The ethics of geometry : a genealogy of modernity

The Ethics of Geometry is a study of the relationship between philosophy and mathematics. Essential differences in the ethos of mathematics, for example, the customary ways of undertaking and understanding mathematical procedures and their objects, provide insight into the fundamental issues in the quarrel of moderns with ancients. Two signal features of the modern ethos are the priority of problem-solving over theorem-proving, and the claim that constructability by human minds or instruments establishes the existence of relevant entities. These figures are combined in the emblematic statement of Salomon Maimon, "In mathematical construction we are, as it were, gods." Construction is the mark of modernity. The disciplines of classical philology, literary interpretation and the history of philosophy and of mathematics are woven together in this volume. Title ......Page 3 Copyright ......Page 4 Contents ......Page 5 Preface ......Page 7 Acknowledgments ......Page 15 I Projection, Construction, and the Idea of Modernity ......Page 17 II "Master of Nature, So to Speak": Consequences of Construction in Vico and Kant ......Page 23 III Two Zoroasters: A Sense of Endings ......Page 32 I Preamble ......Page 41 i Ratios, Multitudes, and Magnitudes ......Page 45 ii Homogeneity ......Page 49 iii Sameness of Ratios ......Page 58 iv Conclusions ......Page 64 III The Powers and Idioms of Construction in Greek Geometry ......Page 65 IV The Establishment of the Kantian Orthodoxy ......Page 66 V Genesis and Knowing ......Page 77 VI The Perfect Imperative ......Page 81 VII The Evidentiary Force of Constructions in Greek Mathematics ......Page 83 i Instrumental Constructions ......Page 86 ii Construction as Operation (Sensory and Imaginative) ......Page 92 iii Noesis and "Mental" Construction ......Page 103 i Introduction ......Page 107 ii Einai and Existence in Aristotle ......Page 109 iii Hyparxis and Deixis in Stoicism ......Page 114 iv Essence and "Existence" in the Medieval Arabic Tradition ......Page 119 v Kant's Relation to Premodern Understandings of Existence ......Page 124 IX The Enigma of the Postulates ......Page 126 X Conclusions ......Page 137 I Preface ......Page 140 II The Art of Origins: Descartes' Fabulous History ......Page 142 i Prelude ......Page 157 ii Reading the Geometry ......Page 158 iii The Structure of the Geometry ......Page 160 iv The Style of the Geometry ......Page 164 v Liberty and Constraints: Why Can Problems Be Solved? ......Page 177 vi The Roots of the Geometry: Unity, Order, and Measure in the Rules ......Page 190 i Reading Notes ......Page 203 ii Pur a Mathesis and Descartes' Geometry ......Page 204 iii The Principal Characteristics of Cartesian Geometry Recapitulation and Extensions ......Page 207 iv Essence and Existence in Cartesian Pura Mathesis ......Page 213 v Morals Par Provision ......Page 216 Notes ......Page 222 Bibliography ......Page 249 Index ......Page 267 In a wide-ranging study of the relationship between philosophy and mathematics, Lachterman discussing the importance of construction from Euclid to Kant and his successors. David Rapport Lachterman. Includes Index. Bibliography: P. 233-250.