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Thomas Bradwardine's De Continuo and the Structure of Fourteenth-Century Learning

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Thomas Bradwardine's De Continuo and the Structure of Fourteenth-Century Learning THOMAS BRADWARDINE'S DE CONTINUO AND THE STRUCTURE OF FOURTEENTH-CENTURY LEARNING EDITH D.SYLLA The late medieval Christian Aristotelian worldview, when pressed to its logical extremes, was incompatible with Euclidean geometry. In the stan­ dard medieval Aristotelian classification of the sciences, natural philosophy and geometry were separate disciplines, each with its own principles and conclusions. Geometry assumed as a principle, without proof, that continua must be infinitely divisible into ever smaller continua, while natural phi­ losophy proved it.1 Geometry included in its subject matter indivisibles such as points, lines, and planes—meanwhile asserting that higher-order continua are not composed of such indivisibles. Natural philosophy asserted that quantities—including indivisibles such as points, lines, and surfaces— really exist in substances as their accidental forms, although they are con­ sidered mathematically in abstraction from such substances.2 But when geometric indivisibles such as points and lines, or purported physical indivisibles such as instants or exact degrees of quality, were sup­ posed to exist in reality {esse in re), paradoxes ensued. In view of these paradoxes, some idiosyncratic scholars opted to compose continua of indi­ visibles at the price of inconsistency with Euclidean geometry and/or Aris­ totelian physics. Mainstream scholars, including William of Ockham, John Buridan, and Thomas Bradwardine, often concluded that geometric indi­ visibles do not in fact exist, thus resolving the apparent paradoxes by deny­ ing the entities that led to them.3 In either case, whether scholars adopted 1 For Thomas Bradwardine's quotation of Averroes's statement of this view, see John Murdoch, "Thomas Bradwardine: Mathematics and Continuity in the Fourteenth Century/' in Mathematics and Its Applications to Science and Natural Philosophy in the Middle Ages, ed Edward Grant and John E. Murdoch (Cambridge, London, New York: Cambridge University Press, 1987), p. 136 n. 62.1 discuss this passage below. 2 For a discussion of John Dumbleton, whose ideas about the relation of mathematics to physics mirror part of the standard view, see Edith Sylla, "The Oxford Calculators and Math­ ematical Physics: John Dumbleton's Summa Logicae et Philosophiae Naturalis, Parts Π and III," in Physics, Cosmology and Astronomy, 1300-1700: Tension and Accommodation, ed. Sabetai Unguru, Boston Studies in the Philosophy of Science, vol. 126 (Dordrecht, Boston, London: Kluwer, 1991), pp. 129-61. 3 For reference to a number of sources on this point, see John Murdoch, "Philosophy and the Enterprise of Science in the Later Middle Ages," in The Interaction between Science and Philosophy, ed. Yehuda Elkana (Atlantic Highlands, N.J.: Humanities Press, 1974), p. 61. It is notable that Bradwardine and the nominalists, although often portrayed as belonging to opposing metaphysical camps, agreed on denying the existence of indivisibles. It should be noted that philosophers who denied the existence of geometric indivisibles nevertheless held the prime movers of the celestial spheres (or angels more generally) and human souls to be indivisible substances. EDITH SYLLA 149 some sort of atomism or denied the existence of geometric indivisibles al­ together, the previously standard Aristotelian understanding of the relation between mathematics and physics broke down: geometry could no longer be understood to deal with quantities existing in, but considered in abstrac­ tion from, physical bodies. In this paper I want to use Thomas Bradwardine's De continuo as a source of evidence about the structure of fourteenth-century learning, par­ ticularly as it involved the relations between natural philosophy, mathemat­ ics, logic, and theology. Thanks to the wide-ranging research of John Mur­ doch, we have extensive knowledge about Bradwardine's text and about the earlier texts of such "atomist" authors as Henry of Harclay and Walter Chatton to which Bradwardine was reacting.4 In addition, in a number of papers Murdoch has presented a broad picture of what he calls "the unitary character of late medieval learning," arguing for the identity of fourteenth- century science and philosophy and for a close connection of natural phi­ losophy and theology through a group of shared tools, tactics, or conceptual schemes ranging from supposition theory in logic, through the mathematics of ratios, to the habit of arguing de potentia Dei absoluta or secundum imaginationem, and so forth.5 What I want to do here is to put these two parts of Murdoch's scholarly contribution more explicitly together—that is, 4 As evidence for my paper I have used both John Murdoch's published work and his un­ published draft editions of a wide range of texts, especially Thomas Bradwardine's De con­ tinuo and Henry of Harclay's question Utrum mundus potent durare in eternum a parte post. In the 1960s Murdoch prepared critical editions and translations of Harclay, Chatton, Odo, Bonetus, and John the Canon, as well as of Bradwardine's De continuo, which were to form the basis of a several-volume study on the problem of the continuum and related issues in the Middle Ages; see John Murdoch and Edward Synan, "Two Questions on the Continuum: Walter Chatton (?), O.F.M. and Adam Wodeham, O.F.M.," Franciscan Studies 26 (1966): 215 n. 10 (hereafter Murdoch and Synan, "Two Questions"). At that time he gave me copies of his editions and translations. It is perhaps appropriate to my topic, if awkward in other ways, that my study is based in part on thirty-year-old manuscript copies of these editions and translations, most of which have yet to appear in print. I have not checked the editions, but I have sometimes made my own translations, because it was simpler to do so and because Murdoch's translations had not been published or were not available for citation. All credit for the texts and the substance of the translations should go to him, while any blame for erro­ neous deviations from the true sense should fall on me. The text of Thomas Bradwardine's De continuo in John Murdoch, "Geometry and the Continuum in the Fourteenth Century: A Philosophical Analysis of Thomas Bradwardine's Tractatus de continuo" (Ph.D. diss., Uni­ versity of Wisconsin, 1957), available in microfilm, is similar but not identical to the edition I am using. 5 John Murdoch, "Mathesis in philosophiam scholasticam introducta: The Rise and De­ velopment of the Application of Mathematics in Fourteenth-Century Philosophy and Theol­ ogy," in Arts libéraux et Philosophie au Moyen Age (Actes du Quatrième Congres Interna­ tional de Philosophie Médiévale) (Montreal and Paris, 1969), pp. 215-54; idem, "Philosophy and the Enterprise of Science," (n. 3), pp. 51-113; idem, "From Social into Intellectual Fac­ tors: An Aspect of the Unitary Character of Late Medieval Learning," in The Cultural Con- text of Medieval Learning, ed. J. E. Murdoch and E. D. Sylla (Dordrecht: Reidel, 1975), pp. 271-339. .
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