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A sixteenth-century Neoplatonic synthesis: Francesco Piccolomini's theory of mathematics and imagination in the Academicae contemplationes


The metamathematical framework of the early modern period is primarily determined by two presuppositions stemming from the Aristotelian tradition: (1) mathematical objects are abstracted from sensible matter; (2) imagination is a reproductive faculty exclusively connected with the sensible realm. The recovery of the works of the Greek commentators confronted the early modern readers with rivalling philosophical–mathematical views that explicitly called into question some of their previously undisputed assumptions. In this article I will argue that Francesco Piccolomini (1523–1607) in his Academicae contemplationes brings about an original fusion of these colliding horizons, by transposing the synthesis established by (?)Simplicius between Aristotelian abstractionism and Neoplatonic innatism into the sixteenth century.

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1 See Claessens Guy, ‘Imagination as self-knowledge: Kepler on Proclus’ Commentary on the First Book of Euclid's Elements’, Early Science and Medicine (2011) 16, pp. 179199, 181–183.

2 The authorship of the De Anima commentary was already questioned by Francesco Piccolomini himself (In tres libros Aristotelis de anima lucidissima expositio, Venice, 1602, f. 216r), and is still disputed. Bossier Fernand and Steel Carlos, ‘Priscianus Lydus en de “In de anima” van Pseudo(?)-Simplicius’, Tijdschrift voor Filosofie (1972) 34(4), pp. 761822, French summary 821–822, attribute it to Priscian. For a different view see Hadot I., Le problème du néoplatonisme alexandrin: Hiéroclès et Simplicius, Paris: Etudes augustiniennes, 1978, pp. 193202. Since Steel's and Bossier's arguments contra Simplicius' authorship are convincing – while their pro-Priscian arguments are still much debated – I refer to the commentary's author as ‘(?)Simplicius'.

3 Kraye Jill, ‘Francesco Piccolomini’, in Kraye (ed.), Cambridge Translations of Renaissance Philosophical Texts, 2 vols., Cambridge: Cambridge University Press, 1997, vol. 1, pp. 6888, 68.

4 Commentarii duo, prior in libros Aristotelis De ortu et interitu, alter in tres libros eiusdem De Anima, Frankfurt, 1602.

5 See the bibliography of Piccolomini in Schmitt Charles and Skinner Quentin (eds.), The Cambridge History of Renaissance Philosophy, Cambridge: Cambridge University Press, 1988, p. 861.

6 Piccolomini Francesco, Stephani Theupoli Bened. F. Patric. Veneti, Academicarum Contemplationum Lib. X. in quibus Plato explicatur & Peripatetici refelluntur, Venice, 1576.

7 Enzo Baldini Artemio, ‘Per la biografia di Francesco Piccolomini’, Rinascimento (1980) series ii, 20, pp. 389420, 399–401.

8 Luigi Lollino, Francisci Piccolominei, Iacobique Zabarellae praestantium nostrorum temporum philosophorum vitae, cc. 34v–35r, quoted in Baldini, op. cit. (7), p. 399.

9 Lollino, op. cit. (8), cc. 34v–35r.

10 Garin Eugenio, Storia della filosofia italiana, Torino: Eunaudi, 1978, p. 659.

11 Baldini, op. cit. (7), p. 400.

12 See, for example, Garin, op. cit. (10), pp. 658–659, who refers to ‘una professione di fede platonica’ and calls Piccolomini a ‘platonico convinto’.

13 Spruit Leen, Species Intelligibilis: From Perception to Knowledge. 2: Renaissance Controversies, Later Scholasticism, and the Elimination of the Intelligible Species in Modern Philosophy, Leiden: Brill, 1995, pp. 239240.

14 See Garin, op. cit. (10), p. 658; and Motta Uberto, Antonio Querenghi (1546–1633): Un letterato padovano nella Roma del tardo Rinascimento, Milan: Vita e pensiero, 1997, p. 22.

15 Piccolomini, op. cit. (6), p. 12.

16 Piccolomini, op. cit. (6), p. 12.

17 Piccolomini, op. cit. (6), p. 12.

18 Piccolomini, op. cit. (6), p. 13.

19 Piccolomini, op. cit. (6).

20 Piccolomini, op. cit. (6), pp. 81–115: I. An Ideae sint. II. Inventores Idearum. III. Quid sint Ideae. IV. Cui conveniant Ideae. V. Quomodo in mente se habeant & consurgant Ideae. VI. Convenientia et differentia Idearum. VII. Quorum non sint Ideae. VIII. Utrum mathematicarum formarum dentur Ideae. IX. Quorum sint Ideae. X. Quos usus praebeant Ideae. XI. An Aristoteles in disputatione de Ideis cum Platone debeat conciliari.

21 Piccolomini, op. cit. (6), p. 100.

22 Piccolomini, op. cit. (6), pp. 100–101: ‘Ex adverso Proclus inter platonicos nemini secundus, in primum Euclidis manifeste asserit, rationibusque confirmat, rerum mathematicarum dari Ideas; & iure quidem: nisi enim daretur Idea recti, Idea circuli, Idea lineae; haec nullibi invenirentur, & evanescerent mathematicae scientiae; cum in materia nec rectum sine obliquitate, nec circulus exacte rotundus, nec linea sine latitudine inveniatur.’ See Proclus, in Eucl., 12, 19–26.

23 Piccolomini, op. cit. (6), p. 101.

24 On Proclus' concept of the geometrical imagination see Nikulin Dmitri, ‘Imagination and mathematics in Proclus’, Ancient Philosophy (2008) 28, pp. 153172.

25 Piccolomini, op. cit. (6), p. 101. Piccolomini probably refers to (?)Simplicius, in de Anima, 233, 7–30.

26 Piccolomini, op. cit. (6), p. 101. The corresponding passage is (?)Simplicius, in de Anima, 277, 1–6.

27 Piccolomini, op. cit. (6), p. 102.

28 Piccolomini, op. cit. (6), p. 102.

29 Piccolomini, op. cit. (6), p. 102.

30 Piccolomini, op. cit. (6), p. 103. Piccolomini adds the sentence ‘and therefore the Ideas are able to display the images of mathematical objects in the imagination’ (& propterea valent Ideae imagines mathematicorum in phantasiam expromere). I cannot, however, explain in which way the Ideas themselves could display projections in the imagination, except indirectly.

31 Piccolomini, op. cit. (6), p. 103.

32 Nikulin, op. cit. (24), p. 158. See Proclus, in Eucl., 6, 7–7, 7.

33 Piccolomini, op. cit. (6), p. 104.

34 Piccolomini, op. cit. (6), p. 118 ff.

35 Both Iamblichus' De secta pythagorica and Syrianus' commentary on Metaphysics M and N were known in the early modern period. Ficino, for example, translated Iamblichus' work (preserved in MS Vat. lat. 4530) probably before 1464; see Celenza Christopher, ‘Pythagoras in the Renaissance: the case of Marsilio Ficino’, Renaissance Quarterly (1999) 52, pp. 667711, 692–693. Syrianus' commentary was translated by Hieronymus Bagolinius: Syriani antiquissimi philosophi interpres in II, XIII et XIV libros Aristotelis Metaphysices commentaries, Venice, 1558. Piccolomini mentions Syrianus' work in his Naturae totius universi scientia perfecta atque philsophica, p. 399 – I follow the second edition, Frankfurt, 1628.

36 O'Meara Dominic, Pythagoras Revived: Mathematics and Philosophy in Late Antiquity, Oxford: Clarendon Press, 1989.

37 Iamblichus, Comm. Math., 63, 23–64, 19. See O'Meara, op. cit. (36), p. 79.

38 Iamblichus, Comm. Math., 64, 4–6.

39 Syrianus, in Metaph., 112, 31 ff. See Dillon John and O'Meara Dominic (trans.), Syrianus: On Aristotle's Metaphysics 13–14, London: Duckworth, 2006, p. 5.

40 Dillon and O'Meara, op. cit. (39), p. 6.

41 O'Meara, op. cit. (36), p. 135.

42 Syrianus, in Metaph., 88, 7–9 and 123, 19–20.

43 O'Meara, op. cit. (36), p. 134.

44 Piccolomini uses the uncommon adjective animarius as equivalent for the Greek ψυχικὸς, since, according to Syrianus, this number belongs to the soul.

45 Syrianus, in Metaph., 133, 4–15.

46 O'Meara, op. cit. (36), p. 133.

47 See Mueller Ian, ‘Syrianus and the concept of mathematical number’, in Bechtle Gerald and O'Meara Dominic (eds.), La philosophie des mathématiques de l'Antiquité tardive, Fribourg: Ed. universitaires, 2000, pp. 7183, 77: ‘Syrianus first points out that the material component of numbers never appears separately from the formal component (just as unformed matter never occurs in the ordinary world)’.

48 Mueller, op. cit. (47), p. 78.

49 Piccolomini, op. cit. (6), p. 268 ff.

50 Piccolomini, op. cit. (6), p. 269.

51 Piccolomini, op. cit. (6), p. 270.

52 Piccolomini, op. cit. (6), pp. 352–353: ‘Mathematica, quae certissima dicitur scientiarum, minime est scientia: tum quia non de intelligibilibus, sed de sensibilibus est: tum insuper, quoniam non traditur per principia scientiae … Promuntur quidem in facultatem imaginandi formae illae, quae a mathematico considerantur, ab anima per rationes; ut cum Proclo etiam Simplicius manifestavit; attamen Mathematicus rationes eas non cognoscit, nec ad eas, & longe minus ad ideas elevatur: aliud enim est cognoscere per rationes, & cognoscere rationes'.

53 Proclus, in Eucl., 55, 5 ff.

54 Proclus, in Eucl., 54, 27–55, 6: ‘For the understanding contains the ideas but, being unable to see them when they are wrapped up, unfolds and exposes them and presents them to the imagination sitting in the vestibule; and in imagination, or with its aid, it explicates its knowledge of them, happy in their separation from sensible things and finding in the matter of imagination a medium apt for receiving its forms’. Cf. Proclus, in Eucl., 141, 4–7: ‘Therefore just as nature stands creatively above the visible figures, so the soul, exercising her capacity to know, projects on the imagination, as on a mirror, the ideas of the figures; and the imagination, receiving in pictorial form these impressions of the ideas within the soul, by their means affords the soul an opportunity to turn inward from the pictures and attend to herself’. All translations from Proclus’ commentary on Euclid are from Morrow Glenn R. (tr.), Proclus: A Commentary on the First Book of Euclid's Elements, Princeton: Princeton University Press, 1992, with some modifications.

55 Piccolomini, op. cit. (6), pp. 344–345.

56 Piccolomini, op. cit. (6), p. 345.

57 Motta oversimplifies matters when he states that Piccolomini simply copies Proclus’ system; see Motta, op. cit. (14), p. 30.

58 (?)Simplicius, in de Anima, 214, 5–11: ‘αὐτὴ ἡ φαντασία ἐγείρεται πρὸς τὴν κατ’ οἰκείους λόγους τῶν φανταστικῶν τύπων προβολὴν οἰκείως τοῖς αἰσθητικοῖς εἴδεσιν, οὐχ ἅπαξ ἢ δὶς ἐνίοτε μόνον τῶν αἰσθητικῶν δεηθεῖσα τύπων ἀλλὰ καὶ πλεονάκις πρὸς τὴν τῶν ὁμοίων προβολήν. ἐγερθεῖσα δ’ οὖν καὶ ἀφ’ ἑαυτῆς προβάλλει τὰ φαντάσματα καὶ τυποῖ καὶ διαμορφοῖ τὸ σχῆμα ἀφ’ ἑαυτῆς ἢ ὁμοίως τοῖς εἴδεσι τῶν αἰσθητῶν ἢ καὶ τὸ ἀκριβὲς προστιθεῖσα, ὡς καὶ τὴν ἀπλατῆ προτείνουσα εὐθεῖαν καὶ σχήματα τὰ ἀκριβέστατα’.

59 Piccolomini, Peripateticae de anima disputationes, p. 146.

60 Piccolomini, op. cit. (59), p. 155: ‘quia ex se cum quadam libertate novas imagines formare valet, sumpto tamen semper initio ab illis quae sub sensum cadunt’: ‘because imagination is able, with a certain liberty, to form new images from itself, albeit always starting from things that fall under sense perception’.

61 See Claessens Guy, ‘Clavius, Proclus, and the limits of interpretation: snapshot-idealization versus projectionism’, History of Science (2009) 47, pp. 317336, 322–326.

62 Claessens, op. cit. (61), pp. 319–322.

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