Aristotle and Mathematics


Book Description

John Cleary here explores the role which the mathematical sciences play in Aristotle's philosophical thought, especially in his cosmology, metaphysics, and epistemology. He also thematizes the aporetic method by means of which he deals with philosophical questions about the foundations of mathematics. The first two chapters consider Plato's mathematical cosmology in the light of Aristotle's critical distinction between physics and mathematics. Subsequent chapters examine three basic aporiae about mathematical objects which Aristotle himself develops in his science of first philosophy. What emerges from this dialectical inquiry is a different conception of substance and of order in the universe, which gives priority to physics over mathematics as the cosmological science. Within this different world-view, we can better understand what we now call Aristotle's philosophy of mathematics.







Mathematics in Aristotle


Book Description

This is a detailed exposition of Aristotelian mathematics and mathematical terminology. It contains clear translations of all the most important passages on mathematics in the writings of Aristotle, together with explanatory notes and commentary by Heath. Particularly interesting are the discussions of hypothesis and related terms, of Zeno's paradox, and of the relation of mathematics to other sciences. The book includes a comprehensive index of the passages translated.




Explorations in Ancient and Modern Philosophy


Book Description

The first of two volumes collecting the published work of one of the greatest living ancient philosophers, M.F. Burnyeat.




Aristotle's Theory of Bodies


Book Description

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. The book is the first comprehensive and rigorous account of the features substances have in virtue of being bodies. It provides an analysis of the concept of three-dimensional magnitude and related notions like boundary, extension, contact, continuity, often comparing it to modern conceptions of it. Both the structural features and the ontological status of body is discussed. This makes it significant for scholars working on contemporary metaphysics and mereology because the concept of a material object is intimately tied to its spatial or topological properties.




The Metaphysics


Book Description

The Metaphysics presents Aristotle's mature rejection of both the Platonic theory that what we perceive is just a pale reflection of reality and the hardheaded view that all processes are ultimately material. He argued instead that the reality or substance of things lies in their concrete forms, and in so doing he probed some of the deepest questions of philosophy: What is existence? How is change possible? And are there certain things that must exist for anything else to exist at all? The seminal notions discussed in The Metaphysics - of 'substance' and associated concepts of matter and form, essence and accident, potentiality and actuality - have had a profound and enduring influence, and laid the foundations for one of the central branches of Western philosophy.




The Metaphysics of the Pythagorean Theorem


Book Description

Bringing together geometry and philosophy, this book undertakes a strikingly original study of the origins and significance of the Pythagorean theorem. Thales, whom Aristotle called the first philosopher and who was an older contemporary of Pythagoras, posited the principle of a unity from which all things come, and back into which they return upon dissolution. He held that all appearances are only alterations of this basic unity and there can be no change in the cosmos. Such an account requires some fundamental geometric figure out of which appearances are structured. Robert Hahn argues that Thales came to the conclusion that it was the right triangle: by recombination and repackaging, all alterations can be explained from that figure. This idea is central to what the discovery of the Pythagorean theorem could have meant to Thales and Pythagoras in the sixth century BCE. With more than two hundred illustrations and figures, Hahn provides a series of geometric proofs for this lost narrative, tracing it from Thales to Pythagoras and the Pythagoreans who followed, and then finally to Plato's Timaeus. Uncovering the philosophical motivation behind the discovery of the theorem, Hahn's book will enrich the study of ancient philosophy and mathematics alike.




Aristotle's Theory of Actuality


Book Description

This is an attack on Aristotle showing that his misplaced drive toward the consistent application of his actualistic ontology (denying the reality of all potential things) resulted in many of his major theses being essentially vacuous.




An Aristotelian Realist Philosophy of Mathematics


Book Description

Mathematics is as much a science of the real world as biology is. It is the science of the world's quantitative aspects (such as ratio) and structural or patterned aspects (such as symmetry). The book develops a complete philosophy of mathematics that contrasts with the usual Platonist and nominalist options.




Philosophy of Mathematics


Book Description

A sophisticated, original introduction to the philosophy of mathematics from one of its leading thinkers Mathematics is a model of precision and objectivity, but it appears distinct from the empirical sciences because it seems to deliver nonexperiential knowledge of a nonphysical reality of numbers, sets, and functions. How can these two aspects of mathematics be reconciled? This concise book provides a systematic, accessible introduction to the field that is trying to answer that question: the philosophy of mathematics. Øystein Linnebo, one of the world's leading scholars on the subject, introduces all of the classical approaches to the field as well as more specialized issues, including mathematical intuition, potential infinity, and the search for new mathematical axioms. Sophisticated but clear and approachable, this is an essential book for all students and teachers of philosophy and of mathematics.