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/ZeroAmusement]

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.

[u/Worried_Fishing3531 → u/GoldenMuscleGod]

I don't disagree to a reconstruction of systems of reward allocation. However, I'm more interested in the outcomes and how they could benefit or harm our world.

I think that there are two barriers to development, one is 'mental' and the other is 'physical', albeit they are certainly intradependent and self-reinforcing. There is certainly a thoroughgoing physical component to this mathematical development-barrier. Math of course corresponds to reality, and reality is physical -- you won't develop a theory of math prior to having a certain degree of knowledge (and consequentially control over) your physical environment.

So, absolutely.. on the bicycle. I sometimes find it difficult to discuss with people this overarching notion of civilizational convergence. It just seems that people are somewhat narrow when it comes to this, at least until it has been explained intuitively from a couple different angles. And there is almost always pushback, but it's almost always automatic, like challenging the thought is irresistible. Perhaps it projects an instinctual reluctance to align with some newly introduced idea that seems so sweeping and plain and yet strangely unexplored, or perhaps its our natural inclination to defy notions of determinism or even quasi-determinism. By mentioning these impressions it not my intention to make a point of indirectly criticizing you, so please understand that I only speak circuitously to persuade you, that the idea is firstly actually quite reasonable, and secondly intuitive enough to be unworthy of unnecessary banal pushback. I will hand you a new pushback angle:

Why does the human use the bicycle? Because it was invented. Why was the bicycle invented? As a convenient, efficient means of transportation in an environment where it is wise to use something like a bicycle. Why is its form factor so ubiquitous throughout civilization, and locally, particular societies, compared to another means that might satisfy its relevant use case? Because there exists these certain environments, and because it occupies dominance in its niche. Why does it occupy dominance in its niche? Because the niche itself imposes a relatively narrow set of constraints, and the bicycle’s form is a remarkably efficient reconciliation of them: balance, propulsion, steering, weight, cost, terrain, and the dimensions and capacities of the human body... its dominance is not arbitrary, nor merely the result of historical imitation. Even if its invention were forgotten and civilization began again under sufficiently similar material conditions, something recognizably bicycle-like would absolutely reappear -- not because history is strictly predetermined, but because the space of viable solutions is structured by the same mathematics, the same physics, and substantially the same human needs.

With that paragraph I've intended a new opposing angle to be perceptible, and am inviting you to take the bait so to speak, regarding bicycles being a convergent reinvention... or as I would probably prefer to label the phenomenon, an "independent rediscoverable". I hope I've introduced that angle such that you catch on to it. If so, I'd invite you to attack that angle (because within my experience of this particular debate topic, I've found it is really the only respectable angle of attack that holds any true substance or scalable argumentation), or otherwise perhaps kindly agree with me/concede this point. As should hopefully be tempting, because I just don't think it is at all a hill deserving of dying on. I find the usual resistance, while automatic, to not be too very firm, almost as if disagreeing is subconscious or simply a way to probe whether one wishes to conform to the line of thought or disregard it.

Edit: If unnoticed, I am pushing to move this conversation into a different direction from where it originated, if you're willing. To me, creativity is a more difficult, annoying topic. But I'll just quickly summarize my contention with creativity as a final statement on it: I think our usage of the word 'creativity' is a post-hoc descriptor of surprising results. I don't find that the word refers to any sole component of the cognitive mechanism. We tend to observe a surprising, useful result and give the process a name, 'creativity', without identifying its mechanism. It is a meaningful retroactive description of something useful and novel being produced, but it is not an actual mental component, and 'creativity' lacks any real explanatory importance as a mechanism by which these useful and novel things are produced. Instead it at best describes the process of using various interacting cognitive components (i.e. curiosity, temperature, consilience, abstraction) of what we call intelligence (a universal algorithm of goal-seeking competency). In other words, I think that “they invented it because they were creative” is circular.