Mathematical Demonstration and Deduction in Descartes's Early Methodological and Scientific Writings

Mathematical Demonstration and Deduction in Descartes's Earl,YMethodological and Scientific Writings DOREN A. RECKER THERE ARE NUMEROUSPASSAGESin Descartes's writings in which he seems to advocate a strong mathematico-deductive approach to physical science (e.g., AT, x: 144; 2: 268; 3: 39, 297-98; 6:19 [CSM, x: 12o-21]; 8-a: 78-79 [CSM, 1: 247]; 9-b: xo-ll [CSM, l: 184-85]; xo: 364-66 [CSM, 1: 1x-x3]; ll: 47 [CSM, l: 97]). ~These and similar passages have fueled many attempts to fit Descartes's scientific program onto one or another logico-deductive or mathematical model. On the other hand, it has long been realized that Descartes also appealed frequently to experiential verification within his scientific works." Further, the context in which Descartes attempted to respond to a number of competing models of the proper foundation for science was complex. Even his more familiar metaphysical and epistemological writings (which have been traditionally taken as buttressing a strict deductivist model of his science) need I would like m thank Ernan McMullin,Janet Broughton, and two referees for theJounm/for comments and criticismson earlier drafts of this paper which helped me to improve the present version considerably. ' References to Descartes's works are to the Adam & Tannery edition (AT), Oeuvresde Descartes , ed. C. Adam and Paul Tannery, new ed., 11 vols. (Paris: Vrin, 1964-1974). Whenever translations of cited passages are provided in John Cottingham, Robert Stoothoff, and Dugald Murdoch, eds. and trans., The PhilosophicalWritings ofDe.scartes, 3 vols. (New York: Cambridge UniversityPress, 1984), I have also provided references to this translation (CSM). , See, for example, Ralph M. Blake, "The Role of Experience in Descartes' Theory of Method,~in Edward H. Madden, ed., Theoriesof ScientificMethod (Seattle:Universityof Washington Press, 196o),74-i o3; Desmond M. Clarke, "The AmbiguousRoleof Experience in Cartesian Science,"PSA (1976): 1: 151-64; and Descartes'PhilosophyofScinw.e(UniversityPark: Pennsylvania State UniversityPress, 1982),Chap. =. [=281 2~4 JOURNAL OF THE HISTORY OF PHILOSOPHY 31"2 APRIL a993 tO be recast in light of increased attention being paid to Descartes's Aristotelian and Neoplatonist-panpsychist rivals. Both his apparent attacks on sense experience and his famous metaphysical dualism, for example, take on new connotations when viewed as responses and challenges to contemporary alternative foundations for science.3 The biggest problem confronting the traditional mathematico-deductivist interpretation, however, is that there are precious few "deductions" or "mathematical demonstrations" (in the usual sense) to be found in Descartes's scientific writings. While Descartes frequently seems to have endorsed the traditional interpretation when talking about his science, he does not seem to have practiced what he preached within his scientific works. Either, then, we must view his methodological pronouncements as inconsistent with what he was in fact able to achieve (indeed as wildly optimistic given his actual scientific results), or we must reconsider the possible meanings of his deductivist-mathematical terminology. I opt here for the second alternative, and argue that a much broader interpretation of 'deduction', 'demonstration', and 'mathematics' is more consistent both with what is found in Descartes's scientific works and with a plausible interpretation of Descartes's use of such terms within his early methodological texts. Further, unlike other attempts to broaden the interpretation of 'deduction', etc., I also argue that Descartes does not even restrict its application to arguments or equations (nor does he restrict 'mathematics' to equations or figures). To prepare the groundwork for this interpretation, and further support the need for such a view, we need first review a variety of earlier readings and compare then with actual examples taken from Descartes's scientific writings. 1. TEXTUAL AND CONTEXTUAL DIFFICULTIES IN DESCARTES'S METHODOLOGICAL WRITINGS The two works most often taken as presenting Descartes's method are, of course, the Discourse on Method (AT, 6: x-78; CSM, l: 1o9- 51), and the Rules sSee, for example, Daniel Garber, "Semel in Vita: The ScientificBackground to Descartes' Meditations," in Amelie O. Rorty, ed., Essays on Descartes' Meditations (Berkeley: University of California Press, 1986),81-116; Gary Hatfield, "The Sensesand the FleshlessEye:The Meditations asCognitiveExercises,"in ibid.,45-79; Richard S.Westfall,The Construction ofModern Science (New York: Cambridge University Press, 1977...