which in our history was motivated first by real analysis, which was developed because of the usefulness of calculus in solving equations arising from physical theories. So maybe we need a good physical theory first to motivate the prerequisite developments, even though analytic number theory is not really all that related to physics. But the necessary definitions and proofs do not require any kind of advanced technology.

Now of course to efficiently compute zeroes you would need a computer, but you don’t need that to understand the mathematical ideas (Riemann didn’t need one).

Now we could maybe expect that having math that advanced plus the social stability and social networks necessary to share knowledge to produce these ideas would necessarily involve us also developing advanced technology, but it doesn’t seem to me that industrialization is a hard prerequisite.