The mathematicians discovered the central mystery of computability, the conjecture represented by the statement P is not equal to NP. The conjecture asserts that there exist mathematical problems which can be quickly solved in individual cases but cannot be solved by a quick algorithm applicable to all cases. The most famous example of such a problem is the traveling salesman problem, which is to find the shortest route for a salesman visiting a set of cities, knowing the distance between each pair. All the experts believe that the conjecture is true, and that the traveling salesman problem is an example of a problem that is P but not NP. But nobody has even a glimmer of an idea how to prove it. This is a mystery that could not even have been formulated within the nineteenth-century mathematical universe of Hermann Weyl.
At a literal level, the above passage contains several howlers (I’ll leave it to commenters to point them out), but at a “deeper” “poetic” level, Dyson happens to be absolutely right: P versus NP is the example par excellence of a mathematical mystery that human beings lacked the language even to express until very recently in our history.
Speaking of P versus NP, I’m currently visiting Sasha Razborov at his new home, the University of Chicago. (Yesterday we had lunch at “Barack’s favorite pizza place”, and walked past “Barack’s favorite bookstore.” Were they really his favorites? At a deeper poetic level, sure.)
One of the highlights of my trip was meeting Ketan Mulmuley for the first time, and talking with him about his geometric approach to the P vs. NP problem. Ketan comes across in person as an almost mythological figure, like a man who flew too close to the sun and was driven nearly to ecstatic obsession by what he saw. This is someone who’ll explain to anyone in earshot, for as long as he or she cares to listen, that he’s glimpsed the outlines of a solution of the P vs. NP problem in the far frontiers of mathematics, and it is beautiful, and it is elegant—someone who leaps from Ramanujan graphs to quantum groups to the Riemann Hypothesis over finite fields to circuit lower bounds in the space of a single sentence, as his hapless listener struggles to hold on by a fingernail—someone whose ideas seem to remain obstinately in limbo between incoherence and profundity, making just enough sense that you keep listening to them.
Now, I get emails every few months from people claiming to have proved P≠NP (not even counting the P=NP claimants). Without exception, they turn out to be hunting polar bears in the Sahara: they don’t even grapple with natural proofs, or relativization, or algebrization, or the lower bounds/derandomization connection, or any the other stuff we know already about why the problem is hard. Ketan, by contrast, might be searching for polar bears with a kaleidoscope and trying to hunt them with a feather, but he’s in the Arctic all right. I have no idea whether his program will succeed within my lifetime at uncovering any of the truth about the P vs. NP problem, but it at least clears the lower hurdle of reflecting some of the higher truth.