#### OK here goes...

But I've only had one cup of tea this morning...

Essentially the problem seems to be looking for a mathematical shortcut. P problems are like E=MC squared. Easy. Fill in the numbers and grab a calculator. NP problems take longer to work out, the more data you have. So, if you have a very big minesweeper grid or a very long encryption key, you need a very big computer and it will take a much longer time. Now if there was a shortcut in maths for cracking a very long key, that is a different algorithm, your internet security would be, to use the technical term, toast. So if P=NP, maths can guess. Or to put it another way, can maths make a guess, using what we might call intuitive reasoning? So really they want to know if maths can emulate human intuition. If P=NP then you could replicate human intuition and make a proper replicant.

I'd posit that the answer is 'no' and that P<>NP because intuitive reasoning is fundamentally different from mathematical reasoning. So full AI cannot happen. You need processors that work so quickly, they can test every possibility and choose, with the appearance of AI. We can pretend that maths can do this sort of thing with fuzzy logic, but fuzzy logic is approximating without intuition. It is probability, smoke and mirrors. It certainly isn't accurate enough to be acceptable amongst chaps who work with grown up algorithms, who may well look down on fuzzy logic types and scorn them in private.

Now the fun bit. Because you cannot mathematically quantify the concept of intuitive reasoning (the way we think and make decisions) you cannot express a proof for this problem in purely mathematical terms (one of the variables being mathematically inexpressible).

This is not to suggests that humans have some special quality beyond science, but that we reason using a technique that differs fundamentally from the way algorithms work in mathematics. It's like trying to convey something to someone in a language that has no words for it and who has no conception of it.

So computers cannot 'cheat' and think intuitively the way we do, although they can pretend to if they can work quickly enough, but people can go some way to emulating computers. At the far end of the autism spectrum, there are folk who can process data far more rapidly than we can, but can have difficulty with the more 'analogue' aspects of day to day life. Which may suggest that human beings typically operate with an intuitive system of reasoning to accommodate emotional interaction.

So, N<>NP and whilst two sides of an algorithm can equate, they will not then express their love for each other.