I have a related question that might be quite silly but let me ask anyway. Suppose that you have a decision problem in NP and you are guaranteed that the problem is in P/poly or specifically that it can be solved by a circuit of size when the input is of size n. The question is how hard it is to find a polynomial size circuit that solves the problem. (In its general case.) Is it, or can it be computationally intractable? Can it be NP-hard or something like that?

1) The general, problem of finding a proof to a mathematical statement can be undecidable, and therefore the proof of Poncare conjecture reflects creativity

2) The size of a proof of a general mathematical statement that has a proof can be huge in terms of the size of the statement. Therefore the proof of Poincare conjecture reflects creativity.

3) Finding a proof of a mathematical statement can be computationally hard even if there is a small length proof. Therefore the proof of Poincare conjecture reflects creativity.

This is an argument why in certain cases P/poly is a better model for “efficient computation” than P. Indeed typically in cryptography, we would like our algorithms to be in P (i.e., explicit algorithm with no large parameters) but want them to be secure even against adversaries in P/poly (even against natural or artificial processes that have large parameters.

Whether you are creative or not, there are no systematic ways to solve intractable questions, or any good chance to stumble upon the solution.

Maybe the issue here is my complexity/cryptogaphy background. I think of an “intractable problem” as a question of mapping inputs to outputs (which in the case of NP have some efficiently verifiable relation between them). The problem is intractable because there is no efficient way that *always* (or in the case of average-case complexity, with good probability) succeeds in mapping an input to the correct output. But even for intractable problems, you may get lucky and manage to find the right output for some input.

You can say that maybe each input defines its own particular problem, and some inputs are intractable and others are easy, but this is a bit problematic, partly because for every particular input there always exists a small circuit that finds the right output (i.e., a circuit with this output hardwired into it), and, as you argue above, it’s not clear if we shouldn’t allow such circuits when we model “efficient computation”.

So, while finding proofs for mathematical statements is hard in general, people still succeed in doing so for particular statements, and when they do so, we feel they are sometimes creative, perhaps exactly because we do not know of any systematic “cookbook” way to find such proofs.

“I don’t think creative people have a systematic way to solve intractable questions, but in some cases (like the proof of the Poincare conjecture), they manage to stumble upon the solution.”

Here we disagree. Whether you are creative or not, there are no systematic ways to solve intractable questions, or any good chance to stumble upon the solution. (This is why they referred to as intractable.) The Poincare conjecture is simply not intractable. Creativity is related to the phenomenon that verifying is easier than finding, but it is not related to intractability.

We can have some ideas about mathematical questions that are indeed probably intractable, like computing the Ramsey number R(50,50) or finding the length of the shorter proof of the prime number theorem in a prescribed proof system. And indeed, these tasks will not be achieved systematically or by chance, by creative people or by other people or by computers.

“Obviously if the output we try to compute is not determined by the input then no computation can help.”

Yes, but this is a huge issue in the larger area of computation, related to modeling, numerical analysis, chaotic behavior etc. We should care about it too!

“For P we assume that the algorithm has a absolute constant..”

That’s correct, but when we apply insights from cc to real-life problems we obtain distinction between huge intractable constants and constants within reach. If you need to recover the (polynomial) algorithm performed by an unknown computer chip and need to go over all the possible chips of the same size, this is intractable (yet constant), and you may have similar issues for algorithms for natural processes.

“I believe that by the time we get a proof, this conjectural picture will only be a small part of the new things we’ll learn about computation, but this of course purely speculative.”

This is certainly a terrific possibility!

OTOH, I also like Anıl’s comment below.

I believe that by the time we get a proof, this conjectural picture will only be a small part of the new things we’ll learn about computation, but this of course purely speculative.

For example, if the P-algorithm depends on a secret that if we don’t know we need exponential search to find.

This seems to apply more to non-uniform computation, which is the class P/poly. For P we assume that the algorithm has a absolute constant description size that does not depend on the length of the input.

Our inability today to give an accurate weather forecast for May 9 2014 is not a computational complexity issue.

Obviously if the output we try to compute is not determined by the input then no computation can help. I don’t know if that’s the case or not for weather prediction. As I commented above in the discussion with Ryan, though I am not an expert on machine learning, I believe an algorithm for SAT will certainly help in many prediction tasks. In particular, I think sometimes the problem is finding which features in the data are relevant for the prediction, something that you could do if you had a SAT algorithm.

There is no reason to think that any aspect of human creativity involves solving computational intractable questions

I don’t think creative people have a systematic way to solve intractable questions, but in some cases (like the proof of the Poincare conjecture), they manage to stumble upon the solution. I agree that computers could also do the same even without P=NP, but would be much better at it with an efficient SAT algorithm.