Philosophy of Mathematics from the Pythagoreans to Euclid

Description

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. Cover Title page Imprints page Philosophy of Mathematics from the Pythagoreans to Euclid Contents Introduction I1 The Development of Greek Mathematics: Detachment from a Practical Context I2 Specific Demarcation of Mathematics in Ancient Greece I3 Relationship between Geometry and Arithmetic I4 Specificities of Greek Mathematics I5 Notion of Numbers 1 Ontology: What Kind of Things Are Mathematical Objects? 1.1 Numbers as the Ultimate Constituents of Things with the Pythagoreans 1.2 Mathematical Objects as Part of the Intelligible Realm in Plato’s Republic and Phaedo 1.3 Mathematical Objects as Underlying the Physical Realm in Plato’s Timaeus 1.4 Mathematical Objects as Abstractions in Aristotle 2 Epistemology: Mathematical Knowledge versus Philosophical Knowledge 2.1 Mathematical Knowledge as a Model 2.2 The Distinction between Mathematical Knowledge and the Highest Form of Philosophical Knowledge in Plato 2.3 The Possibility of Explanation in the Mathematical Sciences 3 Methodology 3.1 Mathematical Deductions as Paradigmatic for Philosophical Proofs 3.2 Reductio ad Absurdum Proofs and Philosophical Paradoxes 4 Central Concepts of Philosophy and Mathematics 4.1 Principles and Starting Points 4.2 Mathematical and Philosophical Notions of Continuity 4.3 Limits – the Distinction between Inner and Outer Limits 4.4 Philosophical and Mathematical Notions of Infinity References Acknowledgements