As for P v NP, I am of the agnostic group: that we do not yet understand how fast problems can really be solved. To give an example, there is an area straddling complexity and algorithms that is often called “exact algorithms” or “exponential algorithms” (I call them “accelerated algorithms”). These algorithms improve on the runtime of brute-force search in a substantial way.

Suppose the number of possible solutions for your favorite NP problem is N, so that brute force search takes O(N*n^c) time for some c > 1. An “accelerated algorithm” for the problem would run in O(N^e*n^c) time, for some constant e<1.

Many NP problems exhibit surprising accelerated algorithms. For example, there is an algorithm for 3-SAT that runs in O(n^c*(4/3)^n) time, where n is the number of variables (this is due to Schoening). New accelerated algorithms for problems are being found every year. While they remain exponential, it is still intellectually interesting that many NP problems can avoid brute-force enumeration, and they make one wonder how far such algorithms can be pushed. We believe that most of the current ideas behind these algorithms (e.g. backtracking, local search, dynamic programming) can only be pushed so far, but it is not clear how to measure this in a general sense.

Yes, all NP-complete problems are linked by reductions, and so in some deep sense they are “equivalent.” But the “equivalent” problems are not all “the same” problem; they are still separate and highly diverse, and that’s what’s crucially important. In attacking some NP-complete problems, we can use specialized tools from graph theory; for other problems there are methods in linear algebra; for still others we can draw on number theory, or even physics. When I say that the “miracle” argument cuts both ways, what I mean is that the thousands of NP-complete problems can be tackled in thousands of different ways; only one of those approaches has to succeed, but so far not one has.

Of course if you already know beyond all doubt that P is not equal to NP, then it’s no miracle or even a surprise that all those techniques fail simultaneously. But that’s putting the cart before the horse.

This whole discussion is very much a repetition of that evening in Muggia, though without the wine and the sunset. Who knows—maybe “Anonymous” is the same mathematician I was sparring with there.

This is a terrible argument. The whole point is that all these problems are (polynomially) equivalent. So you should really say, “Thus for P to be different from NP requires the ‘miracle’ that not even one out of thousands of *equivalent* NP-complete problems has a quick solution…” If you like, I can give you billions more equivalent problems—would that make P=NP more likely to you?

The fact that all these problems are equivalent is *not* any sort of evidence, even intuitive, that P = NP. Rather, they are evidence that the notion of poly-time reducibility and the class NP are useful elements in studying complexity.

http://www.win.tue.nl/~gwoegi/P-versus-NP.htm

A world (like ours?) in which P is not equal to NP is interesting and challenging,
because in this world finding solutions (i.e., doing science!) requires ingenuity, inspiration, and a bit of luck. Verifying a given solution is easy and can be done by a machine or a PhD student.

If P=NP, both tasks could be done by a computer, at least if the proof of P=NP comes
with an efficient algorithm for an NP-complete problem.
This had already been noted by Kurt Goedel in a letter to John von Neumann in 1956
see

http://www.andrew.cmu.edu/user/hardt/godel.html

The worst case, however, would be a non-constructive proof of P=NP…