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"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 unpl

by dsfsdfd 11y ago

"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.

c_lebesgue 11y ago
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.
- TheOtherHobbes 11y ago
  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.
  - foobar2020 11y ago
    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.
drd0rk 11y ago
Or you should no longer consider yourself something else than a computer.
bane 11y ago
Right, it is automated. we are the machine that automates it.
- shizzy0 11y ago
  It's just the time complexity sucks.
nabla9 11y ago
There is zero reason to believe that human level mathematical creativity is computationally significantly more expensive than normal day to day thinking.
- SilasX 11y ago
  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.
SilasX 11y ago
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").

by dsfsdfd 11y ago

c_lebesgue 11y ago

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.

TheOtherHobbes 11y ago
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.
- foobar2020 11y ago
  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.

drd0rk 11y ago

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

bane 11y ago

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

shizzy0 11y ago
It's just the time complexity sucks.

nabla9 11y ago

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

SilasX 11y ago
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.

SilasX 11y ago

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").