Linnean systematics and cladistics are just two different kinds of classifications of dichotomously propagating processes

Linnean systematics and cladistics are just two different kinds of classifications of dichotomously propagating processes: Linnean systematics accepting the fact that the classification of such processes can't be unambiguous, and cladistics instead searching for unambiguous classifications of such processes (ie, not accepting the fact that classification of such processes can't be unambiguous).

These two kinds of classifications are thus just different (orthogonal) opinions on whether classification of dichotomously propagating processes can be unambiguous or not, whereof cladistics is wrong.

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.

The problem with clades

The problem with clades is that each of their internal branches (ie, between nodes) consistently represents two entities: one descendant and one ancestor, because it means that these branches consistently contain changes between character states and thus can possess mutually exclusive properties. This fact is difficult to interpret, but among other things, it means that a "true" clade, like the idea of a "true tree of life", is shorter than the "most parsimonious" such tree. However, since this state is impossible, the practical implication of the fact is instead that there are several just as parsimonious trees. The fact thus means that the idea of a "true" clade in practice is a matter of gray scale rather than of black and white. Every bifurcating process that has to be represented with at least one internal branch does in fact offer several just as parsimonious tree illustrations, all of which are contradictory. There is thus no consistent tree illustration of any bifurcating process that has to be represented with at least one internal branch to be found.

This impossibility may be difficult to understand, but the alternative is to search for something that can't be found.  

On the paradoxical contradiction of the class "clade"

The concept "clade" terms a class of entities (ie, "ancestors including all their descendants"). This class of entities appear "natural" to some biological systematists, but is none the less paradoxically contradictory (ie, the subjective aspect of Russell's paradox).

This fact can be understood by considering that the fact that every class can be a member of another class (like how humans is a member of primates) means that every clade also can be a member of another clade (ie, that every clade contains member clades). Now, if there among all clades should be a single clade that is not a member of any other clade (like the idea "a true tree of life"), then this clade of clades must also equal (be the same as) each and every of its member clades, and thus exclude the possibility of any other clade of clades besides It (ie, exclude the existence of more than one clade of clades), thereby contradicting the fact that there are several clades per definition. Ie, if there are several clades, then there is no single clade, and vice versa.   

The class "clade" thus actually excludes the possibility of single instances of itself by excluding the possibility of clades that are not members of other clades.

The same explanation can be given in terms of entities: the fact that every entity can be a member of another entity (like how a cell in my body is a member of me) means that every clade of entities also can be a member of another clade of entities (ie, that every clade of entities contains member clades of entities). Now, if there among all those clades of entities should be a single clade that is not a member of any other clade (like the idea "a true tree of life"), then this clade of clades must equal (be the same as) each and every of its member clades and thus exclude any other clade of clades besides It (ie, exclude the existence of more than one clade of clades), thereby contradicting the fact that there are several clades per definition.

The class "clade" is thus actually just a mental circularity inside of Russell's paradox, that is, the subjective aspect of Russell's paradox. It actually lacks a single unambiguous solution (but has several ambiguous solutions in orthogonal systems of classification like the Linnean systematics).

On the difference between Linnean systematics and cladistic classification

Both Linnean systematics and cladistic classification classify a bifurcating process in a generic aspect (ie, all specific aspects at the same time). The difference between them is that:

Linnean systematics is a consistent (ie, non-contradictory) orthogonal system, whereas cladistic classification is inconsistent (ie, contradictory).

It means that Linnean systematics can incorporate the generic aspect of a single bifurcating process, whereas cladistic classification can only produce specific (ie, contradictory) classifications.  

On the meeting point between objective and subjective classification (e.g., Linnean and cladistic classification, respectively)

Classification ends in Russell's paradox or the class clade, depending on whether it starts from objects or classes, which thus is the meeting point (i.e., interface) between objectivity and subjectivity. This point is actually the orthogonal opposite to object, and is thus a pure abstraction arising from classification itself. The reason that this point is contradictory is that it is doubly ambiguous, both in time and over time, at the same time, which is contradictory (i.e., a paradox).

The existence of this end point in classfication excludes the existence of a single consistent and unambiguous classification (like, for example, the cladistic idea "The Tree of Life"), instead meaning that classification can only be either ambiguous or contradictory. Objective classification (like Linnean classification) is consistent but ambiguous, whereas subjective classification (like cladistics) is "natural" but contradictory. It means that typology (the belief that classes are real) in practice is an eternal merry-go-round between ambiguous or contradictory alternatives. The difference between classes is in practice fundamentally not qualitative, but quantitative (ie, not a matter of black or white, but of gray scale).

On the choice between the Linnean system of classification and Cladistic classification in Biological systematics

The difference between the Linnean system of classification (like Evolutionary taxonomy) and Cladistic classification in Biological systematics is actually fairly simple to understand. It resides in that classification as an orthogonal system, i.e., wherein each class consists of two orthogonal classes, has two logically consistent, but orthogonal lines of reasonings starting from either the bottom level (i.e., objects, therefore called Objectivity) or one level up (i.e., classes of objects, therefore called Subjectivity). Objectivity is ambiguous with respect to the classified (i.e., objects) and ends in a paradox (in this case called Russell’s paradox), whereas Subjectivity is consistently contradictory (i.e., with respect to both objects, classes and facts, and between assumption and deduction) and likewise ends in a paradox (here called the class clade). The two end points for the two lines of reasoning are actually one and the same point, namely the opposite to their orthogonal starting point, arriving to it from their respective orthogonal line of reasoning.

The problem for us to understand this difference is due to that an orthogonal system also is orthogonally 2-dimensional, that is, an orthogonal series (or orthogonal stack) of 2-dimensional planes, which is infinite. Such orthogonally nested dimensional structure can be simplified into a single 2-dimensional system in two different ways, which are consistent with each of the two orthogonal lines of reasoning respectively: orthogonally as in the Linnean system, which is consistent with Objectivity, and "flat" as in Cladistic classification, which is consistent with Subjectivity.

It means that the choice between the Linnean system of classification and Cladistic classification actually is a choice between being ambiguous with respect to objects (i.e., the Linnean system of classification) or being contradictory with respect to both objects, classes and facts, and between assumption and deduction (i.e., Cladistic classification). In this light, the choice between the Linnean system of classification and Cladistic classification appears fairly given.

Cladistic classification is different from a simple "flat" classification only in applying a consistently contradictory classification of classes. Its reference for this kind of classification to be "natural" is nothing but an appeal to the subjectivity in us all. Unfortunately, subjectivity in this version only leads to the Wonderland (i.e., where everything is up-side-down). Also unfortunately, cladists can’t understand this fact because they presume that classes are real.