There are two traditional problems associated with colors. One is the sort that pseudo-philosophical douchebags take to: “Dude, what if no one really sees the same colors?” The other was very popular in the heyday of classical analytic philosophy: how can we say that Red is Not-Blue analytically if they are empirical & presumably a posteriori data?
Let’s assume for the sake of getting to the real argument that consciousness arises from physical matter in a manner uncontroversial for the materialist. Granting this, why do we all see the same colors, if we do?
The short answer is that we probably don’t. I don’t even see with the same level of clarity that someone with 20⁄20 vision does, at least not without the help of my glasses, which themselves introduce a level of optical distortion not significant to my brain’s processing but certainly significant in a [small] geometric sense.
A quicker way to get at the fact that we probably don’t see quite the same way is to point out that dogs’ eyes aren’t responsive to certain colors which most human eyes can distinguish quite easily. This leads directly to the point that there is probably enough biological variation (& physical deterioration over someone’s lifetime) that we don’t end up with quite the same picture of the world, even though it’s evidently close enough that we all get along all right.
This also leads to the strongest argument (for empirical scientists anyhow) that we do all see roughly the same thing: we’ve got pretty much the same sensory organs & brains to process what is roughly the same data. It seems reasonable to expect that most members of a given species should experience roughly the same picture of the world.
So much for the first problem, at least in brief & from a pragmatic point of view. The skeptical philosopher must admit that this is a silly problem to demand a decisive answer to.
As for the problem of distinguishing between colors analytically, of determining a priori the truth of empirical statements, a mathematical concept is quite helpful, particularly if we’re willing to grant that colors are induced by a spectrum of wavelengths which the eye can perceive. But even if we don’t grant that last fact, introducing the notion of a partition suffices to distinguish the perceived colors (or qualia) inasmuch as it also divides up the spectrum of wavelengths which induce those colors.
Note that this doesn’t help us escape the fact that we require experience to learn of the various colors & the fact that they form a partition, but that isn’t the crux of the problem to begin with. In the same way that we can learn what a round table is & deduce that it is a table analytically, once we become acquainted with the colors & their structure—that is, once we understand the abstract rules governing partitions—we can make analytic claims based only on that structure we understand, and not requiring any further empirical data, or really even the empirical components of the original data.
The well-ordering principle doesn’t really have any effect on canonical orderings, like that induced by the traditional less-than relation on the real numbers.
This doesn’t affect the truth of your claim, but I do think that DanArmak’s point was quite separate from the language he chose. He might instead have worded it as having no real solution, so that any solution must be not-real.