Phantasia and Mathematical Projection in Iamblichus

A. Sheppard: Phantasia and Mathematical Projection113 Phantasia and Mathematical Projection in Iamblichus1 Anne Sheppard Proclus in his commentary on Euclid gives fa?tas?a, "imagination," an important role to play in mathematics. My aim in this paper is to investigate the background to this idea and to see how far it was already present in Iamblichus.2 I shall begin with the relevant passages of Proclus since they are the best known testimony to the connection between fa?tas?a and mathematical projection. In two passages in particular (In Primum Euclidis Librum Commentarius 51.9-56.22 and 78.20-79.2) Proclus expounds the idea that when we are doing geometry the figures about which we are thinking are projections in the imagination of innate intelligible principles.3 If we are thinking about a circle, for example, we are thinking about a figure with extension and shape. Attributes such as extension and shape cannot belong to the intelligible principle of circularity. While such intelligible principles remain the ultimate objects of mathematical thought, ordinary geometry deals neither with these principles nor with the extended, imperfectly circular shapes found in the physical world. Its objects have an intermediate status which Proclus explains by describing them as projections in the fa?tas?a. Another, relatively well-known 1 Translations from Proclus, In Primum Euclidis Librum Commentarius follow the translation by G.R. Morrow (Princeton 1970) except that I render ????? by "principles" rather than "ideas." Translations from other texts are my own. 2 For earlier discussions of this issue, see W. Beierwaltes, Denken des Einen (Frankfurt 1985) 256ff., I. Mueller, "Mathematics and Philosophy in Proclus' commentary on Book 1 of Euclid's Elements" in Proclus, Lecteur et Interprète des Anciens, J. Pépin and H.D. Saffrey, eds. (Paris 1987) 305-18, DJ. O'Meara, Pythagoras Revived: Mathematics and Philosophy in Late Antiquity (Oxford 1989) 132-34, and HJ. Blumenthal in Aristotle and the Later Tradition, HJ. Blumenthal and H. Robinson, eds. (Oxford 1991) 199. 3 For discussion of these passages, see G. Watson, Phantasia in Classical Thought (Galway 1988) 119-21; O'Meara (above, note 2) 166-69. 114Syllecta Classica 8 (1997) passage compares the figures projected in the imagination to images reflected in a mirror: 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 principles 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. It is as if a man looking at himself in a mirror and marvelling at the power of nature and at his own appearance should wish to look upon himselfdirectly and possess such a power as would enable him to become at the same time the seer and the object seen. In the same way when the soul is looking outside herself at the imagination, seeing the figures depicted there and being struck by their beauty and orderedness, she is admiring her own principles from which they are derived; and though she adores their beauty, she dismisses it as something reflected and seeks her own beauty. (In Primum Euclidis Librum Commentarius 141.2-19)4 Fa?tas?a is also compared to a mirror at In Primum Euclidis Librum Commentarius 121.2-7: And thus we must think of the plane as projected and lying before our eyes and the understanding (d?????a) as writing everything upon it, the imagination becoming something like a plane mirror to which the principles of the understanding send down impressions ofthemselves. It has already been pointed out, particularly by O'Meara in Pythagoras Revived: Mathematics and Philosophy in Late Antiquity, that the same ideas are found in Syrianus. Syrianus' commentary on Aristotle's Metaphysics M includes, not surprisingly, a good deal of argument defending the Platonist view that the objects of mathematics exist independently of particulars and are not, as Aristotle thought, simply derived from abstraction (?fa??es??). At 91.11-92.10 Syrianus argues that alongside the Platonic Form of Largeness in the d...