Artificial addition is not intrinsically addition, any more than a particular string of shapes on a page intrinsically means anything. There is no "structure that is addition", but there are "structures" that can represent addition.

What is addition, primordially? The root concept is one of combination or juxtaposition of actual entities. The intellectual process consists of reasoning about and identifying the changes in quantity that result from such juxtaposition. And artificial addition is anything that allows one to skip some or all of the actual reasoning, and proceed directly to the result.

Husserl's Logical Investigations has a lot about the phenomenology of arithmetic. That's where I'd go for a phenomenological ontology of addition. Ironically, through the exactness of its analyses the book played a distant role in launching cognitive science and the mechanization of thought, even while its metaphysics of mind was rejected.

The basic distinction is between intrinsic intentionality and derived intentionality. Thoughts have intrinsic intentionality, they are intrinsically about what they are about; words and "computations" have derived intentionality, they are convention-dependent assignments of meaning to certain physical things and processes. Artificial addition only has derived intentionality. If something has "the structure of addition", that means it can consistently be interpreted as implementing addition, not that it inherently does so.

The problem, of course, is that in the physical world it seems like nothing has intrinsic intentionality; everything is just a pile of atoms, nothing inherently refers to anything, nothing is inherently about anything. But there are causal relations, and so we have theories of meaning which try to reduce it to causal relations. B is about A if A has the right sort of effects on B. I think that's backwards, and superficial: if B is about A, that implies, among other things, than A has a certain causal relation to B, but the reverse does not hold. It's one of those things that needs a new ontology to be solved.

This perspective does not alter Eliezer's point. Even if thou art monadic intrinsic intentionality, rather than "physics", you're still something, and decisions still involve causation acting through you.