A Formal Analysis of Phylogenetic Terminology:

BACKGROUND: For too many years the practice of systematics has been impeded by profound disagreements about the very foundations of this discipline, that is to say the type of information that should or should not be incorporated into a proper classification of life. Two main schools of systematics, both recognizing evolution, oppose each other: cladism states that the classification should only reflect the branching order of the lineages on the tree of life whereas evolutionism states that the length of the branches, that is the degree of modification, should also be taken into account so as to reflect macroevolutionary leaps. The first one forbids the exclusion of any descendant from a group that contain its ancestors, while the second one explicitly requires that the descendants too much different from their ancestors must be classified separately. Moreover, both schools often use the same words, such as “monophyly,” to designate different ideas. This prevents proper communication between the proponents of either side. Consequently, the research in phylogenetics is globally erratic and the taxonomic classification is highly unstable. RESULTS: I rigorously define the terms which designate the phyletic relationships and explore their properties through use of graph theory. I criticize a similar work (Kwok 2011) that was unable to properly catch these notions. This leads me to provide three independent arguments –– one historical, one utilitarian, and one morphosemantic –– in order to retain the original Haeckelian meaning of the term “monophyly” rather than the redefined Hennigian one. I identify some polysemy regarding the term “clade,” and that is why I define two new words, “holoclady” and “heteroclady,” to contrast respectively with “holophyly” and “heterophyly.” I also show that a strictly holocladic or holophyletic classification advocated by cladists is formally impossible. I therefore review and criticize the philosophical postulates subtending such an illogical paradigm. I show that cladism is part of a more general philosophical movement named structuralism, which is mainly characterized by anti-realism and a metaphysical way of thinking. I identify the biologically unrealistic assumptions on which cladism is based and argue that they have been empirically falsified. I therefore defend the use of paraphyletic groups in the scientific classification of life and review the main arguments that have been opposed to this solution. Some of them, such as anthropocentrism or the lack of an objective manner to determine paraphyletic groups, are grossly outdated, while others simply rest upon the difficulty in conceptualizing emergent phenomena. CONCLUSION: Since clades are still useful for methodological reasons, I offer a compromise that should make possible the coexistence of the two main opposing schools of systematics by eliminating competition between clades and taxa for the same names. I propose therefore that in a future revision, the BioCode should approve a dual system by recognizing both a “phyletic arrangement” made of clades and a “phylogenetic classification” made of taxa. Aubert: Formal analysis of phylogenetic terminology 2 “We often discussed his notions on objective reality. I recall that during one walk Einstein suddenly stopped, turned to me and asked whether I really believed that the moon exists only when I look at it.” (Pais 1979) CONTENT Introduction ............................................................................................................................................ 2 1. Genealogical Networks ...................................................................................................................... 3 2. Common Ancestors and Monophyly ................................................................................................. 6 3. Ancestral Groups and Monophyletic Unions ................................................................................... 10 4. Concepts Derived from Monophyly ................................................................................................ 12 5. Partitioning the Different Types of Groups ..................................................................................... 14 6. Phylogeny and Phylogenetic Classification ..................................................................................... 17 7. Notion of Clade ................................................................................................................................ 19 8. Cladograms, Cladification and Cladonomy ..................................................................................... 26 9. Establishment of the Current Paradigm ........................................................................................... 30 10. Analysis of the Cladist Doctrine .................................................................................................... 33 10.1. Structuralist Influence ............................................................................................................. 33 10.2. Paraphyletic Species ............................................................................................................... 34 10.3. Virtual Ancestors .................................................................................................................... 35 10.4. Uniformitarianism and Punctualism ....................................................................................... 35 10.5. Idealist Drift ............................................................................................................................ 37 10.6. Pitfall of Nominalism .............................................................................................................. 38 10.7. Evolutionist Solution .............................................................................................................. 39 Conclusion ........................................................................................................................................... 42 Acknowledgments ................................................................................................................................ 44 Annexe: Mathematical Proofs .............................................................................................................. 44 Literature cited ..................................................................................................................................... 50 INTRODUCTION Systematics has been marked for many decades by vigorous methodological debates concerning the classification of life, especially regarding the status of paraphyletic groups (Mayr 1974; Hennig 1975). These debates have annoyingly never reached a true consensus (Mayr & Bock 2002; Goldenfeld & Woese 2007). These disputes about the topological properties of taxa were interspersed with major terminological controversies (Ashlock 1971; Nelson 1971; Ashlock 1972; Nelson 1973) which plagued the discussions, even today blurring practice and progress in this discipline (Brummitt 1997; Cavalier-Smith 1998; Mayr 1998; Woese 1998; Ghiselin 2004; Brummitt 2006; Hörandl 2007; Podani 2010b; Zander 2011; Schmidt-Lebuhn 2012; Schmidt-Lebuhn 2014; Stuessy & Hörandl 2014; Brummitt 2014). Although both opposing sides have firmly stood their ground over the years, one quickly took the ascendancy over the other in both practice (Dumoulin & Ollivier 2013) and teaching (Lecointre et al. 2008) of systematics throughout the world. This relatively recent paradigm, which is called “cladism”, insists on the purely “genealogical” pattern that a natural classification of living things must have. According to it, taxa must be delimited through a unique formal principle: the inclusion of all descendants of the last common ancestor of its members (Hennig 1966). Cladism probably owes its success in part to the attractiveness of this imperative which is supposed to facilitate the decision making process and free the classification of life from all forms of subjectivity (Hennig 1966; Hennig 1975; Schmidt-Lebuhn 2012). Cladism unfortunately appeared in conjunction with a new method of resolving phyletic relationships: cladistics (Hennig 1950; Hennig 1966). Always presenting themselves as inseparably linked to each other, up to using the same word of “cladistics” to refer to one or the other (Ashlock 1974; Mayr 1974; Hennig 1975), cladism directly benefited from Aubert: Formal analysis of phylogenetic terminology 3 the popularity and effectiveness of this method. The opposing position, evolutionism, has therefore been described as “traditional” or “classical” to emphasize its purported obsolescence. Rejecting cladism would therefore be a synonym of rejecting cladistics, and therefore rejecting progress (Hennig 1975). This claim does not give a faithful image of modern evolutionary systematics which has fully integrated the contributions of cladistics and of subsequent molecular phylogenetic techniques to give birth to many fertile theories (Cavalier-Smith 2002a; Cavalier-Smith 2010a; Zander 2013). The purpose of this paper is primarily to rigorously analyse, with the aid of an appropriate mathematical formalism, the notions of systematics in order to objectively solve the terminological dispute which has been plaguing the theoretical discussions for half a century. On the basis of this common vocabulary, I intend on the one hand to reveal some logical inconsistencies arising from cladism, and on the other hand to decipher the philosophical foundations that underlie them. This will allow me to resolutely advocate abandoning sterile cladism and rallying evolutionary systematics. However, I will also propose a short-term workable compromise which would allow the establishment of a “peaceful coexistence,” or even cooperation, between practitioners of cladonomy and those of taxonomy. 1. GENEALOGICAL NETWORKS It is not the first time that one tries to formally define the phylogenetic concepts. Hennig himself modelled the succession of generations in sexual populations through directed acyclic graphs (Hennig 1966). However Kwok (2011) pointed out that “[w]here mathematical definitions have been presented [...], the properties of such concepts have not been explored”. This fact greatly limited their usefulness. Such an exploration was recently attempted (Kwok 2011), but it failed to adequately describe some key concepts because of the choice of bad axioms as I shall demonstrate. Curiously, the author nonetheless came to the same conclusion as I, concerning the impossibility of consistently reconciling the Linnaean hierarchy with cladism, but without realizing all the implications. In the course of this work, I will therefore emphasize the reasons justifying the better adequacy of my model to the biological reality. I will also insist on the logical consequences that prove the unreasonableness of the cladist imperative and of the assumptions that constitute its framework of thinking. Since this paper focuses on definitions and terminology, and does not have the ambition to unveil novel mathematical properties of directed acyclic graphs, I have relegated in the annexe the proofs of the various lemmas and theorems. Definition 1.1 (Genealogical Network) A genealogical network G is a pair (X, p) where X is a non-empty finite set and p is a binary relation on X such that p is acyclic, i.e. for every sequence x1, x2, ..., xn of elements of X, if n ≥ 2 and if for every i, 1 ≤ i ≤ n – 1, (xi, xi+1) ∈ p, then x1 ≠ xn. This definition of a genealogical network matches the one of a directed acyclic graph (or DAG), a common mathematical object in graph theory. As a consequence of this definition, it can be trivially drawn that (x, x) ∉ p for every x ∈ X. The set X embodies a population and the pair (x1, x2) ∈ p should be read as “x1 is a parent of x2” (see Figure 1). Definition 1.2 (Lineage) Let there be a genealogical network G = (X, p). A sequence x1, x2, ..., xn of elements of X is called a lineage if n = 1, or else if n ≥ 2 and (xi, xi+1) ∈ p for every i, 1 ≤ i ≤ n – 1. In particular, for any x ∈ X, the sequence consisting of the single element x is a lineage. Intuitively, a lineage is thus a line of descendants without gaps. Aubert: Formal analysis of phylogenetic terminology 4 Figure 1. A genealogical network without mergings. Each square represents an individual. The whole of the squares constitute the set X. The whole of the arrows represents the binary relation p defined on the set X (every arrow is an element of p). For example, x1 and y4 are individuals. The individual x1 is a parent of x2 (see Definition 1.1). The sequence x1, x2, x3 is a lineage (see Definition 1.2) and x1 is an ancestor of x3 (see Definition 1.3), but x3 is not an ancestor of x1 (see Lemma 1.6). The individual y1 is an ancestor of y2 which is itself an ancestor of y4, therefore y1 is an ancestor of y4 (see Lemma 1.5). Definition 1.3 (Ancestor) Let there be a genealogical network G = (X, p) and x ∈ X. An individual a ∈ X is an ancestor of x if and only if there exists a lineage x1, x2, ..., xn such that x1 = a and xn = x. Observation 1.4 Let there be a genealogical network G = (X, p), whatever x ∈ X, x is an