fredag 9 december 2011

On the relation between "clade" and "Russell's paradox"

The term (i.e., class) clade is contradictory: if clades can be unambiguously distinguished, then entities (like organisms or biological species) can't, because clades are incompatible with entities. If, thus, clades are entities, then they are contradictory entities, which, thus, can't be unambiguously distinguished. Clades thus can't be unambiguously distinguished if they are entities (and, of course, neither if they are not entities).

This contradiction is defined in Wikipedia in terms of Russell's paradox as (slightly modified by replacing set with clade):
(The axiomatic definitions are that any definable collection is a set (i.e., naive set theory) and that clade is a set). So, let R be the clade of all clades that are not members of themselves (i.e., the ultimate clade of clades, or the True Tree of Life). Now, if R qualifies as a member of itself, it contradicts its own definition as a clade containing all clades that are not members of themselves. On the other hand, if R does not qualify as a member of itself, it qualifies as a member of itself by the same definition. R does thus neither qualify as a member of itself nor as not a member of itself. This contradiction is a true paradox (i.e., with two impossible and contradictory states) called Russell's paradox.

Symbolically:

\text{let } R = \{ x \mid x \not \in x \} \text{, then } R \in R \iff R \not \in R

The term clade thus denotes the same "thing" (i.e., contradiction) as Russell's paradox, that is, the interface between objectivity and subjectivity, An entity of this kind is thus just a passage between objectivity and subjectivity. The difference between it and an entity is analogous to the difference between a passage through a door and the door. Never can they fuse, and the door is, obviously, a paradox. .    

So, what does this paradox tell us? Existentially that history is inherently contradictory, theoretically that our distinctions can't be found, and practically that theorists can screw matters up all the way to contradiction. What it definitely does not tell us is The Truth. On the contrary.   
 

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