The characteristic properties of methods of numerical systematics are considered by its supporters as advantages, while its opponents consider them shortages; the sources of the controversy of these standpoints are briefly discussed below.
The formalized character of its apparatus, which makes it possible to algorithmize the classification procedure, is thought of (or at least presented) by its supporters as the objectivity of both this apparatus and the results obtained with it. In this case, objectivity is understood as an alternative to subjectivity, i.e., as independence from (or minimal dependence on) a subjective factor, including subjective speculations about the background metaphysics. This general declaration is incorrect because of the complete irrelevance of the philosophical category of objectivity to this issue. To begin with, axiomatic foundations of highly formalized methods are arbitrary to a greater or lesser degree and therefore not “objective” [Kline 1980; Perminov 2001]. Then, a strong formalization of the quantitative methods leads to their “emancipation” from cognized reality, which seems true for the results obtained with them, as well. In this regard, it is quite meaningless to apply the category of “objectivity,” in its realistic meaning, to the assessment of the results of numerical systematics, i.e., to speak of their “truth” as a correspondence to what is implied to be “a matter of fact” [Ruse 1973; Rasnitsyn 2002; Pavlinov 2018]. This is just what Albert Einstein asserted, with all his professional comprehension of the subject of mathematical physics: “to the extent that the proposals of mathematics relate to reality, they are not reliable; insofar as they are reliable, they are irrelevant to the reality” (cited after [Vollmer 1975: 28]). Therefore, from a philosophical perspective, as far as the methods and results of numerical systematics are concerned, it is more correct to speak not about their objectivity, but about their intersubjectivity in its rather technical (methodological) meaning [Smaling 1992], i.e., about agreement in the results obtained by different researchers applying similarly organized methods to similarly organized raw data.
The thesis about the empirical nature of numerical systematics is also hardly true. Philosophically, the emphasis on methods makes the entire numerical approach rational rather than empirical. Numerically based research involves analysis not of the natural objects themselves, but their cognitive models in form of a combination of formalized variables (characters). In such an operation of ideation, the natural object disappears, and with it disappears empiricism with its focus on an object as such.
Indeed, a “sample centroid” in numerical systematics, just as an “ideal blackbody” in physics, is but a kind of semblance of a Platonic eidos; more precisely, not an eidos itself, but rather its specific numerical representations.
The formalization and algorithmization of numerical methods are firmly associated with their accuracy and repeatability; here the problems are as follows. The accuracy of each such method is not universal and not “absolute,” but local and “relative”: it is specified only for certain formalizations that serve as a rationale just for this particular method, and may not be so for any other formalizations [Williams and Dale 1965; Shatalkin 1983]. The same is true for repeatability, which is associated with the above-mentioned intersubjectivity: it is fulfilled only under certain standard conditions of solving standard tasks. Thus, both these “advantages” relate to the method as a “thing in itself,” and not to the substantive content of the research tasks, with respect to which at least “accuracy” turns out to be a fake.
The inevitable variety of numerical methods gives rise to a serious methodological problem of the reasonable choice of a particular method as a means of solving a particular research task. An important part of this problem is the requirement to define the basis for this choice; it is the same as a choice among the logical grounds considered above (see Section 3.5). The central point here is the definition of criteria for assessing the consistency (effectiveness) of numerical methods. Proponents of mathematical systematics customarily validate a method by reference to a well-founded mathematical theory underlying it; as was emphasized above, such a rationale makes each method consistent only with respect to its own initial formalizations. However, from a biological (substantive) perspective, this is not enough: “as applied to the natural sciences, any mathematical method makes sense not in itself, but in connection with the purpose for which it is used” [Shatalkin 1983: 52]. As far as the meaningfulness of the entire taxonomic research is determined within the context of some basic substantive theory, the consistency of the method should be assessed within the context of this theory; this is the main provision of the epistemic principle of methodical correspondence (see Section 3.7).
Since different methods lead to different classifications, this generates specific taxonomic uncertainty’: the cumulative result of applying different “exact” methods to the same raw data turns out to be very fuzzy in admitting different particular taxonomic solutions [Sneath and Sokal 1973; Pavlinov and Lyubarskiy 2012]. All this is far removed from the unambiguity that is anticipated by the ordinary users of numerical methods attracted by the slogans of the latters’ proponents.
The orientation of numerical systematics towards the method as such, and thus towards de-ontologization of the taxonomic research, plunges the latter into the already-mentioned fundamental epistemic problem of instrumentalism [Rieppel 2007; Pavlinov 2018] (see Section 3.2.1). The latter means that the quality of classifications is determined not through their compliance with the structure of taxonomic diversity, which they are designed to represent as their cognitive models (the condition of realism), but through the formally substantiated quality of the methods themselves (the condition of instrumentalism). With this, an accentuation on the method as such results in an “inverse correspondence”: it appears that it is the method as such that dictates how an Umwelt should be analyzed, so the properties of the former indirectly shape the properties of the latter.
Leaving aside an “anti-mathematism” inherent in intuitionist researchers and their objections, the following important point should be emphasized. Biological systematics solves substantial tasks; the numerical program develops formal methods for solving these tasks. This means that the numerical theory as such does not have an independent meaning for systematics: certain formal results that it produces, like any other formalisms, are subject to biological interpretation, so it is the latter that should serve as the basis for meaningful taxonomic conclusions [Moss and Hendrickson 1973; Pavlinov 2018]. It is noteworthy that J. Gilmour, one of the first and leading ideologists of the positivist treatment of systematics, pointed out that fascination with quantitative methods may bring the illusion that taxonomic research reaches a conclusion with the results of their application; in fact, it merely begins with them [Gilmour 1961].
Generally speaking, the modern mathematization of biological systematics is much more than just its “digitization” limited to the employment of certain quantitative methods in solving certain classification tasks. As a matter of fact, it plays a very significant role as an epistemic regulator of many aspects of research activity in this discipline; in fact, it leads to the latter’s formalization, operationalization, de-ontologization, de-subjectivation, etc. [Hagen 2003; Sterner 2014]. Considered from this perspective, numerical systematics provides no fewer problems than it pretends to solve, and these problems turn out to be quite fundamental. The current controversy around the numerical research program in systematics is caused by a shift in the scientific paradigm: the post-positivist conception of systematics significantly differs from the positivist one in understanding the structure of cognitive situation, including substantiation of both taxonomic knowledge and its scientific status. Those simple solutions offered by numerical approaches seem hardly adequate to the complexity of the entire problem of comprehending and describing taxonomic reality. Accordingly, at present, the most urgent task is to realize this inadequacy in order to assess properly both the results obtained and possible prospects of the development of this research program of systematics as the biological discipline.