The preceding thought experiment serves to ellucidate several points:

• If a sentient being could be run in a computer, it could be "killed" and restarted many times, living the exact same life over and over.
• A sentient being run in a computer could end up split into several portions of memory or even across several different computer.
• A program containing a sentient being could computed by hand on paper, albeit slowly, and any intelligent output the sentient being could be determined without running it on a computer.
• There may in fact be no difference between predicting a program's output and actually running the program using a mechanical computer.
• Finally, the sentient being may exist whether or not it is ever "run" or even conceived of, since the code that runs it creates universally "true" results, such as one might expect from simple arithmetic procedures like long division.

Towards the end of the discussion I made something of a logical leap - that the medium through which we "run" our program could be reduced not only from a computer to a tracing of code on paper, but to "no medium".  Now that the thought experiment is over, I'd like to go into more detail as to why I think this is true.

Turing Machines

A computer program is, at its basis, a Turing machine.  I've mentioned this before and while you are free to follow the preciding link to the Wikipedia article, I'll attempt to give a layperson's summary.

Turing machines were invented by Alan Turing in 1936 as a mathematician's tool validate mathematical proofs and the like with a greater degree of rigour.  The idea is that any mathematical problem that can be solved by a person can be solved by some Turing machine.   Conceptually the turing machine can be thought of as an infinitely long piece of paper tape with a head that "reads" and "writes" symbols onto it (almost like an audio tape), a "state register" that keeps the current state, an 'action table" that tells the machine what state to move to based on the symbol just read.  However, even though a Turing machine has a very physical-sounding description, it's not an actual device - just a conceptual tool.  You can think of it as a sort of proto-computer with infinite memory.  Or, more appropriately, you can think of a computer as a Turing machine with limited memory!  Any computer that can possibly be made (to the limits of our knowledge) can be simulated with a Turing machine.

The reason this is important is that we know that in Turing machines we have found the ultimate power (that we know of) in computation.  Essentially any problem you were ever given in high school physics can be solved using some algorithm run with a turing machine.  And if we made a Turing machine that had algorithms for the laws of physics, and could throw in all the data from all matter and energy in our universe . . .

Well, this is getting ahead of ourselves.  Let's just be satisfied that we can at least conceive of an infinitely big computer, and that it can "run" pretty much anything.

What Are Numbers?

I previously likened our hypothetical computer program to long division.  To elaborate:  If you want to solve 42 divided by 3, you first divide 3 into 4, getting 1 with a remainder of 1.  Then you move the remainder into the tens place and add in the '2' part of 42, giving you 12.  Finally you divide 3 into this 12 and get 4.  This sits next to the original '1' from when we divided 3 into 4, for a result of 14.

A nice refresher of grade-school long division can be found here.

Long division is a good example because it is at heart an algorithm.  That is to say, it's a set of well-defined instructions which, given an initial state (in this case, 42 divided by 3) terminate in a recognizable end-state (14).  However, we know (at least we are pretty sure) that 42 divided by 3 is 14 whether or not we go through this algorithm.  I said earlier that by the same logic, the sentient being run by our computer program (which is also an algorithm) still thinks the same thoughts whether or not we go through that algorithm.

However, some of you may be thinking something like this:  "Aren't numbers normally just representations of physical objects or properties, and thus 42 divided by 3 only has real meaning with reference those objects and properties?"  That is to say, when humans first came up with the problem "42 divided by 3", wasn't it actually just a representation of something physical, such as, "How many of our 42 sheep will be inherited by each of our 3 children?"  Furthermore, one may contend that with respect to our universe, numbers exist ultimately as a way to describe the magnitudes of the various properites of subatomic particles - mass, for example - and thus argue that algorithms floating in a hypothetical void can't spawn sentience without an actual "thing" for them to describe.

I would counter this (and raise even more questions) by asking what the meaning of "mass" is?  When we're talking about an electron, for example, what do we really mean when we say it has a mass of 9.10938188 x 10^-31 kilograms?  I personally like to think of electrons as glowing blue sparks with a tiny amount of heft to them, but I realize that this is just my brain trying to make it easier for me to understand them.  Unless our physicists find something that tells us otherwise, there isn't much more to electrons other than the raw numbers that describe them.  And as to the properites themselves, what are they anyway?  Is there anything more to something like "mass" than how it causes the particle to behave?  And isn't this behavior something we can process algorithmically (even if the algorithm must include a probability distribution)?

Thus a better picture of an electron might be as a spreadsheet listing a number of different attributes, paired with a set of equations (like the Schrödinger equation) that describe how it acts.  Although there is something called the Heisenberg uncertainty principle that keeps us humans from being able to observe these properties, that is all there really is to them!

If any of this is starting to confuse you, I'd like for you to stop for a moment and try to think what you think something like, say, an apple actually "is".  You know from science class that it's ultimately made from atoms, which are made from protons, neutrons, and electrons.  You might remember diagrams of an atom from high school chemistry or physics, and these diagrams usually make the atom look like a little solar system that you might be able to probe if you had a small enough needle.  But now I want you to really think hard about how each part of this atom really only has a few properties to it, each of which can be described with numbers, and each of these properties always work in defined way according to the circumstances.  It is admittedly difficult to think of the electron as anything other than a glowing spherical orb, but try to dissasociate yourself from this, and think of it instead as being in the same category as, say, a line item in your taxes, or the stats of your favorite baseball player.  It is just numbers.  Once you've got this squared away, think about how when group enough of these properties with a system of mathematical rules, you could get an apple.  And a human being.  And a planet for them to exist on.  And though all of this is merely numbers paired with mathematical rules, the system of mathematics is what causes the human to think, and to label the various groupings of numerical properties with names like "photon", "light", "red", "shiny", "apple", "human", and "planet". 

What Makes a Reality 

Looking at things from this perspective, our reality and the humans who live in it are primarily made of groupings of numbers and a set of rules for how they behave.  Unless we find new information to prove otherwise, there isn't a bunch of magical "stuff" or "goo" making it more real - and if there was, how would we describe it other than by ascribing numerical properties to it and figuring out how they behave using equations?  Our reality might be run on a giant computer, traced on paper by an exceedingly patient alien creature, or none of the above - and we wouldn't know the difference.

It is thus that I argue that we are in the exact same predicament as our hypothetical sentient being, and that to invalidate the idea of this creature existing without a medium to run it's program on is the same as invalidating us - because what's our medium?  In essence our reality is like a program being "run" in a Turing machine, and that it "runs" itself in the same manner that long division "runs" itself.  It is a big congregation of data with some rules that dictate how this data is handled.  These rules dictate that there be organisms (humans) with a nervous system capable of perceiving this reality.

And that, I contend, is all that is needed for a reality to exist.