BMCR 2022.03.15

La méthode de division de Platon à Érigène

, , La méthode de division de Platon à Érigène. Annales de l'Institut de philosophie de l'Université de Bruxelles. Paris: Librairie philosophique J. Vrin, 2021. Pp. 192. ISBN 9782711629695 €18,00.

[Authors and titles are listed below.]

This collection of essays is the fruit of a collaborative project dedicated to the reception of the Platonic method of division. Aside from Plato and his first and foremost critic Aristotle, the schools and authors addressed are Hellenistic and Roman Stoicism, Middle Platonism, Plotinus, the late Neoplatonists Proclus and Damascius, and finally Eriugena, taking us to the threshold of medieval philosophy. The articles are all original contributions and generally of very high quality, written mostly in French, except for two articles in English (see Authors and Titles below). Since brevity is required, I will provide short descriptions for each chapter, with some critical comments here and there.

The basic idea of the method of division (diairesis) is to obtain definitions through a process of step-by-step division of a broad class or genus into more specific kinds until an account of what singles out the definiendum within this genus has been completed. As the two editors, Delcomminette and Van Daele, point out in their introduction to the volume, this method was not only mocked in ancient comedy, but also often dismissed as an idle exercise by modern scholars. In recent years, however, interest in the practice and theory of this method in Plato has noticeably grown. Since it has also found wide application in ancient and medieval philosophy after Plato, it is an important goal for future research to trace the history of its reception, a goal that the present collection aims to serve. In passing, I might mention that it would have been a welcome service to the reader if the introduction had included a sketch of the problems concerning the method of division as used by Plato, not least because we would like to know to what extent the course of its later reception was influenced by its inherent problems. The essays by Crivelli and Zaks alert us to various such issues in Plato, but do not aim to provide an overview.

Is it legitimate to divide a genus into two kinds such that the second kind is defined by negating what characterizes the first (e.g., beautiful vs. not-beautiful, footed animal versus animal with no feet)? Plato seems to make conflicting pronouncements on this issue in the Sophist and Statesman. This is the problem Crivelli tackles. Regarding negative kinds in the Sophist, he argues that a kind not-K corresponds to the class of all things that do not belong to the extension of K, which can also be called the “unrestricted complement of K.” This is based on his reading of Sph. 257b-258b, a passage that conceptualizes negative kinds as species of the genus otherness. (One can object that the examples there seem to suggest restricted rather than unrestricted complements, i.e., restricted to the superordinate genus, comparable to Aristotle’s notion of privation.) Now, whereas the Sophist suggests that negative predicables are legitimate kinds under the genus otherness, the Statesman criticizes the attempt to divide a genus into a kind K and its negative counterpart (here certainly in the sense of a restricted complement), and this has often been understood as amounting to a wholesale rejection of negative kinds. The main novel thesis of Crivelli’s paper is a proposal for how to avoid this apparent inconsistency, taking up a hint at Statesman 263b that prompts us to distinguish between a part and a subordinate species of a genus. According to his argument (a bit too complex to be laid out here), subordinate kinds or species have to be extensionally disjoint, but in many cases this would not be the guaranteed outcome if we were permitted to divide between a species and its complement. The upshot is that Plato allows for negatively defined kinds, but that division has to avoid them whenever this would fail to guarantee disjoint species. Crivelli adds a proposal concerning the notion of “dividing in the middle,” which he takes to mean that the corresponding classes ought to be “roughly equinumerous.” While this fits the examples in Plt. 262e, for many other instances of dichotomous division such a requirement would seem either irrelevant (as in the case of dividing between species in the animal kingdom) or impractical. What would it even mean, say, in the case of wool-working versus fulling (Plt. 282a) to require that the extensions be equinumerous? What are we counting here?

The primary critical response to Plato’s method of division comes from Aristotle. Yet Aristotle also makes extensive use of division and reflects on its metaphysical foundations. The chapter authored by Zaks prudently limits its scope to comparing and discussing those key texts in Aristotle’s logical and biological writings that explicitly reflect on division as a method, identify its shortcomings, and suggest improvements. The first part of the paper presents Aristotle’s objections to the method in its Platonic form. Zaks also quite convincingly points to possible Platonic rejoinders. The second part discusses Aristotle’s suggestion in APo II.5 as to how the method of division could, after all, be put to good use, viz., not as a proof that the proposed characterization captures the essence of the definiendum, but as a method for securing the correct ordering of the elements of the definition and their adequacy and exhaustivity. The author concludes with comments on a significant modification of the method proposed by Aristotle in PA I.3 in response to the practical constraints on division in zoology. He points out that this modification threatens to undermine the unity of the definition according to Aristotle’s own standards.

Division, understood as dividing among concepts, is important in Stoic philosophy as well, especially (but not only) for their ethics. How the Stoics of the Hellenistic era understood the method of division and its relevance is analyzed in the chapter by Gourinat, drawing from a rather meagre body of information on this topic in our testimonia. While this essay is usefully compared with pages 46–58 of the author’s La dialectique des stoïciens (Vrin: 2000), it offers more detail specifically on the Stoic conception of division. The author documents the close link that the Stoics saw between definition and division, notwithstanding the fact that not all Stoic definitions are obtained through division. After a discussion of two Stoic definitions of “definition” (Chrysippus, Antipater) and an explanation of how definition (horos) properly speaking differs from a preliminary account (hupographê) of a concept or preconception, the remainder of the essay deals with the Stoic classification of different kinds of division: division of a genus into its proximate species, anti-division (a form of bipartite division with the help of negation), subdivision, and partition according to different classes of relata. Gourinat detects here an original contribution of the Stoics. One could, however, object that all these kinds of division are not only exemplified in Platonic dialogues that use division and reflect on it, but that it is also likely that these methodological questions were extensively discussed in Plato’s Academy. It seems that the available sources do not allow us to tell whether the Stoic classification amounts to more than the creation of a new technical terminology, something Plato tended to avoid in his dialogues. This does not diminish the novelty and importance of their metaphysical and logical arguments against Platonic concept realism.

The essay by Colette-Duĉić investigates the role of “articulation” (a form of conceptual analysis) and “division” (as one of the methods of articulation) for the Roman Stoics Epictetus and Marcus Aurelius. The great ethical significance of concept-articulation for Epictetus lies in the fact that without it reason will not be able to fulfill its potential for true autonomy. In the case of Marcus Aurelius, the focus turns to dividing the objects of our impressions (i.e., the impressors or phantasta) into their smallest parts so as blunt the effect they might have in triggering “irrational judgments,” i.e., wrongly emotional responses to what we experience.

Bonazzi offers a condensed sketch of the concept of division and its place in dialectic according to the Didaskalikos of the Middle Platonist Alcinous. Drawing also on some other Platonist sources from the same era, the author reaches the surprising result that division is not only not suitable to grasp the highest divine principle, as Alcinous hopes, but does not even attain the separate Forms. Bonazzi also claims that Alcinous’ attempt to incorporate Platonic and Stoic conceptions into his description of division and analysis (the latter substituting, he thinks, for the Platonic notion of collection) leads to a confused result because of the unresolved tension between the underlying Platonic and Stoic ontologies. There are other ways of trying to make sense of the material in chapters 4 and 5 of the Didaskalikos. But one can certainly agree that the eclectic approach of Alcinous, combining ideas and statements from Platonic dialogues with conceptual frameworks taken from the Peripatetic and Stoic traditions, is unlikely to yield a consistent result. The Didaskalikos is nevertheless a very relevant text for the history of reception, not only because of the way in which it runs together Platonic, Aristotelian, and Stoic lines of tradition, but also in view of important echoes in the writings of the later Neoplatonist school.

The Plotinus chapter by Chiaradonna focuses on the very long and difficult treatise On the Classes of Being (VI.1-3). The author provides a thorough and learned discussion of what, according to Plotinus, is involved in distinguishing (1) between the intelligible and sensible domains, (2) among the intelligible realities, and (3) between the classes of sensible things. Against the backdrop of a specifically Aristotelian conception of the conceptual and ontological relations to be traced by the method of division, the author reaches the surprisingly negative result that for Plotinus division plays no important role at all in the classification of beings. One might object that this Plotinian treatise investigates ontological relations, whereas “division (diairesis)” in Plato is an investigative method to be used in the search for definitions of specific kinds. The use of this method is open to different ways of reconstructing the underlying ontological relations between genera and species.

Tresnie traces the complex (but also questionable) strategies used by Proclus in his attempt to integrate Platonic and Aristotelian models of scientific inquiry and to turn the apparent circularity of the Platonic method of division that Aristotle objected to into a kind of virtue. Van Daele’s essay looks at passages in the Philebus commentary and the De principiis of Damascius in order to describe how this Neoplatonist of the early sixth century conceives of the constructive role of division as one of the methods of dialectic, while also highlighting its incapacity vis-à-vis the first principle. One of the new themes that emerge here, according to the author (pp. 160, 169), is how division is no longer just a method of inquiry, but also an ontological process characterizing the human soul. The final contribution to this collection, by Delcomminette, traces the breathtaking philosophical vision of John Scotus Eriugena. This solitary philosophical figure of the Carolingian era elevates division and dialectic from being a method of investigation to being the process of world creation originating from God through the mediation of the human intellect, requiring human collaboration to attain its ultimate soteriological fulfillment.

The collection is sold as a nicely edited and printed paperback for a very affordable price. It should be relevant for everyone interested in the concept and history of the philosophical method of division.

Authors and titles

Introduction / Sylvain Delcomminette & Raphaël Van Daele
Negative Kinds in Plato’s Statesman / Paolo Crivelli
Les réflexions théoriques d’Aristote sur la méthode des division / Nicolas Zaks
La division dans l’ancien stoïcisme et ses applications éthiques / Jean-Baptiste Gourinat
Division es articulation chez Épictète at Marc Aurèle / Bernard Colette-Duĉić
Alcinous on the (Platonist) method of divisions / Mauro Bonazzi
La division dans les traités de Plotin Sur les genres de l’être / Riccardo Chiaradonna
Rôle et place des méthodes de division dans les Commentaires de Proclus / Corentin Tresnie
La dialectique résolutive dans le contexte de la quête des principes chez Damascius / Raphaël Van Daele
La division créatrice dans le Periphyseon de Jean Scot Érigène / Sylvain Delcomminette