Geometry in History PDF
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S. G. Dani · Athanase Papadopoulos Editors Geometry in History Geometry in History S. G. Dani • Athanase Papadopoulos Editors Geometry in History 123 Editors S.G.Dani AthanasePapadopoulos UniversityofMumbai&Department InstitutdeRechercheMathématique ofAtomicEnergy UniversityofStrasbourg,CNRS CentreforExcellenceinBasicSciences StrasbourgCedex,France Mumbai,Maharashtra,India ISBN978-3-030-13608-6 ISBN978-3-030-13609-3 (eBook) https://doi.org/10.1007/978-3-030-13609-3 ©SpringerNatureSwitzerlandAG2019 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthors,andtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional claimsinpublishedmapsandinstitutionalaffiliations. ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG. Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Preface This volume consists of a collection of essays on geometry from a historical viewpoint, addressed to the general mathematical community interested in the historyofideasandtheirevolution. In planning this book as editors we went with the conviction that writing on the basic geometrical concepts from a historical perspective is an essential elementofourscience literature,andthatgeometrico-historicalsurveysconstitute an importantingredient,alongside purely mathematicaland historicalarticles, for the developmentof the subject. We have also held, for many years, that we need sucharticleswrittenbymathematicians,dealingwithtopicsandideasthattheyare directly engaged with and consider as fundamental, and it was our endeavour to put together an ensemble of articles of this variety on a broad spectrum of topics thatconstitutethegeneralareaofgeometry.Itactuallyturnedouttobeanontrivial tasktofindcolleaguescapableandwillingtowriteonthehistoryoftheirfield,but perseveranceeventuallyledtothisvolumecomingtofruition.Wewarmlythankall theauthorsofthisvolumefortheircontribution.Wealsothankallthereferees,who shallstayanonymous,fortheircriticalcommentsandconstructivesuggestionsthat helpedsubstantiallyinimprovingthequalityofthevolume,andElenaGriniarifor hereditorialsupport. The workonthisbookwascompletedduringa stay of thesecondeditoratthe YauMathematicalCenterofTsinghuaUniversity(Beijing). Mumbai,India S.G.Dani Strasbourg,France AthanasePapadopoulos November2018 v Introduction Comprehension of shapes has played a pivotal role, alongside of numbers, in the progressofcivilizations,fromthebeginning.Significantengagementwith shapes, orgeometry,1 isseenintheancientculturesofEgypt,Mesopotamia,India,China, etc.,fromtheveryearlytimes.Asystematicapproachtothesubject,turningitinto a discipline with an axiomatic foundation, was developed by the ancient Greeks, whichservedcruciallyasabasistolaterrewritingsanddevelopmentsatthehands ofArabmathematiciansduringthe Middleagesandin turnto themodernadvent. TheGreeksalsoaddressedvariousphilosophicalissuesassociatedwiththesubject thathavebeenveryinfluentialinthelaterdevelopments. From the point of view of the subject of history of mathematics, there is a need for viewing the historical development of ideas of geometry as an integral whole. The present endeavour is seen by the editors as a limited attempt in that direction, focusing mainly on the historical antecedents of modern geometry and the internal relations within the latter.2 In the overallcontext, the editors also felt theneedtoconcentrateonbringingouttheperspectiveofworkingmathematicians actively engaged with the ideas involved, in their respective areas, as against that of historians of mathematicians viewing developments in mathematics from the outside, a pursuit in which the issues involved and the flavour of the output are differentfromwhatweseektoexploreinthisproject. Mathematiciansbuildupontheworksoftheirpredecessors,whichtheyregularly reshape,refineandreinterpret(sometimesmisinterpret).Therearecountlessexam- ples of ideas discoveredconcurrentlyand independentlyand of others that stayed in the dark until being rediscovered and used much later. This makes the history ofmathematicalideasalivingandintricatetopic,andanyattempttosaysomething 1Althoughthetermgeometryisetymologicallyassociatedwithearthmeasurement, itwasclear sincethetimesofPlatothatthisfield,fromthemomentitbecamemature,ismoreconcernedwith shapethanwithmeasurement. 2Thepresenteditorshadorganizedaconferenceonthesametheme,“GeometryinHistory,”atthe UniversityofStrasbourgduring9–10June2015;thedeliberationsattheconferenceandtheoverall experienceoftheeventhaveservedasaninspirationalprecursortobringingoutthisvolume. vii viii Introduction significantinthisdomainrequires,ontheonehand,adeepknowledgeofthesubject and also, on the other hand, a comprehensive vision of history. In the perception of the editors of this volume, the mathematical community is in acute need of articlespresentingmajorgeometricalideasinarighthistoricalperspective,paying attentionalsotothephilosophicalissuesaroundtheactivity.Thiswasamotivation forbringingoutthisvolume,andwearehappytoseethatmanyoftheessaysinit haveturnedouttobeaconfluenceofmathematics,historyandphilosophy,together withstate-of-the-artmathematicalresearch. The bookis divided into two parts. The first one, consisting of seven chapters, is concerned with topics that have roots in Greek antiquity and ramifications all through the history of mathematics. The second part, consisting of 12 chapters, treats more modern topics. We now describe briefly the themes dealt with in the individualchapters. The opening essay by Stylianos Negrepontis is on a topic from the fourth century BCE, namely, Plato’s theory of anthyphairesis, an idea that would serve asaprecursorofcontinuedfractions.ItmaybeworthwhiletorecallherethatPlato wasaboveallamathematicianandthathisconceptualizationoftheworldisdeeply rooted in mathematics. The goal of this article is to explain the anthyphairetic nature of Plato’s dialectics which is at the foundation of his theories of Ideas, of true Beings, of knowledge and of the distinction he draws between intelligible and sensible Beings. Negrepontis defends the thesis that the whole of Plato’s philosophical system is based on the concept of periodic anthyphairesis. He also providesanexplanationofPlato’spraiseofgeometryandhiscriticismofitspractice bythegeometers.Platocriticizedtheaxiomaticmethod,bywhichmathematicians relyonhypothesesthat,accordingtohim,havenothingtodowithtrueknowledge. He was at the same time critical of the geometers’ use of diagrams and of topics suchasEudoxus’theoryofratios(onwhichthetheoryofDedekindcutsisbased), of Archytas’ theory of quadratic and cubic incommensurabilities and of the use of the notion of “geometric point” in the foundations of geometry. Instead, Plato claims that the method of Division and Collection—a philosophicalexpressionof theperiodicanthyphairesis—shouldbetheonlyonetobeusedintheacquisitionof allknowledgein geometry,inparticularintheconstructionofnumbersandofthe straightline. At the same time, the authoralso proposesan explanationas to why geometrywassoimportanttoPlato. The whole essay is based on a new analysis and a novel reading of several difficult dialogues of Plato, in particular the Parmenides, Sophist, Statesman, TimaeusandPhaedo. Thesecondchapter,writtenbyAthanasePapadopoulos,isanexpositionofRené Thom’s visionary ideas in his interpretation of the work of Aristotle—especially histreatises on biology—froma topologicalpointof view.Thomwas a dedicated reader of Aristotle. With his penetrating intuition as a mathematician, he gave a completelynewexplanationofsomepassagesinthewritingsoftheStagirite,finding theretheideasofgenericity,stratification,boundary,theStokesformulaandother topologicalnotions.HecompletelyadheredtoAristotle’stheoryofformwhichthe latter expanded in his zoological treatises, and he highlighted the importance of Introduction ix these ideas in biology, and more particularly in embryology, namely, the idea of a form tendingto its own realization.Thom’s contributionto the interpretationof Aristotle’swritings,anoutcomeofmanyyearsofreflection,constitutesanewlink betweenscienceinGreekantiquityandmodernmathematics. In Chap.3, Yuri Manin describes a continuous chain of ideas in mathematical thought that connects Greek antiquity to the modern times. He uses this chain as an argumentfor his thesis that the notion of “paradigmshifts”, as an approachin thehistoryofsciencepromotedbythephilosopherThomasKuhn(1922–1996),to theeffectthatradicalchangesintheobjectsofinterestcharacterizethetransitions from one epoch to another, does not apply to mathematics. On the contrary, says Manin, mathematicalthoughtand the mathematicallyinteresting questionsevolve in a continuous manner. The ideas on which his argument is based involve the mathematicsofspace,timeandperiodicity,andhetakesusthroughsomeepisodes fromthehistoryofthissubject,startingwithPtolemy’sdynamicalmodelofthesolar systemuntilthemodernprobabilisticmodelsofelementaryparticles,Schrödinger’s quantummechanicsamplitudeinterferenceandFeynmanintegrals,passingthrough theworkofFourier,andMendeleev’stableofchemicalelements.Inthisdescription ofthehistory,quantummechanics,inManin’swords,becomes“acomplexification ofPtolemy’sepicycles”. In Chap.4, Athanase Papadopoulos reviews the appearances of the notion of convexityinGreekantiquity,morespecificallyintheclassicaltextsonmathematics andoptics,inthewritingsofAristotleandinart.Whileconvexitywastoturninto a mathematical field in the early twentieth century, at the hands of Minkowski, Carathéodory and others, this notion is found in mathematical works all the way fromthoseoftheGreeks.ThethinkinginmanymathematicalargumentsofEuclid, Apollonius, Archimedes, Diocles and others is seen to be based on convexity considerations,evenwhen the concepthad not attained its maturity.In optics, the notion played an important role on account of convex and concave mirrors and lenses, the latter relating also to astronomy.Convexity also features in Aristotle’s theories on a variety of topics and in ancient Greek art. This chapter puts the evolutionofthenotionofconvexityinperspective. Chapter 5, by Arkady Plotnitsky, is again concerned with the question of continuity of ideas in mathematics. Starting with the question of “what is a curve”, and taking as a starting point the curves formed by images drawn 32,000 years ago on the curved walls of the Chauvet-Pont-d’Arc cave in the South of France,theauthorreflectsontheevolutionofmathematics,ontherelationbetween mathematics and art and on the notion of modernity and the difference between “modernity”and “modernism”, both in art and science. Modernism in mathemat- ics, he argues, is essentially a conception characterized by the algebraization of spatiality while adhering to geometrical and topological thinking. The exposition wanders through the works of Fermat and of Descartes—where the notion of algebraization of geometry became essential—through a reflection on the role of experimentationinphysics,andthroughtheideasofHeideggeronmodernscience asbeingessentiallymathematical.Acrisisconcerningthe“irrationality”ofquantum mechanicsandits“rationalization”byWernerHeisenbergiscomparedtothecrisis x Introduction of the incommensurable which traversed ancient Greek mathematics. The author alsocommentsonJohnTate’sprinciple:“Thinkgeometrically,provealgebraically”, extendingitto:“Thinkbothintuitivelygeometricallyandspace-likegeometrically, andprovealgebraically”,andtakingexamplesinthemodernworksofWeyl,Weil, Grothendieck,LanglandsandTate. TheaimofChap.6,byAnnetteA’CampoandAthanasePapadopoulos,istodraw a continuous path from the theory of curves in Greek antiquity until the modern synthetic differential geometry of Busemann–Feller and Alexandrov. The route passesthroughtheworkofHuygensonevolutes,throughthatofEuleroncurvature of surfacesembeddedin 3-space and then throughthe developmentsmade by the French school founded by Monge. These works, together with the backgroundof therespectiveperiodsandoftheauthorsconcerned,arediscussedextensivelyinthis chapter.Theevolutionshowsareturn,inthetwentiethcentury,tothefundamental methodsofgeometrythateschewthedifferentialcalculus. Chapter7,byToshikazuSunada,isahistoricalexpositionofthedevelopmentof geometryfrom ancient times until the modernperiod, with a view of this field as a tool for describing the shape of the universe. The accountinvolvesunavoidably questionsofcosmologyandofphilosophyofspace andtime,andittakesusdeep into the worlds of differential and projective geometry, topology and set theory, discussingnotionsliketheinfinite,infinitesimal,curvature,dimensionandothers. The next two chapters are concerned with configuration theorems, that is, theoremsofprojectivegeometrywhosestatementsinvolvefinitesetsofpointsand arrangementsoflines. Configuration theorems form a coherent subject which is again rooted in Greek antiquity, more precisely in the theory of conics, whose main foundersare Apollonius, Pappus and Ptolemy. The subject continued to grow in the works of Pascal, Desargues,Brianchon,Poncelet,Steiner andothers, and it hasstill a great impactoncurrentresearch.AfamousexampleofaconfigurationtheoremisPappus’ theoremwhichconcernsthealignmentofthreepoints:giventwotriplesofaligned pointsA,B,Canda,b,cintheplane,thethreeintersectionpointsofthethreepairs oflinesAB,Ba,Ac,Ca andBc,Cbarealsoaligned.AnotherexampleisPascal’s theoremstatingthatifwetakearbitrarilysixpointsonaconicandifwejointhem pairwise so as to forma convexhexagon,thenthe three pairsof oppositesidesof this hexagon meet in three points that lie on the same line. In the case where the conicdegeneratesto a pairoflines, Pascal’stheoremreducestoPappus’theorem. Thus,Pascal’stheoremisagoodexampleofatheoremdiscoveredintheseventeenth centurywhichisageneralizationofamuchearliertheorem. Inthisvolume,configurationtheoremsareconsideredinChaps.8and9. Chapter 8, written by Victor Pambuccianand Celia Schacht, is concernedwith the significance of Pappus’ and Desargues’ and other configuration theorems de- rivedfromthemintheaxiomaticfoundationofgeometryandwiththeinterrelation of this topic with algebra and first-order logic. From a historical point of view, it has taken quite long for the wealth of significance of the theoremsof Pappusand Desargues to be recognized. The recognition came only in the twentieth century, Introduction xi spurring with it also a flurry of activity in the general stream of configuration theorems, the genre they represent. The authors consider the article a story of “adventurethroughwhichtheirtrueimportancebecamerevealed”. Chapter 9, by Serge Tabachnikov, is a survey on the impact of configuration theoremsandresultsinspiredbythem inmoderndynamics,namely,in a seriesof iteratedconstructiontheoremsmotivatedbyconfigurationtheorems,inthestudyof mapsofthecircleandself-similarity,inthestudyofactionsofthemodulargroup, in the theory of billiards in ellipses and in the study of caustics. Identities in the LiealgebraofmotionsofEuclideanandnon-Euclideangeometriesareinterpreted asconfigurationtheorems.TheauthoralsosurveystheworkofR. Schwarzonthe pentagram map and the theory of skewers, recently developed by himself, which providesasettingforspaceanaloguesofplaneconfigurationtheorems. In Chap.10, Ken’ichi Ohshika shows how Poincaré’s work on topology and geometry led him to the philosophy of science that he formulated in the various books he published. At the same time, he reviews some important points in Poincaré’s work: the foundation of homology theory, the construction of 3- manifolds by gluing polyhedra or by using complex algebraic equations in three variablesandthefirstresultsonwhatlaterbecameknownasMorsetheory,usedby Poincaréinhisclassificationofclosedsurfaces.OtherquestionsraisedbyPoincaré thatledtothedevelopmentoftopologyarealsomentionedinthischapter.Besides, OhshikaaddressesthequestionastowhymathematiciansofthestatureofPoincaré are interested in the philosophyof science. This serves as an occasion for him to mention,briefly,theapproachesofseveralphilosophersinthisfield,includingKant, Frege, Husserl, Russell and Althusser. The latter talked about the “spontaneous philosophyofscientists”andatthesametimeinsistedonthenecessityofdrawing alineseparatingsciencefromideology. Chapter 11, by Alain Chenciner, is about the applications of the study of the dynamicsofthe iteratesofa mapobtainedby perturbingthegermofthe simplest map of the plane: a planar rotation at the origin. The author recalls how such a study led to the Andronov–Hopf–Neimark–Sacker bifurcation theory, concerned with invariant curves under a radial hypothesis of weak attraction (or repulsion) forgenericdiffeomorphismswithellipticfixedpoints,andthentotheKolmogorov– Arnold–Moser(KAM)theorywhichdealswitharea-preservingmaps.Hebringsout the relation with the so-called non-linear self-sustained oscillation theory of Lord Rayleighand Van der Pol, andthe theoryof normalformsdevelopedby Poincaré inhis1879thesisinconnectionwiththethree-bodyproblem,andhealsomentions therelationwithwhathecallsthe“averagingofperturbations”technique,usedby astronomerssincetheeighteenthcentury,whichgeneralizestothenon-linearsetting the Jordan normalform of a matrix.He concludeswith a result of his own which interrelatesseveralideaspresentedearlierinthechapter. Chapter12,byFrançoisLaudenbach,startswithGromov’sh-principle,aprinci- plethatgivesconditionsunderwhichamanifoldcarryingageometricstructureina weaksense(e.g.analmost-complexstructure,analmostsymplecticstructure,etc.) carries a genuine geometric structure (a complex structure, an almost symplectic structure, etc.,). At the same time, the author highlights a principle formulated
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