# Classification, and its two orthogonal interpretations

The things in a process of continuous dichotomous splitting of such things (as illustrated below), like a cell lineage, can be classified in two orthogonal ways: as kinds OF kinds, eg, Linnean systematics, and as kinds IN kinds, eg, cladistics.

The difference between these two orthogonal ways to classify such a process with regard to the things is generically that OF classifies them consistently, ie, do not conflate “thing” and “class” by not allocating less than two things to any class, whereas IN classifies them inconsistently, ie, do conflate “thing” and “class” by allocating single things to single classes, and specifically in this example that OF thus allocates A and B to one class, C and D to another class, and all of them to one class, whereas IN allocates A and B to one class, A, B and C to another class, D to one class, and all of them to one class, thus making D and C ambiguous (or in the opposite aspect, contradictory) between “thing” and “class”.

The difference between these two orthogonal ways to classify this kind of process with regard to the properties of the things is that kinds OF kinds do not conflate the properties of things (by not conflating “thing” and “class”), whereas kinds IN kinds do conflate the properties of things (by conflating “thing” and “class”).

The conflation in the latter is difficult to understand. The problem is, however, that not only things are hierarchically nested, but also their properties, and the conflation of “thing” with “class” in this kind of classification allows paradoxical contradiction between properties so that a single thing may be, for example, large if it is small and small if it is large (see Russell’s paradox). However, the fact that a single thing can’t be a paradox means that this kind of classification in practice can’t find a consistent solution of the properties of the things. That is, if we know the process, then we can apply this kind of classification (thus including things with paradoxically contradictory properties), whereas if we don’t know the process, then we can’t find any consistent such classification, ie, we’re lost in a consistent inconsistency, as cladists are.

These two kinds of classification differs only when the classified process is asymmetrical. When it is totally symmetrical (ie, when every split gives rise to two splits), they meet in the same classification. Unfortunately, reality refuses to meet this criterion. The problem concerning “evolution” is, however, not only to distinguish reality (ie, things) from our abstractions of it (ie, classes), but also to specify “evolutionary units”, which thus moreover leads into paradoxical contradiction if we succeed.

Classification thus appears to be more of a barrier than a bridge to understanding. Mathematics has however found a way around this problem which can paint reality, but can’t explain it (except mathematically). Unfortunately, this explanation is not expressible in simple terms.