The Answer

It's the question that drives us, Neo. It's the question that brought you here. You know the question, just as I did. 

--Trinity

I have thought of a counterpart to my polynomial-time 3-SAT algorithm.  The purpose of the counterpart algorithm is to attempt to discredit the 3-SAT algorithm, by doing something complicated that looks a lot like exhaustively searching for a problem instance that I don't believe exists.

Unfortunately, this algorithm is at best, even in computer science terms, ridiculous.  Every step of the algorithm contains exponential sub-parts, and the branching factor itself is exponential.  Its failure for a number n proves that my 3-SAT algorithm will succeed for all 3-SAT instances of size n.  Therefore all I have to do is code it and run it on the set of Natural Numbers.  Small problem--more of an issue, really--that set is infinitely large, and therefore impossible to do with a homemade supercomputer, but, ironically, this is the closest I've ever come to proving P=NP.  Wouldn't it be hilarious if NP-Completeness ended up like the Continuum hypothesis?

I am one of those people who religiously cling to the school of thought that science is things that are unprovable theories that we can never know for certain (although you shouldn't mention this around jaded atheists that read Reddit all day), while mathematics is everything that is abstract and provable with absolute certainty.  I cling to this distinction because it is convenient, and because, despite what 1984 claims, even the most hardcore religious extremist or politician can't deny 2+2=4.  Why?  Because you need math to measure the ingredients in your stupid ass bombs that kill people, and you need math to count all the money you bring in from donations/tithes, and you need math to count votes.  You can deny math in some kind of masturbatory game of existentialism, but no one will care, and no one can stand on a pulpit in a church or body of political representatives and convince reasonable people that math is wrong, because it proves itself.  See Bad Wolf.