Mathematical One and Many: Aquinas on Number

Abstract

401 The Thomist 78 (2014): 401-18 MATHEMATICAL ONE AND MANY: AQUINAS ON NUMBER1 D. SVOBODA and P. SOUSEDÍK Charles University Prague Prague, Czech Republic QUINAS’S CONCEPTION of number has attracted relatively little attention from historians dealing with medieval philosophy and mathematics. This lack of interest is due to two reasons. The first has to do with the fact that Aquinas’s conception of number derives mainly from the Aristotelian-Averroan tradition, whose representatives were not as concerned with mathematics and its philosophical problems as were, for example, the Pythagoreans or Plato.2 Aquinas therefore did not pursue this issue in a systematic manner, but only in the context of other, more immediate concerns. The other reason why this issue has not been very much explored thus far is closely related to the first one; as Aquinas worked out no systematic theory of number, we can only rely on marginal notes on this issue, making it difficult to arrive at a precise notion of his conception. Research in this field is by no means conclusive and presents a significant challenge to contemporary researchers of Aquinas’s thought. The present article attempts to respond to this challenge and to reconstruct 1 The work on this paper has been supported by the grant GAČR 13-08512S. 2 See J. A. Aertsen, Medieval Philosophy and the Transcendentals: The Case of Thomas Aquinas, Studien und Texte zur Geistesgeschichte des Mittelalters 52 (Leiden, New York, Cologne: Brill, 1996), 201-4. A 402 D. SVOBODA and P. SOUSEDÍK Aquinas’s conception of number, following up on the (neither very extensive nor intensive) investigation carried out so far.3 The most important accounts of the subject in Aquinas occur (1) in his commentaries on Aristotle’s Metaphysics and Physics, where Aristotle’s conception of the one and the many is laid out; (2) in numerous passages of Aquinas’s theological works, where the nature of unity and multitude is discussed; and (3) in theological treatises on the Trinity and angels, in which Aquinas deals with the problems whether and in what sense numerical terms can be predicated of God and whether angels constitute a multitude.4 In these texts, two mutually related approaches toward the problem may be encountered. Their common point of departure is the logical-ontological framework which Aquinas adopted from Aristotle. According to this framework, real created being divides into ten categories: the first category is being in its basic and primary meaning, that is, substance (ens in se); the other nine categories are being in its derived and 3 The fact that Aquinas did not work out a systematic conception of number probably explains the lack of secondary literature, which is as unsystematic as Aquinas’s reflections on this issue. On Aquinas’s conception of number, see D. Mercier, “L’unité et le nombre d’après Saint Thomas,” Revue Nèosc. de Philos. 8 (1901): 258-75; E. Bodewig, “Zahl und Kontinuum in der Philosophie des hl. Thomas”, Divus Thomas 13 (1935): 55-77. We have also drawn valuable information on Aquinas’s conception of number from L. Oeing-Hanhoff, Ens et unum convertuntur. Stellung und Gehalt des Grundsatzes in der Philosophie des hl. Thomas von Aquin, (Münster, 1953); T. O’Shaughnessy, “St. Thomas and Avicenna on the Nature of the One,” Gregorianum 41 (1960): 665-79; J. Owens, “Unity and Essence in St. Thomas Aquinas,” Mediaeval Studies 23 (1961): 240-59; P. C. Courtès, “L’un selon saint Thomas,” Revue Thomiste 68 (1968): 198-240; R. E. Houser, Thomas Aquinas on Transcendental Unity: Scholastic and Aristotelian Predecessors, (Ph.D. diss., The University of Toronto, 1980); Aertsen, Medieval Philosophy, 201-42; B. Blankenhorn, “Aquinas on the Transcendental One: An Overlooked Development in Doctrine,” Angelicum 81 (2004): 615-37; A. Maurer, “Thomists and Thomas Aquinas on the Foundation of Mathematics,” The Review of Metaphysics 47 (1993): 43-61. 4 See, e.g., IV Metaphys., lect. 2-4; V Metaphys., lect. 7-8; X Metaphys.; III Phys., lect. 5.; VII Physic., lect. 8; I Sent., d. 24, q. 1, aa. 2-3; STh I, q. 11; STh I, q. 30, a. 3 (“Utrum termini numerales ponant aliquid in...