Fundamental facts about classification and systematics (and cladistics)

Bertrand Russell showed 1901 that classification is paradoxically contradictory, ie, internally inconsistent, see Russell's paradox. This inconsistency is immediately due to that classes contain two kinds of classes: finite classes (ie, objects and categories) and infinite classes (ie, abstract types). The reason is that each finite class must correspond to an infinite class (ie, that each real class must correspond to an abstract class), that is, that there must be a one-to-one correspondance between finite classes and infinite classes, because it means that the total number of classes must be even, which, in turn, is impossible, since the relation between finite classes and infinite classes is orthogonal (ie, diametrically opposed), and that their numbers thus differs with one, because it means that their total number always is odd. It is thus impossible to obtain a one-to-one correspondance between finite classes and infinite classes in classification. The situation is like putting a puzzle where the last piece always is redundant.

However, using an orthogonal system of classification (ie, classifying objects into finite classes of infinite classes), like the Linnean system, transfers this internal inconsistency into a an ambiguity between classification and the classified. It means that such system, on the contrary, can't be inconsistent, ie, that every possible such system is consistent. The reason is that the numbers of finite classes and infinite classes in such system differs with one, because it makes their total number odd per definition, which neutralizes the paradoxical contradiction (ie, internal orthogonality) in classification.

These two kinds of classifications are the only kinds of classification there are. It means that the cladistic idea "a single true tree of life" is inconsistent per definition, since it requires that classification is consistent. If we, like cladists, don't acknowledge the fact that classification is inconsistent, but instead erroneously claim that classification can be consistent, then we actually just transfer the inconsistency of classification into our own heads (into our logical reasoning), thereby turning us, ourselves, inconsistent, (if we weren't before) like cladists are.

This internal inconsistency of classification can we not get rid of, but can only transfer into other positions, ie, to between classification and the classified or to our own heads, because it is fundamentally due to, or is the reason for, the ever-changing nature of reality. At this fundamental level it is impossible to distinguish cause from effect. (This, in turn, may be due to that cause and effect are orthogonal, and that a beginning actually is lacking. If so, change just follows the tracks it is bound to follow, but according to certain principles. Principles rule, but they continuously conflict, and the result is a compromise. A beginning is in any case impossible to invoke without transferring the inconsistency of classification into our own heads, ie, to our logical reasoning).

On the limits for classification (and the practical impossibility of cladistic classification)

The fact that classification is orthogonal, ie, that every single class contains several classes within every single set of entities (eg, the class "primates" and all different classes of primates), and that finite class thus is orthogonal to infinite class, means that classification is fundamentally contradictory between single and several, since orthogonalities are contradictory between single and several per definition. Cladistic classification accepts this contradiction, ie, accepts being contradictory, by equalizing the concepts finite class and infinite class.

The only way to avoid this orthogonal contradiction, ie, to achieve consistency, in practical classification is to use an orthogonal system of classification, classifying entities into categories of classes, like the Linnean system, in order to thereby keep finite classes and infinite classes (ie, the orthogonal concepts finite class and infinite class) consistently apart. Such orthogonal system of classification is, however, ambiguous in relation to reality per definition, simply by keeping the orthogonal concepts finite class and infinite class) consistently apart.

It means that classification can only be either internally contradictory or ambiguous in relation to reality. The fundamental reason for this impossibility to achieve unambiguity is that classes can't be (and thus aren't) real per definition, but can only be (and thus are) an artificial invention.