A VIEW OF MATHEMATICS Alain CONNES

1 A VIEWOFMATHEMATICSA lainCONNESM athematicsis thebackboneof modernscienceanda remarkablye cient sourceof newconceptsandtoolsto understandthe\reality"in which we plays a basicrolein thegreatnewtheoriesof physicsof theXXthcenturysuchas generalrelativity, thismentalactivity areoftenmisunderstoodorsimplyignoredeven amongscientistsof useof rudimentarymathematicaltoolsthatwerealre adyknownin theXIXthcenturyandmisscompletelythestren gthanddepthof theconstant was asked to writea generalintroductiononMathematicswhich I endedupdoingfroma ratherpersonalpoint of viewratherthanproducingtheusualendlessli tany\XdidthisandY didthat".Theevolutionof theconceptof \space"in math-ematicsserves as a unifyingthemestartingfromsomeof itshistoricalrootsandgoingtowardsmorerec ent developments in which I have beenmoreor of GeometryandtheDevilof ' MathematicsIt might be temptingat rstto viewmathematicsas theunionof separatepartssuch as Geometry, Algebra,Analysis,Number rstis dom-inatedby theunderstandingof theconceptof \space",thesecondby theartofmanipulating\symbols",thenextby theaccessto \in nity"andthe\continuum" does notdojusticeto oneof themostessentialfeaturesof themath-ematicalworld.

2 Namelythatit is virtuallyimpossibleto isolateany of thatway thecorpusof mathematicsdoes resemblea biologicalentity which canonlysurvive asa wholeandwouldperishif separatedinto disjoint rstembryo of mentalpictureof themathematicalworldonecanstartfromis thatof a networkof bewilderingcomplexity arequitesimpleandaretheresultof a longprocessof\distillation"in thealembicof dictionaryproceedsin a circularmanner,de ninga wordby referencetoanother,thebasicconceptsof mathematicsarein nitelycloserto an\indecompos-ableelement",a kindof \elementaryparticle"of thought witha minimalamountof ambiguity in theirde so forinstanceforthenaturalnumberswherethen umber 3 standsforthatquality which is commonto exactlyafterwe remove oneof itselements,thenremove anotherandthenremove it becomesindependent of thesymbol 3which is justa usefuldeviceto encode useto encode numbersaredependent of thesociologicalandhistoricalaccidents thatarebehindtheevolutionof any language,themathematicalconceptof number andeventhespeci city of a particularnumber such as 17aretotallyindependent of \purity" of thissimplestmathematicalconcepthasbeenus edby HansFreuden-thalto designa languageforcosmiccommunicationwhich he called\Lincos"[39].

3 Thescienti clifeof mathematicianscanbe picturedas a tripinsidethegeographyof the\mathematicalreality"which theyunveilgraduallyin oftenbeginsby anactof rebellionwithrespecttotheexistingdogmati cde-scriptionof thatreality thatonewill ndin \tobemathematician"realizein theirownmindthattheirperceptionof themathemat-icalworldcapturessomefeature swhich donotquite t rstactis oftenduein mostcasesto ignorancebutit allowsoneto freeoneselffromthereverenceto authority by relyingonone'sintuitionprovidedit is backedupby reallyknow,in anoriginaland\personal"manner,a smallpartof themathematicalworld,as esotericas it canlookat rst1, is of coursevitalallalongnotto breakthe\ ld'arianne"which allowstoconstantlykeepa fresheye onwhatever one1my startingpoint was localizationof rootsof , andalsoto goback to thesourceif is alsovitalto always to con neoneselfin arelativelysmallareaof extremetechnicalspecialization,thus shrinkingone'sper-ceptionof themathematicalworldandof thatrespectis thatwhileso many mathematicianshave beenspendingtheirentirescienti clifeexploringthatworldtheyallagreeonits contoursandonitsconnexity: whatever theoriginof one'sitinerary, oneday oranotherif onewalkslongenough,oneis boundto reach a well meetellipticfunctions,modularforms,zetaf unctions.

4 \Allroadsleadto Rome"andthemathematicalworldis \connected".In otherwordsthereis just\one"mathematicalworld,whoseexplorat ionis thetaskof allmathematiciansandtheyareallin exactlyas theexistenceof theexternalmaterialreality seemsundeniablebutis in factonlyjusti edby thecoherenceandconsensusof ourperceptions,theexistenceof themathematicalreality stemsfromitscoherenceandfromtheconsensus of the ndingsof necessaryingredient of a mathematicaltheoryimpliesa much morereliableformof \con-sensus"thanin many otherintellectualor scienti hasso farbeenstrongenoughto avoidtheformationof largegatheringsof researchersaroundsome\religiouslike" scienti pragmaticattitudeandseethemselves astheex-plorersof this\mathematicalworld"whoseexistencethe ydon'thave any wishtoquestion,andwhosestructuretheyunco ver by a mixtureof intuition,notso foreignfrom\poeticaldesire"2, andof a greatdealof rationality requiringintenseperiodsof generationbuildsa \mentalpicture"of theirownunderstandingof thisworldandconstructsmoreandmorepenetra tingmentaltoolsto explorepreviouslyhiddenaspectsof whenunexpectedbridgesemergebetweenpartso f themathematicalworldthatwerepreviouslybe lieved to be veryfarremotefromeach otherin thenaturalmentalpicturethata onegetsthefeelingthata suddenwindhasblownoutthefogthatwashiding partsof a shalldescribe at theendof thispaper onerecent instanceof such a 'lltake theconceptof \space"as a guidelineto take thereaderthrougha guidedtourleadingto theedgeof theactualevolutionof thisconceptbothin al-gebraicgeometryandin noncommutative geometry.

5 I shallalsoreviewsomeof the\fundamental"toolsthatareat ourdisposalnowadays such as \positivity",\coho-mology",\calculus",\a beliancategories"andmostof all\symmetries"which willbe a recurrent themein thethreedi erent partsof is clearlyimpossibletogive a \panorama"of thewholeof mathematicsin areasonableamount of is perfectlypossible,by choosinga precise2as emphasisedby theFrench poet ,to show thefrontierof certainfundamentalconceptswhich play a centralrolein \space"is su cientlyversatiletobe anidealthemetodisplaythisactive evolutionandwe shallconfront themathematicalconceptof spacewithphysicsandmorepreciselywithwhat QuantumFieldTheoryteachesus andtrytoexplainseveralof theopenquestionsandrecent ndingsin SpaceThementalpicturesof geometryareeasyto createby exploitingthevisualareasof wouldbe naive however to believe thattheconceptof \space" develop,is a factas we shallseebelow thisconceptof \space"is stillundergoinga encodea point of theeuclideanplaneor spaceby two (orthree)realnumbersx 2R.

6 Thisirruptionof \numbers"in geome-tryappearsat rstas anactof violenceundergoneby geometrythought of as \actof violence"inauguratestheduality betweengeometryandalgebra,be-tweentheeye of thegeometorandthecomputationsof thealgebraists,which runin timecontrastingwiththeimmediateperceptio nof frombeinga sterileoppositionthisduality becomesextremelyfecundwhengeometryandalg ebrabecomealliesto exploreunknownlandsas in thenewalge-braicgeometryof thesecondhalfof thetwentiethcenturyor as in noncommutativegeometry, two existingfrontiersforthenotionof 'sTheorem:LetPjandQj,j2 f1;2;3gbe pointssuchthatthethree lines(Pj; Qj) :=(Pk; Pl)\(Qk; Ql)are rstbrie ydescribe projective geometrya tellingexampleof theaboveduality themiddleof theXVII thcentury, ,tryingto give a mathematicalfoundationto themethods of perspective usedby paintersandarchitectsfoundedrealprojecti ve geometry. Therealprojective planeof Desarguesis thesetP2(R)of linesthroughtheoriginin threespaceR3. Thisaddstotheusualpoints oftheplanea \lineat in nity"which gives a perfectformulationandsupportfortheempiri caltechniquesof factDesargues'stheorem( gure2) canbe viewed as thebasefortheaxiomati-zationof projective a consequenceof theextremelysimplefouraxiomswhich de neprojective geometry, butit requiresforitsproof thatthedimensionof thegeometrybe therelation\P2L" thelineL, theyare: Existenceanduniquenessof thestraight linecontainingtwo distinctpoints.

7 Two linesde nedby fourpoints locatedontwo meetinglinesactuallymeetin onepoint. Everylinecontainsat leastthreepoints. Thereexistsa nitesetof points thatgeneratethewholegeometrybyiteratingt heoperationpassingfromtwo points toallpoints of dimensionn= 2, Desargues'stheoremis nolongera consequenceof theaboveaxiomsandonehasto addit to theabove dimensionnareexactlytheprojective spacesPn(K) of a (notnecessarilycommutative) thisway in perfectduality withthekeyconceptof algebra:thatof a eld? It is a setof \numbers"thatonecanadd,multiplyandin whichany non-zeroelement hasaninverseso given by the eldQof rationalnumbersbuttherearemany otherssuchas the eldF2withtwo elements or the eldCof eldHof quaternionsof Hamiltonis a beautifulexampleof non-commutative (C) tookitsde nitive formin \LaG eom etrie"of Mongein complexpoints onthesideof therealonessimpli esconsiderablytheoverallpictureandgives a rareharmony tothegeneraltheoryby thesimplicity andgenerality of instanceallcirclesof theplanepassthroughthe\cyclicpoints"a pairof points (introducedbyPoncelet)locatedonthelineat in nity astwo arbitraryconicsany pairof circlesactuallymeetin fourpoints,a statementclearlyfalsein to settleproblemswhoseformulationis purely\real"hadalreadyappearedin theXVIthcenturyfortheresolutionof whenthethreerootsof such anequationarerealtheconceptualformof theserootsin termsof radicalsnecessarilypassesthroughcomplexn umbers.

8 ( to 23 in Cardano'sbook of 1545 Arsmagnasive deregulisalgebraicis). GeometryandtheDevilof (1)GeometryjAlgebraalreadypresent in theabove discussionof projective geometryallows,whenit isviewed as a mutualenhancement, to translateback andforthfromgeometryto al-gebraandobtainstatements thatwouldbe hardto guessif onewouldstay con nedin oneof thetwo bestillustratedby a ,dueto FrankMorley, dealswithplanargeometryandis oneof thefewresultsaboutthegeometryof trianglesthatwas apparentlyunknownto angle,thenyouconsidertheintersectionof consecutive trisectors,andob-tainanothertriangle ( ).Now Morley'stheorem,which hefoundaround1899,says thatwhichevertriangleABCyoustartfrom,the triangle is is analgebraic\transcription"of startwithanarbitrarycommutative eldKandtake three\a ne"transformationsofK. ThesearemapsgfromKtoKof theformg(x) = x+ , where 6= 0. Given such a transformation3exceptpossiblythecommutat ivityxy=y xof 'sTheorem:Thetriangle obtained fromtheintersectionof consecutivetrisectorsof anarbitrarytriangleABCis 2 Kis uniqueandnoted (g).

9 Forg2G,g(x) = x+ notatranslation, 6= 1 onelets x(g) = be theunique xedpointg( ) = groupG(K) ( )calledthe\a negroup"andthealgebraiccounterpartof Morley'stheoremreadsas followsLetf; g; h2 Gbe suchthatf g; gh; hfandf gharenottranslationsandletj= (f gh). Thefollowingtwoconditionsare equivalent,a)f3g3h3= )j3= 1and +j +j2 = 0where = x (f g), = x (gh), = x (hf).Thisis a su cientlygeneralstatement now, involvinganarbitrary eldKanditsproof is a simple\veri cation",which is a good testof theelementaryskillsin\algebra".It remainsto show how it impliesMorley' \ atness"of Euclideangeometry, namely(2)a+b+c= wherea; b; caretheanglesof a triangle(A; B; C) is bestcapturedalgebraicallybytheequalityF G H= 1in thea negroupG(C) of the eldK=Cof complexnumbers,whereFis therotationof centerAandangle2aandsimilarlyforGandH. Thusif we letfbe therotationof centerAandangle2a=3 andsimilarlyforgandhwe gettheconditionf3g3h3= equivalencethus showsthat +j +j2 = 0, where , , , arethe xedpoints off g,ghethfandwherej= (f gh) is a non-trivialcubicroot ofunity.

10 Therelation +j +j2 = 0 is a well-knowncharacterizationof equilateraltriangles(itmeans = j2, so thatonepassesfromthevector ! to ! by arotationof angle =3).Finallyit is easyto check thatthe xedpoint ,f g( ) = is theintersectionofthetrisectorsfromAandBc losestto thesideAB. Indeedtherotationgmovesit to itssymmetricrelative toAB, andfputsit back in provedthatthetriangle( ; ; ) is factwe alsogetforfree18equilateraltrianglesobta inedby pickingothersolutionsoff3= typicalof thepower of theduality betweenontheonehandthevisualperception(w herethegeometricalfactscanbe sortof obvious) ,providedonecanwritethingsin algebraicterms,oneenhancestheirpower andmakes themapplicablein totallydi instancetheabove theoremholdsfora nite eld,it holdsforinstanceforthe eldF4which hascubicrootsof ,passingfromthegeometricalintuitionto thealgebraicformulationallowsoneto increasethepower of theoriginal\obvious"fact,a bitlike languagecanenhancethestrengthof perception,in usingthe\right words".