""Why do I think I have free will?" "Because I do," is a perfectly good answer (assuming free will)."
Disagree. Notice how this answer is only 'good' assuming free will. But our assumption of free will is exactly what we are seeking to understand the cause of. We can assume free will is correct and that this is adequate to justify our answer ('because I do'), but then we have only re-posited the assumption in the consequent.
"No one knows what science doesn't know."
This sort of anthropomorphic bias leads to conceptual errors. 'Science' is the method of acquiring knowledge and the collection of acquired knowledge to which the method is rigorously applied. It is incapable of knowing anything independently of what individuals know; in fact, it can't know anything at all without some knowing individual to practice it. And to be sure, we can know things 'science doesn't know': we know we are in love, that we are happy or sad, that we played baseball for the first time when we were 6 years old at the park in Glens Falls, etc.
"For example, presumably light of different colors is identical to light of different wavelengths."
More specifically, lights of identical wavelengths have identical colors, and vice-versa. Clearly, "waves = colors" is not a valid statement of equality ('color' is an epiphenomenon of wavelengths arising as a percept in a sensory being, while the wavelengths the mind converts into colors exists independently of any observers). A wave is a wave, and a color is a color, and these two properties have a direct relationship upon which the equality or inequality of these properties in some group of objects can be ascertained.
Why does a = a?
The existence of 'reality' is just a logically circular argument of the form: 'what is real exists because it really exists.' There is no reason to prove the existence of reality; we prove or disprove the existence (or non-existence) of things in reality. We are able to falsify the existence of things by this method due to the satisfied precondition of a reality in which things exist,
Of course we could be experiencing some manufactured illusion, but this still necessarily implies some reality in which this illusion can be constructed. Our ability to experience this illusion would suffice to prove that we are real, since all experience is that of an experiencing being or object. This objective experiencing being must exist in relation to some other second existent object of experience. But then we must posit a third object in relation to which both of these two - the object of experience and the experiencing object - are experienced in turn, and so on. The notion of a purely subjective idea-being without an objective reality or existence is absurd - an idea or sensation not inhering in any object in reality has no body in which the subjective being of even an illusion or misapprehension arises. To be aware is to be aware of something.
In fact, the very notion that we could or are living in a false reality to which our minds are inextricably enslaved is more or less religious superstition: all existence as we experience it would require a designer or predetermined purpose who constructs this illusion, all of the sensory objects in this subterfuge, and all of the laws of nature to which the non-existent hallucinations are consistently obedient that this designer must have arbitrarily laid out in advance.
"Tracing back the chain of causality, step by step, I discover that my belief that I'm wearing socks is fully explained by the fact that I'm wearing socks. This is right and proper, as you cannot gain information about something without interacting with it."
Maybe I'm being pedantic on this point, but doesn't the interaction with the socks constitute the act of putting them on which actually fully explains that you're wearing them? Of course, you can go back further along the causal chain to the reason you put them on - perhaps the room was cold, or you had to put on boots to go outside. Regardless, the more we regress backwards on this chain, the farther from the change in state from Socks_off to Socks_on would we find ourselves in the causal sequence of events leading up to your wearing them.
That being said, wouldn't the circular explanation - I'm wearing socks because I'm wearing socks - actually be the explanation arrived at as we approach the causal chain's resulting state of affairs where we're wearing socks instead of not wearing socks? The 'new information' is actually gained by the interaction with the socks preceding our wearing them or our resulting need to explain this phenomenon.
A useful trick to optimize the accuracy of your priors when considering examples is to semantically disentangle each example in the form of a conjunctive statement. This allows you to avoid the conjunction fallacy, where an example, verbally stated, holds only when many smaller statements conjoined by an and or several them are all true.
By doing this, we can also compare examples rather than treating them as all having the same probability value.