Tuesday, June 19, 2007

The structure of qualia space

I have already hinted at the idea of platonic versus access qualia spaces. The access qualia spaces are essentially windowed versions of access spaces where the windowing is carried out both with respect to the intensities as well as the dimensions, or rank-lowering projection operations. The former is a non-linear operation, the latter a linear one.

But one wonders about the structure of the platonic qualia space itself. This would be the mother of all qualia spaces. For example, we know that human vision is trichromatic. But does this mean there are only three different independent color axes in the platonic qualia space itself? As I had previously hinted at, many birds have tetrachromatic vision, so does this mean there is a fourth color axis that is not accessible by us humans? If that is indeed the case, then can there be a fifth, sixth, and so on? Are there a finite number or infinite? And do there exist operators between the various axes of the platonic color space?

The reason for hinting at operators is because, if we take the eigenhues of pitches, there exists a "pitch comparison" operator that orders the eigenhues (in the case of sound, any pure pitch is an eigenhue, and there are an infinite number of them accessible to us humans), and we can also, by corollary, construct eigenhues out of existing ones (like an octave above, a semitone below and so on). The relational operator forms the basis of music theory, and we will not have any sense of melody if this operator were not accessible to us. I call this "operator access", since it is an operator (which should also be present in the platonic pitch space) which is accessible to us on "this" side. I wonder if the eigenhues corresponding to colors have some relational operator on the platonic side, but which is inaccessible to us. Without the operator access of pitch comparison, any melody would be akin to a light show without much meaning.

The very fact that we have the accessible pitch operator implies that the platonic qualia space for pitches is infinite since we can have an infinite number of octaves either way by applying the pitch operator successively. In terms of standard pitch theory, pitches can be viewed as a helix that is a function of frequency that, because of human limitations, run out at the limits of 20Hz and 20KHz. But the mere presence of the operator implies that the platonic helix is infinite in extent, both up, and down.

The pitch operator is akin to a real number operator like multiplication. If an operator were to exist for colors, depending on the exact mathematical structure of the color eigenhues, it may be a discrete one like for modulo arithmetic. But we don't know, since access of such operators, if any, is forbidden, and we don't have any clues at our disposal.