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IMPOSSIBILITY RESULTS: from GEOMETRY to ANALYSIS Davide Crippa

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IMPOSSIBILITY RESULTS: from GEOMETRY to ANALYSIS Davide Crippa IMPOSSIBILITY RESULTS: FROM GEOMETRY TO ANALYSIS Davide Crippa To cite this version: Davide Crippa. IMPOSSIBILITY RESULTS: FROM GEOMETRY TO ANALYSIS : A study in early modern conceptions of impossibility. History, Philosophy and Sociology of Sciences. Univerist´eParis Diderot Paris 7, 2014. English. <tel-01098493> HAL Id: tel-01098493 https://hal.archives-ouvertes.fr/tel-01098493 Submitted on 25 Dec 2014 HAL is a multi-disciplinary open access L'archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destin´eeau d´ep^otet `ala diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publi´esou non, lished or not. The documents may come from ´emanant des ´etablissements d'enseignement et de teaching and research institutions in France or recherche fran¸caisou ´etrangers,des laboratoires abroad, or from public or private research centers. publics ou priv´es. UNIVERSITE PARIS DIDEROT (PARIS 7) SORBONNE PARIS CITE ECOLE DOCTORALE: Savoirs Scientifiques, Epistémologie, Histoire des Sciences et Didactique des disciplines DOCTORAT: Epistémologie et Histoire des Sciences DAVIDE CRIPPA IMPOSSIBILITY RESULTS: FROM GEOMETRY TO ANALYSIS A study in early modern conceptions of impossibility RESULTATS D’IMPOSSIBILITE: DE LA GEOMETRIE A L’ANALYSE Une étude de résultats classiques d’impossibilité Thèse dirigée par: Marco PANZA Soutenue le: 14 Octobre 2014 Jury: Andrew Arana Abel Lassalle Casanave Jesper Lützen (rapporteur) David Rabouin Vincenzo de Risi (rapporteur) Jean-Jacques Szczeciniarz Alla memoria di mio padre. A mia madre. Contents Preface 9 1 General introduction 14 1.1 The theme of my study . 14 1.2 A difficult context . 16 1.3 Types of impossibility arguments . 20 1.3.1 An ancient example . 20 1.3.2 Impossibility in the theory versus impossibility in the meta-theory 21 1.4 Impossibility statements as meta-statements . 26 1.4.1 The unsettled nature of the circle-squaring problem . 30 1.5 Impossibility results in early modern geometry . 39 2 Problem solving techniques in Ancient geometry 44 2.1 Introduction . 44 2.2 Pappus’ division of problems into three kinds . 52 2.2.1 A conjecture about the origin of Pappus’ classification of problems 57 2.2.2 Normative aspects in Pappus’ classification . 60 2.3 Curves and problems of the third kind . 66 2.3.1 The conchoid . 66 2.3.2 The Quadratrix . 68 2.3.3 The Archimedean spiral . 73 2.3.4 The Apollonian helix . 75 2.4 Problems from the third kind of geometry . 78 2.4.1 General Angle Division (Collection, IV, proposition 35) . 78 2.4.2 Problems related to the general angle division . 82 2.5 The rectification problem . 82 5 CONTENTS 6 2.5.1 The rectification of the circumference via quadratrix (Pappus, Col- lection, IV, proposition 27) . 83 2.5.2 The converse problem: to construct a circumference equal to a given segment . 88 2.5.3 Rectification through the spiral (Archimedes, On Spiral lines, prop. XVIII) . 89 2.6 On the quadrature of the circle . 95 3 The geometry of René Descartes 100 3.0.1 Descartes’ geometry and its methodological presuppositions . 100 3.0.2 Descartes’ early methodology of problem solving . 103 3.1 Analysis and Synthesis in Descartes’ geometry . 113 3.1.1 Cartesian analysis as transconfigurational analysis . 113 3.1.2 The constitution of the algebra of segments . 126 3.1.3 The construction of the ‘four figures’ . 131 3.2 Descartes’ construal of geometricity in 1637 . 133 3.2.1 Euclidean restrictions reconsidered . 133 3.2.2 Early instances of geometrical linkages . 146 3.2.3 The determinative character of cartesian algebra . 148 4 Simplicity in Descartes’ geometry 155 4.1 Introduction . 155 4.2 Simplicity in early modern geometry . 157 4.3 Classifications of curves and problems . 160 4.3.1 Ancient and modern classifications . 160 4.3.2 A classification of curves . 161 4.3.3 A Classification of problems . 167 4.4 Construction of third and fourth degree equations . 173 4.4.1 Construction of a cubic equation . 175 4.4.2 The insertion of two mean proportionals . 179 4.5 Easiness versus simplicity . 183 4.5.1 Two solutions compared . 183 4.5.2 Easiness, simplicity and the algebraic ordering of curves . 186 4.6 Impossibility and the interpretation of Pappus’ norm . 203 4.6.1 Impossibility arguments in La Géométrie . 203 4.6.2 A case for unrigorous reasoning . 209 CONTENTS 7 4.6.3 Impossibility results as metastatements . 214 4.6.4 The legacy of the cartesian programme: simplicity at stake . 215 5 Mechanical curves in Descartes’ geometry 221 5.1 Mechanical curves in Descartes’ geometry . 221 5.2 On Geometrical and Mechanical constructions . 227 5.2.1 Constructions by means of twisted lines or strings . 229 5.2.2 Construction of the spiral (Schwenter, Huygens) . 231 5.2.3 Pointwise construction of mechanical curves . 234 5.3 Descartes’ appraisal of string-based mechanisms and pointwise constructions247 5.4 Specification by genesis and specification by property: the case of mechan- ical curves . 255 6 Impossible problems in cartesian geometry 266 6.1 On the cartesian distinction between possible and impossible problems . 266 6.2 Impossibility claims as a meta-statements . 272 6.3 On the significance of early rectifications for Descartes’ meta-statement . 276 6.4 Problems of quadratures and the problem of area . 288 7 James Gregory’s Vera Circuli Quadratura 296 7.1 Introduction: the quadrature of the circle . 296 7.2 The controversy between James Gregory and Christiaan Huygens . 303 7.3 Analyzing the quadrature of the circle . 307 7.3.1 The aims of analysis . 307 7.3.2 Introducing convergent sequences . 314 7.3.3 The convergence of the double sequence . 321 7.3.4 Computing the terminatio . 329 7.4 An argument of impossibility . 334 7.5 Reception and criticism of Gregory’s impossibility argument . 342 7.6 Conclusions . 354 7.6.1 Bibliographical note . 365 8 The arithmetical quadrature of the circle 367 8.1 Introduction . 367 8.1.1 The manuscript of the De quadratura arithmetica . 368 8.1.2 Leibniz’s acquaintance and study of Gregory’s works . 370 8.2 The arithmetical quadrature of the circle, its main results . 373 CONTENTS 8 8.2.1 Looking backward onto Leibniz’s quadrature of the central conic sections . 373 8.2.2 Towards the arithmetical quadrature of the circle: the transmuta- tion theorem . 376 8.2.3 Towards the arithmetical quadrature of the circle: the generation of the ‘anonymous’ curve . 381 8.3 A digression: Mercator-Wallis technique for the quadrature of the hyperbola385 8.4 Leibniz’s arithmetical quadrature of 1674-’75 . 396 8.4.1 Extending Wallis-Mercator technique . 396 8.4.2 The rectification of a circular arc . 401 8.4.3 Leibniz’s fictionalist stance . 403 8.5 The quadrature of the circle in numbers . 406 π 8.5.1 Leibniz’s alternate series for 4 . 406 8.5.2 Oldenburg’s objections and the classification of quadratures . 414 8.6 The impossibility of giving a universal quadrature of the circle . 422 8.6.1 Leibniz’s criticism of Gregory’s arguments . 422 8.7 An impossibility argument . 428 8.7.1 Universal and particular quadratures . 428 8.7.2 The impossibility of the universal quadrature . 430 8.7.3 The impossibility of finding a general quadrature of the hyperbola 438 8.8 Underdeveloped parts in Leibniz’s impossibility argument . 440 8.9 The transcendental nature of curves . 445 8.10 Conclusions . 450 8.10.1 On the limits of cartesian geometry . 450 8.10.2 The constitution of transcendental mathematics and Leibniz’s new calculus . 459 8.10.3 Appendix: primary sources . 464 9 Epilogue 466 9.1 A survey of early-modern impossibility results . 466 9.2 The structure of early modern impossibility results . 470 9.2.1 The role of algebra in XVIIth century . 470 9.2.2 Algebraic proofs of impossibility theorems . 474 9.2.3 Early modern constructions and constructibility . 476 9.2.4 From the constructive paradigm to the conceptual paradigm . 479 9.3 Impossibility arguments as answers to metatheoretical questions . 483 Preface: capturing the unicorn . the fence inside which we hope to have enclosed what may appear as a possible, living creature. O. Neugebauer, The Exact Sciences in Antiquity, p. 177. Otto Neugebauer once recalled, in his masterpiece The exact Sciences in Antiquity, that his endeavour in restoring the mathematics of the past had a simile in the tale of the unicorn, which ended with the miraculous animal captured in a fence and gracefully resigned to his fate. In this dissertation, I have also erected, out of pieces of evidence, conjectures and indirect testimonies, an enclosure in order to capture an elusive but (I think) living subject of research. This subject is provided by the theme of impossibility results in classical and early modern mathematics. I started the inquiry which led to this dissertation out of the following, perhaps naive observation: all the famous impossibility results in geometry (namely, the impossibility of duplicating the cube, trisecting an angle or squaring the circle by ruler and compass) are proved by appealing to a rather sophisticated algebraic machinery. Why mathematicians had turned to algebra in order to prove geometric impossibility results, and what makes algebra such a powerful resource that it could prove the impossibility of solving certain problems in geometry, apparently unprovable by geometric means only? 9 CONTENTS 10 My original questioning was as much interesting to me as it was broad, and perhaps unfit for a discussion within a single dissertation. I then decided to develop my inquiry mainly from a historical viewpoint, and turned to what I considered one of the first examples of algebraic thinking in geometry, namely, Descartes’ epoch-making La Géométrie.
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