This chapter gives an overview of the problems raised by the concept of motion in the period investigated – specifically, its apparent integration of Being and non-Being, and its combination of Time and Space. It then discusses the central notions employed in the present project: the criteria or standards established for philosophical inquiry (the principles of non-contradiction and of sufficient reason, and a criterion termed “rational admissibility”) and the roles of logic and mathematics in establishing natural philosophy. This chapter lays out broad outlines, systematic and historical, of the issues to be discussed in detail in the ensuing chapters, each of which will deal with one thinker or one school. Thus this opening chapter will serve as a first orientation for the project as well as a reservoir for consultation if questions concerning the basic concepts employed arise during the reading of the whole book.

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.

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.