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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 fundamen

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.

whoarewethen 27d ago
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- ActorNightly 27d ago
  There still has to be some logic behind what is the meaning of a proof. Generally, its pointless to explore random equations of numbers and try to prove that this equation holds for every value. That equation needs to have some use, whether its cryptography, or description of a physical process. For example, look at Poincare conjecture proof. As cool as it is, can you name one area where the derivation of that proof or the proof itself has been used (without asking an LLM)?. Note that the core concept, Ricci flow, is used in lots of places, but the application of the proof is largely irrelevant - the homeomorphism of any 3d shape (say like a surface in Blender) to a sphere can be determined in other ways, more efficiently than what Pointcare conjecture states (i.e that every loop can be tightened to a point).
runeblaze 27d ago
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
- publlus_enigma 27d ago
  My read was that "these proofs" was referring to AI generated proofs specifically, not all mathematical proofs.
  - runeblaze 27d ago
    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
    - ActorNightly 27d ago
      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.
      - runeblaze 26d ago
        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

whoarewethen 27d ago

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ActorNightly 27d ago
There still has to be some logic behind what is the meaning of a proof. Generally, its pointless to explore random equations of numbers and try to prove that this equation holds for every value. That equation needs to have some use, whether its cryptography, or description of a physical process. For example, look at Poincare conjecture proof. As cool as it is, can you name one area where the derivation of that proof or the proof itself has been used (without asking an LLM)?. Note that the core concept, Ricci flow, is used in lots of places, but the application of the proof is largely irrelevant - the homeomorphism of any 3d shape (say like a surface in Blender) to a sphere can be determined in other ways, more efficiently than what Pointcare conjecture states (i.e that every loop can be tightened to a point).

runeblaze 27d ago

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

publlus_enigma 27d ago
My read was that "these proofs" was referring to AI generated proofs specifically, not all mathematical proofs.
- runeblaze 27d ago
  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
  - ActorNightly 27d ago
    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.
    - runeblaze 26d ago
      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