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)

Our purpose on the present occasion is to evaluate some ideas the biographers of late antiquity held about the origins of European thought. Speaking about this period we are no longer dealing with the question of transferring of the archaic practices: these practices are indeed long dead. What we encounter can be better defined as the import of ideas. Equally important is a study of the changing attitudes of our authors: rather than passive witnesses, they became active participants of this (...) import. The process is truly fascinating and we hope that the following examples, mostly from Hippolytus, will elucidate this. The best, almost a paradigmatic example is Pythagoras, who in late antiquity had many faces. His biography is an interesting instance of general change of attitude to ancient wisdom, typical for the source utilized by Hippolytus. Looking at a number of peculiar features of Hippolytus’ report which, we hope, will help us to see why the image of Pythagoras and his philosophy, formed by Hippolytus, is somewhat untypical for the period. We will see that Hippolytus’ biographic report, however garbled, shows no signs of so-called ‘Neopythagorean’ biographic development. Admittedly, the later authors frequently combine their sources to make them suitable to their needs, polemical or apologetic. Do we still have a reason to believe that these stories, however doubtful from the historical point of view, may contain the germs of truth? (shrink)