I did not perform the math directly. I cheated. I used a trick--an algorithm I learned--in this case, one I learned in algebra class. But the standard multiplication algorithm is also a trick--it's just as much a cheat. But it's multiplication.
This is because, in mathematics, how you get the answer isn't as important as simply getting the answer in a valid way. There is no single proper way to get the answer; any valid way that you use to get the answer is legitimately performing mathematics.
This is why I'm telling you it's a bad example. Anything that works in a dual space that maps to the problem space, where you use the dual to perform the work, counts as doing the work.
Actually, that's why it's such a good example.
The trick to seeing why is to make sure to always clearly distinguish between the material and the symbolic, the physical computations and the logical computations, the real and the imaginary.
How does real addition happen? Stuff that was separate moves into a group. Or maybe new stuff is created to increase the size of the group. We can observe this happening in all kinds of ways.
Computers simulate this happening, which means they use a different physical system -- their own hardware -- which gets them into the state to, as you say, find the right name.
Now this is a very different kind of thing to do. Real addition isn't part of what the machine is doing. And in fact even what it's doing when we add the human to the system, ending up with the right light pattern on a screen or pattern of ink on paper to make a human brain think of the number five, is quite distinct from real (physical) addition.
So when you say "in mathematics,
how you get the answer isn't as important as simply
getting the answer in a valid way", that's true, but mathematics is about as deep into the symbolic/imaginary side as you can get, so keep in mind that this point is clearly relevant only when you're discussing the "informational" side, but not necessarily the physical side.
I mean, mathematically, time is reversible, but in our lives it's not.
So you're right, the computer comes up with the name -- which is what it's designed to do -- it does not perform physical (real) addition.
And what's interesting, when you think about it, is that the computer isn't even actually simulating real addition... it's simulating mental "addition", which is itself simulating real addition.
The human brain is also in the game of coming up with the name, rather than actually adding things physically.
The brain does that, though, not by having its behavior shaped by a programmer, but by having its behavior shaped by evolution. Which actually isn't a metaphor here, since the process of evolution quite literally determines the physical shape of the brain and all the rest of the body.
And the shape and material of the brain in its environment are the things that determine what it does, how it operates. (Same for a computer, of course, or a kidney.) For these objects, symbols mean nothing... the only "rules" they can be said to follow are the laws of physics.
Over time, the physical channels in your brain which are active when you hear the sound "two plus three" and when you think about the number five, come to overlap to such a degree that the cascade of neural activity that is physically inevitable when that sound hits your ear will at some point include the neural activity that is going on when you think of the number five. (It should do this when you're asleep in many conditions, too.)
This is how the idea of five "occurs to you" or "pops into your head" after you hear the sound "What's two plus three?" It's a matter of neural erosion.
To use the hydraulic metaphor, a flood in the "two plus three" sound area will cause heavy flooding in the "five" number area of the brain, because that's how the pipes are laid out.
This is not the way the computer operates.
So all 3 cases are distinct.
From a mathematical point of view maybe not, but taking that point of view wipes out the entire reason for the exercise, because that view is immersed in one panel of the triptych that we're looking at.
And I might suggest that a failure to move out of the symbol world when considering the physical world may be a big part of the reason why the "man in the world of the simulation" idea still appeals to you.
