1. There is a problem on the borderland of computer science, logic, maths, philosophy, and physics known as “P versus NP”.
2. This concerns the question of whether the class of decision problems whose solutions are quickly verifiable (NP) by a computer is the same as the class of problems that are quickly solvable by computer (P).
3. Historically the problem arose because certain problems seem to be hard to solve.
4. More particularly, they seem to require a lot of time -an exponentially growing amount of time- to solve.
5. An example of a NP problem that seeming takes exponential time is Factoring. While it doesn’t take long to factor 15 or 21, imagine trying to factor the 200 digit integer
6. You can easily check that it divides evenly into the primes
but although it takes a pocket calculator a spit second to do the multiplication, it would take a single 2.2 GHz computer roughly 75 years to do the division.
7. We can prove that there is an arrow of time here – an asymmetry – by considering the Travelling Salesman Problem, which is he problem of whether a salesman can visit a number of cities exactly once and return to a home-city for a certain cost.
8. First we transform TSP into a problem of whether a computer (salesman) can execute some number of instructions (visit some number of cities) which executes every instruction exactly once (visits every city exactly once) before returning to a halt state (home-city) for some maximum cost.
9. An arbitrary computer is therefore working on the problem of whether an arbitrary computer will halt when run with an arbitrary set of instructions, and thus the point will be reached when the evaluation is a self-evaluation, i.e. the point will be reached such that the computer is attempting to determine of itself if it will halt.
10. If we associate to every city an instruction, this self-evaluative point will be reached when the number of cities on the tour is not less than the number of instructions in the program. This leads to a contradiction in the case that the number cities is greater than the number of instructions.
11. It follows that TSP involves a limit on the number of cities, from which it follows that TSP differs from polynomial-time problems, which aren’t sensitive to the size of the input, and that P and NP are not equal.
12. It follows from the arithmetic hierarchy formed by the relative complexity of programs of Turing machines that Factoring is NP-hard.
13. From Shor’s algorithm, it follows that quantum computers can factor integers in polynomial time, and since this involves collapsing the arithmetic hierarchy, that they can solve solve NP-hard problems.
14. From this follow two things of importance.
15. One is that scalable quantum computers cannot be built.
16. Scalable quantum computers cannot be built, for if they could be built, this would collapse the classical/quantum divide.
17. The other is that the conscious mind is not a classical computer.
18. The conscious mind is not a classical computer because a classical computer is as, we have seen, bound by the finitude of its program to operate on things no more complex than this program.
19. From Shor’s algorithm, the conscious mind is -as something that can solve P versus NP and therefore operate on objects of every possible size- a quantum computer, but not something that can be built in, or that exists in the classical domain.
20. But as something that is outside of the classical domain, outside of which the arrow of time doesn’t exist and things can go in either direction with equanimity, the conscious mind is not finite- it is eternal– yet it is contained in a world that is finite/governed by the arrow of time.