Somebody that insists that 1 should be coded as a red light and 0 should be encoded as a green light would claim that the original circuit is not an adder. A view on the topic that recognises that such number-color associations are arbitrary and thus must not be the turning points of the arguments would classify a red-adder and green-adder to be doing the same calculation.

What kind of desirability condition does turing machine fail to not be the correct mathematical representantion of natural computation? Turing machines are very good at describing symbol dances and they don't need to have meanings to be described. It seems to me that if somebody desires something beyond that then the additional bit can't really be about the computation. If you are not satisfied that the word "cat" is made out of c, a and t then you are asking about felines or something rather than anything about words. In the same way if you have "something that brrrrs this way" and zigging that way or zagging that way is not a thing that satisfies you then you are asking about something else than sequences of operations.

This is an excellent post! Thank you for sharing your thoughts! I too am very curious about many of these questions, although I’m also at a half-baked stage with a lot of it (I’d also love to have a better footing here!). But in any case, here are some thoughts (in no particular order).

1. I’ve been interested in the questions you pose around AlexNet for a while, in particular, how much computation is a function of observers assigning values versus an intrinsic property of the thing itself. And I agree this starts getting pretty weird and interesting when you consider that minds themselves are doing computations. Like, it seems pretty clear that if I write down a truth table on paper, it is not the paper or ink that “did” the computation, it was me. Likewise, if I take two atoms in a rock, call one 0, the other 1, then take an atom at a future state, call it 0, it seems clear that the computation “AND” happened entirely in my head and I projected it onto the rock (although I do think it’s pretty tricky to say why this is, exactly!). But what about if I rain marbles down on a circle circumscribed in a square (the ratio of which “calculates” pi)? In this case it feels a bit less arbitrary, the circle and the square chalked on the ground are “meaningfully” relating to the computation, although it is me who is doing the bulk of the work (taking the ratio)? This feels a bit more middle ground to me. In any case, I do think there is a spectrum between “completely intrinsic to the thing” and “agents projecting their own computation on the thing” and that this is largely ignored but incredibly interesting and instructive for how computation actually works.  
2. Relatedly, people often roll their eyes at the Chinese Room thought experiment (and rightly so, because I think the conclusions people draw about it with respect to AI are often misguided). But I also think that it’s pointing to a deep confusion about computation that I also share. The standard take is that, okay maybe the person doesn’t understand Chinese but the “room does,” because all of the information is contained inside of it. I’m not really convinced by this. For the same reason that the truth table isn’t “doing” the computation of AND, I don’t think that the book that contains the translation is doing any meaningful computation, and I don’t think the human inside understands Chinese, either (in the colloquial sense we mean when we don’t understand a foreign language). There was certainly understanding when that book was generated, but all of that generative tech is absent in the room. So I think Searle is pointing at something interesting and informative here and I tentatively agree that the room does not understand Chinese (although I disagree with the conclusion that this means AI could never understand anything).
3. I do agree that input/output mappings are not a good mechanistic understanding of computation, but I would also guess that it’s the right level of abstraction for grouping different physical systems. E.g., the main similarity between the mechanical and electrical adder is that, upon receiving 1 and 1, output 2, and so on. 
4. I get confused about why minds have special status, e.g. “computation is a function of both the dynamics and an observer.” On the one hand, it feels intuitive that they are special, and I get what you mean.  And on the other hand, minds are also just physical systems. What is it about a mind that makes something a computation when it wasn’t otherwise? It’s something about how much of the computation stems from the mind versus the device? And how “entangled” the mind is with the computation, e.g., whether states in the non-mind system are correlated with states in the mind?  Which suggests that the thing is not exactly “mind-ness” but how “coupled” various physical states are to each other.  
5. I also think that the adder systems are far less (or maybe zero) observer dependent computations, relative to the rock or truth table, in the sense that there are a series of physically coupled states (within the system itself) which reliably turn the same input into the same output. Like, there is this step of a person saying what the inputs “represent,” but the person’s mind, once the device is built, does not need to be entangled with the states in the machine in order for it to do the computation. The representation step seems important, but also not as much about the computation itself rather than “how that computation is used.” Like, I think that when we look at isolated cases of computation (like the adders), this part feels weird because computation (as it normally plays out) is part of an interconnected system which “uses” the outputs of various computations to “do” something (like in a standard computer, the output of addition might be the input to the forward-prop in a neural net or whatever). And a “naked” computation is strange, because usually the “sense making” aspect of a computation is in how it’s used, not the steps needed to produce it. To be clear, I think the representation step is interesting (and notably the thing lacking in the Chinese Room), and I do think that it’s part of how computation is used in real-world contexts, but I still want to say that the adder “adds” whether we are there to represent the inputs as numbers or not. Maybe similar to how I want to say that the Chinese room “translates Chinese” whether or not anyone is there to do the “semantic work” of understanding what that means (which, in my view, is not a spooky thing, but rather something-something “a set of interconnected computations”). 
6. Maybe a good way to think of these things is to ask “how much mind entanglement do you need at various parts of this process in order for the computation to take place?”
7. My guess is that computation is fundamentally something like “state reliably changes in response to other state.” Where both words (“state” and “reliably”) are a bit tricky to fully pin down and there are a bunch of thorny philosophical issues. For instance, “reliably” means something like “if I input the first state a bunch of times, the next state almost always follows”, but if-thens are hard to reconcile with deterministic world views. And “state” is typically referring to something abstract, e.g., we say “if the protein changes to this shape, then this gene is expressed,” but what exactly do we mean by “shape”? There is not a single, precise shape that works, there’s a whole class consisting of slight perturbations or different molecular constituents, etc. that will “get the job done,” i.e., express the gene. And without having a good foundation of what we mean by an abstraction, I think talking about natural computation can be philosophically difficult.
8. “Is there anything purely inside of AlexNet that can tell us that 1 in the output node means cat and that 0 means not cat?” I’m not sure exactly what you’re gesturing at with this, but my guess is that there is. I’m thinking of interpretability tools that show that cat features activate when shown a picture of a cat, and that these states reliably produce a “1” rather than a “0.” But maybe you’re talking about something else or have more uncertainty about it than me?
9. I agree that thinking is extremely wild! ‘Nough said.

I wonder if computation makes the most sense in the context of a compiler. Like for the adders, you start with an interest in adding some numbers, and then you compile that into a logic circuit using an engineer's understanding of addition, and then you further compile this logic circuit into either an electrical circuit or a marble machine, which then performs the originally intended computation. When you have a compiler, you end up with clear answers to which algorithm is being computed (whatever is upstream of the compiler) and which system is doing the computation (whatever is downstream of the compiler).

Excellent post Adam!

I presume it is a canonical joke but I had to chuckle at the cat-gorizers =D

If you think about programmes as lambda terms then they have free variables in which you can plug values. Flipping the red and the green light or 0 with 1 becomes less mysterious from this point of view.

Taking a fully applied category/compositionality point of view we could be looking at "open systems" (so systems with input and output ports) and see how they compose.