When we classify things into several levels of classes, ie, in terms of classes of classes and so on, each level is orthogonal to its neighboring level(s) because every other level represents things and classes, respectively. It means that such classification is ultimately paradoxically contradictory between things and classes, because things and classes are orthogonal, and the levels of the classification ultimately conflate them. The classification simply ultimately conflates things and classes by not keeping them apart.
This problem can be avoided by only recognizing classes as levels, ie, an orthogonal system of classification, like the “species”, “genera”, “family” (and so on) levels in Linnean systematics, because such a system avoids conflating things and classes by only classifying classes. This solution does not solve the fundamental orthogonality between things and classes, but just avoids conflating it. (The problem with an orthogonality is that it lacks a consistent middle, and whereas the middle in the naive classification is contradictory, it is instead empty in an orthogonal system of classification. Unfortunately, it can’t be unambiguous, because things and classes are orthogonal.)
This problem is further complicated when some of us inconsistently begin to believe that classes are real (like cladists and particle physicists). Then, the matter is no longer about reaching a consistent classification, but about how to find the non-existing unambiguous middle in the orthogonality between things and classes. For the rest of us, this stupidity turns the matter into a question of how we shall handle these people. They obviously don’t understand the problem, so how shall we handle people that don’t understand the problem and moreover don’t want to understand it? This is the nut we have to crack to save science from being eradicated by belief.