“existence” itself may be a category error—not because nothing is real
If something is real, then something exists, yes? Or is there a difference between “existing” and “being real”?
Do you take any particular attitude towards what is real? For example, you might believe that something exists, but you might be fundamentally agnostic about the details of what exists. Or you might claim that the real is ineffable or a continuum, and so any existence claim about individual things is necessarily wrong.
qualia … necessary for our self-models, but not grounded in any formal or observable system
See, from my perspective, qualia are the empirical. I would consider the opposite view to be “direct realism”—experience consists of direct awareness of an external world. That would mean e.g. that when someone dreams or hallucinates, the perceived object is actually there.
What qualic realism and direct realism have in common, is that they also assume the reality of awareness, a conscious subject aware of phenomenal objects. I assume your own philosophy denies this as well. There is no actual awareness, there are only material systems evolved to behave as if they are aware and as if there are such things as qualia.
It is curious that the eliminativist scenario can be elaborated that far. Nonetheless, I really do know that something exists and that “I”, whatever I may be, am aware of it; whether or not I am capable of convincing you of this. And my own assumption is that you too are actually aware, but have somehow arrived at a philosophy which denies it.
Descartes’s cogito is the famous expression of this, but I actually think a formulation due to Ayn Rand is superior. We know that consciousness exists, just as surely as we know that existence exists; and furthermore, to be is to be something (“existence is identity”), to be aware is to know something (“consciousness is identification”).
What we actually know by virtue of existing and being conscious, probably goes considerably beyond even that; but negating either of those already means that you’re drifting away from reality.
During the next few days, I do not have time to study exactly how you manage to tie together second-order logic, the symbol grounding problem, and qualia as Gödel sentences (or whatever that connection is). I am reminded of Hofstadter’s theory that consciousness has something to do with indirect self-reference in formal systems, so maybe you’re a kind of Hofstadterian eliminativist.
However, in response to this --
-- I can tell you how a believer in the reality of intentional states, would go about explaining you and EN. The first step is to understand what the key propositions of EN are, the next step is to hypothesize about the cognitive process whereby the propositions of EN arose from more commonplace propositions, the final step is to conceive of that cognitive process in an intentional-realist way, i.e. as a series of thoughts that occurred in a mind, rather than just as a series of representational states in a brain.
You mention Penrose. Penrose had the idea that the human mind can reason about the semantics of higher-order logic because brain dynamics is governed by highly noncomputable physics (highly noncomputable in the sense of Turing degrees, I guess). It’s a very imaginative idea, and it’s intriguing that quantum gravity may actually contain a highly noncomputable component (because of the undecidability of many properties of 4-manifolds, that may appear in the gravitational path integral).
Nonetheless, it seems an avoidable hypothesis. A thinking system can derive the truth of Gödel sentences, so long as it can reason about the semantics of the initial axioms, so all you need is a capacity for semantic reflection (I believe Feferman has a formal theory of this under the name “logical reflection”). Penrose doesn’t address this because he doesn’t even tackle the question of how anything physical has intentionality, he sticks purely to mathematics, physics, and logic.
My approach to this is Husserlian realism about the mind. You don’t start with mindless matter and hope to see how mental ontology is implicit in it or emerges from it. You start with the phenomenological datum that the mind is real, and you build on that. At some point, you may wish to model mental dynamics purely as a state machine, neglecting semantics and qualia; and then you can look for relationships between that state machine, and the state machines that physics and biology tell you about.
But you should never forget the distinctive ontology of the mental, that supplies the actual “substance” of that mental state machine. You’re free to consider panpsychism and other identity theories, interactionism, even pure metaphysical idealism; but total eliminativism contradicts the most elementary facts we know, as Descartes and Rand could testify. Even you say that you feel the qualia, it’s just that you think “from a rational perspective, it must be otherwise”.