The root of the problem is that I don't have access to any other phenomenological experiences. I have no direct evidence of how a bird or a tree or a rock experiences their corner of spacetime. It can be described as how to map physical/measurable activity to experiences, but that doesn't actually help if we can't detect/measure the experiences in order to have any clue whether our mapping is correct.
If it's just what someone claims, we have a pretty good mapping - neurons fire, vocal chords vibrate. If it's what it feels like to be them, we don't even know what we're mapping to.
the fact that we have access to our own minds first
I have fairly limited access to my own mind. I give most humans the benefit of the doubt that they do as well, but I can't really be sure.
I mean, I somewhat agree, but I think this is just a less precise way of saying what I said in the post.
Like, I'm trying to explaining how, even if we assume other peoples account of their own experience are faithful, which give us indirect access to others experience (which is different from no access), and allows us to pin down some of the structure of other people subjective experience, the mapping is still impossible to construct even in principle, because the data of the physical world doesn't fully determine either mapping.
This is a somewhat stronger claim than the one you're making.
This feels similar to a gauge transformation in physics, where there seems to be an extra degree of freedom that cannot be eliminated. But even within the same mind there is still a problem with qualia: Take the visual field for example. Imagine reflecting your visual field as in a mirror. Would anything change in the external world? I would argue this would also be a perfectly viable mapping of the external world to phenomenal states, and yet, it would seem to imply that the reflected visual field is isomorphic to the original field. It would seem that this experiment can be done in reality, and although I haven't tried it myself, it is reported that people quickly adjust to this reflected field, and somehow the brain "realizes" the correct orientation of the visual field.
But what does this say about the qualia of spatial orientation? It would seem that I myself would not be able to distinguish whether or not I am seeing a reflected field or the original field. This isomorphism problem seems to lie within the same mind, and not between minds. Both fields seem to map onto a "canonical" representation of spatial orientation that represents all isomorphisms of the same field.
I don't know anything about gauge transformations, but you might be right. The canonical (ha) example I had in mind was category theory, where universal objects are defined up to isomorphism. But where you can have many different realizations of that universal object. Like a vector space over ['red', 'blue', 'orange'] is isomorphic to one spanned by ['red', 'blue', 'green'], but they're not literally the same object.
As for the reflected visual field example, its kind of an instance of what I'm talking about. But, it would have to be done at birth, else its not an automorphism of mental states, because you're not flipping the other mental structures that relate to the visual field.
Like when I'm talking about a map from physical -> phenomenal states, I'm talking about the map from all the stuff making up your body and entering your sense, to your mental states. Not just the part of the map thats determined by what enters your senses.
People talk of the hard problem of consciousness, and have conflicting intuitions. So I've tried come up with a characterization of what I consider to be the "hard part" of the "hard problem of consciousness". The core theoretical issue that makes the issue puzzling and seemingly impossible to solve.
So when were solving the hard problem of consciousness, what are we doing in the most general sense? We're trying to come up with a theory that gives a mapping between physical states* and which phenomenal mental states /qualia arise. Concretely a function:
Now, how do we hope to construct this function? Firstly, we assume peoples reports of their own mental states are correct. This gives a function:
Now, what do we do? We can sample . We can look at the physical state someone is in, and see what they end up talking about.
I think this type of analysis cannot possibly work. The fundamental issue is that his is not enough information to determine f. We can only recover f (and g) 'up to isomorphism' in some sense.
This is where you get thought experiments like "if we have had the same color-qualia except my red was your blue and my blue was your red since birth, would we know?". This alteration would lets call it would apply to f, but it would be cancelled out exactly by being applied to . Eg we'd get ..
This seems to me to be a problem you're gonna get any time you try to account for consciousness purely in terms of physical states, no matter what method you use.
*I think the same characterization is valid for other ontologies, but requires more words to state.
**verbal descriptions
If we assume phenomenal states and physical states are not ontologically different things, but phenomenal states just are a special configuration of physical states, we can sample g and f directly, and the problem disappears.
My response is basically the Democritus quote
Statement: "By convention sweet and by convention bitter, by convention hot, by convention cold, by convention color; but in reality atoms and void."
Response: "Foolish intellect! Do you seek to overthrow us, while it is from us that you take your evidence?"
I think from an epistemic perspective, the fact that we have access to our own minds first, means, even if the reductive physicalist picture was 'true' in some sense, it would still give us have the same isomorphism problem. I can explain more but I'll wait until someone actually raises this objection.