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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 “o

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

ActorNightly 7d ago
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.
- runeblaze 7d ago
  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?
  - ActorNightly 7d ago
    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.

ActorNightly 7d ago

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.

runeblaze 7d ago
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?
- ActorNightly 7d ago
  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.