[bramcohen]

While the author of this paper does not appear to be a crank, nowhere in the entire paper does it discuss why the fundamental barriers of naturalization, algebrization, and relativization don't apply to the work, making it seem unlikely that those barriers have actually been overcome.

[long]

From what little I understand, those barriers prevent only certain proof strategies from working. So for instance, the Razborov-Rudich barrier concerns a class of combinatorial proofs (the so-called natural proofs); this paper uses two techniques - statistical mechanics and model theory - which I gather are out of the province of RR.

[cdavidcash]

"only certain proof strategies" is technically correct, but its closer to "essentially every proof strategy we can conceive of".

And besides, the question is over the entire proof strategy and not the specific techniques involved. It seems plausible that one could give a relativizing proof using some method of calculation from statistical mechanics, for example.

[joe_the_user]

I'd tend to agree...

Taking a methods from a different domain in no way shows that the mechanics of those method don't reduce to the same mechanics as a "natural proof". Further, it makes the actual process much more obscure.

Indeed, statistics in general seems like a hard approach for overcoming the Razborov-Rudich limit, since the limit is on showing that a "typical" function has certain properties and anything statical seems like it would be "typical".

But who knows really, maybe Razborov-Rudich isn't correct, since in fully generality it relies on things that "mathematicians generally believe" (the existence of certain pseudo-random functions) rather than things that are definitely proven - as far as I know.

[long]

Again, I'm no expert, but relativization and algebrization are properties of proofs that invoke oracles, which this paper doesn't appear to do.

[cdavidcash]

Ah, that is not how those "barriers" work. Roughly, the relativization barrier goes like this: Say you have a proof that P!=NP. Does it also prove that P^A != NP^A for any oracle A? If it does, then the proof is flawed, because there <i>does</i> exist an oracle A such that P^A = NP^A!

Such proofs are said to relativize -- i.e., they are still valid relative to any oracle.

[long]

Ah, thanks for explaining!

[joe_the_user]

I would note: "statistical mechanics" is not technique but a subfield of mathematics and physics within which a number of techniques are used.

"Model theory" might be closer to "a" technique or at least a somewhat distinct set of techniques.

And other have noted, there's reason that a proof from statistical mechanics would relativize.