Russell's paradox and Hennig's spin of it (ie, cladistics)

Bertrand Russell demonstrated in 1901 that classification leads to contradiction (called Russell's paradox), which can be exemplified with Barber's paradox.

Barber's paradox read (from Wikipedia):
Suppose there is a town with just one barber, who is male. In this town, every man keeps himself clean-shaven by doing exactly one of two things:

1. Shaving himself, or
2. going to the barber.

Another way to state this is:

The barber shaves only those men in town who do not shave themselves.

All this seems perfectly logical, until we pose the paradoxical question:

Who shaves the barber?

This question results in a paradox because, according to the statement above, he can either be shaven by:

1. himself, or
2. the barber (which happens to be himself).

However, none of these possibilities are valid. This is because:

• If the barber does shave himself, then the barber (himself) must not shave himself.
• If the barber does not shave himself, then he (the barber) must shave himself.

(end of the citation from Wikipedia).

All men in this town thus keep themselves clean-shaven, but whereas some shave themselves, the rest goes to the Barber, and the Barber himself consistently belongs to the other of these two groups if he behaves as defined for any of them. The Barber is thus consistently contradictory between the definitions of the two groups. This is an example of an infinite recursion. Every categorization of the Barber points to the alternative categorization in an infinite loop.

The generic reason for this kind of contradiction (ie, Russell's paradox in a generic sense) is that classification functions by distinction of difference in a similarity, since difference and similarity are diametrically opposed, ie, orthogonal, per definition, because orthogonality is contradictory per definition (in this case between the differences of the similarity). This inherent contradiction of classification is, however, invisible for us as long as it is restricted to only one of the two sides of the fundamental distinction, that is, to either process or pattern, since we then comprehend it as a similarity (rather than a difference and thus a contradiction), but becomes visible as soon as it bridges this fundamental distinction by that it then is not similar (as we intuitively expect), but instead opposite - ie, there is indeed a difference between process and pattern, but this difference is not in a similarity, but in an opposition, ie, what the Barber does is in opposition to each of the definitions of what the Barber can do.

This fundamental orthogonality of classification is also the driving force for the never-ending change in biological systematics. It is the obstacle that hinders it from reaching an unambiguous classification. However, about 50 years ago the German entomologist Willi Hennig escaped this fundamental contradiction by turning the orthogonality up-side-down. He simply asserted that only groups such as the Barber and the categories "those men in town who do not shave themselves" and "those men in town who do shave themselves" are "natural groups". He thus "acknowledged" the "difference part" (ie, contradiction) of the orthogonality and only it (ie, "denied" the "similarity part" of it). By this, he also "acknowledged" the paradoxical applications of this contradiction.

Hennig's irrational move was never accepted for publication by any scientific journal, but was instead published as a book in the 1950-ies, from which it was dragged into biological systematics in a joint effort by Steve Farris and Gareth Nelson in the 1970-ies, then won a popularity contest against consistent traditional science in the 1980-ies, and does today penetrate the thinking in biological systematics (under the name "cladistics") to the degree that the discipline is actually searching for such paradoxes (called clades) in a shared belief in the existence of a single "True Tree of Life". This imaginary "True Tree of Life" is thus actually Russell's paradox in disguise.

So, how does Hennig's move work? The answer emerges when we consider the fact that the Barber in the paradox represents the abstract similarity of a classification, whereas the two categories represent the real different parts of the classification. This consideration allows us understand that Hennig's grouping of the two categories into the Barber folds the difference itself between the real (different) parts of a classification back into its similarity, thus merely running the process of classification backwards, ie, canceling it. By this, it merely legitimates a comprehension that any classification is a "natural group". It thus does not solve the problem (ie, fact) that classification is fundamentally contradictory (and that every classification thus is contradictory per definition), but merely legitimates a comprehension that any classification is a "natural group". It does not change the fact that classification is fundamentally contradictory, but merely acknowledges contradiction, and only contradiction, as "natural groups".

The problem with this approach is that the fact that classification is orthogonal (and thus ultimately leading to Russell's paradox) means that it ultimately leads to infinite recursion of the same kind as that of Barber's paradox, today called clades, wherein every specific classification thus points to another classification in an infinite loop. (This ought not come as a surprise, since only acknowledging contradiction can't, of course, find a non-contradictory solution). The approach is thus merely a spin of classification into infinite recursion, although presently a mass-spin in biological systematics, which in practice is analogous to that like the hamster enter the running around in his wheel instead of using it as a wheel.
A consistent system of classification using an orthogonal arrangement of classes, like the Linnean system, avoids its fundamental contradiction and thus also infinite recursion, although neither it can reach unambiguity in relation to reality. Russell's paradox is namely, unfortunately, a fact we can't change.