The problem with Linnean systematics vs Cladistics (in biological systematics) may be difficult to understand if we start from Darwin’s illustration of “the origin of species”, but becomes clearer if we start from what we actually do, that is, classify reality (in the context of the history of reality).
What we actually see in reality is not “species”, but singular entities we call organisms. These organisms do we sometimes have large difficulties to distinguish, like among coral “colonies”, funghi and different kinds of physically connected “clones”. Exactly how we distinguish them is not important, however, because they only serve as a baseline for further classification of them and their parts, since entities (like them) as a generic presumption consist of entities. It means that the concept entity (kind, or class) is orthogonal (diametrically opposed, or diagonal) to different kinds (or classes) of entities (compositions or parts of these baseline entities), ie, there is no single kind (class) of entities that equals the kind (class) entity, but instead there has to be at least two kinds (classes) of entities to equal the kind (class) entity. Classification simply excludes one kind (class) of entities that equals the kind (class) entity. This exclusion has also been demonstrated by Bertrand Russell as what is called Russell’s paradox.
Now, the concept (class, or kind) species actually denotes a hypothetical kind of entities that equals the concept (class, or kind) organism, that is, the baseline entities. This excluded kind of entities is thus none the less imaginable as a concept that equals the concept organism, that is, the baseline entities, although it actually contradicts this concept. The question is then: what shall we do with this concept?
Linné found the consistent solution to this question by arranging species in a hierarchical orthogonal system of classification wherein every level is orthogonal to the less inclusive level. This system is thus a solution not only to the problem of classification in biological systematics, but moreover to the problem of classification in general, although it is especially useful in biological systematics by the common ancestry of the entities in this discipline.
Willi Hennig, however, appears to have missed all nuances in this problem. Instead, he simply stated that “only monophyletic groups appears to be natural groups”, with monophyletic groups referring to a conflation of entities (in this case organisms) and species. He thus went straight into Russell’s paradox stating that only it appears to be natural groups. The problem with this conflation is that a paradox is an infinite recursion when looking for it. Hennig thus left Linné’s consistent solution of the problem for a dive into the paradox that Linné’s system avoided (ie, Russell’s paradox).
The fundamental problem is thus that classification excludes one kind (class) of entities that equals the kind (class) entity. The difference between Linnean systematics and cladistics does thus just reside in how they handle this fact: Linnean systematics acknowledges and avoids it, whereas cladistics denies and thus enters it.