Infinity:AVeryShortIntroduction

VERYSHORTINTRODUCTIONSareforanyonewantingastimulatingandaccessiblewayintoanewsubject.Theyarewrittenbyexperts,andhavebeentranslatedintomorethan45differentlanguages.Theseriesbeganin1995,andnowcoversawidevarietyoftopicsineverydiscipline.TheVSIlibrarynow

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IanStewart

INFINITYAVeryShortIntroduction

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Icouldbeboundedinanutshellandcountmyselfakingofinfinitespace.

WilliamShakespeare,HamletAct2,Scene2

Contents

Listofillustrations

Introduction1 Puzzles,proofs,andparadoxes2 Encounterswiththeinfinite3 Historicalviewsofinfinity4 Theflipsideofinfinity5 Geometricinfinity6 Physicalinfinity7 Countinginfinity

References

Furtherreading

Publisher’sacknowledgements

Index

Listofillustrations

1Successiverefinementsofastaircase

2Infinitelymanyinfinitelythintriangles

3Allmoveupone,andRoom1isfree

4Howtoaccommodateaninfinitecoach-load

5Themanager’s‘diagonal’order

6Successivestagesintheconstructionofthesnowflakecurve,theareaunderahyperbola,andGabriel’shorn

7Constructingthedigitsof

8PerpendicularbisectorBCofOAmeetsthecircleatC.Ordoesit?

9Thefirsttwostagesinapproximatingacircle

10Areaunderthegraphoffapproximatedbyrectanglesofwidthɛ,andrateofchangeoffunctionfoverasmallintervaloflengthɛ

11ExampleofCavalieri’sprinciple

12Parallelrailwaytracksappeartomeetonthehorizon©123RF/FlorianBlümm

13RamsesII’svictoryattheSiegeofDapurNordiskfamiljebok/WikimediaCommons/PublicDomain

14FlagellationofChristbyPierodellaFrancescaclassicpaintings/AlamyStockPhoto

15AlbrechtDürer,fromUnderweysungderMessungMitdemZirckelunRichtscheyt,inLinien,Nuremberg,1525SLUBDresden/WikimediaCommons/PublicDomain

16Ashipdisappearingoverthehorizon

17Ashiponaplanarocean,relativetothedirectionofthehorizon

18Astheshipmovesfurtheraway,itrisestowardsthehorizonandappearstobecomesmaller

19RepresentationsoftheEuclideanplane

20Howtheartist’seyeprojectsalineintheplane

21Howtofitaninfiniteplaneinsideacircle

22ParallellinesintheEuclideanplanemeetatinfinity

23Agridofsquareswithtwodirections‘pointingtoinfinity’correspondingtosetsofparallellines,andauxiliarydiagonallinesusedtoconstructaperspectiveversion

24Successivestagesinconstructingaperspectivedrawingofthegrid

25Vermeer’sTheConcert,withlinesshowinguseofperspectiveSuperstock/Glowimages.com

26DeviationD(α)intheangleoflighthittingasphericalwaterdroplet

27GraphofD(α)againstα,incominglightiscompressedneartherainbowangle

28Airyfunctionofwaveopticsandray-opticalintensityneartherainbowangle

29Xia’sscenario

30Countingsheepinthreelanguagesandtheplanetsofthesolarsystem

31Graphofaperiodicfunctionwithperiodp

32Aone-to-onemappingfromtheopenunitintervalontoR

Introduction

Itmayseemparadoxicaltowriteaveryshortintroductiontoaverybigconcept,butinfinityisparadoxical.It’salsoremarkablyuseful,andmathematiciansandusersofmathematicswouldbelostwithoutit.However,itcanalsobedangerous,unlesshandledwithconsiderablecare.Philosophersandtheologianshavefacedthesamedilemma,thoughwithdifferentemphasis.Ittookmorethantwothousandyearstolearnhowtohandletheinfinitewithoutitexplodinginourfaces,andeventhen,itcanstillcausetrouble.

ThefirstrecordeduseofaspecificwordfortheinfiniteisgenerallycreditedtoAnaximander,apre-SocraticGreekphilosopherwhoflourishedaround580BC.Histermapeironcanbetranslatedinseveralways—boundless,limitless,indefinite,infinite.Hiscontextwasasearchfortheoriginofallthings,whichheheldtobeanendlessprimordialmass.Beinginexhaustible,apeironcouldgenerateeverythinginexistencewithoutbeingusedup.Exactlywhathehadinmindisunclear,butmanyscholarsconsiderittobeakindofprimevalchaosthatcanbeseparatedintothefourancientelements—earth,air,fire,water—fromwhich,theGreeksbelieved,allelseisformed.

Anaximanderproposedthatorderlyrealityhadbeencreated—extractedmaybeabetterword—fromformlesschaosbypullingoppositequalitiesasunder.Inthisrespect,apeironresemblestoday’squantum-mechanicalexplanationoftheoriginofmatterthroughtheappearanceofparticle–antiparticlepairs,andisreminiscentofGalileo’sparadox—aninfinitesetcanbematchedwithapropersubset—andwiththeshenanigansthatgooninHilbert’shotelwheninfinitelymanyguestschangeroomstoaccommodateanewcomer.Bothcanbeinterpretedasextractingsomethingfromaninfinitesetwithoutanythingbeingusedup.Resolvingthisparadoxwasakeysteptowardsaprofoundadvanceinourunderstandingofinfinity:GeorgCantor’srealizationthatsomeinfinitiesarebiggerthanothers.

Thefirstknownreferencestomathematicalfeaturesoftheinfinitearethecelebratedparadoxesofanotherpre-Socratic,ZenoofElea,wholivedbetweenabout490and430BC.ThemostfamousisthefableofAchillesandthetortoise,inwhichthetortoiseisgivenaheadstart.Achilles,thoughthefasterrunner,cannevercatchthetortoise,becausebythetimehereacheswhereitwas,ithasmovedalittlefurtheron.Sohehastoperforminfinitelymanytasksbeforehecancatchup,whichallegedlyisimpossible.Zeno’sparadoxeshaveadeceptivesimplicity,buttheychallengeourintuitionaboutspace,time,motion,andcausality.

Infinitylurksinthesimplest,mostmundaneareaofmathematics:arithmetic.Whenchildrenfirstlearnaboutnumberstheyoftenwonderwhatthebiggestoneis,usuallysettlingforthebiggestwhosenametheyknow—ahundred,orathousand.Butmostofthemquicklycometorecognizethatthereisnobiggestnumber,becauseaddingonemakesanynumberbigger.Onewaytosaythisis‘thereisnolargestnumber’.Aristotlecalledthiskindofinfinity‘potentialinfinity’.Anotherdescription,morecontentiousbutricherinmathematicalandphilosophicalpromise,is:‘thereareinfinitelymanywholenumbers’.Aristotlecalledthiskindofinfinity‘actualinfinity’,buthedidn’tdistinguishmathematicsfromrealitythewaywedonow,so‘actual’isamisnomer.

Whydoweneedtothinkabouttheinfinite—aconceptweneverencounterdirectly?Therearemanyreasons.Eveninelementarymathematics,weencounteraspectsofinfinity,forexamplewhenwritingthefraction1/3asadecimal.Togetanexactrepresentation,thedecimalmust‘recur’:repeatthesameblockofdigitsforever.Moregenerally,ourmindsseemtorequiretheideathatthingsmight‘goonforever’—inspaceandintime,inthefutureandthepast.Infinityis,perhaps,amentaldefault,anaturalsideeffectofthepattern-seekingabilitiesofourminds.Evolutionhasmouldedustonoticepatternsintheexternalworld,betheyrealorimaginary.Patternshavesurvivalvalue.Goingonforeverwithoutchangingisperhapsthesimplestpatternofall.

Inconsequence,we’rehappytoexplaintimeassomethingthathasalwaysexisted,andthereforehasnoorigin.Wefindthatmorecomfortablethantimesomehowbeginning,eventhoughthat’swhatcurrentcosmologyproposes.Weobjecttotimestartingbyasking‘whatcamebefore?’,failingtograspthatiftimehadabeginning,therewasno‘before’.Weprefertothinkthatspaceisinfinite,andtheuniverseextendswithoutlimit,becauseweimaginethatifnot,theremustbeaboundary—andweask‘whatliesbeyondtheboundary?’We’rewrongontwocounts.Iftheuniverseendssomewhere,there’snothingbeyond,notevenemptyspace.Andtheuniversecouldbefinitebutunbounded.

Infinity—especiallyitstemporalversion,eternity—playsasignificantroleinmuchreligiousthinking.It’sastandardtopicinphilosophy.Ithasintriguedartistsaswellasscientists.Itsoundsimpressive,youcanattributeallsortsofpropertiestoit,andnoonecanproveyouwrongunlessyourlogicisinerror.Morepositively,it’safascinatingconcept,fullofsubtleties,logicalpitfalls,puzzles,andparadoxes.

Oneofthegreatestparadoxesoftheinfiniteisthatit’sturnedouttobeextremelyuseful.Astheinspirationbehindcalculus,it’stakenhumanitytotheMoon,andfliesmillionsofusacrosstheglobeeveryday.Mathematiciansfinditverydifficulttogetanywherewithoutinfinity,eveninareasofthesubjectsuchascombinatorics,whichcountsfinitesetsofobjects.Patternsinthesenumberscanoftenbeneatlypackagedintoasingleinfiniteobjectcalledageneratingfunction,whichcanthenbemanipulatedtoobtainusefulinformationaboutperfectlyfinitethings.

Mathematicianshaveevengiveninfinityitsownspecialsymbol:∞.Therearealsomorerecentsymbolsforspecifictypesofinfinity,suchas andω,whichwemeetinChapter7.Perhapsthemostimportantmathematicalcontributiontoourunderstandingoftheinfiniteistherealizationthatthesameword‘infinity’canhavemanydistinctinterpretations.Thesecanbedefinedrigorously,andtheirsimilaritiesanddifferencescanbededucedlogicallyfromthedefinitions.

Althoughthereexistphilosophicalviewsofwhatmathematicsshouldbethatforbidallreferencetotheinfinite,virtuallyallpractisingmathematiciansworldwidefindtheconceptnotjustuseful,butindispensable.However,therearealsosomeintriguingscientificquestionsaboutphysicalinfinity.Forexample:istheuniversefiniteorinfinite?Whathappensinsideablackhole?Usuallyphysicistsinterpretinfinityasasignthattheirtheoryhasdepartedfromreality,butmanyofthemratherliketheideaofaninfiniteuniverse.I’llexaminethepsychologybehindthisinconsistencyinChapter6.

Infinityisatwo-edgedsword.Usedwithduecaution,itopensupimportantmethodssuchascalculus,uponwhichmostofmodernscienceisfounded.Manyoftoday’stechnologicalwonderswereinventedusingsomeaspectoftheinfinite—evendigitaltechnology,whichoperatesonfinitebinarynumbers,butisbuiltusingmaterialsscience,optics,chemistry,andquantumphysics—allinvolvingthemathematicsoftheinfiniteinessentialways.

Thesetriumphsnotwithstanding,veryminorchangestothewayinfinityisusedcanequallywellleadtononsense.Andit’snotalwayseasytodistinguishadividinglinebetweentheprofoundandtheabsurd.Allofthismakesinfinityoneofthemostfascinatingconceptseverinvented.If‘invent’istheword.

Outlineofthebook

Anintroductioncanopenupsomebasicquestionsandanswers,butitcanonlytouchuponthedeeperissuesbehindthem.Mymainaimhereistogetyouthinkingaboutthoseissues,andtoraiseawarenessofthesubtledistinctionsthatphilosophers,theologians,andmathematicianshavebeenforcedtomakewhencontemplatinginfinity.Myviewpointwillbethatofmodernpuremathematics,whichfocusesonlogicalissues.Physicsandappliedmathematicsoftenmakelessformaluseoftheinfinite,butthisisn’tacomprehensivescholarlytreatise,andI’llonlyskimthesurface.

Wethereforebeginwithawarm-upchapter,introducingninetypicalexamplesofreasoningabouttheinfinite—puzzles,paradoxes,evenafewproofs.Wediscusseachofthembriefly,andanalysewhetherthemethodsortheanswersarelogicallyacceptable.Somedeservefurtherdiscussion,andwe’llreturntotheminduecourse.

Thesecondchapterraisessomecommonmisconceptionsaboutinfinity,andshowshowinfinitynaturallyappearsinelementaryarithmetic.Theaimistoshowhowdeeplyembeddedinfinityis,eveninbasicareasofmathematics,andtoclarifypossibleconfusionabouttopicsthatwethinkweunderstand.

Chapter3focusesonhistoricalattitudestoinfinity,mainlyinphilosophyandreligion,includingZeno’sfamousparadoxes.Infinityisn’tathing,butaconcept,relatedtothedefaultworkingsofthehumanmind.Zeno’sparadoxesappeartobeaboutphysicalreality,buttheymainlyaddresshowwethinkaboutspace,time,andmotion.Acentral(butpossiblydated)contributionwasAristotle’sdistinctionbetweenactualandpotentialinfinity.Theologians,fromOrigentoAquinas,sharpenedthedebate,andphilosopherssuchasImmanuelKanttookupthechallenge.Mathematiciansmaderadicaladvances,oftenagainstresistancefromphilosophers.

Chapter4examinesalogicalcounterpartoftheinfinite:infinitesimals.Thesearequantitiesthatareinfinitelysmall,insteadofinfinitelylarge.Historically,suchquantitiesformedthebasisofcalculus,oneofthemostusefulbranchesofmathematicseverinvented.However,theycausedconsiderablehead-scratching,startinganargumentthattookabouttwocenturiestoresolve.ThiswasachievedusingaversionofAristotle’spotentialinfinity—namely,potentialinfinitesimality,ifthereissuchaword.(Thereisnow.)

OnseveraloccasionsI’verathercasuallycalledinfinityaconcept.It’snot.It’sameta-concept:ajumbleofmoreorlessrelatedideas,masqueradingunderthesamename.Muchofthephilosophicalandmathematicalfuncomesfromtryingtoteasethedifferentmeaningsapart,anddecidingwhichmakesense,andwhy.AclearexampleoccursinChapter5,wherethediscussiontakesasharpleft-handturnintoadifferentrealmoftheinfinite:projectivegeometry.AsEuclidinsistedinoneofhisaxioms,parallellinesnevermeet.ButthepaintersoftheItalianRenaissance,analysingperspective,stumbledacrossarichveinofgeometryinwhichitmakessensetoinsistthatparallelsdomeet—atinfinity.Ifyou’veeverstoodatarailwaystationwatchingthetracksconvergeastheydisappearintothedistance,you’vecaughtaglimpseofgeometricinfinity.

Frommathematicswemovetotherealworld,andChapter6tacklesquestionssuchas‘isspaceinfinite?’Inmanyareasofphysics,thepresenceofaninfinitequantity(oftencalledasingularity)isconstruedasawarningthatthetheoryislosingtouchwithreality.Forinstance,accordingtoclassicalrayoptics,theintensityoflightatthefocusofalensisinfinite.Thephysicalresolutionofthisdifficultyinvolvesreplacinglightraysbywaves.Incosmology,however,thepossibilityofinfinitespaceismorerespectable.

Chapter7returnstothemathematicsofinfinity,discussingCantor’sremarkabletheoryofhowtocountinfinitesets,andthediscoverythattherearedifferentsizesofinfinity.Forexamplethesetofallintegersisinfinite,andthesetofallrealnumbers(infinitedecimals)isinfinite,buttheseinfinitiesarefundamentallydifferent,andtherearemorerealnumbersthanintegers.The‘numbers’herearecalledtransfinitecardinals.Forcomparisonwealsomentionanotherwaytoassignnumberstoinfinitesets,byplacingtheminorder,leadingtotransfiniteordinals.Weendbyaskingwhethertheoldphilosophicaldistinctionbetweenactualandpotentialinfinityisstillrelevanttomodernmathematics,andexaminingthemeaningofmathematicalexistence.

Chapter1Puzzles,proofs,andparadoxes

Togetusthinkingcriticallyandimaginativelyabouttheinfinite,herearesomedeductionsandquestionsthatuseit.Somegivetherightanswer,somedon’t,andsomeareplainbaffling.Thinkaboutthembeforereadingon.Comparethem.Whydosomemakesensebutothersnot?

Nineappealstoinfinity

LargestnumberInfinity(∞)isthelargestnumberthereis.So Subtract∞frombothsidestoget .

DiagonalofasquareImaginearegular‘staircase’alongthediagonalofaunitsquare(Figure1).Thetotallengthofthispolygon—treadsandrisers—is2,becausethetreadsaddto1andsodotherisers.Ifthenumberofstepsbecomesinfinite,andthestepsbecomeinfinitelysmall,thestaircasebecomesthediagonalofthesquare.Thereforethelengthofthediagonalis2.

1.Leftandmiddle:successiverefinementsofastaircase.Right:thelimitwithinfinitelymanysteps.

AreaofacircleAcircleisacurveformedbyinfinitelymanyinfinitelyshortlines.JoiningthemtothecentreasinFigure2createsinfinitelymanyinfinitelythintriangles,eachwithperpendicularheightequaltotheradiusrofthecircle.Eachtrianglehasarea ,wherebisthelengthofthebase,sosummingthemall,theareaofthecircleis timesitscircumference.Thecircumferenceis2πr,sotheareais½r.2πr=πr2.

2.Infinitelymanyinfinitelythintriangles:oneshaded.(Only32shown.)

LightswitchAttime0,alightswitchisoff.AfterhalfasecondIswitchiton.AquarterofasecondlaterIswitchitoff.AneighthofasecondlaterIswitchitonagain.AsixteenthofasecondlaterIswitchitoff,andsoon.Eachsuccessiveintervaloftimebetweenmovingtheswitchishalfthepreviousone.Afteronesecond,isthelightonoroff?

BallsinthebagIhaveinfinitelymanyballs,numbered1,2,3,…,andanemptybag.Attime0,Iputballs1–10intothebagandtakeoutball1.Attime1/2second,Iputballs11–20intothebagandtakeoutball2.Attime3/4second,Iputballs21–30intothebagandtakeoutball3.Attime7/8second,Iputballs31–40intothebagandtakeoutball4,andsoon.Thenumberofballsinthebagincreasesby9eachtime.Soafteronesecond,howmanyballsarethereinthebag?

OnethirdindecimalsIfwetrytoexpress1/3asadecimal,itcanneverterminatebecause10dividedby3is3withremainder1,sothecalculationrepeatedlygives3s:0·333333…goingonforever.Ifwestopatanyspecificplace,say0·333333,thenumberislessthan1/3,becauseonmultiplyingitby3weget0·999999,whichdiffersfrom1by0·000001.Soistheinfinite(recurring)decimal0·333333…smallerthan1/3,orexactlyequaltoit?

SquaresandnumbersThisextractfromGalileo’s1638DiscorsieDimostrazioniMatematicheIntornoaDueNuoveScienze(discoursesandmathematicaldemonstrationsrelatingtotwonewsciences)hasbeenslightlyeditedforlength.

Salviati:Wecannotspeakofinfinitequantitiesasbeingtheonegreaterorlessthanorequaltoanother.Itakeitforgrantedthatyouknowwhichofthenumbersaresquaresandwhicharenot.

Simplicio:Iamquiteawarethatasquarednumberisonewhichresultsfromthemultiplicationofanothernumberbyitself;thus4,9,etc.,aresquarednumberswhichcomefrommultiplying2,3,etc.,bythemselves.

Salviati:Verywell;andyoualsoknowthatjustastheproductsarecalledsquaressothefactorsarecalledsidesorroots.ThereforeifIassertthatallnumbers,includingbothsquaresandnon-squares,aremorethanthesquaresalone,Ishallspeakthetruth,shallInot?

Simplicio:Mostcertainly.

Salviati:IfIshouldaskfurtherhowmanysquaresthereareonemightreplytrulythatthereareasmanyasthecorrespondingnumberofroots,sinceeverysquarehasitsownrootandeveryrootitsownsquare,whilenosquarehasmorethanonerootandnorootmorethanonesquare…Thisbeinggranted,wemustsaythatthereareasmanysquaresastherearenumbersbecausetheyarejustasnumerousastheirroots,andallthenumbersareroots.

Sagredo:Whatthenmustoneconcludeunderthesecircumstances?

Salviati:SofarasIseewecanonlyinferthatthetotalityofallnumbersisinfinite,andtheattributes‘equal’,‘greater’,and‘less’,arenotapplicabletoinfinite,butonlytofinite,quantities.

Hilbert’shotelInalecturein1924DavidHilbertillustratedCantor’stheoryofinfinitenumbers(transfinitecardinals)byimaginingahotelwithinfinitelymanyrooms,numbered1,2,3,….Supposeanewguestarriveswhenallroomsarealreadyfull.Atfirstsight,thenewcomerwillhavetofindanotherhotel,butthemanagerhasabrainwave.Heaskseveryguesttovacatetheirroomandmovetotheroomwhosenumberisonegreater.Thatis,theguestinroom1goestoroom2,theguestinroom2goestoroom3,theguestinroom3goestoroom4,andsoon.Theyallmovesimultaneously.Nowallexistingguestsstillhavearoom,androom1ismiraculouslyfreeforthenewguest.

Inafinitehotelthiswon’twork:theguestintheroomwiththelargestnumberhasnowheretogo.ButinHilbert’shotel,there’snolargestroomnumber.

Youmightliketoconsidertwofurtherquestions:

•Supposethehotelisfullandacoacharriveswithinfinitelymanynewguests,sayinseatnumbers1,2,3,…onthecoach.Canthehotelaccommodatethemallbymovingguestsaround?

•Whatifinfinitelymanyinfinitecoachesarrive?Again,assumethecoachesarenumbered1,2,3,…,andsoaretheseatsineachcoach.

Grandi’sproofoftheCreationIn1703GuidoGrandipublishedQuadraturaCirculaetHyperbolaeperInfinitasHyperbolasGeometriceExhibitata(quadratureofthecircleandhyperbolaexhibitedbyinfinitegeometriccurves),inwhichheconsiderstheinfiniteseries

Bythebinomialtheoremthisequals Setx=1todeducethat

Ontheotherhand,wecangroupthetermsas

Therefore0=1/2,whichGrandiinterpretedasaproofthatGodcancreatetheworldfromnothing.Anothergroupingis

so0=1,equallypuzzling.

Solutionsandcomments

Fromtheviewpointoftoday’smathematics,mostoftheseexamplescanbedealtwithwithoutintroducingtoomanynewideas.Someneedextradiscussion,continuedinsubsequentchapters.

LargestnumberThereasoningisclearlyfalse,butwhy?Oneproblemmightbetheassumptionthatinfinityisanumber,whichinturnraisesthequestion:whatisanumber?Thedeductionwouldbevalidforaconventionalnumber,so can’tbeanumberinanyconventionalsense.However,mathematicianshavedefinedlessconventionalmeaningsinwhichinfinityisa(newkindof)number.Therethestatement ismathematicallyacceptable,althoughadifferentsymbolisnormallyusedtomakethecontextclear.What’snotacceptableissubtracting∞frombothsidesoftheequation,becausesubtractioncan’tbedefinedforinfinitequantitiesifwewanttheusualrulesofarithmetictohold.

DiagonalofasquareTomakesenseofthisexample,mathematiciansrephrasetheargumentintermsofafinitenumbernofsteps,whichtendstoinfinity.That’safancywaytosay‘remainsfinitebutgrowsindefinitelylarge’.Whateverthevalueofn,thelengthofthestaircaseis2.There’sawell-definedlimitingcurve,anditisindeedthediagonalline.However,byPythagoras’stheoremthelengthofthediagonalisnot2,but .It’ssometimesclaimedthatthelimitingcurveisnotthediagonal,butaninfinitelywigglylinethatrepeatedlycrossesit.Notso.Thelengthofthelimitingcurveisnotthelimitofthelengthsofthestaircases.That’sall.

AreaofacircleThemethoddescribedgivestherightanswer,andsomethingverylikeitcanbejustified.ArchimedesdidsousingaGreekmethodcalledexhaustion,butheassumedwithoutproofthatacirclehasawell-definedarea,asexplainedinChapter4.Todayweusuallyresorttocalculusinstead.Theideaistouseafinitenumbernofverythinslices,allexactlythesameshapeandsize,straightenedattheirouterendstoformtriangles.Theydon’tcovertheoriginalcircleexactly,buttogethertheyapproximateitsareaveryclosely.

Theareaofeachtriangleishalfitsbasetimestheperpendicularheight,sothetotalareaishalftheperimetertimestheperpendicularheight.Theperimeterisveryclosetothecircumferenceofthecircle,length2πr.Theperpendicularheightisveryclosetotheradiusr.Sothetotalareaisverycloseto

Byestimatinghowclose,andapplyingthelogicalfoundationsofcalculusasinChapter4,itcanbeprovedthatthelimitofthetotalareaofthetriangles,asthenumberntendstoinfinity,isexactlyπr2.Thislimitisthedefinitionoftheareaofthecircleincalculus,avoidingtheassumptionthattheareaexists.Incompensation,wemustalsoprovethatarea,definedinthismanner,hasalltheexpectedproperties.

LightswitchMathematically,theproceduredefinesthestateoftheswitchforalltimeslessthanonesecond.Thistellsusnothingaboutthestateafteronesecond.Notallinfiniteprocesseshaveasensiblemeaning,andthisisoneofthem.

Physically,wewouldquicklyenduptryingtomovetheswitchfasterthanlight,whichisimpossiblebyrelativity.Beforethat,frictionwouldhavemeltedtheswitch.Beforethat,thelightbulbwouldprobablyhaveblown.

BallsinthebagInfinitelymany?Notsofast!

Pleaseignorethepracticalissueshere.Thisisahypotheticalexerciseinanon-relativisticworld.Thesettingcanbeformalizedtomakemathematicalsense.

Atstagenthebagcontains9nballs,butwecan’tjustletntendtoinfinitytodeducethatthe‘final’numberofballsisinfinite.Thelimitofthenumberofballsinthebagisnotthenumberofballsinthelimitofthebag.Inthisrespect,it’slikethestaircasesandthediagonalofthesquare.

Actually,afteronesecondtherearenoballsinthebag.Toseewhy,observethatballnisremovedatthenthstage,andisneverputback.Everyballgoesintothebag,hangsaroundforatime,andthengetstakenoutagain.Whenitallshakesdown,thebagisempty.

OnethirdindecimalsAninfinitedecimalcanbegivenarigorouslogicalmeaning,seeChapter4.Thatgranted,supposethat

Then

Therefore9S=3,soS=3/9=1/3.

Althoughstoppingatanygivenstageyieldsanumberlessthan1/3,thedifferencedecreasesrapidlythemoredecimalplacesyouuse.Aninfinitesequencethatbecomesarbitrarilysmallhaslimitzero.Theapparentparadoxarisesbecauseitneverreachesthatvalue.

SquaresandnumbersIt’sremarkablethatGalileocamesoearlytoaconclusionthatwasn’tfullysortedoutuntilCantordevelopedhistheoryofcountingforinfinitesetsover200yearslater:seeChapter7.AdoptingCantor’sview,mostofwhatSalviatisaysattheendiscorrect,exceptthat‘equal’,‘greater’,and‘less’canbeappliedtoinfinitequantities.However,theydon’tbehaveexactlyastheydoforfinitequantities,whichyoucouldargueiswhatSalviatireallymeant.

Hilbert’shotelHilbert’sscenarioamountstoaproof,withinCantor’sframeworkforinfinitesets,thatifthe‘numberofelements’inthesetofwholenumbers(atransfinitecardinal)isdenotedbyℵ0,then (ThesymbolℵistheHebrewletter‘aleph’.)TheunderlyingideaistomapthesetNofwholenumberstoitssubsetMofwholenumbersgreaterthan1.Thismapmustbeaone-to-onecorrespondence,meaningthatdifferentelementsofNmaptodifferentelementsofM,andallelementsofMarisefromNinthisway.Figure3showshowthiscanbedone.Thetoplinerepresents thebottomlinerepresentsℵ0,andthearrowsprovethey’reequal.

3.Allmoveupone,andRoom1isfree.

Supposeaninfinitecoacharrives(Figure4).Themanagernowmoveseachexistingguestntoroom2n,theevennumbers.Theneachcoachpassengermcanbeassignedtoroom ,theoddnumbers.Againallguestscanbeaccommodated.InCantor’snotation,thisprovesthat

4.Howtoaccommodateaninfinitecoach-load.

Todealwithinfinitelymanycoaches,themanagerassignsroomnumbersaccordingtoFigure5,followingthediagonalarrowsandreturningtothenextroominthetoprowafterreachingtheendofeacharrow.Allguestscanbeaccommodated,andthisprovesthat

5.Themanager’s‘diagonal’order—thenumbers2–3,4–5–6,7–8–9–10,andsoonslanttotheleft.

Theslightlyweirdarithmeticforℵ0makessenseifweinterpretitnaivelyas‘infinity’.Ifyouadd1toinfinity,doubleit,orsquareit,yououghttogetinfinityagain.Cantor’samazingdiscoveryisthatthearithmeticoftransfinitecardinalsismuchricherthanthat.We’llseewhyinChapter7.

Grandi’sproofoftheCreationLeonhardEulerperformedasimilarcalculationaround1730andwashappywith1/2asthesum.Latermathematicianseventuallydecidedthatinorderforaninfiniteseriestohaveameaning,itmustconverge:getarbitrarilyclosetosomefixednumberifyouaddenoughtermstogether,seeChapter4.Thisseriesconvergeswhen ,butnotwhen .Soletting isillegitimate.

That’snottheendofthestory,though.Euler’svalue1/2istheaverageofthetwonumbers1and0obtainedbystoppingafterafinitenumberofterms,sothere’sasenseinwhichitrepresentstheoverallbehaviourbetterthananyothervalue.Considerationsofthatkindledtoatheoryof‘summability’forseriesthatdon’tconverge,andheretheresultingsumis1/2.

Chapter2Encounterswiththeinfinite

Weusetheword‘infinity’rathercasuallyineverydayspeech,soit’sworthclarifyingafewbasicsbeforeweplungeintosubtleraspectsoftheinfinite.I’llfocusontwoissues:

•Infinityisnotjustasynonymforaverybignumber.Weoftenuseitthatway,forpoeticordramaticreasons,orjustoutofignorance,butinmathematicsandphilosophyinfinityisadifferentconceptaltogether:notaverylargelimit,buttheabsenceofanylimit.

•Infinityisnotjustsomeesotericinventioninadvancedmathematics.Werunintoitquiteearlyonatschoollevel.Thefirstimportantoccurrenceisnottheabsenceofalargestwholenumber.Wedon’treallyneedtoknowthat,andteacherscaneasilysliderounditifasked.Theinfiniteraisesitsheadinamoresignificantmannerwhenwe’retaughtaboutdecimalsandputthemtogetherwithourpreviouslylearnedconceptoffractions.

Finiteandinfinite

I’mnotgoingtostartbydefininginfinity,becausethewordhasmanymeanings,andIwanttoworkmywaytowardsthese.Asaworkingruleofthumb,anumber(whole,fractional,decimal,whatever)isfiniteifit’ssmallerthansomenumberinthefamiliarsequence1,2,3,…,andinfiniteifnot.(Fornegativenumbers,makethempositivefirst.)Anobjectisfiniteifitssizeisfinite,andinfiniteifnot.Soacircleisfinite,butalinethatgoesonforeverisnot.

Therearemanymeasuresofsize,andthesameobjectmaybefinitebyonemeasurebutinfinitebyanother.Acirclehasfinitecircumferenceandarea,butiscomposedofinfinitelymanypoints.Thesnowflakecurveoffractalgeometryisobtainedfromanequilateraltrianglebyrepeatedlyaddingsmallerequilateraltrianglesalongthemiddlethirdofeachedge(Figure6,top).Itslengthisinfinitebutitenclosesafinitearea.Thearcofthehyperbola runningfrom toinfinity(Figure6,middle)hasinfinitelength,andtheareabetweenitandthex-axis(shaded)isinfinite.In1644EvangelistaTorricelliprovedthatwhenthiscurveisspunaboutthex-axistoformasurfaceofrevolution(Figure6,bottom),ithasinfiniteareabutenclosesafinitevolume.Infact,thevolumeisexactlyπ.ThissurfaceiscalledGabriel’shornorTorricelli’strumpet,andatthetimeitposedaseriouschallengetomathematicalintuitionabouttheinfinite.

6.Top:successivestagesintheconstructionofthesnowflakecurve.Middle:areaunderahyperbola.Bottom:Gabriel’shorn,alsocalledTorricelli’strumpet,hasinfiniteareabutfinitevolume.

Notjustabignumber

Standarddictionarynumbersstopwithcentillion.Thisis10303inAmericanandmodernBritishusage,inwhichabillionisathousandmillion.It’s10600intherestofEuropeandbytheolderBritishconventioninwhichabillionisamillionmillion.Suggestionsforextendingthenamesleadtomillinillion,whichis103003.Thisisconsiderablybiggerthanthefamousgoogol,10100,butalotsmallerthanagoogolplex,whichis .Eventhegoogolplexistinycomparedtoinfinity.Infinityisbiggerthananyspecificwholenumber,whatevernotationalsystemweuseandwhatevernewnamesweinvent.Inpracticewerunoutofnamesbeforewerunoutofnumbers.

Archimedesunderstoodthis,andhewrotePsammites(sandreckoner)todisprovetheassertionthatthenumberofgrainsofsandonthesurfaceoftheEarthisinfinite.It’scertainlybiggerthananynumbertheancientGreekscouldnameintheireverydaylanguage,butArchimedesconsideredthattobeevidenceforthepaucityofcommonlanguage.HewroteapamphletaddressedtoGeloII,theKingofSyracuse:

Iwilltrytoshowyoubymeansofgeometricalproofs…that,ofthenumbersnamedbymeandgivenintheworkwhichIsenttoZeuxippus,someexceednotonlythenumberofthemassofsandequalinmagnitudetotheEarthfilledup,butalsothatofthemassequalinmagnitudetotheuniverse.

Archimedesthenderivedafiniteupperboundonthenumberofgrainsofsandthattheuniversecancontain,bycombiningtwoingredients:amodelofcosmology,andhisnovelmethodfornamingverylargenumbers.Heconcludedthat,inourterms,atmost1063sandgrainscanfilltheuniverse.Withtoday’sfigureforthesizeoftheobservableuniverse,thatbecomes1093.Stillfinite.

Otherculturesalsotookaninterestinverylargenumbers.TheJainreligioninIndiawasfoundedaround600BC,takingoverfromVedicreligions.Jainsbelievethatalllivingcreaturesembodysoul,whichisimmortalandperfect.Souls,includinghumanones,migratetonewcreaturesafterdeath.Toescapethisendlesscycleoftransmigration,thedevoteeavoidsanyactionthatharmsalivingcreature.Evenswattingaflyisnotpermitted.Jainismhasnoconceptofadivinecreatorordestroyer,andbelievestheuniversetobeeternal,bothinitspastanditsfuture.

Jaincosmologyinvolvesaverylongperiodof2588years,roughly10177.ButtheJainsacknowledgedacleardistinctionbetween‘verylarge’and‘infinite’.TheAnuyogaDwaraSutra,probablydatingfromabout100BC,discussesahemisphericaltroughwithdiameter100,000yojanna,about1millionkilometres.‘Fillitwithwhitemustardseedscountingoneaftertheother.Similarlyfillupwithmustardseedsothertroughs…Stillthehighestenumerablenumberhasnotbeenattained.’

MathematicianshavedevisednotationalsystemsfornumbersfarlargerthananythingthatArchimedesortheJainscontemplated.Butthese,too,arefinite.

Infinityisbigger.

Infinityinarithmetic

Mathematicalinfinityfirstappearedinconnectionwiththewholenumbers1,2,3,4,…andsoon,oftenalsocalledthecountingnumbers.Mathematiciansgenerallythrow0intothemixaswell,placingitatthefront,whichgivesthenaturalnumbers.Historically,thenextextensionofthenumbersystemledtofractionslike1/2and4/15,followedbytheintroductionofnegativenumbers−1,−2,−3,…and−1/2,−4/15.Thepositiveandnegativenaturalnumbersaretheintegers,andwhenpositiveandnegativefractionsareincludedaswellwegettherationalnumbersorplainrationals.

Allthesesystemareinfinite,inthesensethatthereisnolargestcountingnumber:howeverlargenmaybe,n+1islarger.Theothertwosystemscontainevenmorenumbers,sotheymustalsobeinfinite.However,wedon’trunintotheseinstancesofinfinityinanyessentialmannerwhenwelearnarithmetic.Thefirstmeaningfulencounterwithinfinity,formostofus,iswhenwestartlearningaboutthedecimalsystem.Eventhere,westartwithfinitedecimals,suchas3·14or1·41421(whichhappentobeapproximatevaluesforπand ).Arithmeticwithfinitedecimalsisessentiallythesameasarithmeticwithintegers;wejusthavetolearnwheretoputthedecimalpoint.

Pandora’sboxopensupwhenweputfractionsanddecimalstogether,andaskwhat1/3lookslikeasadecimal.Toagoodapproximation,theansweris0·333.Toabetterapproximation,0·3333.Toabetterapproximationstill,0·33333.Ifyouwantareallycloseapproximation,keepappendingmore3s.However,noneoftheseapproximationsisexact.Toseewhy,multiplythemby3,obtaining

Anexactvaluewouldgivetheanswer1,butthesearesmaller.Theydifferfrom1by

andalthoughthesenumbersrapidlybecomeverysmall,noneofthemiszero.

Althoughfiveorsixdecimalplacesaregenerallyenoughforpracticalcalculations,exactrepresentationsaredesirableformathematicalpurposes.Otherwisethedecimalsystemwouldmissoutalotofinterestingandusefulnumbers.Fortunately,there’sawayroundthedifficultywith1/3:theconceptofarecurringdecimal.Thisrepeatsthesamepatternofdigitsindefinitely,perhapsafterafinitenumberofexceptionsthatdon’tfitthepattern.Onenotationistoputdotsoverthedigitsatthefrontandendoftherepeatingsequence.So

goingonforever,and

Itthenturnsoutthateveryfractioncanberepresentedexactlybyarecurringdecimal,orafiniteoneifthedecimalrepresentationsstops,asitdoesfor .(Youcanconsiderthistobefollowedbyarecurringblockofzerosifyouprefer.)Conversely,everyrecurring(orfinite)decimalrepresentsafraction.Thetwodisplayednumbersare1/3and23/35,forinstance.

Irrationalnumbers

Arecurringdecimalmay‘goonforever’,butwecanprescribeitinitsentiretyinafinitemanner—asIjusthave.ThespannerintheworkswasthrownbyanancientGreek;legendhasitthathewasHippasusofMetapontum.HewasamemberofthePythagoreancult,whichbelievedthateverythingintheuniverserestsonnumbers.Atthattime,thewordimpliedwholenumbers,orfractionsformedfromwholenumbers.OneofthetriumphsofPythagoreanismwaswhatwenowcallPythagoras’stheorem:thesquareonthehypotenuseofaright-angledtriangleisthesumofthesquaresontheothertwosides.Hippasushadbeenmusingaboutthediagonalofaunitsquare.ByPythagoras’stheorem,thesquareofitslengthmustbe sothelengthofthediagonalisthesquarerootof2.Hippasusprovedthatthesquareofarationalnumbercannotequal2.Youcangetclose,indeedascloseasyouwish,butyoucan’trepresentthediagonalexactlyusingrationalnumbers.

ThiswasablowtoPythagoreanism’sdeepestbelief.Accordingtothelegend(forwhichthere’snoevidence)HippasusunwiselyannouncedhisresultonboardaboatcrossingtheMediterranean,andhisfellowtravellersweresoincensedtheythrewhimoverboard.Theresult,however,isundeniable:inmodernterminology,thesquarerootof2isirrational.TheGreeksgotroundthisdifficulty,fortheoreticalandphilosophicalpurposes,byabandoningnumbersandworkinggeometricallywithlengths,areas,volumes,andangles.Whendecimalswereinvented,however,theawkwardstatusof oncemorecametothefore.

ItfollowsfromEuclideangeometry,withabitofextraassistancetofillinsomelogicalgapsinEuclid’spresentation,thateverygeometriclengthcanbeexpressedasaninfinitedecimal.(I’llshowyouhowlater.)Thedecimalexpansionof startslikethis:

There’snoobviouspatternthatrepeatsthesameblockofdigitsforever.Appearancesmightbemisleading,becauseinprinciplethatblockmightbeverylarge,butarecurringpatternrepresentsarationalnumber,whereas isirrational.Sothedigitsof goonforever,butwecan’tspecifythembyastraightforwardrulethatletsuspredictexactlywhat,say,themillionthdigitmustbe.Forrecurringdecimals,wecan:it’s3for1/3andit’s1for23/35.

However,thereisarule.Eachsuccessivedigitisthelargestonethatmakesthesquareoftheresultlessthan2.Wecanusethisruletocalculateasmanydigitsaswewish.Inpracticetherearemoreefficientmethods,butthisoneworks.Thestandardarithmeticalmethodforfindingsquarerootsisreallyjustanefficientvariant.In1768,JohannLambertprovedthelong-standingconjecturethatouroldfriend

isirrational.

Assoonaswereachthestageinourmathematicaleducationatwhichgeometryanddecimalnotationcollide,wefacethenotionofadecimalexpansionthatcontinuesforever,butneednotrecur.These‘infinitedecimals’are(aconceptualnotationfor)therealnumbers.Unlessthenumbercanbewrittenasafractionwithdenominatorapowerof10,stoppingatsomefinitestageisalwaysanapproximation.

Numbersthroughthemicroscope

IclaimedthatEuclideangeometryletsusprovethatanylengthhasaninfinitedecimalexpansion.LetmeshowyouwhatIhaveinmind,using asanexamplesincethatcanbeconstructedgeometricallybyrulerandcompasses.Figure7illustratesthefirstfoursteps.Atthetop,welocate onthenumberlinebetween1and2,correspondingtothefirstdigit,namely1.Inthesecondwemagnifytheintervalfrom1to2tenfold,andlocate onthissubdivisionoftheline.Itliessomewherebetween4and5,correspondingtothenextdecimalapproximation1·4.Inthethirdrowwemagnifytheintervalfrom4to5tenfold,andlocate onthisfurthersubdivision.Itliessomewherebetween1and2,correspondingto1·41.Yetanothersubdivisionlocatesitat1·414,andiftheprocessweretocontinuewewouldobtainsuccessivedigits2,1,3,andsoon.

7.Constructingthedigitsof

Youwon’tfinddecimalsinEuclid,butBookVI,Proposition2oftheElementsisageometricconstructiontosubdividealinesegmentintoanynumberofequalpieces.SothisprocessisconsistentwithEuclideangeometry.

Givenalineoflengthπ,thesameprocesswouldgiveitssuccessivedigits.Althoughπisnotconstructiblewithrulerandcompasses,theGreeksknewotherwaystoconstructsuchaline.Indeed,anylinesegmentleadstoaninfinitedecimalexpansion.Conversely,wecanreverse-engineerthewholeapproachtofindoutexactlywhereonthenumberlineanygiveninfinitedecimalappears.

I’veleftoutoneimportantpoint,whichimpliesthatthedecimalexpansionobtainedinthismannerdeterminesthepointuniquely.Namely,thatgivenanylinesegment,howevershort,somesuchsubdivisionofalineofunitlengthleadstoasegmentthat’sshorter.Thisisequivalenttothestatementthatgivenanyfinitenumber(herethereciprocalofthelengthofthatsegment)somepoweroftenislarger.Thiscanbeprovedbycontradictionprovidedweagreethatanycollectionofnaturalnumbershasasmallestmember.Assumesomenumberexiststhatislargerthananypowerof10.Letnbethesmallestsuchnumber.Thenn−1islessthanorequaltosomepowerof10,say Therefore acontradiction.

Theconditionthatanycollectionofnaturalnumbershasasmallestmemberiscalledthewell-orderingprinciple.Informallyit’sobvious:pickanynumberinthecollection.Ifthatnumberisthesmallest,we’redone.Ifnot,wehaveonlyafinitenumberofcandidatestocheck:thenaturalnumberslessthanourchosenone.Inaformaltreatmentofthelogicalfoundationsofmathematics,wemakethewell-orderingprincipleanaxiom.

Myproofmakesonehiddenassumption:thereciprocalofanynon-zeropositivenumberisfinite.Wehavetotakethatasanaxiomaswell.Inanumbersystemthatincludesinfinitesimals,it’sfalse:seeChapter4.

Discreteandcontinuous

Oneofthegreatdichotomiesinmathematicsisthedistinctionbetweendiscreteobjectsandcontinuousones.Thenaturalnumbersarediscrete:eachisseparatedfromalltheothersbyadefinitegap.There’snonaturalnumberstrictlybetween1and2,orbetween1066and1067,forthatmatter.

Therealnumbersarecontinuous.Givenanypositiverealnumber,howeversmall,wecanfindasmallerone:justhalveit.Anyintervalofrealnumbersthatcontainsmorethanonepointcanbesubdividedintosmallerintervals.Betweenanytwodistinctrealnumbersthereliesatleastonemorerealnumber;indeed,infinitelymany.

Rationalnumbershaveanuneasyexistencebetweenthesetwoextremes.They’renotdiscrete:youcanfindadifferentrationalascloseasyouwishtoanygivenrational.Betweenanytwodistinctrationalnumbersthereliesatleastonemorerationalnumber;indeed,infinitelymany.Despitethat,therational‘numberline’hasgaps.Itfailstocontain andπ,forexample.Soit’snotreallycontinuous,either.

IfwetrytoreworkEuclid’sgeometryusingonlyrationalnumbers,werunintotheproblemthatHippasuspointedout.Infact,werunintosubtleconsequencesthatarecountertoourgeometricintuition.Figure8showscirclecentreO,withradius .ThebisectorofthissegmentisB,andtheperpendicularbisectoristhelineBC,meetingthecircleatC.

8.PerpendicularbisectorBCofOAmeetsthecircleatC.Ordoesit?

HowlongisBC?Pythagoras’stheoremimpliesthat whichequalsOA2,whichis4.Thatis, =4,so However,thereisnorationalnumberwiththisproperty: canbeprovedtobeirrational.Sointheworldofrationalgeometry,inwhichtheonlymeaningfullengthsarerational,thepointCdoesnotexist.Theperpendicularbisectorofaradiuspassesthroughthecirclewithoutactuallymeetingit.It‘squeezes’throughapoint-sizedgapinthecircumferenceofthecircle.

Thisfeatureoftherationalsrevealsasubtletyinthetraditionaldistinctionbetweendiscreteandcontinuousnumbersystems.There,continuityisusuallyconflatedwithinfinitedivisibility:anylinesegment,howevershort,canbesubdividedintosmallerpieces,asintheanalysisofinfinitedecimalsinChapter2.However,infinitedivisibilitydoesnotimplycontinuity.Therationalsareinfinitelydivisible,buttheyhavegaps.

Euclid’sElementstacitlyassumesthiskindofthingcan’thappen.WhenDavidHilbertstudiedthefoundationsofgeometryin1899,hefoundalargenumberofunstatedassumptionsofthiskind.Eucliddidagoodjobforhishistoricalperiod,butbymoremodernstandardshisaxiomatictreatmenthasmanyflaws.

Inthe19thcentury,similarissuesappearedinthetraditionalapproachtorealnumbersandindeedintegers.Inhis1893WasSindundWasSollendieZahlen?(whatarenumbersandwhatshouldtheybe?),RichardDedekindpointedoutthateveryonewasassumingthetruthofstatementssuchasbutnoonehadprovedthem.ThisparticularonecanbepolishedoffusingEuclid’sgeometry,butDedekinddevelopedamoregeneralapproach,basedonarigorousdefinitionofarealnumberasasection:twosetsofrationalnumbersthatdividethelineintoaleft-handregionandaright-handregion.Forinstance,theleft-handregionofthesectioncorrespondingto consistsofallpositiverationalswhosesquareislessthan2;theright-handregionconsistsofalltheotherrationals.

Oncesectionsaredefined,youcandefinehowtoaddthemormultiplythem,andprovethatthestandardlawsofarithmetichold.Inthismanner,Dedekindshowedthatonceyouhaverationalnumbers,youcan

constructrealnumbersfromthem.Ataprice,however.Asectionisapairofsetsofrationals,andthesesetsareinfinite.Whenyoudoarithmeticwithsuchsets,you’reconceptuallyworkingwithinfiniteobjects.Today’smathematicianshavegotusedtothiswayofthinking,andareundisturbedbyitsphilosophicalovertones.Philosophers,onthewhole,founditworrisome,andmostofthemarguedagainstit.Theargumenteventuallystoppedbecausebothsideslostinterest.InChapter3,we’llexaminesomeofthedifficultiesthattheygrappledwith.

Chapter3Historicalviewsofinfinity

Thehistoryofhumanity’sconceptsoftheinfinite,andtheuseswe’vemadeofthem,goesbackover2500yearstoAnaximander’sapeiron.Threemainintellectualareaswereinvolved:theology,philosophy,andmathematics(withrecentinterventionsfromtheoreticalphysics,whichI’llsubsumeunderthe‘mathematical’heading).Acomprehensivetreatmentwouldbeamajorundertaking,soI’llsummarizeafewkeyfiguresandideas.Onthetheologicalside,I’llrestrictthediscussiontoChristianity,andmerelytrytogivesomeflavourofthetheologicalissuesthatweredisputedanddebated.

ThedevelopmentofmathematicssinceAnaximander’stimecanbedividedintofourmainperiodsandlocales.Initially,themainactionwasinGreece,asexemplifiedbyEuclid’sElementsandthephilosophicalworksofAristotle.ForthenextmillenniumitshiftedtoChina,India,andArabia.From1400,followingtheRenaissance,mostmajoradvancesinmathematicsweremadeinEurope.Bythe20thcenturythemathematicalcommunityhadgoneglobal.Thisisaverybroad-brushdescriptionofacomplextrainofevents,andthroughouttherewerecontributionsfromotherregionsandcultures.

Priortothe20thcentury,religionandphilosophyhadasubstantialinfluenceonmainstreammathematicalthinking.Thethreeareaswerecloselyintertwined,oftentoanextentwenowfindsurprising.ChristiantheologyreveredAristotleandmadehisunderstandingoftheinfiniteakeystoneinitsthinkingaboutthenature,andexistence,ofGod.Theinfinitudeofthenaturalnumbersbecamecoreevidenceintheologicaldebates.Philosophersdiscussedfoundationalissuesinlogicandmathematics,andmathematicianstookinspirationfromtheirconclusions.Individualmathematiciansstruggledtoreconciletheirmathematicaldiscoverieswiththeirpersonalbeliefs.

Atthestartofthe20thcentury,withthedevelopmentofanaxiomaticbasisforthefoundationsofmathematics,theselinksbegantofallapart.Axiomaticsettheorybecametootechnicaltoappealtophilosophers,whomostlyabandonedthephilosophyofmathematics.ThemainexceptionswereBertrandRussellandLudwigWittgenstein,whodisagreedwitheachother.IronicallyRussell,ablyassistedbyAlfredNorthWhitehead,wasoneofthemainpeopleresponsibleforthedeeplytechnicalnatureoftheset-theoreticfoundationsofmathematics.Mostmathematiciansceasedtopaymuchattentiontowhatphilosopherssaid,especiallywhen(aswithWittgenstein)theytoldthemathematicianstheyweredoingeverythingwrong.Religionlostmuchofitspoliticalclout;throughoutmostofthedevelopedworldreligiousbeliefdeclined,thoughtodifferingextentsindifferentcountries.Inparticular,mathematiciansnolongerfeltconstrainedbytheteachingsoftheChurch.

Theadoptionofpreciseaxiomaticfoundationsformathematicsclarifiedthelogicalissuesconsiderably,withoutnecessarilyresolvingthemall.Infinitywasstillpuzzling,butatleastweknewwhatweweretalkingaboutandwhyitwaspuzzling.Alongsidethesedevelopmentscameanewviewpointonmathematicalexistence.There’snoneedformathematicalconceptstobedirectmodelsofreality,orindeedtoberelatedtorealityatall.Mathematiciansshouldconsiderthemselvesfreetointroducenewconcepts,providedtheydon’tcreatelogicalcontradictionsandcanberelatedtoexistingconcepts.AsCantorremarked,‘Theessenceofmathematicsisitsfreedom.’

Warning:maycontaininfinity

TheexamplesinChapter1illustratethepoweranddangersoftheinfiniteasaframeworkfortheadvancementofmathematics.Argumentsthatseemalmostidenticalcanbevalidinonecontextbutfallaciousinanother.Historically,subtledistinctionsofthiskindoftenemergedfrommathematicalandphilosophicalcontroversies.Inmathematicstheconceptof‘infinity’isneitherpredeterminednorunique:instead,itdependsonthecontextandisdefinedaccordingtothelogicalrequirementsofthatcontext.Philosophers,too,havedistinguisheddifferentinterpretationsoftheinfinite.

Thecentralmathematicalissueiswhetherfamiliarpropertiesoffiniteobjectsandprocessesremainvalidforinfiniteones.Infinityisnotaloneinthisrespect;negativenumbersareacaseinpoint.ThefirsthistoricalrecordofnegativenumbersistheChineseJiǔzhāngSuànshù(ninechaptersonthemathematicalart),datingfromtheHanDynasty(202BC–220AD)butprobablygoingbackmuchearlier.By400,ChineseandIndianmathematiciansmadefreeuseofnegativenumbers,buttacitlyassumedtheyobeythesamebasicarithmeticallawsaspositivenumbers.Whencomplexnumbers,inwhichminusonehasasquareroot,wereintroduced,thesametacitassumptionwasmade,butitwasallterriblymysterious.Eventuallymathematicianslearnedtodefinetheseextensionsofthenumbersystemabstractly,tolistexplicitlythebasicrulesrequiredtoworkwiththem,andtoprovethattheextendednumbersystemsdidordidnotsatisfyanyparticularrule.Dedekind’suseofsectionsoftherationalnumberstodefinetherealnumbersisanexample.

Logicaltreatmentsofinfinityfollowedthesamepattern.Initially,therewasanaiveassumptionthatbasicfeaturesoffiniteprocesseswouldautomaticallybevalidforinfiniteones.AnexampleisGottfriedLeibniz’s‘lawofcontinuity’,which,inalettertoPierreVarignonin1702,hesummarizedas:‘therulesofthefinitearefoundtosucceedintheinfinite’.Thentherewasaperiodofconfusionwhenitturnedoutthatassumptionsofthiskindweresometimeswrong.Finally,clarityemergedwhentheconceptsweredefinedlogically,therequiredpropertieswerestatedexplicitly,andthesepropertieswereprovedordisproved.

Preconceptionsofthehumanmind

Isuspectthatourcognitiveprocessespredisposeustowardsconceptsoftheinfinite,becausewenaturallyextrapolatesimplepatterns.Thisabilityoffersmanyevolutionaryadvantages—forecastingtheseasonsandtheweather,avoidingpredatorsbyobservingtheirhuntingmethods,understandinghowplantsgrowinordertocultivatethem.Butitalsoleadsustoextrapolatepatternsfromasmallamountofevidence,andinfinityissuchanextrapolation.

Ourmentalimagesofspaceandtimeareaffectedbythiskindofextrapolation,andtosomeextentreflecthowourbrainsprocessimagesandevents.Euclid’sgeometryisfoundedonlinesandpoints,whichareamongthebasicstructuresthatthevisualcortexextractsbyprocessingincomingsensorydata.Becausethere’snoparticularlimittothelengthofaline,thesimplestassumptionisthatthere’snolimit.FortwothousandyearsitwasassumedthatEuclideangeometryisatruerepresentationofnature;indeedthatit’stheonlypossiblegeometry.Infact,it’sneither;butitdoesembodyanidealizationofthesimplepatternsthatourvisualsystempresentstous.

Similarly,oursenseofthepassageoftimeleadsustoarrangeeventsinlinearorder.Weperceivenobeginning;memoriesofthepastjustfadeoutthefurtherbackwego;wealsobecomeawarethateventshappenedbeforewewereborn.Ourpersonalsenseoftimepassingalsoseemstohavenoend,becausewhenitdoesend,weceasetoexist;moreover,weunderstandthatmostthingswillcontinuelongafterwe’redead.Sincewefinditdifficulttocomprehendtimestopping,weassumethatitneverwill.Thebeliefthat‘itmustcontinue’underpinsallreligionsthatbelieveinanafterlife.

Spacegoingonforeverisinfinite;timegoingonforeveriseternal.Theformercarriesconnotationsof‘verylarge’.Thelatterofcoursereferstoanendlessperiodoftime,butithasasecondconnotation:stability.Anythingthatlastsaneternityisconsidered,insomesense,tobeunchanging.Asaresult,ourdefaultimageforinfinitetimeissubtlydifferentfromthatforinfinitespace.Thisdistinctionisreinforcedbecauseobjectsextendacrossspace,butprocesseshappenintime.Anobjectisafinishedthing,existingandcomplete.Aprocessceasestobeaprocesswhenitstops;allthatisleftistheresultoftheprocess.Butaprocesscanalsobeongoing,abletocontinue;and,asalreadyexplained,ifweseenogoodreasonforittostop,thenwenaturallyimagineitgoesonforever,andthinkit’seternal.

Themathematicalformalismofnaturalnumbers,integers,realnumbers,andsoon,explicitlybuildsthisassumptionintothedefinitions.However,it’sopentochallenge,asZenorealized.

Zeno’sparadoxes

ThefirstknownmathematicaltreatmentofinfinityoccurredintheworkofZenoofEleaaround450BC,andithascomedowntoussecond-handthroughSimpliciusofCiliciaandAristotle’sPhysics.Zenodiscussedfourparadoxicalargumentsaboutmotion.Twoexplicitlyrestontheinfinite,andthethirdcanbeviewedasbeingaboutinfinitesimals.Thefourthismoreobscure.

AchillesandthetortoiseAchilleshasaracewithatortoise.Herunstentimesasfastasthetortoise,soforfairnesshegivesitaheadstart—say100metresinmodernunits.Bythetimehereacheswherethetortoisestartedfrom,thetortoisehasmovedafurther10metresahead.Bythetimehereachesthatpoint,thetortoisehasmovedafurther1metreahead.Bythetimehereachesthatpoint,thetortoisehasmovedafurther1/10metreahead…andsoon.Inordertocatchupwiththetortoise,Achilleshastopassthroughinfinitelymanypoints,butit’simpossibletocompleteaninfinitenumberoftasks.

Or,asAristotlesaysinPhysics:‘Inarace,thequickestrunnercanneverovertaketheslowest,sincethepursuermustfirstreachthepointwhencethepursuedstarted,sothattheslowermustalwaysholdalead.’

ThedichotomyInordertotraverseagivendistance,wemustfirsttraversehalfofit.Butbeforewecandothatwemusttraversehalfofthatfirsthalf,andbeforedoingso,halfofthat…Wehavetocompleteaninfinitenumberoftaskseventogetstarted.

Or,asAristotlesaysinPhysics:‘Thatwhichisinlocomotionmustarriveatthehalf-waystagebeforeitarrivesatthegoal.’

ThearrowInorderforanarrowtomove,itmustchangeitsposition.Butatanyinstant,thearrowmustbestationary,sincenotimepasses.Ifit’sstationaryateveryinstant,itcan’tmove.

Or,asAristotlesaysinPhysics:‘Ifeverythingwhenitoccupiesanequalspaceisatrest,andifthatwhichisinlocomotionisalwaysoccupyingsuchaspaceatanymoment,theflyingarrowisthereforemotionless.’

ThestadiumAristotle’sPhysicsagain:

Concerningthetworowsofbodies,eachrowbeingcomposedofanequalnumberofbodiesofequalsize,passingeachotheronarace-courseastheyproceedwithequalvelocityinoppositedirections,theoneroworiginallyoccupyingthespacebetweenthegoalandthemiddlepointofthecourseandtheotherthatbetweenthemiddlepointandthestarting-post.This…involvestheconclusionthathalfagiventimeisequaltodoublethattime.

Thisoneisfarmoreobscurethantheotherthree,andnotobviouslyabouttheinfinite.Moreplausibly,it’saboutmotionwhenbothspaceandtimearediscrete.Aristotledismisseditasanevidentfallacy.KevinDavey’sarticleinFurtherReadinggivesadetailedanalysis,soI’llsaynomoreaboutit.

DiscussionofZeno’sparadoxes

AchillesandthetortoiseSimpliciustellsusthatDiogenestheCynicrefutedZeno’sargumentsbystandingupandwalking.Thisearlyexampleofexperimentaldisproofshowsthatsomethingmustbewrong—eitherwiththeargument,orwithhowtheassumptionsaboutmotionrelatetoreality—butitdoesn’ttelluswhat.

Zeno’sdescriptionoftheparadox,andhisargument,requirebothspaceandtimetobeinfinitelydivisible.Thatis,nospecificsmallestquantityexists,andanynon-zeroquantitycanbemadesmallerwhileremainingnon-zero.Inshort,bothspaceandtimeareconsideredtobecontinuous.(Inmuchofthephilosophy,infinitedivisibilityandcontinuityareconflated,althoughtherationalandrealnumbersystemsshowthey’redifferent,seeChapter2.Iwon’tdisentanglethedistinctionherebecauseitdoesn’taffectthediscussion.)Aristotlepointedoutthatasthedistancetobecovereddecreases,sodoesthetimetakentotraverseit.Inthiscontinuummodelofmotion,amovingbodydoesperforminfinitelymanytasksinafinitetime.

SupposefordefinitenessthatAchillesrunsat10metrespersecond,whilethetortoisemovesat1metrepersecond(fastforatortoise;I’mbeinggenerous).WecandoamathematicalDiogenes:after20seconds,Achilleshasgone200metreswhilethetortoisehasgone20,soAchillesis80metresinfront.Hemusthaveovertakenthetortoisealongtheway.Wecanfindoutexactlywhen.Solvingtheequation

gives Atthatinstant,bothAchillesandthetortoisehavereachedthesamepoint, ofametrefromwhereAchillesstarted.

ThesameanswerarisesifwefollowZeno’sreasoningstepbystep.Achillesmovesadistance

whilethetortoisemoves

Bothtakethesametime:

Fromamodernviewpoint,theseareconvergentseries,whoserespectivesumsare

There’saphilosophicalissuehere,notcompletelyansweredbythiscalculation.Duringthattime,bothcontestantshavemovedinfinitelymanytimes,therebyperforminginfinitelymanytasks.Causalitybecomesawkwardinsuchcircumstances.However,inthiscasewecanarguethatalthougheachtasktakesanon-zerotime,thetimesdecreasesorapidlythatallofthetasksarecompletedinafinitetotaltime.

Wecanavoidsuchconsiderationsbyraisingalogicalobjection.Zeno’sargumentprovesthatAchillesdoesnotcatchthetortoiseataninfinitenumberofspecifictimes.However,thatdoesn’truleoutthepossibilitythathecatchesitatsomeothertime—andacontinuummodelofspaceandtimeshowsthathedoes.Thisresolutionoftheparadoxreliesonmakingsenseofacontinuum,itselfanontrivialtask.Dedekind’sconstructionoftherealnumbersisonewaytogo.

Ifweusethecontinuumofrealnumberstomodelbothspaceandtime,anyintervalofspacecontainsinfinitelymanylocations,andanyintervaloftimecontainsinfinitelymanyevents.Motioncombinesboth.Anymovementofanobjectacrossafinitedistanceinfinitetimerequirespassingthroughinfinitelymanyintermediatelocationsandinfinitelymanyintermediateevents.That’showthemodelbehaves.

ThedichotomyThelogicalstructureissimilartothefirstparadox,butnowtheinstantsoftimeunderconsiderationconvergetothestartratherthantheend.Ifittakes10secondsforthearrowtoreachitsmark,we’reaskedtoconsidertimes10,5,5/2,5/4,….Theseformadecreasinginfinitesequenceconvergingtozero.Providedtimeiscontinuous,wecancalculatethetimeatwhichthearrowreacheseachpoint.

Again,motioninvolvescarryingoutinfinitelymanytasks.Thesetasksarenotorderedlikethepositiveintegers,occurringinturnasthefirst,second,third,butagaintheargumentconsidersonlyspecificintermediatepoints(‘tasks’).Acontinuumcontainsinfinitelymanyothers,inamorecomplexorderthanthatofthepositiveintegersorthenegativeintegers.Theparadoxdoesn’tcorrectlyrepresenttherelationbetweenspaceandtimeinacontinuummodelofmotion.

ThearrowManyresolutionsoftheArrowparadoxhavebeenproposed.Atitsheartistheproblemofrepresentingtimeasbothacontinuouslyflowingvariableandasasuccessionofinstantsofzeroduration.Aristotlewrote:‘Timeisnotcomposedofindivisiblenowsanymorethananyothermagnitudeiscomposedofindivisibles.’

Thisstatementisatvariancewiththemodernviewofthecontinuumofrealnumbers,wheretimeiscomposedofinfinitelymanyindivisiblenows(points),andthesamegoesforspace.Whatmattersishowthesepointscombinetomakeacontinuum,andthisisnotachievedbyorderingthemlikeadiscretesystem,suchasthepositiveornegativeintegers.‘Succession’isthewrongimage.

Thinkingaboutthephysicsofmotion,andhowitrelatestotheusualcontinuummodels,focusesattentionontheassumptionthataninstantaneoussnapshotofamovingobjectisindistinguishablefromthatofastationaryone.Thereareseveralobjectionstothis.One,proposedbyPeterLynds,isthatinstantsoftimeandinstantaneousvaluesofvariablesdon’tphysicallyexist.However,theydointhemathematicalmodelsusedbyphysicists,andtheparadoxisaboutthosemodels.BertrandRussellsuggestedthatmotiondependsonobservinganobjectattwotimesorpositions,notjustone.Atanyfixedinstantitappearstobestationary,butifit’ssomewhereelseatalaterinstant,itmusthavemoved.Furthermore,betweenthoseinstants,itmustoccupyeveryintermediatepointinspace(assumingforsimplicitythatitmovesalongaline).

NickHuggett’sresolutionistoquestionZeno’sassumptionthatanobjectthatisinstantaneouslyinthesamepositionasitwouldbeatrest,mustactuallybeatrest.OnewaytomakesenseofthatistorecasttheargumentinthecontextofHamiltoniansystems,ageneraltheoryofmechanicalsystemsthatSirWilliamRowanHamiltondevelopedfromanearlierideaofJoseph-LouisLagrange.Inthisformulation,thestateofapointparticleorabody,atanygiveninstant,isdeterminednotjustbyitsposition,butalsobyitsmomentum.Momentumismasstimesvelocity.Astationarybodyatagivenpositionhaszeromomentum.Amovingbodyatthesamepositionhasnon-zeromomentum.Aninstantaneoussnapshotshowstheirpositionstobethesame,butcan’tdistinguishtheirmomenta.Todothat,wemustcomparetwosnapshotsatdistinctinstants,andseewhethertheparticlehasmoved.SowhatZenoismissingisthepossibilityofa‘hiddenvariable’thatdiffersfromlocation,anddistinguishesamovingparticlefromastationaryone.ThisresolutioncanbeseenasaformalizationofRussell’sandHuggett’sviews.

Philosophersandtheinfinite

ForalongtimeafterZeno,themathematicsoftheinfinitefadedintothebackground.Itsphilosophicalandreligiousconnotationsdidnot.AkeyfigureinthephilosophyofinfinityisAristotle,whotackledthetopicinPhysicsandMetaphysics.Inthefirst,herelatesinfinitytonature,whosemainfeaturesarechangeandrest.Changeiscontinuous,henceinfinitelydivisible,sonatureleadsinevitablytocontemplatingtheinfinite.Studyingtheinfiniteisthereforejustifiedasanecessaryprecursortothestudyofnature.Theinfinitemustexist,insomesense,forotherwisetherewouldbe‘manyimpossibleconsequences’,suchastimehavingabeginningandanend,orsomelinesbeingindivisible,contrarytoEuclid.

Thekeyquestionisthen:inwhatsensedoesinfinityexist?

Itmightexistinactuality:somethinginfiniteexistingasacompletedobject.Alternatively,itmightexistpotentially:asaprocessthatcanalwaysbeextended,butwhichatanystageremainsfinite.Inbooks4and5ofPhysics,Aristotledemolishes,tohisownsatisfaction,thepossibilityofactualinfinity.Book6polishesofftheargument:infinitycanexistonlypotentially.

Thesimplestexampleofpotentialinfinityistheprocessofcounting.Nomatterhowlargeanumberyou’vereached,there’salwaysanextone.Theprocesshasnolimit,butatnostagedoesitreachaninfinitenumber.

Euclidwasclearlyawareofthisdistinction.InhisElements,thetheoremthatwenowstateas‘thereareinfinitelymanyprimenumbers’appearsinadifferentform.BookIX,Proposition20states:‘Primenumbersaremorethananyassignedmultitudeofprimenumbers.’Thatis,theprocessoflistingprimenumberscanbecontinuedindefinitely.Thisisastatementaboutpotentialinfinity.NowheredoesEuclidcontemplatethe‘object’comprisingallprimenumbers.Hejustprovesthathowevermanyyouhave,youcanconstructanotherone.

However,there’saproblemwithAristotle’sconceptionofpotentiality,ashehimselfadmitted.Herepeatedlymakesstatementslike‘onethingafteranotherisalwayscomingintoexistence’,withtheemphasisonfuturecontinuation.That’sallverywellforcounting,primes,extendinggeometriclinesegments,andsoon.ButitranintotroublewhenconfrontedwithAristotle’sbeliefthattheuniversehasexistedforaneternity,sotimehasnobeginning.Ifso,surelytheeventsthathavehappenedinthepast—allthepastinstantsoftime,thenumberoftimesthecelestialsphereshaverotated—constituteanactualinfinity.Youcan’tgetthembyallowingtimetocontinue;youwouldhavetorunitbackwards.Sotheydon’tfitthedefinitionofpotentialinfinity.

Wedon’tknowwhoraisedthisobjection;itcouldevenhavebeenAristotle,asdevil’sadvocate.Later,around550,JohnPhiloponusarguedagainsttheNeoplatonistProclusinhisDeAeternitateContraProclum(ontheeternityoftheworldagainstProclus).ProclusfollowedAristotleinassertingthattheworldhadnobeginning.Philoponusarguedthatthiswouldmakethepasthistoryoftheworldactuallyinfinite.SimpliciusthenpointedoutthatAristotlehadalreadydemolishedthatclaimbyslidingneatlypastthisobjection.Pasteventscan’tconstituteanythingactual,because‘thepartsthataretakendonotpersist’.They’vecomeandgone.‘Actual’wouldrequirethemtoexisthereandnow.

Evenso,thisobjection,saysJohnBowininAristotelianInfinity,‘caughtAristotlecompletelyoffguard,sincehistheoryofthepotentialinfinitewasclearlydevisedtoexplain[infinitesuccession]’.Aristotle’sanswertacitlyinvokesathirdkindofinfinity,neitheractualnorpotential.Hisdeductionthatactualinfinitiesareimpossibleassumesinfinityiseitheractualorpotential,andthesearemutuallyexclusive.Sohisclaimtohaveprovedthatallinfinitiesarepotentialhasalogicalgap.

Philosopherstradedblowsovertheinfiniteforcenturies,mainlygoingoverthesamegroundinnewways.WilliamofOckhamisfamousfor‘Ockham’srazor’:entitiesshouldnotbemultipliedbeyondnecessity.Hewrote:‘Everycontinuumisactuallyexistent.Thereforeanyofitspartsisreallyexistentinnature.Butthepartsofthecontinuumareinfinitebecausetherearenotsomanythattherearenotmore,andthereforetheinfinitepartsareactuallyexistent.’Hereheseemstobemakingaveryfinedistinctionaboutthemeaningof‘infinite’,ratherthanaboutexistence.Somethingcanbeactual,andinfinite,withoutbeingactuallyinfinite.

Inhis1690AnEssayConcerningHumanUnderstandingJohnLocke,aleadingempiricistphilosopher,statedthatallhumanideasstemfromsensoryperceptions.Sinceoursensesarefinite,somustourperceptionsbe.Sincewecan’tperceiveinfinity,itdoesn’texist:

Theinfinityofnumbers,totheendofwhoseadditioneveryoneperceivesthereisnoapproach,easilyappearsto

anyonethatreflectsonit.But…thereisnothingyetmoreevidentthantheabsurdityoftheactualideaofaninfinitenumber.Whatsoeverpositiveideaswehaveinourmindsofanyspace,duration,ornumber,letthembeeversogreat,theyarestillfinite;butwhenwesupposeaninexhaustibleremainder,fromwhichweremoveallbounds,andwhereinweallowthemindanendlessprogressionofthought,withoutevercompletingtheidea,therewehaveourideaofinfinity.

ThisisAristotle’spotential/actualdistinctionagain.Lockemadethefurtherobservationthatourthinkingaboutinfinityiscontradictory:‘Letamanframeinhismindanideaofanyspaceornumber,asgreatashewill,itisplainthemindrestsandterminatesinthatidea;whichiscontrarytotheideaofinfinity…asupposedendlessprogression.’Thisisacategoryerror.Ourideaofsomethingneednotbethesameasthethingitself.Ourideaofacatisnotacat,butarepresentationofacat.Wecanrepresentinfinitybysomethingfinite.

Kant’sviewwasalmosttheexactopposite.The1781CritiqueofPureReasonlistsfour‘antinomies’:pairsofmutuallycontradictorybeliefs,whichinhisviewarisewheneverthehumanmindattemptstograspreality.Thefirstantinomybearsdirectlyontheinfinite,contrastingtwoopposingviews.Eithertheworldhadabeginningintimeandislimitedinspace,orit’seternalandinfinite.Reality,saidKant,transcendsthemind,whichislimitedbyoursenses.Thereforethemindcannotgraspthetruenatureofreality.Kant’sviewiseasiertounderstandinthecontextofspace:

Spaceisnotanempiricalconceptwhichhasbeenderivedfromouterexperiences.Forinorderthatcertainsensationsbereferredtosomethingoutsideme…therepresentationofspacemustalreadyunderliethem.Therefore,therepresentationofspacecannotbeobtainedthroughexperiencefromtherelationsofouterappearance;thisouterexperienceisitselfpossibleatallonlythroughthatrepresentation.

InKant’sjargon,ourmindshavesyntheticaprioriknowledgeofthepropertiesofspace.Amongthosepropertiesareinfiniteextentandinfinitedivisibility.

Thedangerwiththisviewisthatitpromotesconceptualnotionsofspace,suchasEuclid’s,aboveobservation.KantconsideredEuclideangeometrytobenecessaryanduniversal.Wenowknowthatit’sneither,notevenwithinmathematics.Empiricalobservationhasshownthatrealspaceisn’tEuclidean.

InfinityinChristianbelief

Someversionoftheinfiniteoccursinmanyreligions,butI’llfocussolelyonChristianitytokeepthetopicwithinbounds.AsPhiloponusillustrates,thephilosophyandmathematicsofinfinitybecameintimatelyentwinedwithearlyChristianbeliefs.Inmedievaltimes,thenotionthatGodhasnolimitsbecameentrenched;itwasprettymuchthedefinitionoftheDeity.Manisephemeral,mortal,withlimitedpowersandknowledge;Godiseternal,immortal,omnipotent,andomniscient.

TheBibleprovideslesssupportforthesebeliefsthanwemightexpect.IntheKingJamesversion,‘infinity’neveroccurs,and‘infinite’appearsjustthreetimes.ThemostrelevantisPsalms147:5:‘GreatisourLord,andofgreatpower:hisunderstandingisinfinite.’However,Job22:5reads‘Isnotthywickednessgreat?andthineiniquitiesinfinite?’whichsuggeststhemetaphoricalmeaning‘verylarge’.‘Eternal’appearsmoreoften,butmostreferencesarelegalistic:eternalcovenant,eternalagreement.OnlyafewconcernattributesoftheDeity.Otherwordswithsimilarmeanings,suchas‘everlasting’,‘immortal’,alsooccur,butthesetooarerare.

EarlytheologiansseemnottohaveconsideredGodtobeliterallyinfinite.Around200AD,inDePrincipiis(onfirstprinciples),Origen,thefirstChristiantheologianofrepute,maintainedthatGod’spowerisfinite.Thereasonisthatperfectioncan’thaveblurrededges.Itslimitsmustbesharp.Latinperfectusmeans‘complete’.IfGod’spowerwereinfinite,itwouldbeincomplete,henceimperfect.

TheinfinitudeofGodbecomesexplicitaround395AD,whenEunomiusarguedthatChrist,asthesonofGod,issubordinatetoGod.Thesonwascreated,thereforehadnotalwaysexisted,thereforewasnotdivine.HefurtherarguedthatCreationasawholeisfinite,sothesonisfinite—againnotdivine.TheCouncilofConstantinopleformallycondemnedthisEunomianheresyin381.GregoryofNyssaprovidedalengthycounterargument,whichIwon’tattempttosummarize,todemonstratethatEunomius’sclaimsfailifGodisinfinite.BythetimeofAugustineofHippo,around400AD,theinfinitudeofGodhadbecomefundamentaltoChristiantheology.Heevengaveamathematicalproof:‘Letusthennotdoubtthateverynumberisknowntohim“ofwhoseunderstanding,”thePsalm[147:5]goes,“thereisnosetnumber”.’

EmphasisonGodbeinginfinitewasreinforcedbymedievalattemptstoproveHisexistence.IntheProslogionof1077–8AnselmofCanterburypresentedwhat’snowcalledtheontologicalproofofGod’sexistence.(Ontologyisthephilosophyofpurebeing.)Anselmhimselfhadamorepersonalobjective:hisbookdescribeshow,bymeditating,hebecameconvincedthatGodexists.Hisargumentcanbesummarized,inbroadterms,asfollows.Considerthemostperfectpossiblebeing.Sinceabeingthatexistsismoreperfectthanonethatdoesnot,themostperfectpossiblebeingmustexist.

Thissketchmaynotdofulljusticetothesubtletyofthethinking,butitcapturesthemainlineofargument.TheviewthatGodhasnolimitationsisadirectconsequenceoftheontologicalargument.Themostperfectpossiblebeingcannothaveanyspecificlimitation,forthesamereasonthere’snolargestwholenumber.Ifyoustatealimit,somethinggreaterisconceivable,whichwoulddescribeamoreperfectbeing.

It’sdifficultnottofeelthatAnselm’sargumentgetssomethingfromnothing.Merelycontemplatingahypotheticalbeingleads,withoutanyempiricalevidence,toarealone.KantattackeditasfallaciousinhisCritiqueofPureReasonof1781,arguingthatexistenceisnotalogicalpredicate—apropertythatsomethingcanpossessorlack.Ifitwere,thestatement‘Godexists’becomes‘ThereisaGod,andHehasthepropertyofexistence.’Soundsfine,butbythesamereasoning‘Goddoesnotexist’becomes‘ThereisaGod,andHehadthepropertyofnon-existence’,whichisself-contradictory.

Mathematically,thefallacyisclear.Youcan’tinferpropertiesofanobjectfromitsdefinitionuntilyou’veprovedsuchanobjectexists.Forexample,considerthedefinition‘thelargestpositiveinteger’.Here’saproof,bycontradiction,thatthisintegeris1.Letxbethelargestpositiveinteger,andsupposethatx>1.Then contrarytothedefinitionofx.Thereforex=1.Thefallacyisthatnosuchxexistsinthefirstplace.Whatwe’veprovedisactually‘ifxexiststhen .Inlogic,therearetwowaysfor‘ifPthenQ’tobetrue.OneisthatQistrue.TheotheristhatPisfalse.Sothebestwecaninferisthateither orxdoesnotexist.Anselm’sontologicalargumentsimilarlyallowsustoconcludethateitherthemostperfectbeingexists,orthemostperfectbeingdoesnotexist.Itwouldbedifficultnottoagree,butitgetsuspreciselynowhere.

Suchobjectionsnotwithstanding,manyreligiousdevoteesacceptedtheontologicalargument,anditscorollary:nolimits.Thisledthemtobelievenotjustinaverypowerfuldeity,butanomnipotentone;notaveryknowledgeabledeity,butanomniscientone;notaverylong-liveddeity,butaneternalone;notaveryextensivedeity,butaninfiniteone.

Around1260–70ThomasAquinasofferedadifferentproofoftheexistenceofGod,alsodependingoninfinity.HisSummaTheologiaeandSummaContraGentilesdiscussfivesuchproofs,ofwhichthesecondrelatestocausality.Heassertedthataninfinitechainofcausalityisimpossible,sotheremustbeaFirstCause.ThisisGod.

Supposewetakeanyparticularevent—say,gettingoutofbedthismorning.Amongitspriorcausesisthemanufactureofthebed.Thistracesbacktothefellingofthetreethatprovidedthewood,thentoaseedfromthepreviousgenerationoftrees,andsoon.Aquinasarguedthatthiskindofreversesequenceofcausescan’texplainanythingifitjustgoesbackforever.The‘explanation’wouldhavenobasis:thewholesequenceshouldalsohaveacause.ThisechoesastatementImadeearlier,andit’scloselyrelatedtoZeno’sdichotomyparadox.

Philosophershavearguedforandagainstthe‘FirstCause’proofforavarietyofreasons.Oneobjectionisthateverythinginexistenceissupposedtohaveacause…excepttheFirstCause.Whythespecialpleading?Anotherproblemiscausalityitself:whatisit,andwhydowethinkalleventshavecauses?Athirdistheassertionthataninfinitechainofcausalityisimpossible,whichisassumedasanaxiomwithoutjustification.AfourthistheidentificationoftheFirstCausewiththeGodofChristianity.

Amoremathematicalone,apparentlyunnoticed,isthatevenifaFirstCauseexists,itneednotbeunique.Causalitycorrespondsroughlytoa‘partialorder’,inwhichagivenentitycanbelargerthananother,smaller,orthetwomaybeincomparable.Forapartialorder,minimalelementsneednotexist,butevenwhentheydo,theyneednotbeunique.Withoutuniqueness,there’snorationaleforidentifyingjustoneofamultitudeofFirstCauseswithanyparticularentity,realorhypothetical.

Minimalelementsalwaysexistforapartialwell-ordering,inwhichanydescendingsequencemuststopafterfinitelymanysteps(sothere’snochainofcausalityreachinginfinitelyfarintothepast—Aquinas’saxiom),butminimalelementsstillneednotbeunique.Uniquenessdoesholdforatotalorder:givenanytwoelements,oneofthemisgreaterthantheother.Butcausalityisn’tatotalorder.

Modernera

Today’smathematiciansthinkabouttheinfiniteinaratherdifferentway,andseldomdrawAristotle’sdistinctionbetweenactualandpotential.Mathematicsisconceptual,bothinitsobjectsanditsprocesses.Psychologically,thesearedistinct;mathematically,they’retwosidestothesamecoin.Today’snotionofmathematicalexistenceisnotthesameasthatofphysicalexistence.

Aristotleputmathematicalinfinitiessuchasnumbersintothesamecategoryas‘allthementhathaveeverlived’.Totheancients,manymathematicalconceptswere‘real’—inanidealizedPlatonicsense.Plato’stheoryofformsassertsthatthehighestkindofrealityconsistsofabstractforms,orideas,andthatthematerialworldisanimperfectimageoftheidealone.Euclid’sgeometrywasthoughttobethetruegeometryofspace,albeitusingidealperfectformssuchaspointswithlocationbutnotsize,andcirclesthatwereperfectlyround,drawnwithlinesofzerothickness.Reality,inkedonpapyrusorscratchedinsand,wasapaleshadowoftheideal.Propertiesofrealobjectscouldbededucedbyconsideringtheiridealversions,whichweresimpler.(Imaginetryingtodefinewhichgrainsofsandconstitutethepointofintersectionoftwolinesdrawnasslightlywobblygrooves;especiallysincethelinesandpointsareplaceswherethesandisnolongerpresent.)EverythingAristotleandmanyofhissuccessorssaidabouttheinfinitewasmixedupwiththisconfusedviewofmathematicalreality.

Today’smathematiciansdon’tconsiderthedistinctionbetweenactualandpotentialinfinitytobeimportant,becausemathematicalobjectsare‘actual’onlyonaconceptuallevel.Infinityisn’ttheproblem,thoughitdoesaddtotheconfusion.Whatabout‘two’?Icanshowyoutwocatsortwochairs—butIcan’tshowyouthenumber‘two’.Holdingupapieceofpaperwith‘2’onitdoesn’twork;that’sasymbolforthenumber,anumeral.Nottheactual(!)number.Thatsaid,mathematicianswouldanswerthe‘pasttime’objectiontopotentialinfinitybyallowingprocessestorunbackwardsaswellasforwards.Amathematicalprocessisasequenceofsteps,each‘following’theotherinlogical,nottemporal,succession.Intheprogressionofyears‘2016,2015,2013,2012,…’eachyearsucceedstheotherinthesequence,butprecedesitinhistoricaltime.

Evenifyouinsistonmaintainingtemporalorder,similarreasoningapplies.Thenumberofeventssinceanyspecificpasttimecanbemadebiggerbystartingearlier.Ifthingscomeoneaftertheother,thentheothercomesbeforetheone.InpracticethisisprettymuchthepositionthatAristotleadopted.

Couldtherebealargestnumber?

Loftyphilosophicalargumentsandmathematicalabstractionssometimeslosecontactwithempiricalreality.Inboth,wefondlyimaginethathoweverbiganumbersomeonewritesdown,wecanalwayswritedownabiggerone.However,asapracticalmatter,thisisn’ttrue.

Thegoogol,10100,is1followedbyahundred0s.Ittakesonlyacoupleofminutestowritethisoutinfullinbase-10notation.It’seasytowritedowntheresultofaddingone,ortoputanextrazeroattheendtomultiplyitbyten.Wereallycanwritedownabiggernumber.Butcomparethegoogoltoitsbigbrother,thegoogolplex whichis1followedbyagoogolof0s.Thehumanlifespanistooshorttowritethisoutinfullinbase-10notation,oreventomakeasignificantdentinthetask.Moreover,theentireglobalsupplyofpaperandink,fromnowuntiltenbillionyearshencewhentheSunexpandsintoaredgiant,wouldbeinadequatetorecordthatnumber.Ofcoursewecanwrite butforanypre-specifiednotationalsystem,therecomesapointwhenit’snotpossibletowriteabiggernumberdown.There’snotenoughtime,ornotenoughroom.Ourfiniteworld,thoughgigantic,can’tcontinuetheprocessaswenaivelyimagine.

Ouruseofinfinitedecimals—indeed,verylongfinitedecimals—alsofallsfoulofreality.Ourmentalimageofspaceisaninfinitelydivisiblecontinuum.Anyinterval,howevershort,canbesubdividedintotenshorterones,asinChapter2.Butmatterstartstobebecomeindivisiblewhenwegetdowntothescaleofanatom,andinquantummechanics,spaceisindivisibleonthescaleofthePlancklength,whichis

metres.Ontheoreticalgrounds,nomeasurementsmallerthanaboutonetenththatsizeispossible.So,inthecontextofactualphysicalmeasurement,numberswithmorethan35or36digitsafterthedecimalpointhavenosensiblemeaning.

Theseremarkshaveaninterestingimplication.Allofthenumbersthatanyonehaseverused,beitformathematics,science,medicine,orbuyingfood,usinganynotationyetinvented,aresmallerthansomespecificnumber.Ihavenoideawhatitis,andwritingitdownwouldimmediatelydestroythatproperty,butitmustexist.Soinpractice,onlynumberssmallerthanthatboundhaveeverbeenneeded.Absolutelynoactivitiesthatdependonnumbers,inthewholeofhumanhistory,wouldchangeifwehadlimitedourselvestothisfiniterangeofnumbers.

Sowhydomathematiciansinsistthattherangeofnumbersmustbeinfinite?Onereasonistheunconsciousassumptionthatifasimplepatternpersistsforalongtime,itmustpersistforever.Asecondisthatwhenmathematiciansstartedtoformalizetheprocessesofcountingandarithmetic,theyrealizedthateverythingissimplerifweassumefromthestartthatcertainarithmeticalrulesareuniversal.Oneoftheseisthatn+1isalwaysbiggerthann.Ifweabandontheconventionthatthereareinfinitelymanywholenumbers,thetraditionalrulesofarithmetic,hencealsoalgebra,don’twork.

It’snotimpossibletosetupafiniteversionofarithmeticwithaverybiglargestnumber,butit’sinelegantanddifficulttoworkwith.Mathematiciansprefertheirpatternstobeuniversalinscope,sotheyembracetheinfinitudeofthewholenumbers.Infinityissimplerthansomespecificbutinexplicitverylargenumber.

Chapter4Theflipsideofinfinity

Wenowturnfromtheinfinitelylargetotheinfinitelysmall.ThreeexamplesinChapter1(diagonalofasquare,areaofacircle,onethirdindecimals)areinthiscategory.Eachdescribesaprocessinwhichageometricobjectoranumberisrepeatedlysubdivided,orapproximatedevermoreaccuratelybyfinerandfinerstructures,andtheresultisthenmadeexactbyconsideringaninfinitelyfine—infinitesimal—subdivision.

TheancientGreekswereexcellentlogicians,andrecognizedthatthismethodisfallaciouswhenexpressedinsuchterms.However,theyfoundarigorouswaytomakesenseofit,whichtheycalledexhaustion.Eudoxususedthismethodtoputthetheoryofproportiononasoundlogicalbasiswhenthelengthsinvolvedareincommensurable—ineffect,todealwithirrationalnumbers,althoughtheGreekspreferredtoreasonintermsoflengthsoflines,notnumericalmeasuresofthoselengths.

We’lltakeabrieflookatexhaustion,andthenprogress,bywayofcalculus,tothemodernconceptofalimit,whichabolishedinfinitesimals.Thenwe’llseehowtheywerereinstated.

ProofofArchimedes’stheorem

Archimedesdidn’tuseπexplicitly.Instead,heprovedthattheareaofanycircleisequaltoitsradiusmultipliedbyhalfthecircumference.Ifwedefineπastheratioofthecircumferencetothediameter,thisresultisequivalenttotheusualformulaπr2.Itcanbemotivatedbycuttingthecircleintoever-thinnerslices,andthinkingaboutalimitingcaseof‘infinitelymanyinfinitesimalslices’asinChapter1.Butthisapproachlackslogicalrigour.Instead,Archimedesusedexhaustion,basedonsequencesofapproximatingpolygonswhoseareasandperimeterswereknown.Onesequenceapproximatesthecirclefrominside,theotherfromoutside.

LetAdenotetheradiusmultipliedbyhalfthecircumference.Thenthefollowingstatementsaremutuallydistinctandtogetherexhaustallpossibilities:

(1)TheareaofthecircleisgreaterthanA.(2)TheareaofthecircleislessthanA.(3)TheareaofthecircleisequaltoA.

Insteadoftryingtoprove(3)directly,themethoddisprovesboth(1)and(2),usingproofbycontradiction.Logically,only(3)remains.

The‘outside’sequenceofapproximatingpolygonsisdefinedbycircumscribingaregularhexagonroundthecircle,andrepeatedlybisectinganglestocreatecircumscribingregularpolygonswith12,24,48,96…sides.Figure9(a)showsthefirsttwostagesinthisprocess—laterstagesaretooclosetothecircletodrawclearly.Thegeometricaldetailsofthedisproofof(1)arecomplicated,butthebasicideaissimple.If(1)holds,theareaofthecircleexceedsAbyaspecificamount .Eachexternalpolygonhasgreaterareathanthecircle,soitsareaisalsogreaterthan .However,ifthenumberofsidesissufficientlylarge,theareaofthepolygoncanbeprovedtobelessthan .Thiscontradictiondisproves(1).Asimilarargumentusingthe‘inside’sequenceofpolygonsdisproves(2),Figure9(b).Nowstatement(3)follows.

9.Thefirsttwostagesinapproximatingacircle.(a)fromtheoutside.(b)fromtheinside.

ThemainpracticaldeficiencyofexhaustionisthatyouhavetoknowthecorrectanswerAtosetupthetrichotomy.Themaintheoreticaldeficiencyisthatyouhavetoknowthatthequantityyou’reseekingexists.TheGreeksassumedthateveryshape—inparticular,everycircle—hasawell-definedareaandawell-definedperimeter(orcircumference).Muchlateritturnedoutthatthisassumptioninvolvessubtlefeaturesofanalysis,butthatwithenoughcareandeffortareasandlengthsofmanyshapes—thoughnotall—canbedefinedandvarioussensiblepropertiesproved.Butbythetimethiswassortedout,bettermethodsthanexhaustionhademerged.

Calculusanditsprecursors

Wenolongeruseexhaustiontostudyareasandvolumes,becauseasimplerandmoregeneraltechniquewasdeveloped:calculus.Thehistoryofcalculusiscomplicated,withaseriesofprecursorsandafull-blowncontroversyoverwhodeservesthecreditforbringingthesubjecttofruition:LeibnizorIsaacNewton.(Theconsensusis‘both’.)Initially,thelogicalformulationwasrathervague,makingintuitiveuseofinfinitesimalswithoutprovidingcleardefinitions.Thiswastypicalofmathematicsuntilthe1800sinanycase—notjustforinfinitesimals,butalsofornumbers,functions,andotherlessesotericconcepts.

LeibnizandNewtonunifiedtwodistinctbranchesofthesubject:

•Integralcalculus,whichcalculateslengths,areas,volumes,andsimilarquantities.•Differentialcalculus,whichcalculatestheinstantaneousrateofchangeofsomequantity;forexample,accelerationis

therateofchangeofvelocity.Geometrically,theaimistofindthetangenttoacurveatagivenpoint.

Integralcalculus(sayforanarea)proceedsbydividinganapproximationtotheareaintopieceswithsimpleshapes,calculatingtheareaofeachpiece,addingtheresults,andthenmakingthepiecesarbitrarilysmallandtheirnumberarbitrarilylargetoremoveanyerror,asinFigure10(left).Differentialcalculusdividesthechangeinthequantityconcerned,overasmallintervaloftime,bythelengthofthatinterval;thentheintervalismadearbitrarilysmall,asinFigure10(right).Sobothprocessesinvolve‘infinitesimal’quantities,andintegralcalculusalsoinvolvesinfiniteones(thenumberofpieces).I’lldiscusstheseprocessesinmoredetailshortly,butweneedthegistofthemnowtoseehowthehistoricalideasrelatetothefinaloutcome.

10.Left:areaunderthegraphoffapproximatedbyrectanglesofwidthε.Right:rateofchangeoffunctionfoverasmallintervaloflengthε.

Precursorstocalculusabound.Democritus,aGreekphilosopherwhoflourishedaround400BC,ismainlyrememberedforthetheorythateverythingismadefromindivisibleatoms.Buthewasalsoamongthefirsttodiscoverthatthevolumeofaconeisonethirdtheareaofitsbasetimesitsheight.Heprobablyobtainedthisresultbyslicingtheconeintoinfinitesimallythickcircularsectionsparalleltothebase,treatingeachasaverythincylinder,andaddingtheirvolumes.Unlikemostofhissuccessors,Democritushadreservationsaboutthelogicofthisprocedure,butitgavetherightanswer.

Inthemedievalperiod,themaincontributionscamefromChina,India,andtheMiddleEast.Around500ADZuGengzhistatedthatiftwobodieshavethesamecross-sectionswhenslicedbyequidistantparallellines(planes),thentheirareas(volumes)areequal,whichisavariantofDemocritus’smethod.Around1000ADAlhazen(Abūal-Haytham)madeanotherprediscoveryofconceptswenowassociatewithintegralcalculus.Heusedformulasforsumsofsquaresandsumsoffourthpowersofintegerstofindthevolumeofaparaboloid—ineffect,calculatingintegrals.

Inthe14thcenturyMadhavaofSangamagrama,theleadingfigureintheKeralaschoolofmathematics,developedthetechniqueofexpressingafunctionasapowerseries,discoveringwhatlaterbecameknownastheTaylorseriesofafunction,creditedtoBrookeTaylor.Heappliedhismethodtostatepowerseriesexpansionsfortrigonometricfunctions.Powerserieslaterbecameanimportantapplicationofcalculus,andthebasisforanalysis,especiallywithregardtocomplexnumbers.

In1635BuonaventuraCavalieripublishedGeometriaIndivisibilibusContinuorumNovaQuadamRationePromota(geometrydevelopedbyanewmethodthroughtheindivisiblesofthecontinua).LikeDemocritusandZu,hetreatedareasandvolumesassumsofinfinitelymanyinfinitelythinparallelslices,whichhecalledindivisibles.Heusedthisideatofindtheareaunderthecurve anearlyexampleofintegration.Hismethodiscalled‘Cavalieri’sprinciple’,andhemadeeffectiveuseofit.

Atypicalexampleoftheprincipleshowsthattheareaofatriangleishalfthebasetimestheperpendicularheight.First,observethatthisistrueforaright-angledtriangle,becausetwocopiesfittogethertoformarectangle(Figure11,left).Cavalieri’sprincipleextendstheresulttoanarbitrary

triangle.Sliceitintoinfinitelymanyhorizontallinesandslidethemsidewayssothattheirleft-handedgesformalineatrightanglestothebase(Figure11,right).Thisconvertsthetriangleintoaright-angledonewiththesamebase,perpendicularheight,andarea.Euclidgaveadifferent,lesscontentious,proofofthisresult.

11.ExampleofCavalieri’sprinciple.Left:areaofright-angledtriangleishalfthatoftherectangle.Right:slidetheslicestochangetheshapetoaright-angledtrianglewiththesamearea.

Cavalieri’sprinciplealsogivescorrectformulasfortheareasandvolumesofpolygons,circles,cylinders,cones,spheres,andmoreesotericobjects.However,ithastobeusedwithcare.Forexample,aswellasslidingthelinestothelefttocreatearight-angledtriangle,wecouldalsomovethemdownwardsbyhalvingtheirheightabovethebase.Everylineintheoriginaltrianglestillmatchesauniquelineofthesamelengthintheright-angledtriangle,buttheareaishalved.

PierredeFermat,famousforhisconjectureinnumbertheory,tookCavalieri’sideasfurther,definingaconceptthathecalledadequality:differingonlybyaninfinitesimalerror.Inthe17thcenturyJohnWallis,IsaacBarrow,andJamesGregoryputthetwobranchesofcalculustogether.By1670BarrowandGregoryhadproved(thoughnotrigorously)thatintegrationisthereverseofdifferentiation,aresultoftencalledthefundamentaltheoremofcalculus.

TherigorousjustificationofCavalieri’sprinciple,carefullystated,ismostsimplyobtainedthroughcalculus.LeibnizandNewtonputalloftheseideastogetherintoasystematicpackage.Theyuseddifferentnotationbuttheirmainresultsandconceptswereverysimilar;notbecauseeitherstoletheother’sideas,aswaslaterclaimed,butbecausethesubjectnaturallyfitstogetherinonlyoneway.Today,elementarydifferentialcalculusmostlyusesLeibniz’snotation forthederivative(rateofchange)ofquantityywithrespecttoquantityx,and forthesecondderivative.ButtracesofNewton’snotationstillremain,suchas and forthesamethings,or forthederivativeofafunctionf.Theterm‘calculus’isLeibniz’s;Newtoncalledthesubject‘fluxions’.

Leibniztooka‘puremathematical’viewpointandwasmainlyinterestedinthephilosophicalimplicationsofcalculus.Newton’sapproachgavebirthtotheoreticalphysicsandappliedmathematics.Hisformulationreliedonphysicalintuition,andheusedcalculustoanswerawiderangeofbasicquestionsinthephysicalsciences.Ironically,theextensiveapplicationsofcalculustophysicsoverthenextcenturyweremainlydiscoveredincontinentalEurope,notBritain,andusedLeibniz’snotation.

Limits

It’spossibletobecomeproficientincalculusbylearningalotofrulesandpractisingthem,butunderstandingwhythoserulesarecorrectisanothermatter.Wecanlearnbyrotetherulethatthederivativeofthefunction is Wecanevenusethatfacttosolvepracticalproblems.Butwhyisittrue?

Figure10(right)illustratesthedefinitionofthederivativeforageneralfunctionf,whichwenowtaketobe Thecalculationof considersasmallincrementfromxtox+ε,sothatthefunctionchangesfromx2to Thenthedifferenceinthex-valuesisε,whilethedifferenceinthefunctionis

Theirratio(‘averagerateofchange’overtheintervalε)is

whichisnotexactly2x.

Nowcomesthesleightofhand.Ifεbecomesverysmall,theexpression getsverycloseto2x.Theeasiestwaytoseethisistoset ,inwhichcaseonly2xremains.However,BishopBerkeleypointedout(withsomeheat)thattheprevioussteptheninvolvedthefraction0/0,whichismeaningless.

Asimilartrickisusedinintegralcalculus,withasimilarobjection.Tofindtheareaunderthegraphofafunction,approximateitbyaseriesofrectanglesofwidthεandthenletεbecomeverysmall(Figure10,left).Berkeleywouldobjectthatifεisnotzerotheareaiswrong,butwhenε=0eachrectanglehasareazero,sothetotaliszerotoo,whichisalsowrong.

Leibnizattemptedtodealwiththeseissuesbyconsideringεtobeinfinitesimal,aconceptthatheexplainedindetail.Forthederivative,itcouldthenbeneglectedtoleave2x.Somethingsimilardealtwiththeintegral.Newtonusedaphysicalimageinstead:εisnotafixedquantity,butonethatflowstowardszerowithouteverreachingit.Then flowstowards2x.

Itprobablydidn’thelpthatNewtonusedthesymbolowhereI’veusedε.ThisallowedBerkeley,ineffect,toaccusehimofconfusingowith0.Inhis1734bookTheAnalyst,Berkeleyscathinglyreferredtooasthe‘ghostofadepartedquantity’,claimingthatcalculusobtainedcorrectresultsthroughcompensatingerrors.Inaway,hewasright,buthewastoobusyindulgingintheologicalpoint-scoringtoaskhimselfamoreimportantquestion:whydotheerrorsalwayscompensate?Ifyoucanexplainthat,withastrongenoughguarantee,they’renoterrorsatall.

MathematiciansgenerallyignoredBerkeley,notbecausehewaswrongbutbecausetheyfoundtheentireargumentirrelevant.Theresultsemergingfromcalculusincludedpowerfulinsightsintoheat,sound,elasticity,gravity,electricity,magnetism,andfluidflow.Evenifthereweretinyerrorsinthesums,theycouldbemadealotsmallerthanmeasurementerrorsinexperiments.

Whenarigorousformulationofcalculuswasdevisedinthe19thcentury,thesubjectbecameknownasanalysis.Augustin-LouisCauchyverynearlysortedoutthelogicalfoundationswhenhewasdevelopingatheoryofanalysisusingcomplexnumbersandfunctionsinplaceofrealones.Inhisview,aninfinitesimalisavariablequantitythatapproaches(butneednotreach)zero;ineffect,asequenceofnumbersanthatbecomesarbitrarilysmallifnislargeenough.Anexampleis whichisneverzero,butcanbemadeassmallasweplease.Then,writingεforthissequence,theexpression becomesthesequence Thisdiffersfrom2xbyan,whichisinfinitesimalinCauchy’ssense,so‘inthelimit’weget2x.Whatwedon’tdoisjustset .

Theconceptofavariableisitselfinformal,however,sothisapproachfallsshortofmodernlevelsofrigour.EventuallyBernhardBolzanoandKarlWeierstrassdevisedtheformulationweusetoday.Aquantityg(x)tendstoalimitLasxtendstoafixednumberaiff(x)canbemadearbitrarilyclosetoLbymakingxsufficientlyclosetoa.Toformalizethisstatementwespecifyhowclose.Letεbeanypositivenumber.Thentheremustalwaysexistsomepositiveδ,dependingonε,suchthatwhenever wehave This‘ ’definitionofalimitisprecise,makesnoexplicituseofphysicalimagerysuchas‘flowing’,andmakesnomentionofinfinitesimals.Everythingthatappearsisanordinaryrealnumber.Wedon’teveninsistthatεorδissmall.That’sjustwherethemainimplicationscomeintoplay.Ifδworksforsomeε,italsoworksforanythinglarger.

Aristotlewouldrecognizeacunningapplicationofpotentialinfinity(moreaccurately,potentialinfinitesimality)here.Wedon’tmakeεinfinitesimal.Wetakeittobeanypositiverealnumber.Wethinkofεasbeingsmall,butthemainpointisthatwhateversizeitis,wecanalwaysmakeitsmaller.Thenwemustmakeδsmallertoo,butthat’spermitted.Ithastobe;ifwespecifiedδonceandforallatthestart,wecouldmakeεsosmallthattheconditiononLfailed.

Withdefinitionssuchasthese,BolzanoandWeierstrassturnedcalculusintoanalysisinitsmodernsense.Theinfinitesimalwasbanished,evenasinformalmotivation.Initsplacewasacomplicatedformofwords,liberallysprinkledwith‘forall’and‘thereexists’quantifiers,affectionatelycalled‘epsilontics’byirreverentmathematicsstudents.Withaneffort,youcouldlearntomasterthelanguage,andanalysisfittedtogetherlogicallyanditallmadesense.Asgenerationupongenerationofstudentswentthroughtheprocessofgettingusedtoit,thebadolddaysoftheinfinitesimalfadedfrommathematicalmemory.

Infiniteseries

Withtherigorousformulationoftheconcept‘limit’,calculuspoweredaheadtobecomejustonepartofamuchbroaderareaofmathematics,analysis.Limitsresolvedseveralotherbasicissuestodowithinfinityandinfinitesimals,byrecastingtheminfiniteterms.Aristotlewouldhavebeenproud,becausetheessenceofthisresolutionisamovefromactualtopotentialinfinity.The definitionofalimitisbasedonaprocessthatassignstoanyspecificfinitepositiveεaspecificfinitepositiveδ.

Limitsalsomakesenseofinfiniteseries.Thedefinitionismodifiedslightly;inplaceofarealnumberδwhosemainroleinvolvesbeingsmall,infiniteseriesinvolveanaturalnumbernwhosemainroleistobelarge.Specifically,aninfiniteseries

convergestoalimitLif,foranypositivenumberεthereexistsNsuchthat

whenevern>N.Inwords:thesumoffinitelymanytermsoftheseriesbecomesascloseaswewishtoLifthenumberoftermsislargeenough.

Forexample,wecandefinetherecurringdecimal tobethesumoftheinfiniteseries

ThelimitLofthisseriesisexactly1/3.ThedifferencebetweenLandthesumofthefirstntermsis

Givenε>0,wecanmaketheright-handsidelessthanεbytaking because3.10n>n.

Thesametypeofcalculationprovesthattheseriesrepresentinganyinfinitedecimalconvergestoarealnumber,andthateveryrealnumbercanbeexpressedasapossiblyinfinitedecimal.Thisjustifiestheuseofinfinitedecimalsasaconceptualnotationforrealnumbers.

IfnosuchLexists,theseriesdiverges,andtheinfinitesumcan’tbegivenameaningasalimit.Grandi’sproofofCreationfromnothinginChapter1isfallaciousbecauseitusesadivergentseries.However,somedivergentseriescanbegivenasensiblemeaningbyinventinganew(technical)definitionofthesum.Mathematicianseventuallydidthat,foraclassofdivergentseriessaidtobesummable.ThisjustifiesGrandi’sclaimthat

isameaningfulresultinaspecifictechnicalcontext,butnothisinterpretationthatsomethingcanbecreatedfromnothing.

Infinitesimalsrevenant

Limitsresolvedtheparadoxicalissuesaboutinfinityandinfinitesimalsinanalysis,buttheghostoftheinfinitesimaldidn’tfadecompletely.Inmathematics,it’sunwisetoabandonaninterestingideajustbecauseit’swrong.It’salsounwisetokeeppushinganincorrectideawithoutchangingitoracknowledgingtheerror,butincorrectideascansometimesbereformulatedsothattheywork.Infinitesimalsareacaseinpoint.

Wheneveryonethoughtthatmathematicalnumberswererealityitself,albeitinanidealizedform,thenumbersystem(basically,therealnumbers)wastheonlyonepossible.(Complexnumberswereacceptedwithreluctanceatfirst,andthenmaderespectablebythinkingofthemaspairsofrealnumbers.)So,ifyoudefinedaninfinitesimalas‘apositivenumberthatissmallerthananypositivenumber’youwereintrouble.Ithadtobesmallerthanitself.

Ifyoustopthinkingthattheonlypossiblenumbersaretherealnumbers,however,there’sawayout:defineaninfinitesimalas‘apositivenumberofsomenovelkindthatissmallerthananypositiverealnumber’.Sinceit’snotarealnumber,theargumentthatitmustbesmallerthanitselffails.Makingthisideaworksensiblyisn’tstraightforward,however.Onceyouthrowinanewinfinitesimalnumber—callitε—thenyouhavetomakesenseofallalgebraicexpressionsinvolvingε,suchas and1/ε.Todoanalysis,youneedtodefinesinε,cosε,logε,eε,andsoon.Thenyouhavetoprovethattheseextendedconceptshavealloftheusualpropertiesthatweexpect,andthattheentirestructureislogicallyconsistent.

Assumingthiscanbedone,yournewnumbersystemalsocontainsinfinitenumbers.Bydefinition,forallnaturalnumbersn>0.Therefore forallnaturalnumbersn>0,so1/εisinfinite.

In1877PaulduBois-Reymondbegantodevelopjustsuchanumbersystem.InÜberdieParadoxendesInfinitär-Calcüls(ontheparadoxesoftheinfinitarycalculus)hewrote:

Theinfinitelysmallisamathematicalquantityandhasallitspropertiesincommonwiththefinite…Yetwhenonethinksboldlyandfreely,theinitialdistrustwillsoonmellowintoapleasantcertainty…Amajorityofeducatedpeoplewilladmitaninfiniteinspaceandtime,andnotjustan‘unboundedlylarge’.Buttheywillonlywithdifficultybelieveintheinfinitelysmall,despitethefactthattheinfinitelysmallhasthesamerighttoexistenceastheinfinitelylarge.

Anotherpioneerofameaningfulnotionofinfinitesimal,atmuchthesametime,wasOttoStolz.Heextractedthekeyfeaturethatexcludesinfinitesimalsfromtheusualrealnumbers,namingittheArchimedeanpropertybecauseArchimedesstateditasanaxiomwhenapplyingexhaustioninOntheSphereandCylinder.Thepropertyconcernedappliestoanysystemwithasensibleconceptof‘lessthan’(thatis,satisfyingsomereasonableaxiomsthatIwon’tstatehere).Itcanbeformulatedaseitheroftwoequivalentstatements:

•Everynumberxislessthansomenaturalnumbern.•Ifanumberx>0,then forsomenaturalnumbern.

Thefirststatementsaysthatthesystemcontainsnoinfinitenumbers,thesecondthatitcontainsnoinfinitesimals.

Todayweencapsulatethisideaas:therealnumbersareanArchimedeanorderedfield.Thatis,theusualoperationsofarithmeticcanbedefinedandhavetheusualproperties;anotionof‘lessthan’canbedefinedandhasalltheusualproperties;finally,theArchimedeanaxiomapplies.Indeed,RistheonlyArchimedeanorderedfield,exceptfortrivialchangesinnotation(‘uptoisomorphism’).StolzandduBois-Reymonddiscoveredthat,incontrast,thereexistmanydifferentnon-Archimedeanorderedfields.Bydefinitionthesecontainbothinfiniteandinfinitesimal‘numbers’.DuBois-Reymondconstructedanaturalexamplein1875:allreal-valuedfunctionsofarealvariablex,orderedbytheir‘asymptotic’behaviourforlargex.Thelogarithmicfunctionrepresentsaninfinitesimalelement,andtheexponentialfunctionrepresentsaninfiniteelement.

Non-standardanalysis

Non-Archimedeanorderedfieldscanbeusedtojustifymanyfeaturesofanalysis,suchasthedefinitionofthederivativeincalculus,usinggenuineinfinitesimals.Buttodothissystematically,wemustdefineanaloguesofstandardfunctionssuchaslog,exp,sin,andsoon.Onewaytodothisiswithpowerseriesexpansions,butplentyofusefulfunctionscan’tbedefinedbypowerseries.Sowehavetoestablishexactlywhichpropertiesoftherealnumbershavesensibleanalogues.

Muchbetterwouldbetoconstructthefieldsothateveryimportantpropertyoftherealnumbers(exceptthoselike‘beingArchimedean’)automaticallyhasasensibleanalogue.Thisiseasiersaidthandone,andnooneexpectedittobepossible,butintheearly1960sAbrahamRobinsondiscoveredthatitis.Akeystepwastodistinguishpropertiesthatstillwork(suchas )fromthosethatdon’t(‘thefieldisArchimedean’).His1966Non-StandardAnalysisstates:

Theideaofinfinitelysmallorinfinitesimalquantitiesseemstoappealnaturallytoourintuition…Leibnizarguedthatthetheoryofinfinitesimalsimpliestheintroductionofidealnumberswhichmightbeinfinitelysmallorinfinitelylargecomparedwiththerealnumbersbutwhichweretopossessthesamepropertiesasthelatter…ItisshowninthisbookthatLeibniz’sideascanbefullyvindicatedandthattheyleadtoanovelandfruitfulapproachtoclassicalanalysisandtomanyotherbranchesofmathematics.

Robinson’sdiscoveryemergedfromabranchofmathematicallogicknownasmodeltheory,whichexaminestherelationbetweensystemsofaxiomsandmathematicalstructuresthatsatisfythem.Usingmodeltheory,heprovedtheexistenceofanon-Archimedeanorderedfield,havingallthepropertiesoftherealnumbersthatcanbeexpressedbylogicalstatementsthathave‘boundedquantification’—atechnicalrestrictionontheuseofthequantifiers‘thereexists’and‘forall’.‘BeingArchimedean’can’tbeexpressedinthatmanner.

Anynon-Archimedeanorderedfieldofthistypeisknownas‘the’hyperreals,denotedR*.Suchfieldsarenotunique,butanyofthemcanbeusedasamodelforpropertiesoftherealnumberswithboundedquantification.Nowwecanprovethetransferprinciplethatanybounded-quantificationpropertyisvalidforRifandonlyiftheanalogouspropertyisvalidforR*.Thisimplies,forexample,thatstandardfunctionssuchaslog,exp,sin,andcoscanbedefinedinR*,sothatalloftheirusualproperties,suchaslog ,remainvalid.

SincethesetR*ofhyperrealsformsanon-Archimedeanorderedfield,itcontainsinfinitesimals,andtheirreciprocalsareinfinities.Everyfinitehyperrealcanbedecomposeduniquelyasarealnumberplusaninfinitesimal.Therealnumberiscalleditsstandardpart.Nowclassicallimitingprocessescanbecastinthelanguageofinfinitesimals,providedwetakethestandardpartattheend.Thederivativeofafunctionfcanbedefinednotasalimit,butas

whereεisinfinitesimalandstisthestandardpart.

Bysuchmeans,allusesoflimitsinanalysiscanbereplacedbyintuitivereasoningaboutinfinitiesandinfinitesimals,alonglinesthatgorightbacktoNewton,Leibniz,andindeedpredecessorssuchasFermat,Cavalieri,andArchimedes.Theonlymissingideasweretheexistenceofhyperreals,theproofofthetransferprinciple,and—aboveall—takingthestandardpartintheusualformulas.

Someeducatorshavearguedthatnon-standardanalysisprovidesaneffectivenewwaytointroduceanalysistoundergraduates,whooftenhavedifficultieswithlimits.Themethodhasbeentriedoutinuniversityclasses,withsomesuccess.Onepsychologicalbarrieristhatthedefinitionofhyperrealsisn’tconstructive.Itdoesn’tprovideaspecificmodelforR*inthewaythatR2doesforEuclideangeometryandinfinitedecimalsdofortherealnumbers.Anotheristhatthemodel-theoreticproofsareveryabstract.Sotheclassicallimitapproachtoanalysisisstillthedefaultformostmathematiciansandmostundergraduatelecturecourses.

Importanttheoremshavebeenprovedusingnon-standardanalysis,inareassuchasprobabilitytheoryandfluiddynamics.Bythetransferprinciple,standardproofsofthesetheoremsmustexist,butthey’renotactuallyknown.Inpracticethemainimpactofnon-standardanalysishasbeenaphilosophicalone:itprovidesalogicalframeworkforinfinitesimals.

Chapter5Geometricinfinity

Ifyoustandnexttoalongstraightrailwayline,yougetastrongimpressionthattherailwaytracksmeetonthehorizon(Figure12).Thishappensbecausethetracksareparallel.Theparalleledgesoflongstraightroadsbehaveinthesamemanner.

12.Parallelrailwaytracksappeartomeetonthehorizon.

InthefamiliargeometryofEuclid,parallellinesplayaspecialrole.Bydefinition,twolinesareparallelifthey’realwaysthesamedistanceapart,sotheycan’tmeethoweverfarthey’reextended.However,ifanobserverstandsonaninfiniteplanebetweentwoparallellines,thenthefurtherawaytheyget,theclosertheyseemtobecome.Insomesense,thelinesappeartomeet‘atinfinity’—adiscoverythatinspiredoneunknownschoolboytostatethat‘infinityiswherethingshappenthatdon’t’.

Hewasright.MathematiciansfoundawaytoextendEuclid’sgeometry,byaddinganextra‘lineatinfinity’torepresentthehorizon.Ordinarylinesaresimilarlyextendedbyequippingeachofthemwithanextra‘pointatinfinity’.Thisidealedtoanewandextremelyfruitfulkindofgeometrycalledprojectivegeometry.Historically,amajorsourceofmotivationwasthevisualarts.TheartistsoftheItalianRenaissancewantedtopaintthree-dimensionalobjectsandscenessothattheylookedrealistic.

Linearperspective

Theapparentconvergenceofparallelrailwayslinesisoneofthesimplestexamplesoflinearperspective:theaccuraterepresentationofthree-dimensionalgeometricformsonaflat,two-dimensionalcanvas.Distantobjectsappeartobesmaller.YoucancovertheMoonwithyourthumb;thesheepacrossthefieldlooksalotshorterthantheonestaringatyouoverthefence.Thesefamiliareffectsareaconsequenceofthephysicsoflightraysandthestructureofthehumanvisualsystem.However,it’snotenoughtomakesheepintheforegroundlargerthanthoseinthebackground.Theentiregeometryofthepaintinghastofittogetherinasystematicwaythatrepresentshowobjectsinthreedimensionsappeartotheeye.

BeforetheRenaissance,artistseitherignoredthisissueorgotitwrong.AncientEgyptianart,forexample,ignoresit:thesizeofapersoninareliefislargelydictatedbytheirsocialimportance.Pharaohsandgodstoweroverordinarymortals;servantsaresmallerthantheirmasters;wivesareoften(butnotalways)smallerthantheirhusbands.InthereliefofFigure13,RamsesIIisdepictedasbeingtallerthanhishorses,andfarbiggerthananyoneelse.Theartist’sneedtofitalotofdetailintoalimitedspaceisanotherfactor.Inparticular,thefortificationattherightisshowninastylizedform,notinperspective.(Theseremarksaren’tintendedascriticism:theEgyptiansmadeadeliberatestylisticchoice.Otherreliefsareremarkablyrealistic,especiallydepictionsofbirdsandotherlivingcreatures.)Medievalartistssometimestriedtodepictbuildingsinrudimentaryperspective,butfailedtorelatethemtoeachotherinacoherentway.

13.RamsesII’svictoryattheSiegeofDapur.FromareliefinhistempleatThebes.

Renaissanceartistsbenefitedfromadeeperunderstandingofthegeometryofperspectivedrawing.TheearliestsystematicmethodwasprobablythatofAmbrogioLorenzetti,employedinhis1344Annunciation,whichincludesaveryaccuratetiledfloor.FilippoBrunelleschitookabigstepforwardintheearly15thcenturywithimagesoftheFlorentineBaptisteryandthePalazzoVecchio.AmajoradvancecamewithLeonBattistaAlberti’sDellaPittura,begunin1435.Thebasicideasarosefromacombinationofartisticimperatives,experimentalobservations,andgeometricreasoning.

Figure14showsoneofthemostfamousoftheearlyperspectivepaintings,PierodellaFrancesca’sFlagellationofChrist.Theartistflauntshismasteryofperspectivetoconsiderableeffect.Theinterpretationofmanyfeatures,notablythelargefiguresintheforeground,iscontroversial,butthetiledfloor,withitschequeredpattern,isimpressivelyaccurateyetunderstated,asarethebuildings.

14.FlagellationofChristbyPierodellaFrancesca.

Thetechniquesofperspectivecamefromanidealizationofthegeometryofhumanvisionthatdoesn’tcorrespondexactlytoreality.However,thisidealizationhasconsiderablemathematicalinterest.Itwasahugeimprovementonpreviouspracticeinthevisualartsandwasanearlysteptowardsthescientificunderstandingofvision.Theseartisticendeavoursalsohadasignificanteffectonmathematics,motivatinganewkindofgeometry.It’scalledprojectivegeometry,becauseitcentresonhowimagescanbeprojectedfromoneplanetoanother,muchasaframeofamovieisdisplayedonacinemascreen.

Intheabstract,thisprocesssetsupatransformationfromoneplanetoanother.Chooseafixedbutarbitrarypointnotlyingoneitherplane,thecentreofprojection.Givenapointononeplane,drawthestraightlinethroughthatpointandthecentreofprojection.Thismeetstheotherplaneinauniquepoint,theimageofthefirstundertheprojection.InFigure15,AlbrechtDürerisexperimentingwithaprojectionfromtheplaneofatable,supportingalute,toaverticalscreenthathashelpfullybeenhingedopentoshowtheresultingimage.

15.AlbrechtDürer,fromUnderweysungderMessungMitdemZirckelunRichtscheyt,inLinien,Nuremberg,1525.

Anyoneusingavideoprojectorrunsintooneannoyingfeatureofthistechnique:‘keystoning’,inwhicharectangularpictureisdistortedsothatitlookslikethekeystoneatthetopofanarch,wideratthetopthanatthebottom.Videoprojectorsusuallyhavecontrolsettingstoeliminatethiseffect.Insteadoftryingtogetridofdistortionsintroducedbyprojection,projectivegeometryrevelsinthem.Itscoreobjectiveistofindgeometricpropertiesthatarenotaffectedbysuchdistortions,justasmuchofEuclideangeometryisreallyaboutfeaturesofshapes,suchaslengthsandangles,thatremainunchangedafterrigidmotions.

Beyondthebluehorizon

OneofthesimplestproofsthattheEarthisn’tflatistheexistenceofahorizon.Asashiptravelsfurtherawayfromharbour,itbeginstodisappearfromviewwheretheoceanmeetsthesky.First,thelowestportionsareobscured,thenhigherones,untileventuallyonlythetipofthemastcanbeseen.Thenthat,too,dipsbelowthehorizon,andtheshiphasvanishedentirely.

Realityismorecomplex:atmosphericeffectscandistortthepathstakenbylightraysbetweenshipandeye,variationsinheatcancreatemirages,hazecancausetheshiptofadeslowlyfromview,andthesurfaceoftheoceangoesupanddownaswavespassacrossthelineofsight.Ignoringthesesubtleties,Figure16showsacircularEarthintheplane,across-sectionofasphericalEarthinspace.Thevanishingactisgovernedbyastraightlinefromtheobserver’seye,tangenttothesurfaceoftheocean.AsthecurvatureoftheEarthtakestheshipbelowthisline,thelowerpartsoftheshipceasetobevisible,obscuredbytheocean.ThepointofcontactofthetangentlinewiththesurfaceoftheEarthisthehorizon.Inthreedimensions,onaperfectlysphericalEarth,andobservinginalldirectionsfromthesamepoint,thehorizonformsacirclecentredontheobserver’seye.

16.Ashipdisappearingoverthehorizon.

TheEarthisfinite,butsomethingsimilaroccursifoursphericalplanetisreplacedbyaEuclideanplaneofinfiniteextent.Theshipnolongerdisappears,butthere’sstillawell-definedhorizon.Incross-section,thedirectiontothehorizonisshowninFigure17asasolidarrowparalleltotheoceansurface.

17.Ashiponaplanarocean,relativetothedirectionofthehorizon.

Thedottedlinesarelinesofsighttothetopofthemainmast.Thefurtherawaytheshipis,thecloserthislinegetstothesolidone.Sothetopoftheshipappearstorisetowardsthedirectionofthehorizon.Thebottomoftheship(lineofsightnotshown)alsorises,andtheanglebetweenthesetwolinesshrinks,sotheshipbecomessmallerandsmallertotheeyeasitsangularheightdecreases(Figure18).

18.Astheshipmovesfurtheraway,itrisestowardsthehorizon(dottedline)andappearstobecomesmaller.

Iftheobserver’slineofsightisabovethesolidline,itfailstomeetthesurfaceoftheocean(althoughitsbackwardextensionwouldmeetit,behindtheobserver,iftherewereoceaninthatdirectionandnothingobscuredit).Ifit’sbelowthisline,itmeetstheoceanatsomespecificpoint.Thelineitself,beingparallel

totheoceansurface,doesn’tmeetit,butdemarcatestheboundarybetweendirectionsthatmeettheoceanandthosethatdon’t.

ThisboundaryisshowndottedinFigure18.Everydirectionbelowtheboundaryterminatesatsomepointoftheplane.However,theboundaryitselfdoesnotcorrespondtopointsontheEuclideanplanedefiningtheocean.Thelineinthedirectionofthehorizonisparalleltotheplane,socan’tmeetit.

Thisexampleisalsoaboutprojection,thistimefromtherealworldtotheretinaoftheobserver(moreaccurately,aplaneheldinfrontoftheretina).Wehavethuscometotheparadoxicalconclusionthatinageometricallyaccurateprojectedimageofaplane,thereexistsalinethatisnottheimageofanylineontheplane.Thiscuriousfactremainstrueevenifweallowlinestobeextendedbackwards.Nowpointsabovethehorizoncorrespondtopointsoftheplanebehindtheobserver,pointsbelowthehorizoncorrespondtopointsoftheplaneinfrontoftheobserver,butpointsonthehorizondon’tcorrespondtoanypointoftheplane.

Nevertheless,thehorizonisauniquelydefinedstraightlineintheimage,withaspecificgeometricmeaning.

Asimilareffectarisesforastraightrailwayline.Totheobserver,thelinesseemtomeetonthehorizon.Thisisn’tliterallytrueonourroundplanet—parallellinesdon’texistonasphere—butitisexactlytrueofaplaneinEuclid’sgeometry.Thereseemstobeasenseinwhichparallellinesdomeet,butnotatanypointintheplane.Projectivegeometryhandlesthiseffectbyaddinganextra‘lineatinfinity’totheEuclideanplane.Butittookawhileforthispointofviewtoemerge,andevenlongertoreplaceitbysomethingbetter.

Thelineatinfinity

Mathematiciansdislikeexceptionstootherwisegeneralrules.Parallellinesareanexample:anytwodistinctlinesmeetatexactlyonepoint…oops,sorry:exceptifthey’reparallel.That’sespeciallyuglygiventhatanytwodistinctpointscanbejoinedbyexactlyoneline.‘Parallelpoints’thatcan’tbejoineddon’texist.Whyarepointswellbehaved,whereaslinesbehavebadly?

Ittookmanycenturiesbeforethemathematicalworldrealizedthatexceptionscanoftenberemovedbyartificiallyextendingtheunderlyingstructure.Minusonehasnosquareroot:noworries,justdefineanewkindofnumbertosupplyone.Primefactorizationisnotuniqueforalgebraicnumbers:justdefineanewkindofnumbersothatitis.Historyisfullofwordslike‘imaginary’and‘ideal’,associatedwiththisprocedure,butnowadaysthrowinginnewingredientstofixupanexceptionisconsideredfairgame,justassensibleasanythingelseinmathematics.Soit’snaturaltoupgradetheEuclideanplanebyplugginginalineatinfinity,andtodeemparallellinestomeetthere.

Ittooktimeformathematicianstofeelcomfortablewithsuchprocesses,whichinitiallywererathermysterious.Eventually,asuitablelogicalframeworkemerged,withinwhichtheconstructionsmadesenseanditwaspossibletoprovethattheyworked.

Toillustratehowdifficulttheseideasseemedbeforetheirlogicalstatuswasclarified,I’mgoingtointroducethelineatinfinityinanintuitivemannerfirst,andclarifythemathematics—andthemeaningofinfinityinthiscontext—afterwards.Ifthismakessomeoftheargumentsconfusingandmysterious,don’tworry:I’mjustputtingyouintheshoesofthemathematicianswhohadtograpplewithsimilarconfusions.

Thehorizonisagoodexampleofanannoyingexception.Therealhorizonexists,becauseshipscansailoverit.Butthat’sbecausetheEarthiscurved.IftheEarth,anditsocean,wereaninfiniteflatplane,theartistwouldstillseeawell-definedhorizon,eventhoughnoshipcouldeverreachit.It’sasifEuclid’splanehasalinemissing,whereastheimagedoesnot.Somathematiciansdecidedtogivetheplaneanextralinetoplugthegap,andtheycalleditthelineatinfinity.Intuitively,it’swhattheboundaryoftheplanewouldbeifithadaboundary.Headoffinanygivendirection,walkforever,andwhenyou’vearrived,you’vereachedthelineatinfinity.

Figure19isaschematicofthisidea.TheentireEuclideanplaneissquashedinsideacirculardisc.Theplaneistheinteriorofthedisc,andthelineatinfinityisitsboundary;thatis,thecircleitself,asintheright-handfigure.AlineL1intheEuclideanplanefailstomeetthecircle,becausetheboundaryisnotpartoftheEuclideanplane.(Theendpointsofthelinesegmentsintheleft-handfigurearenotpartofL1andL2.)However,ifL1isextendedinthenaturalway,byaddingpointsA1andB1,itmeetsthelineatinfinityatthosepoints.Startattheorigin,headoffalongL1,andeventually,afterwalkinganinfinitedistance,yougettoA1,onthelineatinfinity.WalktheotherwayandyougettoB1.Weobtainanextensionofthelinelyinginanextensionoftheplane—bothbeyondthescopeofEuclid.

19.Left:representationoftheEuclideanplaneasadisc(shaded)withoutboundary,andtwostraightlinesthroughtheorigin.Right:addinganextralineatinfinity(darkcircle).Theextrapoints(whitedots)attheendofeachlineareidentifiedindiametricallyoppositepairs,sothatlinesmeetatonlyonepoint.

IntheEuclideanplane,thelinesL1andL2meetatonlyonepoint:theorigin.Noextrapointatinfinityliesonbothofthem,sotheydon’tmeeteachotheratinfinityeither.That’sgood,becausethey’renotparallel.(I’llcometoparallellines,andwheretheymeet,shortly.)However,takingthefigureliterally,L1meetsthelineatinfinityintwopointsA1andB1,oneateachend.ThesamegoesforL1.Aswellasbeinginelegant,thispropertymakesthelineatinfinityexceptional,whichisexactlywhatweweretryingtoavoidwhenweintroducedit.

Togetroundthat,we’reforcedtodosomethingthatatfirstsightseemsverystrange.Wemustidentify

thetwopointsA1andB1.ThesamegoesforanyotherlineintheEuclideanplane.Thatis,wemustredefinetheconceptofpoint—fortheseextrapointsatinfinityonly—as‘pairofdiametricallyoppositepoints’.Thismeansthateverypointonthelineatinfinityseemstoappeartwiceinthepicture.Asaconsequence,sodoesthelineatinfinityitself.StartatA1andwalkclockwisealongthelineatinfinityuntilyougettoB1.Nowyou’rebackatthestartingpoint,becauseB1isthesameasA1afteridentification.Ifyoucontinuewalkingclockwise,youcoverthesamegroundasecondtime,untilyougetbacktoA1.Sothelineatinfinityisn’treallyacircle;it’sasemicircle.Exceptthatthetwoendsofthesemicircle,A1andB1,arethesamepoint,sothesemicircle‘wrapsround’anditsendsjoinup.Soactuallyitisacircle,topologicallyspeaking.Justnottheonethatismostevidenttotheeye.

Thismayseemtomakethelineatinfinityspecial,somethingI’vebeentryingdesperatelytoavoid.Butnowthatwehavethoseextrapointsatinfinity,Euclideanlinesalsowrapround.Startattheorigin,headoffalongL1untilyougettoinfinity;thatis,A1.Youneednotstopthere,becauseA1isthesameasB1,andfromB1youcancontinuealongL1togetbacktotheorigin.SoL1,plusitsnewpointatinfinity,alsoclosesuponitselftoformatopologicalcircle.Democracyprevailsamongthelines.Infact,anyline,Euclideanoratinfinity,canbeprojectedtoanyotherline,whilepreservingallgeometricfeatures.

Thelineatinfinityisveryweird:it’scircularandstraightatthesametime.It’sstraightinthesensethatinwhicheverdirectiontheartistlooks,thehorizonintheimageisstraight.Butit’sacircleinthesensethat,iftheartistslowlyspinsroundthrough360degrees,thatperfectlystraighthorizonkeepsextendingandextending…untileventuallyitjoinsupwithitself.OrdinarylinesinEuclid’sgeometrydon’tdothat.Theyheadoffforeveralongonedirection,andtheyheadoffforeveralongtheexactlyoppositedirection.Thelongeryoumaketheline,thefurtherapartitsendpointsbecome.Theycertainlydon’tstarttocloseup.

However,whenweaddthelineatinfinitytotheplane,wecreateaknock-oneffectonordinarylines.Anyordinarylineintheplaneistransformedintoalineintheimage,whichtypicallymeetsthehorizon.Sincethis‘intersectionpoint’liesonthehorizon,itdoesn’tcorrespondtoanythingontheplane.Althoughtheimageofanordinarylineseemstocrossthehorizon,there’snopointonthatlinewhoseimageisthecrossing-point.Nevertheless,thecrossing-pointdoescorrespondtoapointonthelineatinfinity.Itariseswhenweequipanordinarylineontheplanewithanextrapointatinfinity.

Thegeometryofperspectivedrawingmakessenseofmyapparentlyarbitrarydecisiontoidentifythetwo‘ends’ofaEuclideanline.Asafinitelineismadeeverlonger,theimagesofitsendpointsbothapproachthehorizonfromdifferentsides.Figure20showsanartist,drawnasaneyeonastick,standingontheplane(shaded).Thedashedlinewitharrowsisparalleltotheplane,andalsotothelineintheplanethatrunsthroughpointsP1andP2,oneinfrontoftheartist,theotherbehind.Theartist’sviewofthehorizonisthedottedlineonthecanvas(whiteparallelogram).TheimagesofP1andP2onthecanvasareQ1,belowthehorizon,andQ2,aboveit.Thedifferenceisthattheartist’seyeisbetweenP2andQ2,butnotbetweenP1andQ1.

20.Howtheartist’seyeprojectsalineintheplane.

Whatwe’redoinghereismakingmathematicalsenseoftheintuitionthatacircleofinfiniteradiusisastraightline.However,we’vemodifiedintuitionslightlytogetsomethingsensible.Ourcircleofinfiniteradiusisactuallyastraightlineplusanextrapoint.Thatclosesituptoformacircle.

Theweirdnessdoesn’tstopthere.Iftwocirclesmeet,theygenerallydosoattwopoints(unlessthey’remutuallytangent).Soshouldn’tanordinaryline,suitablyaugmentedbyapointatinfinity,meetthelineatinfinityintwopoints?Theansweris‘no’,andthereasonisthatalthoughanever-expandinglinehastwoends,theirimagesbothapproachexactlythesamepointonthehorizon.

Parallels

Toseewhathappenstoparallellines,Ineedtospecifyhowtorepresenttheentireplaneastheinteriorofacircle,tomakesenseofFigure19.Oneway—thereareothers—isshowninFigure21.HereahemispherewithcentreCsitssothatittouchestheplaneatonepoint,saytheorigin.Itsverticalprojectionisadisc,showninpaleshading.TakeanypointPintheplaneandprojectittoQonthehemisphere.ThenprojectQverticallytoR.NoweverypointPintheplaneismappedtoapointRinteriortothecircle,becausetheequatorofthesphereliesinaparallelplane.Theboundaryofthehemisphere,projecteddowntotheplane,becomestheboundaryofthecircle.

21.Howtofitaninfiniteplaneinsideacircle.ProjectpointPtoQonahemisphere,centreC.ThenprojectQverticallytoR.

Wenowhavearecipethatturnsgeometryontheplaneintogeometryontheinteriorofthecircle.Justtransformeachpointoftheplaneaccordingtotheprocedurejustoutlined.Alineintheplane,throughtheorigin,transformsintoasemicircleonthehemisphere,whichishalfofagreatcircle.Thisprojectsbackdowntoformaradiallinesegment,adiameterofthedisc.Astheoriginallinetendstoinfinity,itsimageapproachestheboundarycircle.Itapproachestheboundarycircleattwosuchpoints,oneforeachdirectionalongtheline.Intheprojectiveplane,thesetwopointsareidentified.Sonowwehaveageometricmodelfortheprojectiveplane,obtainedusingEuclideanthree-dimensionalgeometryasanintermediary.

Alinenotpassingthroughtheoriginalsotransformsintoagreatsemicircleonthehemisphere,becauseitliesontheintersectionofthehemispherewiththeplanethroughthelineandC.Itsverticalprojectionishalfanellipse.

Wecannowfindouthowparallellinesbehave.Figure22(left)showsfiveparallellinesintheEuclideanplane,transformedasjustdescribed.Althoughtheyappear,totheeye,tomeetontheboundaryofthedisc,theboundarycorrespondstothelineatinfinity,nottoanythingintheEuclideanplane.Whenthelineatinfinityisadjoined,asinFigure22(right),allfivelinesmeeteachother,andthelineatinfinity,inasinglepoint—thesameoneateitherend,despiteappearances,becauseoftheidentificationrule.

22.ParallellinesintheEuclideanplane(left)meetatinfinity(right).

IsaidthatinEuclideangeometryanytwolinesmeetatexactlyonepoint;adding‘oops,sorry:exceptifthey’reparallel’.Butinourspankingnewgeometrywithanextralineatinfinity,there’sno‘oops’.Anytwolinesmeetinasinglepoint.Yes,itmaybeatinfinity,butlet’snotbepicky.That’sfarmoreelegantthanmakinganexception.Parallellinesfitintothispicturebeautifully.Theirimagesbothpassthroughthesamepointonthehorizon,sotheirextendedversionsbothpassthroughthesamepointatinfinity.Inshort:parallellinesmeetatinfinity.Theschoolboywasright.

Directions

Todayweapproachthewholetopicinadifferentway.Thekeyconceptisnotnewfictitiouspointsonaplane,butexistingdirectionsinspace.Ifyoulookbackatthediscussion,theideaoftheartistlookinginagivendirection,orparallellinesallpointinginthesamedirection,keepscomingup.Thisisaclue.Butwhatisadirection?Theeasywaytoderiveagoodmodelfromordinarygeometryistostartinthree-dimensionalspace,withadistinguishedpoint,theorigin.That’swheretheartistconceptuallyplacesherall-seeingeye.Adirectionisthendeterminedbyalinethatpassesthroughtheorigin.Anylineparalleltothatone‘pointsinthesamedirection’.

Theendresultofabout600yearsofeffort(plussomeevenearlierworkthatwenowseefitsintothesameframework)dispelsthemysterysurroundingpointsatinfinity.Theentireset-upcanbeinterpretedintermsofstandardEuclideangeometryinthreedimensions,buttaking‘points’tobelinesthroughtheorigin,notEuclideanpoints.Reducingthedimensionbyintersectingwithaspherecentredattheorigin,projectivepointsbecomediametricallyoppositepairsofEuclideanpoints.ThehemisphereinFigure22isanoptionalextra;makingitclearhowtheEuclideanplaneembedsintheprojectiveplane.Andthelineatinfinityisnolongerafictitiousobjectintroducedtorepresentwhatanartistsees:it’stheequatorofafiniteobject,thesphere.

Perspectivedrawing

I’mnotgoingtotrytoteachyouperspective,butyououghttobeshownsomepayoffinthevisualarts.Thelineatinfinityyieldsasimplesolutiontowhatotherwisewouldbeaverycomplicatedproblemingeometry.Renaissancepainters,andmanylaterones,madeextensiveuseoffloorsdecoratedwitharegulargridofsquaretiles.JohannesVermeerevenusedthemwhendepictingbuildingsthatdidn’tactuallyhavethem,andhewasn’talone.Onereasonmaybethattiledpatternsarevisuallyimpressive,andveryhardtogetrightwithoutprojectivegeometry.Anotheristhattheycreateagrid,givingvisualcluesaboutdepth.

Themainideascanbeseeninasimpleinstance:drawinga4×4patternofsquaretilesinperspective.Figure23showsthepattern,togetherwithsomearrowsthatpoint‘toinfinity’.Eventhoughthelineatinfinitydoesn’texistinEuclid’sgeometry,we’llusethecorrespondingpointsonittodrawaperspectiveversionofthegrid.

23.Left:agridofsquareswithtwodirections‘pointingtoinfinity’correspondingtosetsofparallellines.Right:auxiliarydiagonallinesusedtoconstructaperspectiveversion.

Thestagesofconstruction(Figure24)golikethis:

24.Successivestagesinconstructingaperspectivedrawingofthegrid.

1.Drawahorizon(dottedline)correspondingtothelineatinfinity.Drawthreecorners(blackdots)andthetwolowersidesofthegrid(thicklines).Thedotscanbeatanylocation,otherthanhavingallthreeinastraightline,becauseanysetofthreepointscanbeprojectedtoanyothersetofthreepoints.Thischoicecorrespondstolookingdownonthegridfromaslightangle.

2.Continuethesolidlinestomeetthelineatinfinity(shadeddots).3.Todrawtheothertwoedgesofthegrid,observethatthey’reparalleltotheiroppositeedgesandsomeetthoseedges

onthelineatinfinity,atthepointsalreadyconstructed—oneforeachsetofparallels.Theintersectionoftheselinesgivesthefourthcornerofthegrid(blackdotattop),andtheothertwosidescanbefilledin(solidlines).

4.Drawthediagonalsofthesquareandfindthepointwheretheycross(whitedot).5.Intheoriginalgrid,twogridlinespassthroughthecentreofthesquare(whereitsdiagonalsmeet).Intheimage,the

correspondinglinesdothesame.Sincethey’reparalleltoappropriateedgesofthesquare,theyalsopassthroughthesamepointatinfinity,whichdetermineswheretodrawthem.

6.Subdividethefoursquaresofthegridnowconstructedbydrawingtheirdiagonalsandfindingtheirintersections:compareFigure24(bottom).Theremaininggridlinespassthroughtheseintersectionsandtheappropriatepointatinfinity(greydot)forthesamereasonasinstep5.

Bydrawingmorediagonals,it’spossibletosubdividethegridinto16squares,then64squares,andsoon.It’salsopossibletoextendsuchatilingbyaddingmorerowsandcolumnsalongitsedge.Todoso,weusethefactthatallmembersofasetofparalleldiagonallinesmeetatthesamepointonthelineatinfinity.Thesehelptolocatethecornersofthenewtiles.Figure25showsVermeer’spaintingTheConcert,withaddedgridlinestobringouttheunderlyinggeometry.

25.Vermeer’sTheConcert,withlines(grey)showinguseofperspectiveand(dotted)thehorizon.Whichishorizontal.

Somethingthatdoesn’tactuallyexistcanstillbeuseful.

Chapter6Physicalinfinity

Infinityisvirtuallyindispensableintoday’smathematics,butthereit’saconceptualentity,notarealone.Philosophersthinkabouttheinfinite,andargueaboutwhetheritexists,andifso,inwhatsense.Religionsoftenclaiminfinityasanattributeoftheirgodorgods,andpeoplehavebeenexecutedfordenyingit,buttodayit’sgenerallyagreedthattheexistenceofadeityisamatteroffaith,notobjectiveevidence.

Doesinfinityexistoutsidethehumanmind?Canitbereal,notjustinthesenseofreligiousfaith,butinthesensethatrivers,trees,cats,androcksexist?Nophilosophicalhair-splittingaboutwhat‘real’and‘exist’mean:cansomeoneshowusinfinity?

Invirtuallyeveryareaofthephysicalsciences,infinityisanembarrassment.Atheorythatpredictsinfinitiesiswrong.Thatdoesn’tmeanit’suseless,butitneedstweakingtogetridofthosepeskyinfinities.However,there’soneareaofphysicsinwhichanactualinfinity—physical,notconceptual—isnotjusttolerated,butpresentedasapossibletruth:cosmology.

I’llstartwithmoremundaneoccurrencesoftheinfiniteintheoreticalphysics.Here,infinitequantitiesareusuallyreferredtoassingularities,andtheirpresenceisevidenceofdefectsinthemodel.However,itmaystillbeveryaccurateawayfromsingularities.I’lldiscusssingularitiesinthreephysicalcontexts:optics,Newtoniangravity,andAlbertEinstein’srelativity.Thenwe’lltakeaquicklookatwhethertheuniverseisinfinite.

Infinityinoptics

Manyofnature’sgloriousspectaclesarecreatedbyunusualeffectsinvolvinglight.Therainbowisthemostfamiliar,anarrowmulticolouredarcacrossthesky.Anotheristheglory,inwhichapersongazingintomistwiththeSunbehindthemseesarainbow-likehaloroundtheshadowoftheirownhead.Someonestandingnexttothemseesmuchthesamething,butnowthehaloisroundtheirhead.Theglorymayhaveinspiredthetraditionofdepictingholyfigureswithhalos,whichgoesbackatleastto1st-centuryBuddhism.CompletecircularhalossometimessurroundtheSunortheMoon,withrarereffectssuchaslightpillars(averticalcolumnoflightabovetherisingorsettingSun)andsundogs(brightspotseithersideoftheSun).

TheseeffectshappenwhenlightfromtheSunisreflectedand/orrefracted—bent—bywaterdropletsoricecrystals.ArainbowappearswhentheSunisbehindtheobserverandrainfallingfromcloudsisinfrontofthem.Sunlighthitseachdropofrainandisrefracted;thenitreflectsoffthebackofthedroplet;finallyitisrefractedonthewayout(Figure26).Theangularradiusoftherainbowisabout42·5°,anditscoloursareorderedbywavelength.Adimmerrainbow,outsidethemainone,isoftenvisible,causedbyafurtherreflectionandrefraction;itsangularradiusisabout52°.Thegloryissimilarlycreatedbysunlightpassingthroughdropsofwaterinmist,butforcomplicatedphysicalreasonsitalmostreversesitspath,whichiswhyitappearstoemanatefromtheshadowoftheobserver’shead.Itconsistsofacomplexseriesofcolouredringsofdifferingbrightness.Halos,bothsolarandlunar,arecausedbyicecrystalsintheupperatmosphere.Thesimplestformisacircularringwithangularradiusabout22°,whichisrelatedtothegeometryoficecrystals.

26.DeviationD(α)intheangleoflighthittingasphericalwaterdroplet.

Allthesephenomenaaremulticolouredbecauselightofdifferentwavelengthsproducesdifferentcolours,andtheSun’swhitelightisamixtureofcolours;indeed,‘allthecoloursoftherainbow’,alongwithotherwavelengthsthehumaneyecan’tdetect.Therainbowforlightofasinglewavelengthisanarcofacircle,centredatapointdiametricallyoppositetheSun,anditsangularradiusistherainbowangleforthatwavelength.Differentwavelengthshavedifferentrainbowanglesinacontinuousrange,whencethebandofconcentricarcsofdifferentcoloursthatweseeinthesky.

‘Wavelength’isaconceptinwaveoptics,wherelightisconsideredtobeawave.Butthefirstseriousmathematicaldescriptionoflightwasrayoptics,inwhichlighttravelsalongstraightlinesinanymediumwithconstantrefractiveindex.Whenweuserayopticstocalculatetheintensityoflightattherainbowangleforlightofasinglewavelength,theansweris‘infinity’.Let’sseewhy.

Figure27(top)showshowthedeviationD(α)varieswithα.Foraspecificshadeofredlight,thiscurvehasaminimumat ,andthecorrespondingdeviationis137·5°.Callthistherainbowangle.Itexceeds90°becausetherayreversesdirection,sotheangularradiusofthearcisthedifference180°−137·5°,whichis42·5°.Figure27(bottom)showswhathappenstoanincomingbandoflightrays.Theyemergeatarangeofangles,shownbytheshadedregions.Ifthebandentersneartheminimumofthegraph,it’scompressedintoaverysmallrangeofoutgoingangles.Sotheoutgoinglightisconcentratedinalmostthesamedirection,makingitbrighter.Iftheincomingraysareveryclosetotheminimum,theintensityoftheemerginglightbecomesarbitrarilylarge,proportionaltothereciprocaloftheslopeofthecurve.Attherainbowangle,theslopeiszeroandthetheoreticalintensityis1/0:infinite.

27.Top:graphofD(α)againstα.Bottom:incominglightiscompressedneartherainbowangle.

Thiscalculationisfarfromuseless.Itgivestherainbowangleveryaccurately,anditgoessomewaytowardsexplainingwhythelightattherainbowangleismuchbrighterthanatanyotherangle.That’swhyweseeasharplydefinedarc.Ontheotherhand,thepredictionofinfiniteintensitycan’tbetheliteraltruth.Sorayopticspredictsasingularityinintensity,whichphysicistshavelearned,withgoodreason,toreject.

Rayopticalcalculationsbreakdownneartherainbowangle,ageometricsingularity.Theproblemwasresolvedbythediscoverythatlighttravelsnotasaray,butasawave.Theintensityaroundtherainbowanglecouldthenberecalculated.ItturnedouttobesomethingcalledanAiryfunction,whichoscillates,andislargebutfiniteattherainbowangle(Figure28).Theoscillationsrepresentdiffractionfringes,awave-opticaleffectthatgivesparallellightanddarkstripes.

28.Airyfunctionofwaveoptics(solidline)andray-opticalintensity(dashed)neartherainbowangle.Theintensitycalculatedusingwaveopticsisfinite.Itspeakisnear,butnotequalto,theray-opticalrainbowangle.

InfinityinNewtoniangravity

AmoredramaticexampleofinfinityoccursinNewtoniangravitation.RecallNewton’sLawofGravity:anytwobodiesintheuniverseattracteachotherwithaforceproportionaltotheirmassesandinverselyproportionaltothesquareofthedistancebetweenthem.Appliedtotwobodies,thelawpredictsthattheyorbitinellipsesabouttheircommoncentreofmass.Forthreeormorebodies,nosimplegeometriccurveorformulaexists;indeed,theorbitsareoftenchaotic.

Around1900,HenriPoincaréandPaulPainlevéasked:whichsingularitiescanariseinasystemofparticlesobeyingNewtoniangravity?Thatis,whendosolutionsceasetoexistaftersomefinitetime?Anobvioussingularityoccurswhenparticlescollide,sothemainissuewaswhetheranythingelsecanhappen.In1895Painlevéprovedthatforthreebodies,allsingularitiesarecollisions.Hethenaskedwhathappenswithfourormorebodies.In1908EdvardvonZeipeluppedtheantebyprovingthatifnon-collisionsingularitiesoccur,particlesbecomeinfinitelydistantinfinitetime.Thisseemedsoweirdthatmathematiciansstoppedthinkingabouttheproblemforthenextfiftyyears.

DonaldSaaritookitupagainin1967,andby1973hehadprovedthatnon-collisionsingularitiesmustalsoinvolveparticlesoscillatingarbitrarilyfast.Particlesmustapproachotherdistantparticlesarbitrarilyoften,andgetarbitrarilyclosetothem.Thatmayseemevenweirder,butin1988ZhihongXiaproveditcanhappenforfiveormorebodies.Inconsequence,thereexisttrajectoriessuchthatallofthebodiesdisappeartoinfinityinafiniteperiodoftime.ThisisasingularityintheNewtonianmodel:mathematically,thesolutiontotheequationdescribingthemotionofthebodiescan’tbecontinuedpastsomeparticularfinitetime.

InXia’sset-up(Figure29),twopairsofbodiesorbittightlyroundeachotherinhighlyellipticalorbits,inplanesatrightanglestoalinealongwhichthefifthparticlemoves.Thepairsplaycelestialtenniswiththefifthparticle,whichshuttlesbackandforthbetweenthematanever-fasterrate.Thepairsmoveawayfromeachotheralongthestraightline,andsinceenergy,momentum,andangularmomentummustbeconserved,thebodiesineachpairgetcloserandclosertogether.Everythingspeedsupsorapidlythatafterafiniteperiodoftime,allfiveparticlesdisappeartoinfinity.

29.Xia’sscenario.

Later,similarbehaviourwasfoundintwo-dimensionalspaceaswell.

Clearlythisbehaviourisn’tphysical.Inarelativisticmodel,itcan’thappenbecausenothingcanexceedthespeedoflight,butthat’snottheonlyissue.InaNewtonianmodel,there’snocosmicspeedlimit.Sowheredoesthemodelfailtomatchreality?Theansweris:theuseofpointparticles.Norealbodyisamathematicalpoint.Usually,though,thissimplificationdoeslittleharm.Newtonprovedthatthegravitationalfieldoutsideasphericallysymmetricbodyisthesameasitwouldbeifallofthemasswereconcentratedatthecentralpoint,soformanypracticalpurposesaplanetcanbemodelledasapointmass.

Sometimes,though,thisassumptionmakesnosense.Withbodiesofnon-zerosize,Xia’sscenariocan’thappen.Whentheygetcloseenough,theytouch.Sothefive-bodydisappearingactisanartefactoftheassumptionofnon-physicalpointparticles.Indeed,thegravitationalpotentialofapointmassisinverselyproportionaltodistance,sothepotentialtendstoinfinityasthedistancetendstozero.Thesingularityinthefive-bodydynamicsisaconsequenceofthissingularityintheNewtoniangravitationalpotential.It’sanartefactofthemathematicalmodel.

Infinityinrelativity

Inspecialrelativity,singularitiesoccurwhenmattertravelsatthespeedoflight.Timegrindstoahalt,lengthsshrinktozero,andmassbecomesinfinite.However,theseeffectsdon’tcorrespondtoaphysicalsingularitybecausenorealbodycanattainthespeedoflight.Theenergyneededtoaccelerateittothatspeedwoulditselfbeinfinite.Ofcourse,lighttravelsatthespeedoflight,butapropertreatmentoflightrequiresquantummechanics.

Generalrelativityinvolvesamoreintriguingsingularity.Einsteinintroducedgeneralrelativitytoincludegravityinhistheoriesofspace,time,andmatter.InNewtonianphysics,gravityisaforce,actingbetweenanytwobodies.Newtondidn’tspecifyhowaforcecanactacrossemptyspace.Hewasawareofthephilosophicalproblemofactionatadistance,buthetookapragmaticviewandignoredit.Einsteinreplacedthisforcebythecurvatureofspace-time.Aplanetorbitingastarfollowsacurvedorbitnotbecauseofanattractingforce,butbecausethestarwarpsspaceandtheplanetisaffectedbythewarp.

GeneralrelativityexplainsmanygravitationaleffectsthatareinconsistentwithNewtonianphysics,suchasaslowchangeinthepositionofMercuryinitsorbitwhenit’snearesttheSun.GPSsatnavsystemsprocesstheirtimingsignalsusinggeneralrelativity,becausethey’dgivethewrongpositioniftheydidn’tcompensatefortime-warpingbytheEarth’sgravitationalfield.

Oneofthesurprisingconsequencesofgeneralrelativityistheexistenceofblackholes.Whenaverymassivestarcontractsunderitsowngravity,itcanbecomesodensethatitstheoreticalescapevelocity—theinitialvelocityneededtopropelabodyfastenoughtogetaway—exceedsthespeedoflight.Sincethat’simpossible,soisescape.Evenlightremainstrapped.Evidencefortheexistenceofblackholeshasbeenaccumulatingsteadily,andwhiletherearesomeseriousdifferencesofopinionaboutquantum-mechanicalfinepoints,astronomersgenerallyacceptthatsomethingverylikethepredictedstructureexists.

Thesimplestkindofblackhole,astaticone,issurroundedbyasphericalshell,itseventhorizon.Bodiescanescapeifthey’reoutsidetheeventhorizon,butnotifthey’reinside.It’sthereforeimpossibleforanexternalobservertofindoutwhathappensinsideablackhole.Theoretically,acollapsingstarshouldcontinuecollapsinguntil,afterafinitetime,allofitsmassisconcentratedatasinglepointofinfinitedensity.This,ifitactuallyhappened,wouldbeagenuinephysicalsingularity.Mostphysiciststhinkthatsomethingelsehappens,avoidingatruesingularity,butthey’redividedonwhatthatmightbe.Iftheblackholerotates,thesingularitybecomesacircle,butthedensityofthematterfromthestarisstillinfinite.

Infiniteuniverse?

Virtuallytheonlycontextinwhichscientistsdon’tconsideraninfinitequantitytobeasignthattheirtheoriesarewrongiscosmology.Atvarioustimesithasbeenentirelyrespectabletoassertthattheageoftheuniverse,oritssize,isinfinite.

Thesimplestwaytoexplaintheoriginoftheuniverse—bothinspaceandtime—istomaintainthatitneverhadone.Iftheuniversehasalwaysexisted,wecanstopworryingabouthowitcameintobeing.It’saseductivelineofreasoning,althoughAquinaswouldn’thaveapproved,sinceherejectedtheabsenceofaFirstCause.ButitwasgoodenoughforFredHoyle,formingthebasisofhissteadystatetheory,inwhichaninfinitelylargeandinfinitelyancientuniversecontinuallyexpandsbythegradualcreationofnewparticle–antiparticlepairsinthedarkbetweenthestars.Hoyle’stheorywaswidelyacceptedinthe1950s,butbythe1970smostphysicistsandcosmologistshadabandoneditinfavouroftheBigBang:bothspaceandtimecameintoexistencefromapointsingularity13·8billionyearsago.

AsAquinasargued,thistypeofexplanationisn’tentirelysatisfactory.Itexplainstheuniverseaway,ratherthanexplainingit,bybeggingasimplesupplementaryquestion:whyhasitalwaysexisted?There’snomathematicaldifficultyincontemplatinganinfinitelyolduniverseandstillasking‘wheredidallofthatcomefrom?’Pushingaproblemawaytominusinfinitydoesn’tentirelygetridofit,andinanycasethecurrentconsensusamongastronomersandcosmologistsisthattheuniversehasn’talwaysexisted.However,ittookawhilefortheBigBangtogainacceptance.Hoylegaveitthatnametopokefunatit.Wehadtroublegettingitoutofourheadsthattheuniversehasalwaysexisted—thatitspastistemporallyinfinite.Thisviewkeepstryingtosneakback,withquestionslike‘whathappenedbeforetheBigBang?’

Wehaveevenmoretroublegettingitoutofourheadsthatthespatialuniversemustgoonforever.Indeed,somecosmologistsareconvincedthatitdoes.Itmightseemthatwhenwegazeheavenwardsonastarrynight,we’relookingintotheinfinite.There’snoobviousboundary;theuniverseextendsawayfromusforvastdistances.However,there’sadefinitelimittothesizeoftheobservableuniverse.Currently,thisisconsideredtobeasphere,centredbydefinitionontheEarth,ofradius46·6billionlightyears,or

metres.

Thisfigureappearsparadoxical,becausetheuniversehasexistedforamere13·8billionyears,andlighttravelsatonelightyearperyear.Since46·6/13·8=3·37,thelightnowreachingusfromtheedgeoftheobservableuniverseseemstohavetravelledatjustoverthreetimesthespeedoflight,sothespeedoflightisthreetimesitself.Theparadoxisresolvedwhenwerememberthattheuniverseisexpanding.Whenthelightnowreachingusfromtheedgeoftheobservableuniversefirststartedout,theregionthatisnowobservablewasmuchsmaller.Calculationsindicateitwasonly42millionlightyearsinradius.

Wecanalsotestthetheoreticalfigureexperimentally.In2003NeilCornishandco-workersusedobservationsofthecosmicmicrowavebackground(CMB)bytheWilkinsonMicrowaveAnisotropyProbetodeducethatiftheuniverseisfinite,itsdiameterisatleast78billionlightyears.Theyassumedthattheuniversehasaclosedtopology,withoutanyboundary,‘wrappinground’onitselflikeasphereoratorus.Wrappingroundcreatespairsofcirclesinthesky,diametricallyoppositeeachother,correspondingtolightfromthesameregionoftheuniversereachingusfromtwodifferentdirections(roundthefrontandroundtheback,sotospeak).ThesecirclescanbedetectedbecausethetemperaturepatternoftheCMBisthesameonboth.Theyfoundnocircleslargerthan25°inangularradiuswithstronglycorrelatedtemperaturepatterns.Thisleadstothe78billionlight-yearlowerbound.In2012Cornishandothersextendedtheestimateto92billionlightyears.

Curvature

Thecurvatureoftheuniverseisoftenproposedasawaytodistinguishfinitefrominfinite,onthereasonableassumptionthatthelarge-scalecurvatureisthesameeverywhere.Mathematiciansrecognizethreedifferenttypesofconstant-curvaturegeometry:Euclidean,elliptic,andhyperbolic.ZerocurvaturegivesEuclideangeometry.Positivecurvature,likethesurfaceofasphere,givesellipticgeometry,sotheellipticanalogueoftheplaneisfinite.Negativecurvature,likeasaddleoramountainpass,giveshyperbolicgeometry,andthehyperbolicanalogueoftheplaneisinfinite.Soitseemsasthoughwecouldusecurvaturetodistinguishfinitespacefrominfinite.Ifthecurvatureispositive,it’sfinite;ifzeroornegative,it’sinfinite.

Youcanstillfindthisargumentpresentedtoday,butunfortunatelyit’sfalse.Cosmologistsrepeatedthemistakefromthe1930stothe1990s.Thecorrectstatementisthattherearethreedistinctspace-timemetricsofconstantcurvature,correspondingtothethreegeometries.Butspaceswithdifferenttopologiescanhavethesamemetric,andsomeofthemcanbefinitewhenthecorrespondinggeometryisn’t.IfyoutakeEuclid’splaneandrollitupintoacylinder,themetricdoesn’tchange.Geometryonthecylinderisthesameasontheplane,unlessdiagramsgetsobigthattheywraproundandoverlapthemselves.

Thisisstillaninfinitespace,butasimilartrickleadstoafiniteonecalledaflattorus.Theeasiestwaytovisualizeitistostartwithasquare,andthenidentifyoppositeedges—addamathematicalrulethattheymustbeconsideredtobethesame.TheflattorushasthesamemetricasEuclideanspace,butdifferenttopology,andit’sfinite.Sozerocurvaturedoesnotimplyinfinitesize.Ironically,KarlSchwarzschild,whoseworkongeneralrelativitycontributedtothediscoveryofblackholes,pointedthisoutin1900.Mathematicianshavealsofoundsimilarconstructionsinhyperbolicgeometry;insteadofasquareyouusespecialpolygons,andtherangeofpossibilitiesisfarricher.AleksandrFriedmanntoldcosmologistsnottoassumenegativelycurvedspacesmustbeinfinitein1924,butfewlistened.Despitewhatitsaysonmanywebsites,thesemathematicalspacesshowthatyoucan’tdistinguishafiniteuniversefromaninfiniteonebymeasuringitscurvature.

Eveniftheuniversereallyisinfinite,it’sdifficulttoseehowthiscanbeverifiedscientifically,sinceanythingfurtherawaythan46·6billionlightyearsisunobservable.Ofcoursetheremightbesomewaytoinferthatit’sinfinitewithoutobservingit,justaswecaninferthetemperatureatthecentreoftheSunwithoutgoingthere.Buteventhen,it’shardtoseehowwecoulddistinguish‘genuinelyinfinite’from‘very,verybig’.

Chapter7Countinginfinity

Tomymindthemostprofounddiscoveryabouttheinfinite,onethatallthephilosophers(andallpreviousmathematicians,forthatmatter)missed,wasmadebyCantorin1874.Hedemonstrated,logicallyandrigorously,thatevenwithintherealmofnumbers,infinitycomesindifferentsizes.Specifically,theinfinitudeofallrealnumbersisgreaterthanthatofthenaturalnumbers1,2,3,….Bythishedidn’tmerelymeanthatsomerealnumbersarenotnaturalnumbers,whichistruebutobvious.Hisproofshowedthatit’simpossibletomatcheveryrealnumbertoacorrespondingnaturalnumber,insuchawaythatdistinctrealnumberscorrespondtodistinctnaturalnumbers.Thenaturalnumbersarecountablyinfinite,buttherealnumbersareuncountablyinfinite.

Cantorreturnedtothistheoremin1891,givingadifferentproof,hiscelebrated‘diagonalargument’describedlater.Butin1874hewasaftermoreclassicalgame.Mathematicianshaddistinguishedalgebraicnumbers,thosethatsatisfysomepolynomialequationwithintegercoefficients,fromtranscendentalnumbers,thosethatdon’t.Itwaswidelybelievedthatspecialnumberssuchaseandπaretranscendental,butproofsoftheseresultsweresomewayinthefuture.Infact,foratimeitwasn’tevenknownwhethertranscendentalnumbersexist.JosephLiouvilleprovedthattheydo,byanexplicitconstructionbasedonapproximationstoalgebraicnumbers.Cantorprovedtheyexistwithoutconstructingany.

ThiswayofthinkingledCantortoabroadtheoryofthenumberconcept,foundedinwhathecalledMengenlehre:thetheoryofcollections.Wenowcallitsettheory.Hisworkonthistheorywasmainlypublishedbetween1874and1884,withfurthercontributionsuntil1897.It’snowseenastheculminationofacenturyofefforttodefine‘number’logicallyandprecisely,andasalogicalfoundationforthewholeofmathematics.However,settheoryissobasicthatitsconceptshardlylooklikemathematicsatall,anditcertainlydidn’tinCantor’sday.It’sabstract,anditfeelsalientoanyonewiththetraditional19th-centurymathematicalupbringing.Cantorwaswellawareofthis,writing:‘IrealisethatinthisundertakingIplacemyselfinacertainoppositiontoviewswidelyheldconcerningthemathematicalinfiniteandtoopinionsfrequentlydefendedonthenatureofnumbers.’

Hisideasweretoorevolutionaryformany;severalleadingmathematiciansdenouncedthemasnonsense,ofteninrobustterms.TheinfluentialLeopoldKroneckerpubliclycalledCantorascientificcharlatan,arenegade,andacorrupterofyouth.ButKroneckerhadanaxetogrind;hewasanumbertheoristwitharatherextremistviewofwhatwaspermissibleinmathematics,famouslystatingthat‘Godmadetheintegers,allelseistheworkofMan.’TodaynoaspectofmathematicsisconsideredtobeGod-given,andthelogicaldifficultiesthatbesetthefoundationsofmathematicsalreadyoccurfortheintegers.Ifthemathematicsoftheintegersislogicallyconsistent,thensoisthatoftherealnumbers,complexnumbers,andindeedCantor’stheoryofsets.

EveninCantor’sday,severalleadingmathematicianshadenoughimaginationtograspthemagnitudeandimportanceofCantor’sinnovation.ThemostprominentwasDavidHilbert,whostatedthat‘NoonewillexpelusfromtheparadisethatCantorhascreated.’EventuallyCantorwontheargumenthandsdown,butbythenhewasdead.Hesufferedfromchronicdepressionfrom1904onwards,anddiedinasanatoriumin1918.

Countingandmatching

TheroadtoCantor’sparadisebeginswiththemethodweusetofindouthowmanythingswehave:wecountthem(seeFigure30).Ashepherdwithasmallflockofsheeppointsattheminsuccessionandchants‘one,two,three,four,five,six,seven’.Runningoutofsheep,andbeingcarefulnottocountanyofthemtwice,sheconcludesthatshehassevensheep.

30.Countingsheepinthreelanguagesandtheplanetsofthesolarsystem.

AMaori,facedwiththesametaskandthesameflock,wouldcountthemtoo,butthechantwouldbedifferent:‘ta’i,rua,toru,’ā,rima,ono,’itu’.Shehas’itusheep.

GeoffreyChaucerwouldhavecounted‘oon,two,thré,fowre,five,syxe,sevene’.Chaucerhadsevenesheep.

There’snodisagreementhere;justdifferentnamesforthesamethings.Anastronomercouldcountthesheepbychanting‘Mercury,Venus,Earth,Mars,Jupiter,Saturn,Uranus’.

Nowsupposeeverysheepiswearingexactlyonecollar.Howmanycollarsarethere?Noonebotherstocount.Instantlytheanswersringout:‘Seven!’‘’Itu!’‘Sevene!’‘Uranus!’Thenumbers—wherethechantstops—arethesame.Why?Becauseeverynumbermatchesasheep,andeverysheepmatchesacollar;soeverynumbermatchesacollar.Theoldmathematicalnameforthisprocedureisone-to-onecorrespondence.That’sabitofamouthful,soschoolsnowteachyoungchildrenaboutmatchingsetsinthehopethatthiswillhelpthemunderstandnumbers.Mathematicianscalltheprocedureabijectionorone-to-oneontomapping.

Theprincipleappliesmoregenerally.Iftheshepherdknewshehad150sheep,andeverysheepwaswearingexactlyonecollar,shecouldbeconfidentthattheyhad150collars.Evenifshedidn’thaveacluehowmanysheepshehad,butkneweverysheepwaswearingexactlyonecollar,shecouldbesurethatthenumberofcollarswasthesameasthenumberofsheep.Thismaysoundtrite,butit’sphilosophicallyandfoundationallydeep.Youcanbeconfidentthattwonumbersareequalwithoutknowingwhattheyare.There’sasenseinwhich‘havingthesamenumber’ismorebasicthan‘howmany?’

Around1880GottlobFregedevelopedadefinitionof‘number’basedonthesameprinciple.Hefeltthatusingsomestandardsequencesuchas‘one,two,three,four’wasarbitraryandinelegant.Anymatchingsequencewoulddothesamejob.Facedwiththedifficultyofselectingaspecificsettorepresent‘seven’,hehitonthecunningideaofusingallofthem.Anumber,hesaid,isthesetofallsetsthatmatchagivenset.Thentheyallmatcheachother,andnosetthatmatchesanyofthemgetsleftout.It’sdemocratic,inclusive,andunique.

Also,logicallyflawed.In1903,justafterFregepublishedthesecondvolumeofhismasterworkGrundgesetzederArithmetik(basiclawsofarithmetic),BertrandRussellfamouslydemolishedakeyassumptionbehindthisapproach:thatphrasesoftheform‘thesetofallsetssuchthat…’aremeaningful.Hisexamplewas‘thesetofallsetsthatdonotcontainthemselves’.Ifitdoes,itdoesn’t;ifitdoesn’t,itdoes.Containitself,thatis.ThisistheRussellparadox,amathematicalversionofthebarberwhoshaveseveryonewhodoesn’tshavethemselves,butmorepreciselystatedandimmunetocleverget-outclauseslikefemalebarbers.

Theseniceties,aside,Fregehadagreatidea.Noneedtodiscarditjustbecauseitdoesn’tprovideausefuldefinitionof‘number’.Itdoesprovideausefuldefinitionof‘samenumber’,withoutanyneedtosaywhatanumberis.EventuallyCantormadeitthefoundationofthistheoryoftransfinitecardinals.

Cantor’stranscendenceproof

Cantordidn’tdevelopsettheoryandhistheoriesoftransfinitecardinalsoffthetopofhishead.Theyemergedfromhisresearchintostandardmathematicalquestions,andittookhimseveralyearstosortouttheunderlyingideassystematically.

Itallstartedin1874,whenhefoundanewwaytosolveaquestionabouttranscendentalnumbers.Recallthatalgebraicnumberssatisfyapolynomialequationwithintegercoefficients;transcendentalnumbersdon’t.Forexample, satisfiestheequation withintegercoefficients1and−2,so isalgebraic.Noonehadfoundanalgebraicequationforπore,sotheywerethoughttobetranscendental.Wenowknowtheyare,butatthetimetheirstatuswasconjectural.Thefirstbreakthroughcamein1844,whenLiouvilleshowedthattheerrorinanyrationalapproximationtoanalgebraicnumbermustbelarge,inatechnicalsense.Thereforeanumberforwhichtheerrorsaresmallmustbetranscendental.Hisexampleswerenumberslike

where1appearsonlyinpositions1,2,6,24,…successivefactorials.Thisprovedthattranscendentalnumbersexist,buttheexamplesweresomewhatartificial.

Cantorsolvedtheprobleminadifferentmanner:heprovedthattherearemoretranscendentalnumbersthanalgebraicones,inastrongbutradicallynewsense.Asawarm-up,I’llprovethattherationalnumbersarecountable:theycanbematchedwiththenaturalnumbers.Theproof,whichisessentiallyCantor’s,issimilartothediscussionofHilbert’shotelinChapter1.Forsimplicity,I’llrestrictattentiontopositiverationals,butit’sstraightforwardtoextendittoallrationals.Theideaistoarrangethepositive(whichimpliesnon-zero)rationalsinorderofcomplexity,wherethecomplexityofp/qis .Onlyfinitelymanypositiverationalshaveagivencomplexity,anditdoesn’tmuchmatterinwhichorderwearrangethose,butfordefinitenesswecanstartwiththesmallestpandworkup.Finally,weuseonlyfractionsinlowestterms—pandqhavenocommonfactorsgreaterthan1—toavoidduplication.

Theliststartslikethis:

Thenwematchthemuptowholenumberslikethis:

andsoon.Everyrationalnumbercorrespondstoexactlyonenaturalnumber,andconversely.Withrespecttothisorder(notthenaturalorderbynumericalvalue)wecanmeaningfullyrefertothenth(positive)rationalnumber.

Toestablishtheexistenceoftranscendentalnumbers,Cantorfirstprovesthat,liketherationals,thealgebraicnumberscanbematched,one-to-one,withthenaturalnumbers.Hisproofisbasedonanotionofcomplexity,determinedbythelargestpowerofxintheequationdefiningthealgebraicnumber,andthesizesofthecoefficientsintheequation.Onlyfinitelymanyofalgebraicnumbersoccuratagivenlevelofcomplexity,sowecanarrangethealgebraicnumbersinalist,startingwiththosethatsolveanequationofcomplexity1,thenthosethatsolveoneofcomplexity2,thencomplexity3,andsoon.Thenwecanassignanaturalnumbertoanyalgebraicnumberbycountingalongthelist.

Finally,Cantorclinchesthedealbyprovingthattherealnumberscan’tbelistedinthatway.Assume,foracontradiction,thattheycan.It’snoweasytoconstructasequenceofintervals,eachinsidethepreviousone,sothatthenthintervaldoesnotcontainthenthalgebraicnumber.Theintersectionofthoseintervalscontainssomerealnumber.However,thiscan’tbeonthelist:ifitwere,itwouldbethenthalgebraicnumberforsomen,butthisisexcludedfromthenthinterval,soitcan’tlieintheirintersection.Butallegedlyeveryrealnumberisinthelist.Thereforenosuchlistexists,andtheremustbearealnumberthat’snotalgebraic.

This‘counting’proofwasajaw-droppingmomentforCantor’scontemporaries.Eitheryougetitoryou

don’t.Ifyoudon’t,youthinkit’snonsense.Ifyoudo,youthinkit’samazinglyoriginalandclever.Itprovestheremustbearabbitintheconjurer’shat—withoutshowingyoutherabbit.

Fourieranalysis

Cantor’searliestmathematicalpublicationswereonnumbertheory.ThenhetookaninterestinamajorunsolvedprobleminFourieranalysis.WhenJosephFourierwasdevelopingamathematicaltheoryofheatflow,hedevelopedatechniqueforstudyingperiodicfunctions—functionsthatrepeatthesamevaluesoverandoveragain(Figure31).Thisdistancebetweensuccessiverepetitionsistheperiod.

31.Graphofaperiodicfunctionwithperiodp.

Themostfamiliarperiodicfunctionsarethesineandcosinefunctions,withperiod2π.Fourier’sideawastowriteaperiodicfunctionwithperiod2πasasumofinfinitelymanysineandcosinefunctions:

Healsoderivedaformulaforthecoefficientsa0,a1,a2,b1,b2intermsofintegrals.Itwasn’tatotallyoriginalidea—EulerandDanielBernoullihadbeenarguingaboutitsfinerpointsforsometime—butFouriermadeeffectiveuseofitinhistheoryofheat,whichwasnew.Heclaimedthatany2π-periodicfunctioncanberepresentedbysuchaseries,butEulerandBernoullialreadyknewthisisn’tcompletelycorrect.TodaywegiveFouriercreditforhisvisionaryideas,butcriticizehislackofrigour.

Themainissueistheconvergenceofthetrigonometricseries:whetherithasawell-definedsum,andifso,whatpropertiesthathas.Cantor’sresearchledhimtoconsiderthesetSofzerosoftheseries—valuesofxforwhichitssumvanishes.Thesetofzeroscanbeinfinite,andverycomplicated.Itcontainsaspecialsubset,calledthederivedsetS1,whichconsistsofalllimitpointsofS.ThesearethepointsthatarelimitsofconvergentsequencestakenfromS.CantorconstructedarelatedFourierserieswithzero-setS1.HenoticedthatS1hasitsownderivedset,S2,andtheprocesscontinueswithS3,S4,andsoon.

Crucially,itdoesn’tstopthere.TheintersectionSωofallofthesederivedsets—thesetofpointsbelongingtoallofthem—neednotbeempty.Ifso,italsohasaderivedset ,andsoonwith , .Cantorfoundanexamplewherethisprocesscontinuedmuchfurther.Insomesensethestructurehereisgovernedby‘infinitenumbers’followingafterallthefiniteones:

Cantorwasintriguedbythisstructure.It’snotjustfantasy:it’sunavoidableifyouwanttounderstandtheconvergenceanduniquenesspropertiesofFourierseries.Itleadsnottotransfinitecardinals,buttoarelatedconcept,transfiniteordinals.I’llcomebacktothoseshortly.

Settheory

Cantor’sresearchonFourierseriesnaturallyledhimtobasicideasofsettheoryandpoint-settopology,inordertosolveaproblemthatEulerandBernoulliwouldhaveconsideredmainstreammathematics.Itwas,butatthetime,thetoolsneededtoansweritweren’t.

Thebasicingredientsofsettheoryaresosimplethatitdoesn’tlookmuchlikemathematics.Asetisacollectionofobjects;inprincipleanyobjects,butinpracticemathematicaloneslikenumbersortriangles.Theseobjectsareitsmembersorelements.Setscanbecombinedandmanipulated;forexampletheunionoftwosetsiswhatyougetwhenyoumergethem,andtheintersectionisthesetofallmembersthattheyhaveincommon.

Afinitesetcanbespecifiedbylistingitsmembers,enclosedinbraces{}:forexample

isthesetofallwholenumbersrangingfrom1to6,and

isthesetofallprimesintherangefrom2to12.Theirunionis

andtheirintersectionis

Aninfinitesetcan’tbedefinedlikethat,butitcanbespecifiedbystatingwhatpropertiesitsmembersmusthave.Forexample

specifyinfinitesets.Bytheway,R2istheset-theoreticspecificationofEuclid’splane,basedoncoordinates.

Cantorworkedoutmanyofthebasicsofsettheory;notasanabstractexerciseinformalreasoning,butbecauseheneededitinhisworkonFourierseriesandtranscendentalnumbers.Themorehedelvedintothisnewtopic,themorefascinatingandunorthodoxhisviewpointbecame.Toounorthodoxformany,andwecansympathizewiththem,becauseitreallydoesrequireanewmindsetandtherejectionofingrainedphilosophicalprinciples.Tomakemattersworse,settheorywasintimatelyboundupwithahostofotherissuesthatmathematicianshadneverreallysortedoutlogically.Itwasaperiodofgreatintellectualconfusion.

Transfinitecardinals

Aswellasdevelopingtheformalismofsettheory,Cantorgeneralizedthenotionofcardinalnumbertoarbitrarysets.Theset has3members;itscardinal|S|is3.ThesetNofnaturalnumbershasinfinitelymanymembers;itscardinal|N|is…what?Infinity,insomesense—butwhatsense?

FromFrege,andothersofhisperiod,Cantortookonekeyprinciple.Twofinitesetshavethesamenumberofmembersiftheycanbematchedinaone-to-onefashion.Thelovelythingaboutthisstatementisthatyoudon’tneedtoknowwhatthenumberofmembersis.Justasyoucancheckthattwostickshavethesamelengthbylayingthemsidebyside,withoutmeasuringtheactuallengths.

Cantorrealizedyoucandothesamethingforinfinitesets.Heintroducedanewkindofinfinitenumber,oftensaidtobetransfinite,anddefinedittobethecardinalofN.Hegaveitasymbol:not∞,whichwaspotentiallyconfusinggiventhehugevarietyofdistinctusagesofthatsymbolinmathematics,butℵ0.Hereℵis‘aleph’,thefirstletterintheHebrewalphabet.FollowingFrege’slead,anysetthatcanbeputinone-to-onecorrespondencewithNisalsoassignedthecardinalℵ0.Theintegers(positiveandnegative)areanexample.Onewaytodefinethecorrespondenceistointerleavepositiveandnegativeintegerslikethis:

Anotherexampleisthepositiverationals:listtheminorderofcomplexity,asalreadydescribed.Toincludenegativerationalsaswell,interleavethemwiththepositiveonesinasimilarmannertothenegativeintegersabove.YetanotherisGalileo’sremarkaboutperfectsquares(seeChapter1).ThesetofallsquarescanbematchedtoN,soitalsohascardinalℵ0.

Arethereanysetsthatareneitherfinitenorhavecardinalℵ0?Fromhisworkontranscendentalnumbers,Cantorknewthereis:thesetRofrealnumbers.Heexpectedthistobethenextcardinalbiggerthanℵ0,inwhichcaseitcouldbedefinedtobeanewcardinalℵ1,nodoubtfollowedbyℵ2,ℵ3,ℵ4,andsoon—anendlessseriesoftransfinitenumbers.Indeed,heprovedthatthere’snolargestcardinal.Thesetofallsubsetsofanygivensetmusthavealargercardinalthanthesetitself.Thatis,there’snowaytomatcheverysubsettoamemberofthesetinaone-to-onemanner.Theproof,ironically,isavariationontheideabehindtheRussellparadox.

OnereasonwhymanyfoundCantor’sideascounterintuitiveisthatwhenyoumatchtwosets,mostoftheirtraditionalfeaturesareirrelevant.Thenaturalorderofnumberscanbejumbledup,forexample.Thisiswhywecanmatchrationalstonaturalnumbers.Thedimensionsofspacesareanunnecessaryencumbrance,whichiswhythereallineRandtheplaneR2match—aresultthatastonishedevenCantorwhenhefirstprovedit,becausetheplanelookssomuchlargerthantheline.Order,dimension,algebraicoperations,andthelikeareextramathematicalsuperstructureattachedtobaresets.Thesuperstructurecanitselfbedefinedusingsettheory,butit’snotautomaticallybuiltintotheunderlyingset.

Cantorusedoperationsonsetstodefinearithmeticaloperationsontransfinitecardinals—sum,product,exponential.Heestablishedtheirbasicproperties.Heandothermathematiciansalsodefined‘greaterthan’and‘lessthan’.HealreadyknewfromhisworkontranscendentalnumbersthatthecardinalofR—callitc—isgreaterthanℵ0,buthecouldn’tprovethatcisthesmallestcardinalwiththatproperty.Isthereacardinalstrictlybetweenℵ0andc?Ifnot,itmakessensetodefineℵ1tobec.PlentyofsetscontainNbutarecontainedinR—theintegers,rationals,algebraicnumbers,positiverealnumbers,transcendentalnumbers,andanendlesshostofothers.Thesearesensiblecandidatesforanintermediatecardinal,butineverycasethecardinaliseitherℵ0orc.Theintegers,rationals,andalgebraicnumbersallhavecardinalℵ0;thepositiverealsandthetranscendentalnumbershavecardinalc.

Obviouscandidatesturnedoutnottowork.Forexample,surelytherearefewerrealnumbersbetween0and1thantherearerealsintotal?Itseemsplausible:thelengthoftheintervalfrom0to1(excludingtheends,fordefiniteness)is1;thelengthofRisinfinite.However,one-to-onecorrespondencesarenorespectersoflength.Thegraph showninFigure32,mapstheunitinterval(0,1)inone-to-onemannerontothewholeofR.

32.Aone-to-onemappingfromtheopenunitintervalontoR.

Thestatementthat becameknownastheContinuumHypothesis.Hilbertlisteditamonghisfamoustwenty-threeunsolvedproblemsin1900.Somemathematiciansbegantosuspectthatthestatementwasrelatedtofoundationalissuesinmathematicallogic.Gödelprovedin1940thatthetruthoftheContinuumHypothesisislogicallyconsistentwiththestandardaxiomaticformulationofsettheory,knownastheZermelo–Fraenkelaxioms.Finally,in1963,PaulCohenprovedthatthefalsityoftheContinuumHypothesisisalsologicallyconsistentwiththeZermelo–Fraenkelaxioms.It’sastunningexampleofastatementthatisindependentoftheusualaxioms.ThereareversionsofsettheoryinwhichtheContinuumHypothesisistrue,buttherearealsoversionsofsettheoryinwhichtheContinuumHypothesisisfalse.

PrecursorstoCantor

Cantor’sworkisprofoundbecausehesetupalogicallyrigorousframework,definedinfiniteanaloguesofcountingandnumbers,andprovedthattheseconceptshavespecifiedproperties.However,hewasn’tthefirstpersontosuggestthatinfinitycancomeindifferentsizes.Thathonour,asfarashistoriansofmathematicsareaware,goestoanunknownIndianmathematicianormathematiciansaround400BC.ThesuggestionisdocumentedinSuryaPrajnapti,aJainmathematicaltext.

WesawinChapter2thatlikemanyIndianreligions,Jainismwasfascinatedbyverylargenumbers.Moreover,theywereawarethatnocountingnumber,howeverlarge,isinfinite.Thathonourtheyreservedforthesmallestuninnumerablenumber.Beyondthis,theyasserted,stretchevenlargerinfinitenumbers.Theyclassifiednumbersintothreetypes,eachwiththreesubtypes:

Enumerable:lowest,intermediate,andhighest.Innumerable:nearlyinnumerable,trulyinnumerable,andinnumerablyinnumerable.Infinite:nearlyinfinite,trulyinfinite,andinfinitelyinfinite.

Theyalsodistinguishedfivedifferentmeaningsof‘infinity’:infiniteinonedirection,infiniteintwodirections,infiniteinarea,infiniteeverywhere(thatis,involume),andperpetuallyinfinite.

Cantorhadasimilarvision,butheflesheditoutwithrigorousdefinitions.Hisconclusionsweresimilar,butwithimportantdifferences.Forexample,inhisformulationaline,aplane,andavolumeallhavethesamenumberofpoints.Thisdoesn’tmeantheJainswerewrong.Theywerethinkingaboutsubtlydifferentideas.Theyweretwomillenniaaheadoftheirtime,buttheirviewsontheinfiniteweresomewhatmystical,andnotformulatedwiththelogicalprecisionwenowrequire.

AftertheJains,thenextsignificantmathematicalcontributiontoourunderstandingoftheinfinitewasprobablyGalileo’s.Chapter1includesanextractfromTwoNewSciences,inwhichSalviati,theirritatingknow-all,arguesthat‘thereareasmanysquaresastherearenumbers’,because‘everynumberistherootofsomesquare’.Simplicio,thedunce,agrees;hewouldn’tstandachanceifhedidn’t.Sagredo,thestraightman,feedsSalviatiusefullineswhentheperformancestartstoflag.

ThephilosophicalissuethatGalileowasaddressinghereisthebeliefthat‘thewholeisgreaterthanthepart’.Thereareobviouslymorenumbersthansquares,becauseeverysquareisanumberbutsomenumbersarenotsquare.Let’stakealook,markingsquaresinboldface:

Theproportionofsquaresgetssteadilysmaller,exceptwhenwemeetthenextone.Amongthenumbersupto100thereare10squares;upto10,000thereare100squares;uptoamillionthereareonlyathousandsquares.

Salviati’spointisthatalthoughsquaresgetthinnerandthinnerontheground,theyneverrunoutaltogether.Soonerorlater,anotheronecomesalong.Sowecanmatchthenumberstothesquares:

Ifwestopatsomefinitelimitn,therearealotmorenumbersthansquares.Butifwedon’tstop,everynumbermatchesexactlyonesquare,andconversely.Sothepartcanmatchthewhole.Cantor’sideasexplainedSalviati’sobservation.

Transfiniteordinals

Therearetwowaystoviewthewholenumbers:ascardinals,numericalmeasuresofhowbigsomethingis,orasordinals,whichplaceobjectsinorderbyrunningthroughthesequence1,2,3,….Forfinitenumbersthisdistinctionisabitnit-picking,andmakespreciouslittledifference.Whenwecometoinfinitenumbers,however,there’sabigdifference.Toeachtransfinitecardinaltherecorrespondinfinitelymanydifferenttransfiniteordinals.CantorfirstranintotransfiniteordinalsinhisworkonFourierseries.

Transfinitecardinalssatisfythewell-orderingprinciple:anysetofcardinalshasasmallestmember,necessarilyunique.There’saparalleltheoryofwell-orderedsets,inwhichtheone-to-onecorrespondenceisrequiredtopreserveorderaswell.Nowcardinalsarereplacedbyordinals,andthesmallestinfiniteordinal,correspondingtoN,iscalledω.Anywell-orderedsetthatcanbeplacedinone-to-onecorrespondencewithN,withoutdisturbingtheorder,hasordinalω.AnexampleisGalileo’ssetofsquares:thecorrespondencekeepsthemintheirnaturalorder.Anotheristhesetofallprimes.

Ordinalshavearichstructurewithsomestrangefeatures.

Incardinalarithmetic, isequalto andbothareequaltoℵ0.SupposewetakethenaturalnumbersNandaddanewelementX,differentfromanynaturalnumber.Becausetheorderingisnotimportantforcardinals,wecanmatchthislargersettoNbyshiftingeverythinginNalongonespace,andputtingXatthebeginning,justlikeaccommodatinganewguestinHilbert’shotel:

Thisreasoningappliestoboth and againbecauseorderisunimportant.Transfinitecardinalssatisfythecommutativelawofaddition .Bysimilarreasoning,wecanprovetheoremssuchas:

andsoon.However,somearithmeticaloperationsleadtolargercardinals;forinstance,

Indeed,

Whenitcomestoordinals,therulesareverydifferent.Forinstance, isnotequaltoω,butlarger.Asetwithordinal canbeconstructedbytakingthenaturalnumbersNandappendinganewelementX,deemedtobegreaterthananynaturalnumber;thatis,comingafterthemallintheordering.Sothisorderedsetlookslike

withXtaggedonasanafterthought.Thisislessartificialthanitmightappear,sincethesetofallcardinalsuptoandincludingℵ0,inorderofsize,lookslike

Indeed,wecanwritethesequenceofallpossibledistinctcardinalsasℵα,whereαrunsthroughallordinalsinascendingorder.

Whendealingwithordinals,wecannolongerdoaHilberthotelandmatchthislargersettoNbyshiftingeverythinginNalongonespaceandputtingXatthebeginning,becausenowwehavetokeepeverythinginthesameorder.SinceXisthelargestelement,wecan’tmoveittothefront,whereitwouldbecomethesmallest.Wecan’tputitanywhereinthemiddle,somewhereinsideN,forthesamereason.Ithastoremainwhereitis.So isdifferentfromω.Infact,it’sthenextbiggestordinalafterω.

Thesequenceofinfiniteordinalsgoeslikethis:

andsoon.Ifweignoretheorder,thecorrespondingsetscanallbemadetomatchN,sotheyallhavecardinalℵ0.Buteventuallywereachthefirstuncountableordinal,denotedbyε0.

Ontheotherhand,1+ωisthesameasω,becausenowwecanaddanextraelementXatthefrontandshiftNalongonespacewithoutdisturbingtherelativeorder,whichishowthefirstextraguestgotaroominHilbert’shotel.Soadditionofordinalsdoesn’tsatisfythecommutativelaw,unlikecardinals.

WecannowreturntothefirstexampleinChapter1:themeaningof .Itdependsonhowweinterpretthesymbol∞.Ifit’sthetransfinitecardinalℵ0,then Ifit’sthetransfiniteordinalω,then but Ifit’s1/ε,whereεisCauchy’sinfinitesimalsequence(1/n),then

Eachmeaningmustbeinvestigatedinitsownright.

CantorandWittgenstein

Cantorrepeatedlyemphasizedthatsettheorywasaboutactualinfinity.HeexplicitlycontrasteditwithAristotelianpotentialinfinity,andhediscussedvariousphilosophicalviewsaboutinfinityinhiswork.Buttheissuesarerathersubtle.

IfyouinterpretCantor’sideasliterally,theyrefertoa(conceptual)actualinfinity.WethinkofthesetNofallnaturalnumbersasanobject,notasaprocess.InCantor’sview,aset,beitfiniteorinfinite,isavalidmathematicalobject.Thesetofallnaturalnumbers‘exists’inthesamewaythat{1,2,3}exists.Cantoraimedtoassociatewitheachinfinitesetacardinal,determininghowmanymembersithas.It’sdifficulttoseehowtostatetheseideasusingonlythelanguageofpotentialinfinity.Atbest,anyattemptwouldbehopelesslycontrived.

Thisisespeciallyclearforasecondproofthattherealnumbersareuncountable,whichCantorgavein1891.It’smoreelementaryandavoidsassertionsaboutnestedsequencesofclosedintervals.ItstartsbyassumingthesetRofrealnumbersiscountable,andderivesacontradiction.Thisisobtainedbyfirstreducingtheissuetoasimilarstatementaboutrealnumbersgreaterthan0andlessthan1,whichisroutineusingFigure32.Havingdonethis,everynumberinthatrangehasadecimalexpansion

Suchanexpressionisn’tquiteunique;forexample0·199999…isequalto0·2.(Peopleoftenthinkthesearedifferent,byaninfinitesimalamount,butinconventionalmathematicsthey’rethesame.Justas1/2and2/4lookdifferent,butrepresentthesamefraction.)Toremovetheambiguity,forbidinfiniterecurringsequencesof9s.

SupposeRiscountable.ThenthecountingnumbersNcanbematchedtoR:

Byassumption,everyrealnumberoccurssomewhereinthelist.Nowweconstructonethatdoesn’t.Wedefinesuccessivedecimalplacesx1,x2,x3,…ofthisnumberxasfollows:

andsoon.Ingeneral,makexneither0or1,anddifferentfromthenthdigitoftherealnumbercorrespondington.

Byconstruction,xdiffersfromeverynumberonthelist.Itdiffersfromthefirstnumberinitsfirstdigit,fromthesecondnumberinitsseconddigit;ingeneral,itdiffersfromthenthnumberinitsnthdigit,soit’sdifferentfromthenthnumber,nomatterwhatvaluenhas.However,weassumedthatthelistexists,andeveryrealnumberappearsonit.Thisisacontradiction,andwhatitcontradictsistheassumptionthatsuchalistexists.Thereforenosuchlistexists,andRisuncountable.

Wittgensteindespisedthediagonalargument.InLecturesandConversationsheofferedtoputtheproofinsuchawaythat‘itwillloseitscharmforagreatnumberofpeopleandcertainlywillloseitscharmforme’.InRemarksontheFoundationsofMathematicshedisputedHilbert’s‘paradise’remark.Asfornoonebeingexpelled,hisLecturesontheFoundationsofMathematicsclaimedthat‘you’llleaveofyourownaccord’.WithCantorlongdead,Wittgensteincontinuedtoexpresshisprofoundphilosophicaldissatisfaction,complainingthatmathematicswas‘riddenthroughandthroughwiththeperniciousidiomsofsettheory’.

Setting‘pernicious’aside,itwas,andstillis.NooneleftCantor’sparadise.Afewdecidednottoenter,butthosethatdidfoundlittletojustifyWittgenstein’sscepticism.Cantor’snew-foundfreedomhastaken

mathematicsfromstrengthtostrength.Hilbertwasright.

Mathematics,philosophy,andreligion

Cantor’sapproachdoesraiseamajorphilosophicalissue.Itrequiresustothinkofthesetsinvolvedasspecificobjects,notprocesses:actualinfinity,inAristotle’ssense.‘Actual’inaconceptualmanner,ofcourse,asforallmathematicalconcepts,adistinctionthatwasn’tfullyunderstoodinAristotle’stime.Cantorhadadeepinterestinphilosophy,andwaswellawareoftheexplosivenatureofthisview,buthefounditimpossibletoavoid.

Atthattime,lessthan150yearsago,theconceptofactualinfinitywascommonground(oftenbattleground)formathematics,philosophy,andreligion.EuropewasintenselyChristianandbeliefinGodwasthedefaultview,althoughatheismandagnosticismwerealreadybeginningtogainground.Christianssawtheirdeityasaperfect,infinite,eternalbeing;indeed,astheuniqueactuallyinfinitebeing.TheyhadnoqualmsaboutAristotle’spotentialinfinity,butassertingtheexistenceofanotheractualinfinitywastheologicaldynamite—evenwhenthe‘actual’infinityconcernedwasamathematicalabstraction.

Religion’spoliticalgraspwasslipping,sothechurchesdidn’tmakeasmuchfussastheyhadinthe17thcenturyaboutinfinitesimals,butthestatusofmathematicalinfinitywasaseriousissueforthereligious.WhenKroneckersaid‘Godcreatedtheintegers’hewasn’tspeakingmetaphorically.Cantorwasalsoreligious,andhewenttoconsiderablelengthstoexplainhow,inhisview,transfinitecardinalscouldbereconciledwithGodastheuniqueabsoluteinfinity.Hecouldprovethereisnolargesttransfinitenumber.Paradoxically,thisimpliesthatthesetofalltransfinitenumbers(surelyameaningfulandindeedimportantsetinhistheory)issobigthatitdoesn’thaveacardinal.Impressivelyinfinitethoughanygiventransfinitecardinalmaybe,itcan’tapproachtheabsoluteinfinityofGod.ThisjustificationhasclearechoesofAugustine’sproofthatGodisinfinite(seeChapter3).

Cantorstatedthat‘thetransfinitespeciesarejustasmuchatthedisposaloftheintentionsoftheCreatorandHisabsoluteboundlesswillasarethefinitenumbers’,neatlyturningthetablesonsomecritics.IfyouclaimthattransfinitecardinalsareonaparwithGod,thenyou’resayingtherearelimitstoHispower,whichistheologicallysuspect.CantorbelievedthathisknowledgeoftransfinitecardinalshadcomedirectlyfromGod,anditwashisChristiandutytotelltheworldaboutthem.Hecorrespondedwithdistinguishedphilosophersandtheologians,andpublishedthecorrespondence.HeevenwrotetoPopeLeoXIII,andsenthimseveralpamphletsonthetopic.

Thismayseemextreme,butsuchwasthespiritoftheage.Itdidn’thelpthatfoundationalissuesinmathematicshadstimulatedseveralschoolsofthoughtthatrejectedtheinfinite,oraccepteditonlywhenaspecificconstructioncouldbegiven.Kroneckerwasaconstructivistofthiskind.Liouville’sworkontranscendentalnumbers,whichconstructedaspecificexample,wasacceptable;Cantor’sallegedproofthattranscendentalsexistdidn’tactuallyconstructone,soitwasrubbish.Infact,eventhelogicalbasisofmathematicswasunderattackbytheIntuitionistschoolofLuitzenBrouwer,whichrejectedproofbycontradictionaswell.

Aversionofconstructivismstillexists.FoundedbyErrettBishopinhis1967FoundationsofConstructiveAnalysis,itseeksconstructiveanaloguesofbasicmathematicaltheorems.Forexample,theintermediatevaluetheoremofstandardmathematicsstatesthatifacontinuousfunctionisnegativeatsomepointandpositiveatanother,thensomewhereinbetweenitmustbezero.Constructiveanalysisrejectstheusualexistenceproof,insistingitmustbereplacedbyanalgorithmicprocedurethatdefinessuchapointexplicitly.However,there’sapricetopay:theconstructiveanalogueof‘continuous’hastobeamuchstrongerpropertythanthetraditionalone,sotheanalogyisincomplete.Mostmathematiciansviewconstructiveanalysisasavalidbutspecializedareaofmathematics,focusedonconstructionsandalgorithms,provinganaloguesofbasictheoremsinwhichstrongerhypothesisgivestrongerresults.Constructivists,however,tendtoseeitasareplacementforexistingmathematics;itstheoremsarenotanalogues,buttheonlylogicallyvalidwaytoproceed.Thisviewhasnotmademuchheadway.

Theideathatwecan’tbesuresomethingexistsunlesswe’retoldexactlyhowtofindithasadefiniteappeal.IfDavidLivingstonehadcomebackfromAfricasaying‘I’veprovedthatthesourceoftheNileexists’,noonewouldhavebeenimpressed.Theywantedtoknowwherethesourcewas,andthat’swhatLivingstonetoldthem.(Hewaswrong,butnomatter.)However,existenceargumentscutthroughalotofirrelevantdetail,andweusethemmorethanweimagine.Ifafriendiswalkingalongtheroadtoyourhouse,andyouheadoffalongthereverseroutetointercepthim,you’reconfidentyou’llmeetup,eventhoughyoucan’tpredictpreciselywhere.

Viciousnarrow-mindedoppositioncausedCantormuchgrief.In1904JuliusKöniggavealectureatthe3rdInternationalCongressofMathematicians,claimingthattransfinitenumbersandsettheorywerebasedonanerror.Königwaswrong;ErnstZermelofoundamistakeinhisallegedprooftheverynextday.

ButCantorhadbeenhumiliatedinpublic,andwassodistressedthatheevenbegantodoubthisfaith.Thiseventmayhavetriggeredhisrecurrentdepression.HeretiredjustbeforeWorldWarI,livedinpoverty,anddiedtenmonthsbeforethewarended.

Ironically,Cantor’sownviewofinfinitesimals,asformalizedbyduBois-ReymondandStolz(Chapter4),wasjustasextreme.Hecalledinfinitesimalsthe‘cholerabacillusofmathematics’.Heprobablyobjectedtotheapproachtoinfinity,whichdifferssignificantlyfromhisown.Thereciprocalofaninfinitesimaldoesn’tsitcomfortablywithcounting.TheverdictofposterityisthatCantorwaswrong;heignoredhisownviewsonmathematicalfreedomandheforgotthat‘infinity’inmathematicscanhavemanymeanings,notjusttheonehewasadvocating.

Processesandthings

Settheoryhabituallyreifiesprocessesasobjects.Forexampleamathematicalfunctionwastraditionallyconsideredtobeaprocess:arulefortransformingan‘input’intoarelated‘output’,bothusuallybeingnumbers.The‘square’functionturnsanyinputintoitssquare.Thiscanbeconsideredaninstanceofpotentialinfinity,becausetheruleitselfcanbestatedinfiniteterms:‘multiplythenumberbyitself’,anditcanthenbeappliedtoanyfinitecollectionofnumberswithoutinvokinganythinginfinite.

However,thestandarddefinitionofthisfunctioninsettheoryis:thesetofallpairsoftheform(x,x2).Thisisaconceptualtableofallvaluesofthefunction,justlikenormallogarithmicortrigonometrictables,exceptthatallpossiblevaluesofxarelistedandbothxandx2arearbitraryrealnumbers,determinedtoinfiniteprecision.Thelististhoughtofasacompleted‘actual’object,notastheprocessdefiningthatobject.Theprocessisreplacedby‘lookitupinthetable’.Fromafoundationalviewpoint,allfunctionsarenowdefinedinthismanner.

AsWittgensteinsaw,manyusesofthisconstructioncanbereducedtopropertiesoftheprocessitself,sotheycanberephrasedintermsofpotentialinfinity.Forexample,inCantor’stheory,thesetofallwholenumbersandthesubsetofallevennumberscanbeputintoone-to-onecorrespondencebytakingtheinputntotheoutput2n.Thisconstruction(whichweusedtoaccommodateacoachloadoftouristsinHilbert’sjam-packedhotelinChapter1)wasthenviewedasaproofthatthewholecanequalthepart.InPhilosophicalRemarks,Wittgensteincorrectlyrealizedthatactualinfinityisn’tessentialtothisparticularconstruction,saying:‘Doestherelation correlatetheclassofallnumberswithoneofitssubclasses?No.Itcorrelatesanyarbitrarynumberwithanother,andinthatwaywearriveatinfinitelymanypairsofclasses,ofwhichoneiscorrelatedwiththeother.’

Fairenough:he’spointingoutthatallyouneedistheruleforgettingoutputmfrominputn.Butfewmathematicianswouldconcurwithwhatfollows:‘butwhichareneverrelatedasclassandsubclass.Neitheristhisinfiniteprocessitselfinsomesenseorothersuchapairofclasses…Inthesuperstitionthat

correlatesaclasswithitssubclass,wemerelyhaveyetanothercaseofambiguousgrammar.’

Wittgensteinhatedsettheorywithapassion,declaringthatithad‘completelydeformed’philosophicalandmathematicalattitudes.Whateverthemeritsofhisviews,hewasdisappointed.Mathematiciansignoredhisadvice,withgoodreason.Evenifhewasright,andanyapparentuseofactualinfinitycanberecastinfiniteform,that’snotanargumentforsettingasidetheinfinite.Onthecontrary,ittellsusthattheinfiniteintroducesnologicalinconsistenciesthataren’talreadypresentinthefinite.Thetwoapproachesarenotlockedintoaconflictthattearsthemapart:theyconstituteanalliancethatstrengthensboth.Moreover,translatingproofsfromthelanguageoftheinfinitetothelanguageofthefiniteoftenturnssimple,transparentstatementsintotortuous,laboriousones.Settheoryisathinkingtool,andaverypowerfulone,becauseitmakescomplexideassimple.

Thinkaboutturningthediagonalproofintoapotentialinfinityproofusingprocesses.Eachrealnumberisaninfinitedecimal—aprocess.Thehypotheticallistisaprocessoperatingonthoseprocesses.Themissingrealnumberobtainedbylookingalongthediagonalisaprocessoperatingonahypotheticalprocessappliedtoprocesses.Thecontradictioncomesbycomparingthiswitheachprocessinvolvedintheprocessthatrepresentsthelist.Theresultmightjustbephilosophicallypurer,butitwouldalsobeutterlyincomprehensible.

Forreasonslikethis,settheoryhasnowcompletelytakenoveradvancedmathematics,bothpureandapplied,includingapplicationstothephysicalsciences,biology,eveneconomics.Notjustasawayofformulatingconcepts,orasanotation,butinfully-fledgedCantoriansplendour.Differentsizesofinfinity,especiallythedistinctionbetweencountableanduncountableinfinity,arevitaltohugeareas,andindispensableinappliedscience.Probabilitytheoryrestsontheconceptofcountableadditivityofameasure.Thatis,ifyouaddupageneralizationofareaforacountablyinfinitecollectionofdisjointsets,yougettheareaoftheirunion.Uncountableadditivity,ontheotherhand,canbeprovedself-contradictory.PartialdifferentialequationsreceivedahugeboostfromStefanBanach’sintroductionofinfinite-dimensionalspacesofoperators.QuantummechanicsdependsonHilbertspaces,keyexamplesofBanach’snewconcept.

WittgensteinmadethesamecategoryerrorasLocke.The‘actualinfinities’ofsettheoryaremathematicalconcepts,notrealobjects.They’re‘mathematicallyactual’whenthemathematiciancontemplatesthemascompletedthings,notasprocesses.They’re‘mathematicallypotential’whenthemathematiciancontemplatesthemasprocesses.Whatdistinguishesthetwoviewpoints,inanyspecificpieceofmathematics,ishowthey’reused.Notwhatthey‘really’are.Theyaren’tanythingreal.

Mathematicalexistence

Formostmathematicians,makingsenseofinfinityisnotaboutthemeaningoftheinfinite;it’saboutthemeaningofmathematicalexistence.There’sastrongconsensusthatmathematicsisn’treality;itjustresemblesrealityinusefulways.Amathematicalobjectorprocessexistsifitdoesn’tleadtologicalcontradictions,whichistheviewpointthatCantorpromoted.Itsexistence,inthatsense,canbeprovedbyconstructingitwithinthenormalframeworkofknownmathematics,orbyshowingthatitsnon-existenceleadstoalogicalcontradiction.

Thisviewofmathematicalexistenceisproblematicinonerespect:itassumesmathematicsitselfislogicallyconsistent.Ifnot,thecriterionimpliesthatmathematicsfailstoexist.However,Gödelprovedthattheconsistencyofmathematicscanneverbeprovedwithinanyaxiomaticframework…unlessit’sfalse,inwhichcaseanythingcanbeproved.GerhardGentzenprovidedaconsistencyproofin1936basedontransfiniteordinals,butthatmethodisofcourseopentophilosophicaldoubts.

Thesecondtypeofexistenceproof—non-existenceleadstoacontradiction—isnon-constructive.Somephilosophicallymindedmathematiciansobjecttosuchproofs.However,evenineverydaylifewemakecommonuseofnon-constructivearguments—usuallywithoutnoticing.Iwonderhowconstructivistswouldreactifthepolicegavethemaspeedingticketbecausetheiraveragespeed,measuredoveraparticularstretchofroad,exceededthespeedlimit.Ifitcametocourt,thepolicewouldarguethatsuchanaverageprovestheexistenceofsometimeatwhichtheaccused’sspeedexceededthelimit.Asufficientlycommittedconstructivistwouldbeobligedtoarguethatunlessthepolicecanestablishaspecifictimeatwhichtheyexceededthelimit,there’snocasetoanswer.

Idon’twanttoleaveyouwiththeimpressionthatmathematicshasexplainedeverypuzzleaboutinfinity.Noteventhatmathematiciansclaimtohaveexplainedeverypuzzleaboutinfinity.Therearestillplentyofunsolvedproblems,especiallyinaxiomaticsettheory.Butmathematicianshaveputtogetheralogicalframeworkinwhichwecanunderstandthosequestions,answermany,andmakedistinctionsbetweendifferentinstancesofinfinity.Thatframeworkhasledtodramaticnewdiscoveries,enrichingmathematicsandleadingtonewapplications.

Welcometothebizarrebutbeautifulworldoftheinfinite.

References

ExtractfromGalileo’s1638DiscorsieDimostrazioniMatematicheIntornoaDueNuoveScienze:GalileoGalilei,DialoguesConcerningTwoNewSciences,translatedbyHenryCrewandAlfonsodeSalvio,Macmillan,NewYork1914.

QuotationfromArchimedes’Psammites:JamesR.Newman,TheWorldofMathematics,SimonandSchuster,NewYork1956.

QuotationsbyAristotleaboutZeno’sparadoxes:Aristotle,Physics,translatedbyR.P.HardieandR.K.GayefromTheCompleteWorksOfAristotle(editorJonathanBarnes),PrincetonUniversityPress,Princeton1984.

QuotationbyPaulduBois-Reymondabouttheinfinitelysmall:PaulduBois-Reymond,ÜberdieParadoxendesInfinitär-Calcüls,MathematischeAnnalen11(1877)150–67.

ExtractfromImmanuelKant,CritiqueofPureReason,translatedbyPaulGuyerandAllenWood.Cambridge:CambridgeUniversityPress,1998.

Furtherreading

Chapter1:Puzzles,proofs,andparadoxesBrianClegg.BriefHistoryofInfinity:TheQuesttoThinktheUnthinkable,Robinson,London2003.RevielNetzandWilliamNoel.TheArchimedesCodex,Weidenfeld&Nicolson,London2007.RudyRucker.InfinityandtheMind:TheScienceandPhilosophyoftheInfinite,PrincetonUniversityPress,Princeton

2004.

Chapter2:EncounterswiththeinfiniteEugeneP.Northrop.RiddlesinMathematics:ABookofParadoxes,Penguin,Harmondsworth1960.IanStewartandDavidTall.TheFoundationsofMathematics(2nded.),OxfordUniversityPress,Oxford2015.DavidFosterWallace.EverythingandMore:ACompactHistoryofInfinity,W.W.Norton,NewYork2004.

Chapter3:HistoricalviewsofinfinityJohnBowin.Aristotelianinfinity,OxfordStudiesinAncientPhilosophy32(2007)233–50.KevinDavey.Aristotle,Zeno,andthestadiumparadox,HistoryofPhilosophyQuarterly24(2007)127–46.MichaelHellerandW.HughWoodin(eds.).Infinity:NewResearchFrontiers,CambridgeUniversityPress,Cambridge

2011.JoeMazur.Zeno’sParadox:UnravelingtheAncientMysteryBehindtheScienceofSpaceandTime,Plume,NewYork

2008.

Chapter4:TheflipsideofinfinityAmirAlexander.Infinitesimal:HowaDangerousMathematicalTheoryShapedtheModernWorld,Scientific

American/Farrar,StrausandGiroux2014.MikhailKatzandDavidSherry.Leibniz’sinfinitesimals:theirfictionality,theirmodernimplementations,andtheirfoes

fromBerkeleytoRussellandbeyond,Erkenntnis73(2013)571–625.H.JeromeKeisler.ElementaryCalculus:AnInfinitesimalApproach,UniversityofWisconsin,Madison2000.AbrahamRobinson.Non-StandardAnalysis(2nded.),PrincetonUniversityPress,Princeton1996.

Chapter5:GeometricinfinityKirstiAndersen.TheGeometryofanArt:TheHistoryoftheMathematicalTheoryofPerspectivefromAlbertitoMonge,

Springer,NewYork2007.H.S.M.Coxeter.IntroductiontoGeometry.JohnWiley&Sons,NewYork1969.MorrisKline(ed.).MathematicsintheModernWorld,W.H.Freeman,SanFrancisco1968.

Chapter6:PhysicalinfinityNeilJ.Cornish,DavidN.Spergel,GlennD.Starkman,andEiichiroKomatsu.Constrainingthetopologyoftheuniverse,

PhysicsReviewLetters92(2004)201302.TimPostonandIanStewart.CatastropheTheoryandItsApplications,DoverPublications,NewYork1996.(Reprintof

1978Pitmanedition.)DonaldSaariandZhihingXia.Offtoinfinityinfinitetime,NoticesoftheAmericanMathematicalSociety42(1995)538–

46.PascalM.Vaudrevange,GlennD.Starkman,NeilJ.Cornish,andDavidN.Spergel.Constraintsonthetopologyofthe

universe:extensiontogeneralgeometries.PhysicalReviewD86(2012)083526.

Chapter7:CountinginfinityJosephWarrenDauben.GeorgCantor:HisMathematicsandPhilosophyoftheInfinite,PrincetonUniversityPress,

Princeton1979.L.C.Jain.SettheoryintheJainaschoolofmathematics,IndianJournalofHistoryofScience8(1973)1–27.GeorgeG.Joseph.TheCrestofthePeacock:Non-EuropeanRootsofMathematics(2nded.),Penguin,London2000.NavjyotiSingh.JainTheoryofActualInfinitiesandTransfiniteInfinities,NationalInstituteofScience,Technologyand

DevelopmentStudies(NISTAD),NewDelhi1987.IanStewartandDavidTall.TheFoundationsofMathematics(2nded.),OxfordUniversityPress,Oxford2015.

Publisher’sacknowledgements

Wearegratefulforpermissiontoincludethefollowingcopyrightmaterialinthisbook.

ExtractfromGalilei,Galileo(1954)[1638].Dialoguesconcerningtwonewsciences.TranslatedbyHenryCrewandAlfonsodeSalvio(NewYork:Dover2003).pp.31–3,withpermission.

Thepublisherandauthorhavemadeeveryefforttotraceandcontactallcopyrightholdersbeforepublication.Ifnotified,thepublisherwillbepleasedtorectifyanyerrorsoromissionsattheearliestopportunity.

Index

AAchillesandthetortoise 2,37,38–41actualinfinity 2,43–4,45,51,63,121,123–4,127–8seealsosettheoryadequality 59Airyfunction 95Alberti,LeonBattista 72aleph(ℵ) 114,119algebraicnumbers 103–4Alhazen(Abūal-Ḥaytham) 58analysis 62–3,66non-standard 67–9

Anaximander 1–2,32AnselmofCanterbury 47–8apeiron 1–2,32Aquinas,Thomas 49,99,100Archimedes 13–14,21–2,55,66–7,69Aristotle 2,33,37–9,41,42–5,50–1,63,121,123–4Arrowparadox 37,41–2art,perspectivein 71–5,85–90assumptions 34–6AugustineofHippo 47,124axiomaticsettheory 33

Bballsinthebagpuzzle 10,14–15Banach,Stefan 129Barrow,Isaac 59Berkeley,George(Bishop) 61–2Bernoulli,Daniel 111,112Bible,referencestoinfinity 46BigBangtheory 100bijection/one-to-oneontomapping 105–7,114–16Bishop,Errett 125blackholes 98–9,102Bolzano,Bernhard 62–3Bowin,John 44Brouwer,Luitzen 125Brunelleschi,Filippo 72

Ccalculus 4,14,56–61,63Cantor,Georg 2,11,15–16,18,34,103–5,108–16,117–26cardinalnumbers,transfinite 113–16,119,124Cauchy,Augustin-Louis 62,121causality,chainof 49–50Cavalieri,Buonaventura(Cavalieri’sprinciple) 58–60,69Chinesemathematicians 34Christianity 33,46–50,124circlesarea 55–6perpendicularbisectors 29–30puzzle 8–9,13–14

Cohen,Paul 116complexnumbers 34,62,65constructivism 125,130continuousanddiscreteobjects 28–31ContinuumHypothesis 116continuummodel 39,40–1,44Cornish,Neil 101cosmicmicrowavebackground(CMB) 101cosmology 91,99–102

counting 105–7Creation,proofof 12,18,64–5curvature,universe 101–2

DDavey,Kevin 38decimals,recurring 10,15,23–7,52,63Dedekind,Richard 30–1,34,40Democritus 57–8diagonalargument 103,128diagonalofasquareproblem 8–9,13,25differentialcalculus 57,59–60dimensions 85–6directions 85–6discreteandcontinuousobjects 28–31DuBois-Reymond,Paul 66–7Dürer,Albrecht 74–5

EEarth,curvature 76–7,79Egyptianart 71–2,73Einstein,Albert 98Euclid 6,43,59Euclideangeometry 25,26–7,29,30,35,46,70,75,78,84Euclideanplane 79–84,102

Eudoxus 54Euler,Leonhard 18,111,112Eunomius 47eventhorizons 99exhaustion 14,54–6,66existencearguments 125–6mathematical 129–30propertiesof 47–50

exponentialfunction 67extrapolation 35

FFermat,Pierrede 59,69finiteness 20,22rules 34–5

‘FirstCause’proofofexistenceofGod 49–50,99flattorus 102Fourier,Joseph(Fourieranalysis) 110–12,113,119Francesca,Pierodella 72,74Frege,Gottlob 107,114Friedmann,Aleksandr 102fundamentaltheoremofcalculus 59

GGabriel’shorn 20,21Galileo 2,10–11,15–16,114,117–18,119generalrelativity 98–9,102Gentzen,Gerhard 129geometricinfinity 6glory(sunlighteffect) 92God,infinityandexistence 46–50,124Gödel 116,129googolandgoogolplex 20,51–2Grandi,Guido 12,18,64–5gravity,NewtonianLaw 96–8Gregory,James 59GregoryofNyssa 47

HHamilton,SirWilliamRowan(Hamiltoniansystems) 42Hilbert,David 30,105,116hotel 11–12,16–18,119–21,123

HippasusofMetapontum 25,29

horizon,perspectivetowards 75–84Hoyle,Fred 99–100Huggett,Nick 41–2hyperbolicgeometry 102hyperreals(R*) 68–9

IIndianmathematicians 34,117infiniteseries 63–5infinitesets 2,15–16infinitesimals 54,56–7,58–9,61–3,65–9,126infinitydefinitions 4,19differentsizes 117–18earlyuseofterm 1–2existence 42–6,91interpretations 34–5

integers 23,28–9,30,104,114integralcalculus 57,58–9,61irrationalnumbers 25–6

JJainism 22,117

KKant,Immanuel 45–6,48keystoning 74–5König,Julius 126Kronecker,Leopold 104,124,125

LLagrange,Joseph-Louis 42Lambert,Johann 26largestnumberpuzzle 8,13lawofcontinuity 35Leibniz,Gottfried 35,56–7,60,61,68,69light 92–5lightswitchpuzzle 9–10,14limits 63–5lineatinfinity 70,78–90linearperspective 71–5,85–90Liouville,Joseph 103–4,108,125Locke,John 44–5logarithmicfunction 67logic 34–5,48Lorenzetti,Ambrogio 72Lynds,Peter 41

MMadhavaofSangamagrama 58mathematicalexistence 129–30mathematicsassumptions 34–5relatedtophilosophyandreligion 32–4,123–6

mentalrepresentations 45modeltheory 68–9momentum 42motionparadoxes 2,37–42

Nnaturalnumbers 23,28,66,103nature,studyof 42negativecurvature 101,102negativenumbers 34Newton,Isaac 56–7,60,61,69LawofGravity 96–8

numbersystems 23–4,34,65,103–4,114–16,118–21numbers

countingwith 105–7infinite 43largest 8,13,20–3limitations 51–3

OOckham’srazor(WilliamofOckham) 44ontologicalargument 47–8optics 92–5orderedfields 67–8ordinalnumbers,transfinite 118–21Origen 47

PPainlevé,Paul 96paradoxesofinfinity 1–2,3,8–9,36–42parallellines 70–1,78–90patterns,cognitiveaffinityfor 35perceptions 35–6limitations 44–6

periodicfunctions 110–11perspectiveinart 71–5,85–90towardsthehorizon 75–84

Philoponus,John 43–4,46philosophyofinfinity 42–6relatedtoreligionandmathematics 32–4,123–6

physicalinfinity 91–102pi(π) 26,27,55Plancklength 52Plato 50Poincaré,Henri 96polygons,area 55–6,59positivecurvature 101potentialinfinity 2,43–4,45,51,63,121,124,128powerseries 58,67primenumbers 43probabilitytheory 128–9processes 36,126–9Proclus 44projectivegeometry 6,70,74–5,78proofs 12,18puzzles 8–12,13–18Pythagoras’stheorem 25,29

Rrainbows 92–5rationalgeometry 29–30rationalnumbers 23,25–6,29–31,34,108–10,114rayoptics 93–5realnumbers(R) 26,28–9,30–1,40,41,65,66–7,68–9,103,110,114–15,122–3relativity 98–9,102religion,relatedtophilosophyandmathematics 32–3,123–6religiousbeliefs 22,46–50Renaissanceart 70,72,74,86Robinson,Abraham 67–8Russell,Bertrand 33,41,107

SSaari,Donald 96sandgrainsintheuniverse 21–2Schwarzschild,Karl 102settheory 104–8,112–13,115–16,121–9Simplicius 44singularities 91–2,95,96,98–9snowflakes 20,21space 38–9,45–6divisibility 52infinity 36,100

specialrelativity 98

squarerootof2 25–7squarednumbers 10–11,15–16staircasepuzzle 8–9,13steadystatetheory 99–100Stolz,Otto 66–7symbolsforinfinity 4

TTaylor,Brooke(Taylorseriesofafunction) 58technology 4theology 33,46–50timedivisibility 41eternity 3,36perceptionof 35–6

Torricelli,Evangelista 20,21transcendentalnumbers 103–4,108–10,113–16,125transferprinciple 68–9transfinitecardinals 113–16,119,124transfiniteordinals 118–21trianglearea 59

Uuniqueness 49–50universe 3curvature 101–2origin 43,99–100size 22,100–1

VVermeer,Johannes 86,89–90volume 20,21

WWallis,John 59waveoptics 95Weierstrass,Karl 62–3well-orderingprinciple 28,50,119Whitehead,AlfredNorth 33Wittgenstein,Ludwig 33,123,127–9

ZZeipel,Edvardvon 96ZenoofElea 2,36,38–41Zermelo,Ernst 126Zermelo–Fraenkelaxioms 116zerocurvature 101,102ZhihongXia 96–7ZuGengzhi 57–8

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