Imagine science advances to the point where medical science can replace individual brain cells with perfect robot replicas. These robot cells interact with surrounding cells in exactly the same way as the real ones and can be initialized with the state of the real cell they are replacing. They're small though, so all the real computation work they may need to do is offloaded to a computer (over radio signals perhaps, or whatever.) Perhaps we can even optimize their operation such that communication with other robot cells that may happen to be next to it takes place inside the computers modelling both cells, instead of actually firing off the connection "for real".

Preform this operation once, and few would argue that the mind in question has died, been replicated, ceased to operate and been replaced with another... anything. It's just the same mind.

If you are willing to accept that (and if you're not, we'll just have to disagree), then preform the operation again. And again. And again.

This is what sodiumphosphate was referring to by "metamorphosis", and it does not require dualism.

http://en.wikipedia.org/wiki/China_brain

http://en.wikipedia.org/wiki/Ship_of_Theseus#Human_body

Replacement with the subtly different is interpreted by our minds as "change". You no more "die" when the cells or atoms in your brain are replaced than a river dries up when all the water molecules you could see at one point in time flow away.

"recognizing" it is irrelevant and poorly defined.

Simply assuming it is the case because you want it to be the case is not science, it is religion.

> It is an open question whether there can be actual deterministic physical processes that, in the long run, elude simulation by a Turing machine, and in particular whether any such hypothetical process could usefully be harnessed in the form of a calculating machine (a hypercomputer) that could solve the halting problem for a Turing machine amongst other things. It is also an open question whether any such unknown physical processes are involved in the working of the human brain, and whether humans can solve the halting problem.

http://en.wikipedia.org/wiki/Halting_problem

Being able to solve the halting problem for programs generally seen in real life programming does not imply in the slightest that you can solve the halting problem for arbitrary programs. Can you look at arbitrary Turing machine tapes (large enough to eliminate the possibility that you are simulating it without realizing it of course) and non-algorithmically decide if they will terminate? Can you in fact do anything non-algorithmically? That is in essence what you are suggesting, the non-algorithmic nature of the human mind.

You are of course not the first to suggest a non-algorithmic mind. The likes of Roger Penrose have also suggested similar things (see his The Emperor's New Mind and later Shadows of the Mind). His reputation is enough to give me serious pause, and the fact that he does not seem to be a dualist further inclines me to listen to what he has to say.

If you want to delve further into this subject I suggest you check out Penrose's work, then read the criticism of it from his peers. Penrose has mounted the most qualified defence of the non-algorithmic mind of which I am aware, but even so acceptance of his thesis is rather rare in academia. It is not light subject matter but the general consensus among experts in the fields which he touches is that in order to support his thesis he made numerous errors. In absence of their expertise, I must defer to their assessment of Penrose's work.

Does this halt? Assume all variables are arbitrary precision integers.

/* http://en.wikipedia.org/wiki/Perfect_number#Odd_perfect_numb... */

  void searchForOddPerfectNumber() {
    int n = 1, sumOfFactors, factor;

    while (1) {
      sumOfFactors = 0;
      for (factor = 1; factor < n; factor++)
        if (n % factor == 0)
          sumOfFactors += factor;
      if (sumOfFactors == n) break;
      n += 2;
      sumOfFactors = 0;
    }
  }