Vera Entia : The Nature of Mathematical Objects in Descartes

Vera Entia: The Nature of Mathematical Objects in Descartes GREGORY BROWN L As DESCARTESBEGINSthe fifth of his Meditations, he is fast at work upon the groundwork for the ontological argument; he soon unveils a doctrine that forms the basis of his position concerning the nature of the objects of mathematics. Reflecting upon the ideas that he discovers in himself, Descartes claims to find among them countless "id6es de certaines choses, qui ne peuvent pas 6tre estim6es un pur n6ant, quoique peut-~tre elles n'aient aucune existence hors de ma pens6e, et qui ne sont pas feintes par moi, bien qu'il soit en ma libert6 de les penser ou ne les penser pas; mais elles ont leurs natures vraies et immuables. ''1 As an example of such an idea, Descartes considers that of the triangle, asserting that encore qu'il n'y ait peut-dtre en aucun lieu du monde hors de ma pens6e une telle figure, et qu'il n'y en ait jamais eu, il ne laisse pas n6anmois d'y avoir une certaine nature, ou forme, ou essence d6termin6e de cette figure, laquelle est immuable et 6ternelle, que je n'ai point invent6e, et qui ne d6pend en aucune fa~on de mon esprit; comme il paralt de ce que l'on peut d6montrer diverses propri6t6s de ce triangle, ~ savoir, que ses trois angles sont 6gaux h deux droits, que le plus grand angle est soutenu par le plus grand c6t6, et autres semblables, lesquelles maintenant, soit que je le veuille ou non, je reconnaise tr~s clairement et tr~s 6videment 6tre en lui, encore que je n'y aie pens6 auparavant en aucune fa~on, lorsque je me suis imagin6 la premi6re fois un triangle; et partant on ne peut pas dire que je les aie feintes et invent~es. 2 In considering the first of these passages, it is perhaps not altogether clear what it is that Descartes takes to be the possessor of the "nature vraie et immuable" that he mentions , the text being ambiguous between the idea and the thing that the idea represents? Anthony Kenny, however, has argued that it is the thing, and not its idea, that Descartes proposes as the owner of the true and immutable nature. 4 That this is the correct interpretation is surely borne out by what Descartes says in the second of the two passages quoted above--for it is clear that it is not the idea, but the triangular figure, which is said to possess the "essence d6termin6e." Further support for this interpretation is proOeuvres de Descartes, ed. Charles Adam and Paul Tannery, 12 vols. (Paris: Cerf, 1897-1910), 9, pt. 1:51 (hereafter cited as AT); The Philosophical Works of Descartes, ed. E. S. Haldane and G. R. T. Ross, 2 vols. (Cambridge: Cambridge University Press, 1931), 1: 179-80 (hereafter cited as HR). 2AT9, pt. I:51;HR 1:180. 3 As Anthony Kenny has observed, "the Latin and French texts are both ambiguous as to whether it is the ideas, or the things, that have true and immutable natures; hut is clearly of res, and not of ideae, that Descartes says "non tamen dici possunt nihit esse"'" (Descartes [New York: Random House, 1968], p. 150). Ibid. See atso Kenny, "The Cartesian Circle and the Eternal Truths," Journal of Philosophy 67 (Oct. 8, 1970) :692. [23] 24 HISTORY OF PHILOSOPHY vided by something Descartes says in response to an objection that was raised by Burman . In commenting upon the passages from the fifth Meditation, Burman had remarked : "Sed sic nec chimaera erit ens fictum, cum etiam de ea varias proprietates demonstrare queam. ''5 Descartes's response is revealing: Quicquid distincte et clare in chimaera concipi potest, illud est ens verum; nec est fictum, quia habet veram et intellectualem6 essentiam; et illa essentia tam a Deo est, quam actualis aliarum rerum. Dicitur autem tum ens fictum, cum nos illud existere supponimus. Sic etiam omnes demonstrationes Mathematirorum versantur circa vera entia et objecta, et sic totum et universum Matheseos objectum, et quicquid ilia in eo considerat, est verum et reale ens, et habet veram ac realem naturam, non minus quam objectum ipsius Physices...