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Comment by ActorNightly | Hacker News Reader

The thing about these math proofs is that they really add nothing of value. Its a cool tech demonstration on how LLMS can search the trained space, but fundamentally, LLMS haven't "discovered" anything groundbreaking.

For NS equations, they are trying to model something that is discrete (i.e molecules colliding) in a continuous manner. You can easily think of a condition where they fail - imagine a vaccum where there is sufficient space between air molecules, so that collisions aren't always possible. NS won't be able to predict the state of the fluid in every single point in space.

In practice, when you do CFD, no package uses direct differential simulation of NS equations, you usually have simpler approximations that are good enough for the space you are working for. And if you want accuracy, you usually do something like LBM which simulates particle collisions using probability distributions.

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I can't wait to tell my pure maths professors that their most of their research adds nothing of value. I mean I am sure most of them would agree to some extent, but like, dude, have some more faith in the utility of pure maths, esp. centuries down the line

My read was that "these proofs" was referring to AI generated proofs specifically, not all mathematical proofs.

sure I get that, but like, my field has plenty of counterexample as proofs. we have had non-constructive proofs like probabilistic arguments. i don't think we can play the game of "oh this proof is useful that proof is not useful" well

Generally, the proofs that are in the form of "here is a single contradiction to an established statement that proves that its not always true", are generally useless. We can prove that newtons laws don't apply when you start considering relativity, but because they still apply for a large domain, they are still used.

Same with NS equations. Who cares if you can find a singularity.

And if you want an example of something novel that is worth pursuing - Its highly likely that the modern transformer architecture is sub optimal, you probably don't need to do full matrix multiplies in the transformers. There potentially could be a higher level mathematical formulation of minimal math operations that are needed without having to do trial and error - especially because all of the math involves linear combination passed through smooth activation functions.

But coincidentally, there hasn't been any research in terms of point LLMS to self optimize in this way, because there isn't enough human math literature on the LLMs to train on.

i really think we are opening a can of worms with these “who cares if you find a single counter example as disproof” arguments. i think the better version is “ok any lemmas or techniques we can generalize from this” or “what did we learn about maths through this” and use this as a basis to say LLM proofs are not useful

like say if god lets me find a single counter example to P=NP and thus disproving it — I think we can learn tons about complexity theory from this counter example by studying it. we should not have the hubris of assuming “oh a single counterexample is generally useless” — why, how. this is the same hubris imo that produced like “number theory is useless” until it is not

Again, context matters.

P vs NP is a more fundamental problem that if proven, will have insane consequences, perhaps more than anything else out there. For starters, you would be insantly able to design an an actuall all knowing AGI.

The NS equations are far,far,far less meaningful. Like I mentioned earlier, if you actually want accurate CFD, you dont even use them.

i am not a complexity theorist but I am a CS academic by training (I never was a good one, but welp), and during my PhD it is often said that maybe P vs. NP an initial proof/disproof to the statement is not that practically important, e.g., if P=NP, maybe the NP -> P reduced algorithm is very very cosmic. P=NP by itself hardly proves that one would instantly design an AGI whatsoever. Often the downstream potential theoretical/practical insights/results seems more exciting;

> The NS equations are far,far,far less meaningful. Like I mentioned earlier, if you actually want accurate CFD, you dont even use them.

Sure. Consider this: in algorithm research often the most optimal algorithm in big-O is not the one used IRL; examples are numerous: matrix multiplication, LCA data structures, many variants of shortest paths.

An academic can work two years on faster-in-theory matrix multiplication that no one expects to be used in practice (in our currently imaginable univese). Do you consider that less meaningful than working on faster matmul kernels?

Essentially, every task that AGI can do is the same as the traveling salesman problem - you have some system with dynamics where you can make decisions and to get to the right outcome, you are trying to find an optimal path. The best it can do at this moment is to create parallel simulations (i.e AlphaZero/MuZero MCTS - the tree represents the dynamics and evolution of the game board and it effectively does a simulation of the game prior to making a move). To do this with something as complex as the world, including all the humans within it, most likely means that you have to have more compute than reality allows - otherwise you are taking a hit in accuracy, which given chaotic effects creates problems where you can be way off on the predictions.

On the flip side if P=NP, that means that instead of dedicating compute to running branching simulations, An AGI can dedicate compute to just solving directly the actions it needs to do for any given outcome. This is a shortcut to reality, which means that reality in itself is compressible.

There is plenty of work that goes on in the math field that is super important. But any field has research that is irrelevant. So far, all the proofs done by ai have been largely irrelevant.

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