> He points out that we can with some actual mathematical rigor observe that the mapping itself can be said to be doing all the work.
There is no mathematical rigor whatsoever in that portion of the paper. He just conjectures that there would be no efficient algorithm for encoding (e.g.) chess positions as states of a waterfall. This actually seems fairly unlikely. You only need to be able to specify a set of physical states that compose in such a way that you can, e.g., incrementally push items onto a stack, and you're basically done. How can we possibly be confident that there is no state description of a waterfall that makes that possible? Note that we have to allow the state descriptions to be very complex, since the physical states corresponding to the computational states of a microprocessor are also extremely complex. This complexity happens to be easier for us to deal with because microprocessors have been designed to make it easy for us to put them into particular computational states. But the relevant states of a waterfall, while much more difficult for us to manipulate, will not obviously be any more complex. And to make the key point, we need only find one suitable inanimate physical system. It really doesn't seem so unlikely that there are a few of them out there.
On top of all this, why should it matter if there's no efficient encoding algorithm? It would obviously lead to an infinite regress if we say that a physical state has to have been "encoded" in order to count as a bona fide computational state (since then the input to the encoding algorithm would itself need to have been encoded).
Right - there's also no efficient algorithm for encoding chess positions into the electrical and chemical states of a couple of pounds of active neurons, and yet living humans seem to be able to play chess.
"He just conjectures that there would be no efficient algorithm for encoding (e.g.) chess positions as states of a waterfall. This actually seems fairly unlikely."
I think you're answering a different question than Scott. You seem to be answering "Can I construct a water-based computer that looks like a waterfall that could solve a chess problem, set up the initial parameters, and read the answer off the bottom?" This is conceivable. It is likely it would involve a highly implausible object of highly implausible accuracy and reliability, but if such an object existed it would still be a polynomial calculation to produce it. (Or, at least, a polynomial calculation could produce an object, if not the optimal one. The lack of solution to the Navier-Stokes equations might prevent you from finding an optimal one. And you might find yourself having some trouble with quantum effects. But such are the problems in the real world.) And our polynomial approximation is still not going to look like a waterfall as it will inevitably need looping constructs where state flows "back" uphill... if you want it to be one-way like a waterfall that is going to be exponentially large.
Scott's question is a different one. Given a real world waterfall that you have not constructed, but simply come across, can it be said to be computing the solution to a chess problem? Is a random rock that is sitting on the ground, jiggling away with atoms in constant motion that all the computers in the world could not possibly provide enough computational power to fully simulate, able to be said to be computing something with all that power? Could a boulder be sitting there simulating a human mind? I don't mean in the pan-psychic sense, I mean, literally, is it simulating a human mind? It has the computational capacity, when considered in the raw.
This is where Scott's argument kicks in, where the mapping is doing all the work.
Further evidence of this is that even if you want to sit here and argue that this boulder that is sitting in front of us (metaphorically) really and truly is calculating the state of your mind as well, well, with another equally sensible (and exponential and impossible) mapping, it's also calculating mine, it's also calculating Attila the Hun's, and it's also calculating the brain states of the final human being to ever live, and also deer's brains and the solutions to world hunger and pretty every other interesting thing ever, really. The contortions required to create the mapping of boulder state to human brain state are such that you can fit literally almost anything into it, and therefore, it is reasonable to point out that it is meaningless. It's an interesting argument that provides a surprisingly rigorous line that allows us to say that, no, that waterfall is not solving a chess problem or anything else... it really is, well, a waterfall.
> Given a real world waterfall that you have not constructed, but simply come across, can it be said to be computing the solution to a chess problem?
Yes, that's the question I was addressing. I'm really can't figure out why you thought otherwise. There's nothing in my original comment about constructing artificial water-based computers.
>This is where Scott's argument kicks in, where the mapping is doing all the work.
It is his conjecture that there is no computationally simple mapping, but I see no reason to believe that this conjecture is correct. He certainly doesn't give any reason to think so in the paper. And to echo jameshart's point, he gives no reason to think that the mapping from chess positions to brain states would be any simpler than the mapping from chess positions to waterfalls.
If you find his conjecture plausible for some reason, that's fine, but you're quite wrong if you think that the paper presents any rigorous mathematical argument. The discussion is much less rigorous and sophisticated than the existing philosophical literature on the topic. (After all, Putnam's original paper contained a proof.)
>Further evidence of this is that even if you want to sit here and argue that this boulder that is sitting in front of us (metaphorically) really and truly is calculating the state of your mind as well, well, with another equally sensible (and exponential and impossible) mapping, it's also calculating mine, it's also calculating Attila the Hun's, and it's also calculating the brain states of the final human being to ever live, and also deer's brains and the solutions to world hunger and pretty every other interesting thing ever, really.
Yes, that's the problem. That's why the computational theory of mind seems to be a bit of a non-starter. Since pretty much every physical system computes pretty much every function, it's difficult to see how thought can arise merely as a consequence of the brain computing a particular function.
There is no mathematical rigor whatsoever in that portion of the paper. He just conjectures that there would be no efficient algorithm for encoding (e.g.) chess positions as states of a waterfall. This actually seems fairly unlikely. You only need to be able to specify a set of physical states that compose in such a way that you can, e.g., incrementally push items onto a stack, and you're basically done. How can we possibly be confident that there is no state description of a waterfall that makes that possible? Note that we have to allow the state descriptions to be very complex, since the physical states corresponding to the computational states of a microprocessor are also extremely complex. This complexity happens to be easier for us to deal with because microprocessors have been designed to make it easy for us to put them into particular computational states. But the relevant states of a waterfall, while much more difficult for us to manipulate, will not obviously be any more complex. And to make the key point, we need only find one suitable inanimate physical system. It really doesn't seem so unlikely that there are a few of them out there.
On top of all this, why should it matter if there's no efficient encoding algorithm? It would obviously lead to an infinite regress if we say that a physical state has to have been "encoded" in order to count as a bona fide computational state (since then the input to the encoding algorithm would itself need to have been encoded).