One of the things that we understand about P vs NP is that it may take us 2,000 years to finally solve it. But I recently finished reading Gregory Chaitin'sThe Limits of Mathematics: A Course on Information Theory and the Limits of Formal Reasoningand I'm more skeptical than even that.

Irrational and Transcendental numbers sets are larger than the Reals set. We know this from Cantor. But we know next to nothing about transcendental and irrational numbers. In fact, our understanding of numbers is, to put it lightly, only in its infancy. With the work of Leibniz and Liouville we even learned of numbers and entire functions that can't be used or calculated through algebra. Chaitin goes one step further and shows us there are numbers that can't even be described or conceptualized in any way. We simply lack the formal reasoning to understand these numbers.

P vs NP is deeply integrated into our understanding of mathematics. Even if we pretend to have solved it for our present knowledge, we can't claim it. We know the problem is still unsolved. P vs NP domain is the entire domain of problem solving techniques, including those we don't know about yet or those we haven't invented yet.

We are in a constant search for new and better mathematics. But heck, in order to finally put P vs NP to rest we need to reach a plateau of mathematical understanding in which we just know there is nothing new to invent in this area. Highly unlikely.

As we have been shown since as far back as ancient times, algebra isn't enough to deal with the largest sets of numbers. Although it has been refined and expanded through the ages, it also helped expose an hidden mathematical universe we have yet to tap into. We will eventually develop new axioms. New fields of mathematics will eventually one day be able to explore everything, from irrationals and hyperreals to transcendentals, in more formal ways. But what new unreachable secrets will they reveal?

Seems likely we will always think we live in a NP universe with P always tagging behind as we progress and evolve. If and when we reach the answer one day, our intellects may be too evolved to even consider P vs NP more than an ancient curiosity of our more single minded ancestors. Also, we may forget to adjust the 1 million dollar prize to inflation.