Michael White, The Continuous and the Discrete: Ancient Physical Theories from a Contemporary Perspective. Oxford: Clarendon Press, 1992. $72.00. ISBN 0-19-823952-1.
Reviewed by Dirk t. D. Held, Connecticut College.
Michael White's The Continuous and the Discrete offers a detailed analysis of ancient concepts of spatial magnitude, time, and motion in relation to the opposing ontological principles of continuity and discrete quanta. Rather than presenting a comprehensive survey of these topics in antiquity, White focuses on three major conceptualizations. Two thirds of the book is devoted to Aristotle's well documented model in which magnitude, time, and motion are understood as continuities. White then examines the atomistic or quantum theories of Epicurus and Diodorus Cronus, and ends with a somewhat speculative reconstruction of a Stoic physical theory that combines Aristotelian continuity with the absence of limit entities such as boundaries and edges, resulting in a very unAristotelian ontological indeterminacy. Throughout the discussion, the ancient conceptual models are played off against modern mathematical perspectives on spatial magnitude, time, and motion. The result is an unusually productive approach to the history of philosophy, one that combines historical reconstruction through detailed technical analysis from within the perspective of the ancient theories, with what Richard Rorty called a "Whiggish" account that makes ample use of later knowledge of a topic.1 Far from cultivating injudicious anachronism however, White uses this contrast of ancient and modern perspectives to clarify the ancient views and amplify the reasoning behind them. White goes beyond simply setting out the assumptions underlying a particular notion. He is as eager to uncover the ramifications of arguments as he is their philosophical underpinnings, and often teases out possibilities that even if remote from the original intent can illuminate the full scope of a concept. His rationale is that "the prudent pursuit of the history of possible philosophy" enriches our philosophical understanding not only of the "history of actual philosophy ... but also of contemporary philosophical and scientific developments" (p. vii).
Aristotelian analysis of the physical world rests on the applicability of geometry to physical phenomena. Geometry concerns itself with magnitudes, that unlike plurality (which is divisible into discrete parts), were seen as continuous from one limit to another, and this continuity was infinitely divisible. Aristotle holds as a fundamental principle that nothing continuous and infinitely divisible can be composed from separate, or discrete, entities. White presents several Aristotelian arguments against the possibility of continuity being built from any such "points", showing that for Aristotle divisions of anything continuous must yield parts of positive size, all of which are themselves continuous, and therefore in principle able to be further divided. (The parts cannot have null size, thereby precluding modern point-set analysis, and the "general tendency of contemporary mathematics ... [towards] arithmetizing magnitude" [p. 31]). He then describes the role of Aristotelian metaphysics in this question, specifically the principle of potentiality (dynamis) which is the key to Aristotle's understanding of infinite divisibility. Aristotle never holds that a finite magnitude is actually composed of an infinite collection of homeomerous magnitudes, only that every magnitude carries an inherent potentiality of being infinitely divided. What results is called by White a "foundationless form" of infinite divisibility. There are no "ultimate" parts in any spatial magnitude.
Aristotle's analyses of time and motion will obviously be closely linked to his analysis of magnitude. Just as magnitude cannot be composed of any bits or "chunks", so time cannot be constituted by discrete temporal points or "nows". Of course, both time and motion present an asymmetry to the observer that magnitude does not have. Time gets its "underlying topological structure as a linear continuum" (p. 95), the basis for Aristotle's "metrical conception of time", from the recurrent and eternal processes of the heavens. But apart from this Urzeit as White calls it, time and motion are overtly directional. White points out the indeterminacy of the future contrasted to the fixed condition of the past, and motion's occuring between a terminus a quo and terminus ad quem.
White argues that Aristotle's basic ideas of spatial magnitude, time, and motion look familiar to moderns, and indeed they are in terms of the formal properties of infinite divisibility and continuity. Disagreements arise between Aristotle's mathematics and our own with the arrival of point-set analysis of continuity. White says that this "completed the eradication of the Aristotelian distinction between magnitude and plurality." This development affected the modern understanding of continuous motion, which under Aristotelian kinematics could never be broken down into a series of locations of a moving body. White adduces the modern idea of the "at-at" ontology of motion: a moving body is at a different place at a different time. Modern point-set analysis of continuity can accommodate this analysis, whereas for Aristotle a body was never in motion, but only in a state of having finished a motion. On White's argument, motion for Aristotle had to be construed through the metaphysical category of potentiality and could not be reduced to "non-motions in the form of instantaneous spatial positions of the moving body" (186). Aristotelian motion is "ontologically primitive ... an irreducible potentiality" for reaching an actual terminus ad quem (195).
The Hellenistic models of Epicurus and Diodorus Cronus assumed matter was composed of discrete quanta or "chunks" in contrast to Aristotelian matter that was continuous and homeomerous. White adroitly recognizes the quantum theories as metaphysical alternatives to Aristotelian dependence on "latent" and hence mysterious potentialities. This attempt to be rid of the unsatisfactory concept of dynamis led the Hellenistic philosophers to reject the entire Aristotelian account of the formal structure of magnitude, time, and motion. Several arguments presented themselves in favor of quanta. Lack of ultimate parts led to the paradox of measure: assume a line-segment, which will have an infinite number of parts, of either zero or positive magnitude; if the parts have zero magnitude, the line made from them will have zero magnitude; if the parts have positive magnitude, the infinite number of them in the line will make the line infinite in length. Moreover, Aristotle's finitist cosmology created inconsistencies, in which he accepts limit entities or boundaries within a continuous magnitude that exist only potentially, while externally the limits were actual: surfaces, the end-point of motion, and the finite cosmos.
The advantage of the quantum model was in its explanatory power. Non-locomotive change could now be explained "reductively in terms of the unchanging, 'primary' properties of atoms" and their relations to one another (281). The objectivity of measure was preserved, and motion, rather than being eliminated as it was under the Aristotelian account, could be analyzed into a series of positions of a body at successive times, the ancient version of the "at-at" ontology of motion (282).
The last case White examines is the Stoic model, whose removal of limit-entities from the world he enterprisingly scrutinizes in terms of modern "fuzzy-set" theory. Essentially what this approach does is to remove any sharp delimiting boundaries in favor of indeterminate ones, for example surfaces and edges which blend into one another. By this means White examines possible Stoic responses to such puzzles as motion kata athroun or all-at-once, and the conic section puzzle, famous since Democritus. (If you bisect a cone with a plane, is the circular face beneath the top piece the same size as the circular face atop the bottom piece? If the same size, the cone should become a cylinder; if different, the sides of the cone will be stepped rather than smooth.)
One obvious conclusion of the book, built from many rigorous, subtle, and suggestive analyses, is that the principles of the continuous and the discrete have a wide range of significance for ancient and modern physical and mathematical analysis. In the process of arriving at this conclusion, White impresses upon the reader the varying relations that can hold between mathematical analysis and physical reality. White has not disguised the intrinsically technical nature of his material, but he has explicated it clearly and usually succinctly. A modest amount of logical and mathematical notation is used, and his rigorous analysis is supported by a scattering of geometrical drawings and graphs. All of this is kept to a minimum though, and a technical background is not necessary for any thoughtful reader to learn a great deal from White's study. A thorough index is included. Overall, one might say that the "problem set" White surveys might seem constricted to some but the results are bold and responsive to important philosophical issues.
 "The Historiography of Philosophy", in Rorty, Richard, Schneewind, J.B., and Skinner, Quentin, edd., Philosophy in History, Cambridge 1984, p. 56.