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This Element looks at the very beginning of the philosophy of mathematics in Western thought. It covers the first reflections on attempts to untie mathematics from its practical usage in administration, commerce, and land-surveying and discusses the first ideas to see mathematical structures as constituents underlying the physical world in the Pythagoreans. The first two sections focus on the epistemic status of mathematical knowledge in relation to philosophical knowledge and on the various ontological positions ancient Greek philosophers in early and classical times ascribe to mathematical objects – from independent and separate entities to mere abstractions and idealisations. Section 3 discusses the paradigmatic role mathematical deductions have played for philosophy, the role of mathematical diagrams, and mathematical methods of interest for philosophers. Section 4, finally, investigates a couple of individual concepts that are fundamental for both philosophy and mathematics, such as infinity.
This book explores a distinctive feature of ancient philosophy: the close relation between ancient ethics and the study of the natural world. Human beings are in some sense part of the natural world, and they live their lives within a larger cosmos, but their actions are governed by norms whose relation to the natural world is up for debate. The essays in this volume, written by leading specialists in ancient philosophy, discuss how these facts about our relation to the world bear both upon ancient accounts of human goodness and also upon ancient accounts of the natural world itself. The volume includes discussion not only of Plato and Aristotle, but also of earlier and later thinkers, with an essay on the Presocratics and two essays that discuss later Epicurean, Stoic, and Neoplatonist philosophers.
In contemporary discussions, ethics and inquiry into the natural world are often treated as two completely independent fields of study. By contrast, many ancient thinkers took them to be intimately connected. This volume aims to shed light on the various ways in which ancient thinkers drew connections between these two fields. We human beings are in some sense part of the natural world, and live our lives within a larger cosmos, but yet our actions are governed by norms whose relation to the natural world is up for debate. The chapters in this volume discuss how these facts about our relation to the world bear upon both ancient accounts of human goodness and also ancient accounts of the natural world itself. The chapters focus primarily on Plato and Aristotle. But we have also included some discussion of earlier and later thinkers, with a chapter on the Presocratics and a couple of chapters that at least in part point ahead to later Epicurean, Stoic, and Neoplatonist philosophers.
This chapter addresses the puzzling question of why, in his Timaeus, Plato combines two very different topics: a cosmogony and account of the universe, on the one hand, and a story about the moral actions of ancient Athens, Atlantis, and Egypt, on the other. Sattler argues that the key to understanding the relation between these two parts is recognition that, in Plato’s view, they confront us with a structurally similar problem: how we are to account for the intelligibility of processes in the phenomenal world. Sattler shows that Plato no longer chooses to solve this problem by tying intelligibility to complete uniformity, as he did in the Republic, but by tying intelligibility to a rule – to norms and laws for actions in the human cultural realm, and to ratios and descriptive rules for the motions of the heavenly bodies in the natural realm. While Plato also accounts for the concerns specific to ethics and physics, the attempt to understand processes raises similar problems for him in both realms. Recurring natural catastrophes, such as floods and fires, appear as one kind of natural regularity in this Platonic account.
Plato’s cosmology in the Timaeus can be seen as being framed in biological terms, since it claims the universe as a whole to be a living being, more specifically, a created god. In this chapter, I show that the central assumption that leads Plato to understand the created cosmos as a living being is the idea that the world is as good as possible. In a second step, I want to show that this assumption of the world’s bestness is also responsible for two important twists to the biological framing Plato uses. First, being as good as possible also implies that the world is self-sufficient, which means that many of our common biological notions are of no relevance for an account of the cosmos as a living being. Secondly, I show that while Plato gives an account of all kinds of living beings, his assumption of the bestness of the world leads him to be ultimately interested only in rational living beings. Accordingly, what starts out in biological terms turns into a form of rational psychology and rational theology. This will finally lead to a discussion whether Plato works with a consistent notion of life in the Timaeus.
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.