chapter 8 Theon of Smyrna: Re-thinking Platonic Mathematics in Middle Platonism Federico M. Petrucci Broadly speaking, an interest in mathematics characterizes the history of Platonism.1 However, it manifests itself in different ways. The most familiar approach involves the use of the “potentialities” of mathematical objects as ontological items. Indeed, this is a typical element of Platonic instruction from the Old Academy to Neoplatonism, rooted both in philosophic historiogra- phy and in various Platonic passages (of Parmenides, Republic, Timaeus): from Xenocrates to Eudorus, from Plutarch to Proclus, ontological principles were often identified with notions somehow borrowed from mathematics, such as the One and the Dyad. This does not imply, however, that such notions main- tained their mathematical status: rather, in consequence of their “re-use” as ontologically defined items, they were eventually transfigured, while their fundamental properties (e. g. extreme simplicity of the one/monad) were con- sidered in a new perspective. Accordingly, this use of mathematics (including arithmological speculation)2 is somehow non-mathematical, and will not be the focus of this paper.3 However, two approaches clearly show a direct com- mitment to mathematical (i. e., arithmetical, geometrical, musical and astro- nomical)4 doctrines as such, whose goal was to outline (if at different levels) a 1 See O’Meara (1989); Dörrie & Baltes (1993), 266–79; Fowler (1999); Giardina (1999); Radke (2003). 2 Arithmological speculation, somehow arising in ancient Pythagoreanism and recovered in the Old Academy, underwent fundamental developments in the Hellenistic age, e. g. Mans- feld (1971), was practiced by later Platonists, and in some context was used in the exegesis of Plato’s psychogony to discover the “potentialities” of the numbers employed by Plato. See Petrucci (2012), 408–27. 3 Exceptions involve those authors, such as Nicomachus and Theon, who ascribed to numbers (qua numbers) a specific ontological status; in this case, arithmology could be a means to indicate “qualities” instantiated by number-intelligibles. 4 It is noteworthy that classifications of mathematics in antiquity – not only those based in the quadrivium-structure – chiefly depend on Plato’s account in Rep.VII. See Vitrac (2005). For mathematics based in the quadrivium-structure see Hadot (2006). © Koninklijke Brill NV, Leiden, 2018 | doi 10.1163/9789004355385_010 144 Petrucci “Platonic system of mathematics”: some Platonists5 (including Nicomachus of Gerasa) wished to establish a certain model of mathematics by applying their own Platonic perspective to the task of writing technical treatises. Others (in- cluding Theon of Smyrna) took the dialogues (especially Timaeus) as starting points to develop such a system exegetically. In this paper, priority is given to the latter group, for reasons which shall emerge in due course. I Nicomachus: “Platonically Orientated” Mathematics In the Old Academy a deep and variegated interest in mathematics found fertile ground, from both a theoretical and a technical point of view. While the latter is exemplified by Eudoxus’ seminal researches, it is likely that arith- mogeometry was also systematically formulated in this context:6 numbers are geometrically structured aggregates of monads, and are provided with related properties and reciprocal relationships. Traces of such an approach virtual- ly disappear until the first centuries of our era, when Nicomachus of Gerasa wrote his Introductio Arithmetica. This work, which soon became a reference text, is at face value a comprehensive handbook of arithmogeometry, although this entailed further important topics, such as an exposition of ancient theory of ratios and proportions.7 Its historical relevance mainly depends on the fact that it establishes a complete and systematic survey of (partly non-Euclidean) arithmetic, namely arithmogeometry. In this way, Nicomachus locates himself in an established tradition: the Old Academy’s legacy had probably been recov- ered at least as early as Moderatus of Gades,8 and Theon of Smyrna opens his Expositio with an arithmogeometrical survey. In the next centuries Platonists wrote works inspired by Nicomachus’ (such as Iamblichus’ In Nicomachi Arithmeticam, Domninus’ Encheiridion, Asclepius’ and Philoponus’ Commen- taries).9 But Nicomachus applied a similar approach also to music: he wrote 5 I shall avoid the category of “Neo-Pythagoreanism”. See Centrone (2000). 6 See Burkert (1972), 401–47, esp. 446–7. 7 Most relevant studies are D’Odge (1938); Bertier (1976); O’Meara (1989), 14–23; Dillon (1977/1996), 352–61; Mansfeld (1998), 82–91; Giardina (1999), 36–52; Radke (2003), pas- sim; Helmig (2007). As to its posterity, besides related writings, it is telling that according to Marinus (Vit. Procl. 28) Proclus considered himself to be a reincarnation of Nicomachus. 8 See Centrone and Macris (2005) for a survey. Important mathematical interests must be ascribed also to Thrasyllus, on whom see Tarrant (1993). 9 On Domninus see Riedleberger (2013); on Iamblichus’ In Nicom. see Vinel (2014); on Philo- ponus’ Commentary see Giardina (1999)..