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Comment by dsfsdfd | Hacker News Reader

"But another reason we believe P≠NP is that otherwise mathematical creativity could be automated!"

I have a problem with this. Just because the thought is unpleasant does not make it true or false. Besides, perhaps this is the very difference between automation and intelligence, perhaps the point at which you can ask this of a computer is the point at which you should no longer consider it a computer.

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There is zero reason to believe that human level mathematical creativity is computationally significantly more expensive than normal day to day thinking.

In terms of the computations performed in the brain by the mathematician, at the time the insight is achieved, it's not.

But appropriately accounted for, I think it is.

The relevant metric would be, "what do I have to increase in order to get more [interesting, new] theorems proved?" Is it as easy has having mathematicians work more hours? Having more people start working on them?

Intuitively, it is not so easy -- each insights need exponentially more work as time progresses. An exponentially-small fraction of humans is capable of producing novel results, and so on.

This, I think, is what Aaronson is getting at: if P = NP, if proving is as easy as verifying, then coming up with hard mathematical insights should be as easy as following a cookbook, or any other routine, mundane mental task.

But that doesn't seem to describe the world we live in, where getting mathematical insights has rapidly diminishing returns per unit resource invested.

Sure, it is certainly possible that mathematics can be automated, but mathematics is also over 2000 years old. We are not much closer to automating it than the Greeks were. P=NP does not only break the Complexity theory (which is a relatively new invention), it also has profound consequences to all of human reasoning.

We already have commoditised math solvers for any number of (relatively) trivial problems.

What we haven't automated is mathematical creativity, which is a completely different class of problem.

The whole difficulty of the hard complexity questions such as "Is P=NP?" hides in the non-trivial, "corner" cases. Sure, we have SAT solvers that are by many practical standards really good. P=NP would imply that we can relatively easily find polynomially long proofs to many hard propositions, such as the P=NP problem itself! Or many other open mathematical problems.

Creativity is an anthropomorphic, abstract concept, a way of describing human ways to find solutions. What really matters is the proofs.

Or you should no longer consider yourself something else than a computer.

Right, it is automated. we are the machine that automates it.

It's just the time complexity sucks.

It's not an appeal to consequences ("it would be horrifying if this were true") but to absurdity ("believing this would require us to believe all these implausible things").

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