If you can show a causal role for pure experience, that would be something else, but instead you speak of the "causal role they appear to play."
I was actually talking more about the deduction that experiences are causally downstream from physical stimulation of sense organs, and causally upstream from voluntary motor action. This deduction is made because the physical brain is in that position; the physical causal sequence matches up with the subjectively conceived causal sequence "influences from outside me -> my experiences -> my actions"; so one supposes that experiences are in the brain and relevant to "physical" causality.
If you posit abstractly characterized structural entities, you are still left with the problem regarding what makes that configuration give the appearance "blue."
To say that these entities have abstract structure, is not to say that that is the whole of their being. I am only emphasizing how qualia, and things made out of qualia, can be part of a mathematically characterized fundamental physics. The mathematical theory would talk about a causal network of basic objects characterized with the abstruseness typical of such theories - e.g. as combinations of elements of an algebra - and some of those objects would in reality be qualia.
If you were then to ask "what makes one of those objects blue? what makes it look blue?" - those are questions which could not be answered solely on the mathematical level, which doesn't even talk about blue, only about abstracted structural properties and abstracted causal roles. They could only be tackled in a fuller ontological context, where you say "this entity from the theory is an experience, this property is the property of being blue, this process is the experiencing of blue", and so on.
It's like the difference between doing arithmetic and talking about apples. You can count apples, and numbers can be calculational proxies for groups of apples, but apples aren't numbers and talking about numbers isn't really the same thing as talking about apples. These abstracted propositions would only belong to the mathematical part of a theory of causally efficacious physical qualia, and that's not the whole theory, in the same way that arithmetic statements about how many apples I have, are not my whole "theory of apples".
The "non-mathematical part" doesn't just include a series of verbal stipulations that "these abstractly characterized entities are 'experiences'". It implicitly also includes a bit of phenomenology: you would need to be able to single out various aspects of your own experience, and know that those are what is meant by the corresponding terms in the theory. You should be able to look at something blue and think, "OK, that's blue, that's property X from the formalism, and my awareness of blue, that's property X'...", and so on, for as far as theory and thought can take you.
That is a long-term ideal for a physical theory of consciousness; nothing we have right now measures up.
[Cross-posted.]
1. Defining the problem: The inverted spectrum
A. Attempted solutions to the inverted spectrum.
B. The “substitution bias” of solving the “easy problem of consciousness” instead of the “hard problem.”
2. The false intuition of direct awareness
A. Our sense that the existence of raw experience is self-evident doesn’t show that it is true.
B. Experience can’t reveal the error in the intuition that raw experience exists.
C. We can’t capture the ineffable core of raw experience with language because there’s really nothing there.
D. We believe raw experience exists without detecting it.
3. The conceptual economy of qualia nihilism pays off in philosophical progress
4. Relying on the brute force of an intuition is rationally specious.
Against these considerations, the only argument for retaining raw experience in our ontology is the sheer strength of everyone’s belief in its existence. How much weight should we attach to a strong belief whose validity we can't check? None. Beliefs ordinarily earn a presumption of truth from the absence of empirical challenge, but when empirical challenge is impossible in principle, the belief deserves no confidence.