I examine the reasons Aristotle presents in Physics VIII 8 for denying a crucial assumption of Zeno’s dichotomy paradox: that every motion is composed of sub-motions. Aristotle claims that a unified motion is divisible into motions only in potentiality (δυνάμει). If it were actually divided at some point, the mobile would need to have arrived at and then have departed from this point, and that would require some interval of rest. Commentators have generally found Aristotle’s reasoning unconvincing. Against David Bostock (...) and Richard Sorabji, inter alia, I argue that Aristotle offers a plausible and internally consistent response to Zeno. I defend Aristotle’s reasoning by using his discussion of what to say about the mobile at boundary instants, transitions between change and rest. There Aristotle articulates what I call the Changes are Open, Rests are Closed Rule: what is true of something at a boundary instant is what is true of it over the time of its rest. By contrast, predications true of something over its period of change are not true of the thing at either of the boundary instants of that change. I argue that this rule issues from Aristotle’s general understanding of change, as laid out in Phys. III. It also fits well with Phys. VI, where Aristotle maintains that there is a first boundary instant included in the time of rest, but not a “first in which the mobile began to change.” I then show how this rule underlies Aristotle’s argument that a continuous motion cannot be composed of actual sub-motions. Aristotle distinguishes potential middles, points passed through en route to a terminus, from actual middles. The Changes are Open, Rests are Closed Rule only applies to actual middles, because only they are boundaries of change that the mobile must arrive at and then depart from. On my reading, Aristotle argues that the instant of arrival, the first instant at which the mobile has come to be at the actual middle, cannot belong to the time of the subsequent motion. If it did, the mobile would already be moving towards the next terminus and thus, per Phys. VI 6, would have already left. But it cannot have moved away from the midpoint at the very same moment it has arrived there. This means that the instant of arrival must be separated from the time of departure by an interval of rest. I show how Aristotle’s reasoning applies generally to rule out any continuous reflexive motion or continuous complex rectilinear motion. On my interpretation, however, the argument does not apply to every change of direction. When, as in the case of projectile motion, multiple movers and their relative powers explain why the mobile changes directions, distinct sub-motions are not involved. Aristotle holds that such motions cannot be continuous, not because they involve intervals of rest, but because they involve multiple causes of motion. My interpretation of the Changes are Open, Rests are Closed Rule allows us to make better sense of Aristotle’s argument than any previous interpretation. (shrink)

David Bostock; III*—Aristotle, Zeno, and the Potential Infinite, Proceedings of the Aristotelian Society, Volume 73, Issue 1, 1 June 1973, Pages 37–52, https://.

Die "Paradoxien des Unendlichen" sind ein Klassiker der Philosophie der Mathematik und zugleich eine gute Einführung in das Denken des "Urgroßvaters" der analytischen Philosophie. Das Unendliche - seit jeher ein Faszinosum für die philosophische Reflexion - wurde in der Zeit nach der Grundlegung der Analysis durch Leibniz und Newton in der Mathematik zunächst als Problem betrachtet, das sich nicht vollkommen widerspruchsfrei behandeln lässt. Bernard Bolzano, der heute als "Urgroßvater der analytischen Philosophie" (Michael Dummett) gilt, zeigt in diesem klassisch gewordenen Text (...) jedoch, dass es beim Nachdenken über das Unendliche, sofern man begrifflich differenziert argumentiert und klare Definitionen zugrunde legt, keine echten Widersprüche gibt und sich die "Paradoxien des Unendlichen" auflösen lassen. Bolzano setzt sich zunächst mit den Unendlichkeitsbegriffen der Mathematik und der (idealistisch geprägten) Philosophie seiner Zeit auseinander, bevor er sich Schritt für Schritt die analytischen Grundlagen einer konsistenten Theorie des Unendlichen erarbeitet. Bereits einige Jahrzehnte vor Georg Cantors bis heute maßgeblicher mathematischer Unendlichkeitsdefinition gelingt Bolzano mit diesem Werk, dessen logische Klarheit und dessen Grad an philosophischer Durchdachtheit bis heute Beispielcharakter haben, die Rehabilitation des Unendlichen für die Philosophie der Mathematik. Die "Paradoxien des Unendlichen" wurden zuerst 1851 aus dem Nachlass herausgegeben. Diese Neuausgabe beruht auf dem der Erstausgabe zugrundeliegenden Manuskript. (shrink)

Christian Pfeiffer explores an important, but neglected topic in Aristotle's theoretical philosophy: the theory of bodies. A body is a three-dimensionally extended and continuous magnitude bounded by surfaces. This notion is distinct from the notion of a perceptible or physical substance. Substances have bodies, that is to say, they are extended, their parts are continuous with each other and they have boundaries, which demarcate them from their surroundings. Pfeiffer argues that body, thus understood, has a pivotal role in Aristotle's natural (...) philosophy. A theory of body is a presupposed in, e.g., Aristotle's account of the infinite, place, or action and passion, because their being bodies explains why things have a location or how they can act upon each other. The notion of body can be ranked among the central concepts for natural science which are discussed in Physics III-IV. (shrink)

Richard Sorabji here takes time as his central theme, exploring fundamental questions about its nature: Is it real or an aspect of consciousness? Did it begin along with the universe? Can anything escape from it? Does it come in atomic chunks? In addressing these and myriad other issues, Sorabji engages in an illuminating discussion of early thought about time, ranging from Plato and Aristotle to Islamic, Christian, and Jewish medieval thinkers. Sorabji argues that the thought of these often negelected philosophers (...) about the subject is, in many cases, more complete than that of their more recent counterparts. “Splendid. . . . The canvas is vast, the picture animated, the painter nonpareil. . . . Sorabji’s work will encourage more adventurers to follow him to this fascinating new-found land.”—Jonathan Barnes, Times Literary Supplement “One of the most important works in the history of metaphysics to appear in English for a considerable time. No one concerned with the problems with which it deals either as a historian of ideas or as a philosopher can afford to neglect it.”—Donald MacKinnon, Scottish Journal of Theology “Unusually readable for such scholarly content, the book provides in rich and cogent terms a lively and well-balanced discussion of matters of concern to a wide academic audience.”— Choice. (shrink)

A survey of Euclid's Elements, this text provides an understanding of the classical Greek conception of mathematics and its similarities to modern views as well as its differences. It focuses on philosophical, foundational, and logical questions — rather than strictly historical and mathematical issues — and features several helpful appendixes.

The essence of the theory of minima naturalia is the contention that a physical body is not infinitely divisible qua that specific body. A drop of water cannot be divided again and again and still maintain its “wateriness”. There are several statements in Aristotle's Physics which suggest such an interpretation, and the theory of minima naturalia is commonly considered to have originated in the thirteenth century as an interpretation of these statements. The present paper is a preliminary presentation of the (...) role of Ibn Rushd in the evolution of the theory, hitherto neglected. His theory developed not only as an elaboration on the “suitable” statements of Aristotle, but mainly as an attempt to solve the difficulties raised by Aristotle's thesis that body and motion are continuous, infinitely divisible entities and are associated qua such. According to Ibn Rushd's interpretation, body and motion are associated not qua being continuous but qua having indivisible minimal parts. It seems that Epicurus' and Ibn Rushd's theories of minima developed as responses to Physics VI and offer modifications of classical atomism and of classical Aristotelianism , which to a certain extent reduce the gap between these two systems. (shrink)

In _Aristotle’s Ever-turning World in _Physics _8_ Blyth analyses the reasoning in Aristotle’s explanation of cosmic movement, with detailed evaluation of ancient and modern commentary on this central text in the history of ancient and medieval philosophy and science.

These selections from Le système du monde, the classic ten-volume history of the physical sciences written by the great French physicist Pierre Duhem (1861-1916), focus on cosmology, Duhem's greatest interest. By reconsidering the work of such Arab and Christian scholars as Averroes, Avicenna, Gregory of Rimini, Albert of Saxony, Nicole Oresme, Duns Scotus, and William of Occam, Duhem demonstrated the sophistication of medieval science and cosmology.

Diese fünfbändige Aristoteles-Ausgabe in griechischer Sprache ist (mit Ausnahme von Bd III) ein fotomechanischer Nachdruck der maßgeblichen Aristoteles-Ausgabe von 1831-1870. Band I und II enthält die Werke Aristoteles. In Band III wird die durch O. Gigon besorgte Bearbeitung und Ergänzung der Fragmente des Aristoteles wiedergegeben. Band IV bietet eine Auswahl der bedeutendsten Stücke aus den antiken Kommentaren zu Aristoteles, sowie eine Konkordanz mit den Commentaria in Aristotelem Graeca. In Band V ist der Index Aristotelicus von H. Bonitz nachgedruckt.

What is the relation between time and change? Does time depend on the mind? Is the present always the same or is it always different? Aristotle tackles these questions in the Physics. In the first book in English exclusively devoted to this discussion, Ursula Coope argues that Aristotle sees time as a universal order within which all changes are related to each other. This interpretation enables her to explain two striking Aristotelian claims: that the now is like a moving thing, (...) and that time depends for its existence on the mind. (shrink)

This book presents a detailed analysis of three ancient models of spatial magnitude, time, and local motion. The Aristotelian model is presented as an application of the ancient, geometrically orthodox conception of extension to the physical world. The other two models, which represent departures from mathematical orthodoxy, are a "quantum" model of spatial magnitude, and a Stoic model, according to which limit entities such as points, edges, and surfaces do not exist in (physical) reality. The book is unique in its (...) discussion of these ancient models within the context of later philosophical, scientific, and mathematical developments. (shrink)

Bowin begins with an apparent paradox about Aristotelian infinity: Aristotle clearly says that infinity exists only potentially and not actually. However, Aristotle appears to say two different things about the nature of that potential existence. On the one hand, he seems to say that the potentiality is like that of a process that might occur but isn't right now. Aristotle uses the Olympics as an example: they might be occurring, but they aren't just now. On the other hand, Aristotle says (...) that infinity "exists in actuality as a process that is now occurring" (234). Bowin makes clear that Aristotle doesn't explicitly solve this problem, so we are left to work out the best reading we can. His proposed solution is that "infinity must be...a per se accident...of number and magnitude" (250). (Bryn Mawr Classical Review 2008.07.47). (shrink)

Aristotle is said to have held that any kind of actual infinity is impossible. I argue that he was a finitist (or "potentialist") about _magnitude_, but not about _plurality_. He did not deny that there are, or can be, infinitely many things in actuality. If this is right, then it has implications for Aristotle's views about the metaphysics of parts and points.