Chapter 2 presents the challenge that Parmenides’s philosophy presents for a scientific treatment of motion and change. It lays out the criteria for philosophy that we find established in Parmenides’s poem under his particular interpretations: consistency, rational admissibility, and a principle of sufficient reason. A careful examination of his use of negation shows that negation for him is a separation operator that indicates the extreme opposite to the thing negated. The counterpart to this understanding of negation is a connection operator that expresses absolute identity. A further step explains how Parmenides’s operators and his criteria for philosophy make it impossible to give any account of motion and change. Finally, it is shown that the cosmology in the doxa part of Parmenides’s poem should be understood as his attempt to expound a best possible cosmology and its short-comings – the rationale being that if even the best possible cosmology cannot fulfil the criteria for philosophy, no one else’s cosmology needs to be considered.

In Aristotle’s Physics we find for the first time motion and speed implicitly measured in terms of time and distance covered, as the discussion of book VI, chapter 2 shows. Aristotle’s explicit account of measurement, however, which he gives in Metaphysics Iota and with which this chapter starts, understands measure not only as homogeneous with the measurand, but also as one-dimensional only. Accordingly, the explicit measure of motion is simply time in the Physics, as we see from examining Aristotle’s understanding of time as the measure and the number of motion. For a full account of motion and speed and a complete response to Zeno’s challenge, however, a complex measure is needed, one that takes account of both time taken and distance covered. The chapter shows that this is exactly what Aristotle implicitly develops in his Physics, when he compares motions of different speed and responds to Zeno’s paradoxes of motion. But it is not what he can accommodate in his theory of measurement.

Chapter 6 examines Plato’s introduction of mathematical structures in order to explain the natural world. It contrasts his ‘mathematical approach’ to nature with that of the Pythagoreans, and shows how his use of mathematics enables Plato to make motion intelligible in itself to a certain degree. For this, the idea of measuring motion in temporal terms is crucial. However, Plato’s treatment of measurement in the Timaeus does not include measuring the distance covered by a motion. And Plato’s treatment of time and space (the receptacle) as entities of fundamentally different status, taking time to be intelligible in a way in which space is not, prevents him from connecting time and space in an account of speed. It is shown that Plato instead reduces speed to the time a motion takes. The chapter finishes by spelling out the problematic consequences of this reduction – that it only allows for restricted comparability of different motions and that in certain cases it can lead into inconsistencies.

This chapter discusses the central question why Aristotle, in spite of having everything required to conceptualize a complex measure of speed in terms of time and space, did in the end not explicitly develop such a measure. It is first investigated whether contemporaries of Aristotle may have worked with such a complex measure of speed, and concluded that it cannot be found in either of the two thinkers most likely to have done so, namely Eudoxus and Autolycus. The second part of the chapter investigates what made Aristotle cling to a simple measure and suggests that there are mathematical and metaphysical reasons: metaphysically, Aristotle cannot explicitly accommodate a relation as a measure of motion, since relations are derivative and problematic for him; mathematically, the principle of homogeneity which derives from the realm of Greek mathematics makes it impossible to combine of different dimensions in a single measure in the way needed for measuring speed in a mathematically informed physics such as Aristotle’s.

This chapter discusses the atomistic account of motion, as an example of the first reactions to the Eleatic challenge by succeeding natural philosophers. The atomists are shown to change the logical basis by implicitly employing a different conception of negation that allows them to understand Being and non-Being as on a par. It also enables them to build a different ontological basis in which non-Being qua void plays a central role in natural philosophy. This new ontological basis allows the atomists to integrate the phenomenal world into their philosophy and to deal with the mereological problems bequeathed by Zeno’s paradoxes. The starting point for this development is the idea that what truly is must in some way also be responsible for the appearances of the phenomenal world. Generation on the phenomenal level is now understood as the combination and separation of aggregates of atoms, while change consists in the rearrangement of some parts of the aggregate. Although the atomists’ notion of the void can be seen as a predecessor to a notion of space, they do not in fact react to the spatio-temporal paradoxes of Zeno.

The main object of this book is to study how the understanding of physical motion in ancient Greek thought developed before and up to Aristotle. It investigates which logical, methodological, and mathematical foundations had to be in place to establish a fully fledged concept of motion that also allows for comparing and measuring speed.1 Given that physical motion is the core concept of natural philosophy, this study thereby also seeks to reconstruct in rough outlines how natural philosophy came to be established as a proper scientific endeavour in ancient Greece.2

The basis of Aristotle’s strategy for bringing together time and distance covered in an account of motion is his understanding time, distance, and motion as continua. Against the background of Parmenides’ notion of continuity and the atomistic understanding of magnitudes, this chapter investigates Aristotle’s notion of continuity. It demonstrates the extent to which Aristotle adopts a mathematical notion of continuity according to which divisibility without end is crucial, and it expounds his appropriation of this mathematical notion for the physical realm. The main characteristics of Aristotle’s account of continuity turn out to be a new part-whole relation (in which the whole is prior to the parts since it is only by dividing the whole that we gain parts), a new account of limits, a new understanding of infinity, and a careful distinction between potential divisibility and actual division. We see how this new understanding of continuity helps counter the mereological problems of Zeno’s motion paradoxes by making it illegitimate to define time, space, and motion in terms of an additive part-whole-relation.

In the third chapter I show how Zeno takes up and advances Parmenides’s criteria for philosophy by developing the genre of paradoxes. The paradoxes are based on Parmenides’s understanding of the principle of non-contradiction but allow Zeno to start with the position of his opponent in order to show how this position will yield inconsistencies. The mereological problems raised by the plurality paradoxes will be discussed, but the main focus will be on Zeno’s paradoxes of motion which seem to show motion to be self-contradictory. They confront natural philosophers with two kinds of problems, mereological ones and spatio-temporal ones. I argue against viewing the paradoxes as simply awaiting their solutions in modern mathematics to be solved. Instead, I interpret them as questioning the very consistency of the notions of time, space, and motion. In this way Zeno’s paradoxes will be shown to constitute one of the severest attacks on any project of conceptualizing motion. The paradoxes will therefore act as a touchstone for whether natural philosophy can develop in such a way as to meet Parmenides’s challenge.

Chapter 5 examines the development of the logical basis required for natural philosophy in Plato. In particular it shows how Plato in the Sophist develops further understanding not only of negation and the connection operator, but also, in connection with this, the principle of non-contradiction. These developments allow for connecting Being and non-Being, which is necessary for making sense of motion without falling into inconsistencies. The chapter then examines Plato’s employment of the principle of sufficient reason and the criterion of rational admissibility in the Timaeus. He develops the principle of sufficient reason further by distinguishing for the first time between necessary and rational reasons. And rational admissibility is taken up by Plato in the way used by the atomists: that is, the basic ontological constituents not only have to be testable by our own reason, but they also have to explain the phenomena. These requirements together emerge as Plato’s standards for natural philosophy and cosmology by being the positive criteria an eikôs mythos has to fulfil.

This monograph is the first study to assess in its entirety the fourth-century CE Latin translation of and commentary on Plato's Timaeus by the otherwise unknown Calcidius.The first part examines the authorial voice of the commentator and the overall purpose of the work; the second part provides an overview of the key themes; and the third part reassesses the commentary's relation to Stoicism, Aristotle, potential sources, and the Christian tradition.

This chapter examines the role of matter as an independent principle in the universe, and shows how Calcidius' borrows elements from Aristotle, the Stoics, and the "Pythagorean" Numenius to posit a minimal dualism.