Infinity and Continuity in Ancient and Medieval Thought (review)

BOOK REVIEWS 399 Norman Kretzmann, Editor. Infinity and Continuity in Ancient and Medieval Thought. Ithaca and London: Cornell University Press, 1982. Pp. 367 . $27.5 o. During the two thousand years of classical antiquity and the Middle Ages authors dealt with the problems of infinity and continuity in such a way that "that development does not belong simply to the history of science" but also to metaphysics, logic, theology and mathematics (7-8). In order to study that development scholars from different disciplines gathered at Cornell University in April, 1979, for a conference on those problems, which has resulted in this important volume. Its first five chapters deal with Aristotle's theories, worked out in response to Leucippus and Democritus, and subsequent Greek reactions to Aristotle himself. David J. Furley in "The Greek Commentators' Treatment of Aristotle's Theory of the Continuum" considers Alexander of Aphrodisias, Porphyry, Themistius, Olympiodorus , Simplicius and John Philoponus. From studying Zeno, Leucippus and Democritus, Plato, Xenocrates, Aristotle himself, Strato, Plutarch of Chaeronea, Damascius and Islamic authors after 8oo A.D., Richard Sorabji in "Atoms and Time Atoms" concludes that only Diodorus Cronus and Epicurus seemed to have accepted the notion of time atoms and motion "by jerks." While studying the "nihilistic" and "atomistic" horns of the dilemma of divisibility, Fred D. Miller in "Aristotle Against the Atomists" offers a valuable exegesis of Aristotle's De Generatione et Corruption, I, 2, 317al sqq., and Physics, VIII, 8. In "Infinity and Continuity: The Interaction of Mathematics and Philosophy in Antiquity" Wilbur R. Knorr investigates the mathematical aspects of Zeno's paradoxes , the origin in Hippocrates of Chios, Eudoxus and Euclid of the method of exhaustion, and Archimedes' heuristic analysis by indivisibles, and he concludes (vs. Tannery, Szabo, Luria, Mau) that "alleged philosophical factors actually had minimal significance for the development of pre-Euclidean geometry" (142), which arose rather from "geometers' commitment to the ideology of strict demonstration" (x43). According to Ian Mueller's "Aristotle and the Quadrature of the Circle" a contemporary of Aristotle (Dinostratus) used the curve invented by Hippias of Elis to square the circle but Aristotle mentions only the problematic quadratures of Antiphon, Bryson and Hippocrates and apparently believes that quadrature is impossible. In the last six chapters attention is paid to the fourteenth century, where "work of most historical importance and most intellectual interest tends to be concentrated (9). John E. Murdoch in "William of Ockham and the Logic of Infinity and Continuity" relies upon Ockham's still unedited Expositio Physicorum (generous portions of it are reproduced in Murdoch's footnotes) and Quaestio Quodlibetalis, II, to set forth the logical elements in Ockham's theory: "propositional analyses" (177), part of which is "the application of the successive verification of contradictory propositions" (179); "exponential propositions" (x91), and "positional difference" (195). Eleonore Stump concludes her study, "Theology and Physics in De Sacramento altaris: Ockham's Theory of Indivisibles," with the statement that even though Ockham's theory is open to objections (e.g., it is basically negative) it is "more sophisticated and less open to objection than it has been made to seem [by Murdoch]" (229). 400 HISTORY OF PHILOSOPHY In "Infinite Indivisibles and Continuity in Fourteenth-Century Theories of Alteration " Edith Dudley Sylla reflects on the problems which alteration of qualities brings to Aristotle's definition of continuity as exemplified in treatises of Walter Burley and Richard Kilvington. Calvin G. Normore concentrates also upon Walter Burley in "Walter Burley on Continuity" but largely from the perspective of his (and Aristotle's) difference from Dedekind. In "Continuity, Contradiction and Change" Norman Kretzmann distinguishes between the "Quasi-Aristotelianism" of Henry of Ghent, Hugh of Newcastle, John Baconthorpe and Landulf Caraccioli and the orthodox "Aristotelianism" of Francis of Marchia, John the Canon (see 275 ) and Richard Kilvington (284ff.). This latter proves to be an orthodox Aristotelian "by devising apparent paradoxes involving simultaneous contradictories in the temporal interval beginning at the instant of transition and adapting the elements of the Aristotelian analysis to the resolution of those paradoxes" (296) . In the final chapter Paul Vincent Spade takes a more benign view than does Kretzmann on "Quasi-Aristotelianism" by arguing that although such a position is neither "authentically Aristotelian...