Mathematics in the 21st Century: 6th World Conference, Lahore, March 2013 PDF
Preview Mathematics in the 21st Century: 6th World Conference, Lahore, March 2013
Springer Proceedings in Mathematics & Statistics Pierre Cartier A.D.R. Choudary Michel Waldschmidt E ditors Mathematics in the 21st Century 6th World Conference, Lahore, March 2013 SpringerProceedingsinMathematics&Statistics Volume 98 This book series features volumes composed of selected contributions from workshops and conferences in all areas of current research in mathematics and statistics, including operation research and optimization. In addition to an overall evaluationof the interest, scientific quality, and timeliness of each proposalat the handsofthepublisher,individualcontributionsare allrefereedto thehighquality standards of leading journals in the field. Thus, this series provides the research community with well-edited, authoritative reports on developments in the most excitingareasofmathematicalandstatisticalresearchtoday. Moreinformationaboutthisseriesathttp://www.springer.com/series/10533 Pierre Cartier • A.D.R. Choudary Michel Waldschmidt Editors Mathematics in the 21st Century 6th World Conference, Lahore, March 2013 123 Editors PierreCartier A.D.R.Choudary InstitutdesHautesÉtudes AbdusSalamSchool Scientifiques(IHÉS) ofMathematicalSciences Bures-sur-Yvette,France Lahore,Pakistan MichelWaldschmidt UniversitéPierreetMarieCurie(ParisVI) ParisCedex05,France ISSN2194-1009 ISSN2194-1017(electronic) ISBN978-3-0348-0858-3 ISBN978-3-0348-0859-0(eBook) DOI10.1007/978-3-0348-0859-0 SpringerBaselHeidelbergNewYorkDordrechtLondon LibraryofCongressControlNumber:2014950900 ©SpringerBasel2015 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerptsinconnection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. 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Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) Foreword: Mathematics for a New Century The6thWorldConferenceon21stCenturyMathematics2013tookplaceinLahore (Pakistan) from March 6 to 9, 2013, in the AbdusSalam Schoolof Mathematical Sciences(ASSMS).Itwasasuccessfuleventbringingtogethermanyscientistsfrom all over the world and a large audience of local students and colleagues. Despite the recurringpoliticalunrestin Pakistan,itwas a peacefulevent,underthe expert guidanceofDr.A.D.RazaChoudary. Itakethisopportunitytocommentonthedevelopmentsofmathematicalsciences in the past century. Everyone associates the second International Congress of Mathematicians (ICM), that took place in Paris, August 1900, with the famous address of David Hilbert (the 23 problems of Hilbert). The audience was mostly European (French, Germans, Italians, British) and the towering figures were Poincaré from France, and Hilbert from Germany. Both were universalscientists, andtheir work extendsfrommechanicsto philosophy,throughalgebra,geometry, numbertheory,andanalysis.Theyparticipatedinstrongdebatesaboutthenatureof mathematicalobjects, PoincarébeingbasicallypragmatistandHilbertformalist— reflectingperhapsthedifferentphilosophicaltraditionsoftheirrespectivecountries. Bothwereawareoftheimportanceofthecreation,byGeorgCantor,ofthetheory ofsets. Letusalso mentionthata youngassistant to theBritish embassyin Paris, Bertrand Russell, followed with great interest the various lectures: this was the beginningofhiscareerasalogicianandaphilosopher.Atthattime,settheorywas plaguedbytheso-calledparadoxesorinconsistencies:Russellinventedthetheory oftypesasanalternativetosettheory—moreaboutthatlater. As I already mentioned,science was Europeanat the time, even dominated by scientistswithaGermanculturalbackground(whetherGermans,Danes,Swedish, Hungarians, even Russians). There were very few scientists both in northern and LatinAmerica;afewJapaneseintheMeijierausedGerman,andafewIndiansused Englishtocommunicate.Acenturylater,mathematicsistrulyinternationalandwe rememberICM2002inBeijing,ICM2010inHyderabadandICM2014inSeoul. The number of world practitioners in mathematics increased in a century from maybeathousandtohundredthousands.MathematicsinVietnam,forinstance,did v vi Foreword:MathematicsforaNewCentury not exist in 1970, and in 2012 the joint meeting of the mathematical societies of FranceandVietnamgathered500Vietnameseand100Frenchparticipants.Science in Latin America faredwell in the last 50 years, after a modeststart in Sao Paulo around1950(rememberthevisitsofAndréWeil,OscarZariski,JeanDieudonnéand Alexander Grothendieck). This trend has been supported by various international organizations,amongthemtheInternationalMathematicalUnion(IMU)recreated after the Second World War, the SISSA1 in Trieste created by Abdus Salam, and the CIMPA2 (created in Nice at the request of UNESCO for organizing summer schools all over the world). As a note of comfort, let us mention the quick and ratherunexpecteddevelopmentofscienceinIndia(theTataInstituteofFundamental Research,Mumbai,andnowanumberoforganizationslikeNISER3)aswellasin China(duetotherebirthofthisbigandancientcountry).Thechallengesfacingus nextare in the Middle East (includingPakistan and its neighbours)and in Africa, especiallytropicalandEastAfrica.Itakegreatcomfortbyobservingmanyprojects inthisdirection. So, mathematics at the turn of the twenty-first century is truly international. Another welcome development is the increasing number of women studying mathematics.Ihavebeensurprisedbythenumberofwomenattendingmylectures in Pakistan, as well as in Kurdistan and in Algeria. I have been told of similar patternsinIran.Eveninaso-calledadvancedcountrylikeFrance,theprogresshas beenslow:mymotherdidnothaveapersonalcheckbookbefore1948;mymother- in-law,awidow,wasnotthe legalguardianofherdaughter,mywife;thesections forboysandforgirlsoftheÉcoleNormaleSupérieuremergedin1990only,andin themostprestigiousÉcolePolytechnique,thereareapproximately20%offemale students!ItisalsorecentlythatawomanhasjoinedtheFieldsmedallists! Anotherimportantdevelopmenthasbeenthegradualchangeofemphasis:what isreallyimportantinmathematics?AsHilbertstateditrepeatedly:“Nooneshould take us outside Cantor’s paradise”. One of the initial successes of Hilbert was his book on geometry, where he revised Euclids’ axiomatics for geometry, by taking into account the critical study (by Pasch, Peano, etc.) motivated by the advent of non-Euclidean geometry. The dream of Hilbert was a complete exposition of mathematicsviatheaxiomaticmethodandtheuseofsettheory.Therewereafew initial successes, like Hausdorff book on topology, followed by Banach and his normed vector spaces, and even more importantly, the “Modern Algebra” by van derWaerden(calledsimply“Algebra”inlatereditions).ThephilosophyofHilbert isbestdescribedbytheinscriptiononhistomb(inGöttingen): Wirmüssenwissen, Wirwerdenwissen4 1ScuolaInternazionaleSuperiorediStudiAvanzati. 2CentreInternationaldeMathématiquesPuresetAppliquées. 3NationalInstituteofScienceEducationandResearch. 4Wemustknow,weshallknow. Foreword:MathematicsforaNewCentury vii Hilbert was convinced that, using the axiomatic method, every mathematical problemcouldbesolved.Inhislistof23problems,thesixthiscalled“Axiomatics ofphysics”,andhisownversionof Einstein’sgeneralrelativityis presentedasan axiomatictheory.Wearelessambitiousnow,especiallyafterGödel’sdiscoveryof the incompleteness of all formal axiomatic systems, and more dramatically after Cohen’sproofoftheundecidabilityofthecontinuumhypothesis. These limitations did not hamper Bourbaki’s enterprise, whose goal was the materializationofHilbert’sdream.Inapproximately50years(from1934to1983), the group of 10–15 (mostly) French mathematicians, with varying membership, published an encyclopedic treatise with eight complete series (ranging from set theory to Lie groups) and the beginning of two more. The recent reprint consists of 30 volumes, totalling slightly less than 10,000 pages. The initial ambition was to cover all existing mathematics; too big! But after the foundations (from set theorytoLebesgueintegration),Bourbakipublishedtwoverysuccessfulserieson “CommutativeAlgebra”and“LieGroupsandAlgebras”.Thiswassupposedtobe thestartingpointfordevelopmentsinalgebraictopology,differentialgeometryand also algebraic number theory.Despite many unpublisheddrafts on these subjects, the momentum was exhausted after 50 years and led to the advent of a fourth generationofcollaborators.Tocompensateforthisshortage,anumberofimportant booksintheBourbakispiritwerepublishedundertheirownnamesbymembersof Bourbaki,orlaterbydisciples.5 ThefamousBritishhistorianEricHobsbawmpublishedabookentitledTheShort 20thCentury (1914–1991)describinginhisownterms:“a centuryofideologies”. Thisfitsquitewellwiththedevelopmentofmathematics:thetwentiethcenturywas anepochofformalismandaxiomatics.Thereweregreatsuccessesdespiteanumber oflimitations,andIwilltrynowtodescribethebirthofanewera. There are a number of challenges, each one requiring a sharp turn. The first is coming from the inside, with the birth and development of category theory. This was created by S. MacLane and S. Eilenberg around 1940, as a tool to be used in algebraic topology. It was enormously developed by Ch. Ehresmann and A. Grothendieck in the 1960s and it is now one of the most vigorous branch of mathematics.Theoretically,itcanbedescribedassomekindofexoticextensionof grouptheory: a category is a set with a partially defined inner operationallowing manyunits.Afunctorisnothingbutahomomorphismbetweentwoobjectsofthis kind.Thereisaninterestingcombinatorialtheorydevelopedalongtheselines,but the most useful applications transcend this narrow domain. Among the so-called inconsistencies of set theory, the Russell paradox is paramount, and rests on the illegal assumption of a set of all sets. But, practitioners of category theory have no hesitation at mentioning the category of all sets, and worst, the category of all categories. Grothendieck invented a beautiful escape from this tangle, namely universes. A universe is a set U such that the collection of U sets (that is the 5LetusmentionR.Godement’sTheoryofsheavesandA.Weil’sBasicnumbertheoryamongmany othersbyChevalley,Serre. viii Foreword:MathematicsforaNewCentury elements of U) obeys all the properties attributed to sets. The category of all U sets is now a legitimate object. All that is required is to assume the existence of such universes; but this is tantamount to assume the existence of so-called large cardinals.Herewepenetrateintothemuddywatersofsettheory,thelogicalmarsh. Itcouldverywellbethatsomepropertyofcategoriesistrueforoneuniverse,but notforall.6Veryfewpeople(GrothendieckhimselfinSGA4,aswellasDemazure andGabrielintheirtreatiseonalgebraicgroups)putthisburdenontheirshoulders, thepricetobepaidbeingheavy.Inviewofthebigsuccessesofcategorytheoryin itsmanyapplications,mostpractitionersofmathematics,especiallytopologists,are temptedtofollowtheadviceofCharlesdeGaulle:“L’intendancesuivra”.7 History of mathematics teaches us to be hopeful and brave. In the eighteenth century,mathematicianslikeEuler,Lagrangeandmanyothersdevelopedcalculus, differential geometry, and mechanics, using mathematics with shaky foundations. Berkeleyhadalreadypinpointedthe inconsistenciesofthe notionof infinitesimal. Forthecure,wehadtowaituntilthemiddleofthenineteenthcenturywithCauchy, WeierstrassandDedekind.Nevertheless,theconceptofinfinitesimalisstillwidely used by many physicists and engineers. One of the greatest innovations in the eighteenthcenturywasthecalculusofvariations.Therethefoundationswereeven moreshaky,andHilbertmentionedthisamonghis23problems.Aftertwocenturies of struggle, we have reliable foundations for ordinary calculus and calculus of variations, and a very large part of the discoveries of Euler and his followers has beensalvaged.Sowecanbehopeful. A possiblecureforthisdiseasecouldbeofferedbythetheoryoftypes,created by Bertrand Russell. To explain the difference between sets and types, I will use the parableof the green cats. In the fairy tale version, the set of green cats exists becausethekingwasabletogatheralloftheminabigroom.Intheentomologist’s version, the type of green cats is a box in a museum with the proper name, to accommodate all green cats to be caught. In more serious terms, a set is closed, defined by the collection of all its members; a type is open ended waiting for the creation or discovery of new members. Recently, Voevodsky in Princeton made a serious attempt by developinghis theory of homotopicaltypes. Each type is open endedaswellasthecollectionoftypes.Themaindifficultyistodefineequalityof two types: it cannot be static, it has to be dynamic; that is, an equality statement isanevolvingproof.Thisviewiscomfortedbyarecentdiscoveryinlogics—-that proofsandprogrammesarevirtuallythesameobjects. Thiskindofproblemsiscloselyconnectedwiththeadventofcertifiedsoftware. This is a practical problem. In modern technology, many big systems have been fullyautomatized.Runninganuclearplant,controllingaspaceship,monitoringthe flightsinanairportorthetrainsintherailwaysystem,requireshugeprogrammes. I have been told that two million instructions are not uncommon. Who can write such a programme knowing that any serious bug is a threat to safety? For 6Thisistobeexpected,aftertheincompletenesstheoremofGödel! 7Thesupplyshallfollowthefighters! Foreword:MathematicsforaNewCentury ix mathematicians,asimilarchallengeistheexistenceofmonstrousproofs:thefour- colourproblem,Wiles’proofofFermat’slasttheorem,thespherepackingproblem, and the classification of finite simple groups. The printed version of Wiles’ proof runsover600pages,butitisa“human”proofnotusingcomputersinaseriousway. The classification of finite simple groups consists of more than 10,000 published pages. The four-colour problem and the sphere-packing conjecture use extensive computercalculations,bothcombinatorialandnumerical. Of course, the dream of a mathematician is to have a beautiful and easy-to- follow proof, which could be printed as one of the “Proofs from the Book”.8 Everyonehopesthat the problemsmentionedabovewill receivesuch a proof,but I am doubtful about such a possibility. Anyhow, refusing to accept that kind of proof would seriously hamper the development of mathematics, since obviously we will have more and more of such proofs in coming years. Maybe the duty of mathematicianswillbetocreatesomekindofastronomicalclocks,imitatingnature andmindoperations,toberunandwatched.Apossiblecureexistsalreadywiththe existenceofproofassistants,like9theFrenchversionCOQ.Theambitionistohave acertifiedencyclopediaofmathematics:itcontainsacore(ornucleus)ofabout500 instructionswritteninC++,rathereasytocheckbyhumanmeans,containingallthe basicsyntacticalrules.Thenitdevelopslikeonionrings—eachlevelreferringtoits innerlevel.Ihavebeentoldthatmathematicsatthelevelofseconduniversityyear isalreadyavailableinsuchsystems.Soonerorlater,weshallwriteourproofsinour standardhalf-formalway, but the refereeswill use such proofassistants to certify ourpaper.SoHilbert’sdreamofamechanizableaxiomaticsystemmaycometrue. Sofar,thesolutionofthefour-colourproblem,aswellasthe250-page-longproofof thetheoremofFeitandThompson(“everyfinitesimplegrouphasanevenorder”) havebeencompletelycheckedbysuchmethods. Comingbacktotypes,theyarealreadyastandardtoolincomputerscience:even inold-fashionedlanguageslikeFORTRAN,wewoulddeclare realx integerp, etc. The recent systems are based on the typed (cid:2)-calculus, like LISP and the followers.It seems to me that a seriousproposalfor revisionof the foundationof mathematicswouldbetoreplacesettheorybytypetheory.Forinstance,declaring atypeSET,atypeCAT(category),atypeCAT/CAT(categoryofallcategories)is withoutflawaslongasyoudonotinsistongatheringallsetsinasingleroom,etc. ThetoposofGrothendieckcanbeviewedasunorthodoxmodelsofsettheory.Their flexibilityshouldallowthemtohelpinthissearchfornewfoundations.Maybewhat is at stake is to develop a purely syntactical mathematics, withoutany underlying ontology. 8TouseErdo˝s’terminology:IreferyoutoAigner’sbook(publishedbySpringer)withthesame title. 9ImentionalsoHOLIGHTdevelopedbyT.Hales.
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