BRIANCLEGG is the author of the highly acclaimedLightYears andThe First Scientist (published byConstable, 2002). While working for British Airways he set up the Emerging Technologies Group,responsible for researching cutting-edge technologies. He currently runs his own creative consultancybusiness.

PraiseforLightYears

Inthisextraordinarybook...Cleggsaccessiblewritingstylemanagestoencapsulatethelivesoflightsdiscipleswithhumorousandinterestinganecdotes...quiteawesome.

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Afascinatingbookonafascinatingsubject.SirPatrickMooreCBE

PraiseforTheFirstScientist

Thisisbothafascinatingbiographyandavividpictureofthemediaevalworldinwhichadvancesweremadeandcontroversiesflourished.

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CleggsbookstrikesagoodbalancebetweenwhatisknownaboutBaconslifeandwhatisinterestingabouthiswork.

FinancialTimes

Cleggdoeswellto...presentBaconasthegreatintellectualhewas.BenjaminWoolley,Guardian

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ABRIEFHISTORYOFINFINITY

TheQuesttoThinktheUnthinkable

BrianClegg

ROBINSONLondon

Constable&RobinsonLtd5556RussellSquareLondonWC1B4HP

www.constablerobinson.com

FirstpublishedintheUKbyRobinson,animprintofConstable&RobinsonLtd,2003

CopyrightBrianClegg2003

TherightofBrianCleggtobeidentifiedastheauthorofthisworkhasbeenassertedbyhiminaccordancewiththeCopyright,DesignsandPatentsAct1988

Allrightsreserved.Thisbookissoldsubjecttotheconditionthatitshallnot,bywayoftradeorotherwise,belent,re-sold,hiredoutorotherwisecirculatedinanyformofbindingorcoverotherthanthatinwhichitispublishedandwithoutasimilarconditionincludingthis

conditionbeingimposedonthesubsequentpurchaser.

AcopyoftheBritishLibraryCataloguinginPublicationdataisavailablefromtheBritishLibrary

ISBN978-1-84119-650-3eISBN978-1-47210-764-0

PrintedandboundintheEU

109

ToGillian,RebeccaandChelsea,andalsotoNeilSheldonattheManchesterGrammarSchoolforhavingpatienceoverthematteroffrogs

andlily-pads.

CONTENTS

1ToInfinityandBeyond2CountingonYourFingers3ADifferentMathematics4ThePowerofNumber5TheAbsolute6LabellingtheInfinite7PeekingundertheCarpet8TheIndivisibleMystery9FluxionWars10ParadoxesoftheInfinite11SetinStone12ThinkingtheUnthinkable13OrderversustheCardinals14AnInfinityofInfinities15MadnessandSanity16InfinitesimallySmall17InfinitytoGo18EndlessFascination

ReferencesIndex

ACKNOWLEDGEMENTS

ManythankstoallthosewhoprovidedinformationandassistanceinputtingthisbooktogetherincludingthestaffatSwindonCentralLibraryandtheBritishLibrary,ForrestChanofLiPACE(OpenUniversityofHong Kong), Professor David Fowler of the University of Warwick, Dr Jeremy Gray of the OpenUniversity,ProfessorDaleJacquetteofThePennsylvaniaStateUniversity,ProfessorEberhardKnoblochof the Institut fr Philosophie in Berlin, Professor Shaughan Lavine of the University of Arizona,ProfessorRudyRuckerofSanJoseStateUniversity,PeggyBrusseauandJamesYolkowski.

Thanksalsotomyagent,PeterCox,inhisconsistenteffortstoencouragepublishersthatfinanceandthe infinite are not incompatible, and to all thosewho have been involved at Constable&Robinson,particularlyJanChamierandPeteDuncan.

1TOINFINITYANDBEYOND

Inthisunbelievableuniverseinwhichwelivetherearenoabsolutes.Evenparallellines,reachingintoinfinity,meetsomewhereyonder.

PearlS.Buck,ABridgeforPassing

THEINFINITEISACONCEPTSOREMARKABLE,sostrange,thatcontemplatingithasdrivenatleasttwogreatmathematiciansovertheedgeintoinsanity.

IntheHitch-hikersGuidetotheGalaxy,DouglasAdamsdescribedhowthewritersofhisimaginaryguidebookgotcarriedawayindevisingitsintroduction:

Space,itsays,isbig.Reallybig.Youjustwontbelievehowvastly,hugely,mind-bogglinglybigit is.Imean,youmaythinkitsalongwaydownthestreettothechemist,butthatsjustpeanutstospace.Listen...andsoon.Afterawhilethestylesettlesdownabitanditstartstellingyouthingsyouactuallyneedtoknow...1

Infinitymakesspaceseemsmall.Yetthisapparentlyunmanageableconceptisalsowithuseveryday.Mydaughterswerenoolderthan

sixwhentheyfirstbegantocountquickerandquicker,endingwithablurofwordsandatriumphantcryofinfinity!Andthoughinfinitymayintruthmakespaceseemsmall,whenwetrytothinkofsomethingasvastastheuniverse,infiniteisaboutthebestlabelourmindscanapply.

Anyonewhohasbrokenthroughtheboundsofbasicmathematicswillhavefoundthelittlesymbolcreepingintotheirwork(thoughwewilldiscoverthatthisdrunkennumbereightthathasfallenintothegutterisnottherealinfinity,butaghostlyimpostor).Physicists,withacarelessnessthatwouldmakeanymathematicianwince, are cavalierwith the concept.When Iwas studying physics inmy last years atschool, a commonsayingwas the toast rack is at infinity.This referred to anearbybuilding,partofManchesterCateringCollege,builtintheshapeofagianttoastrack.(Theresemblanceisintentional,arareexampleofhumourinarchitecture.Thecompanionbuildingacrosstheroad,whenseenfromtheair,lookslikeafriedegg.)Weusedthebricksonthisimaginativestructuretofocusopticalinstruments.Whatwe reallymeant by infinitywas that the buildingwas far enough away to pretend that it is infinitelydistant.

Infinityfascinatesbecauseitgivesustheopportunitytothinkbeyondoureverydayconcerns,beyondeverythingtosomethingmoreasasubjectitisquiteliterallymind-stretching.Assoonasinfinityentersthestage,itseemsasifcommonsenseleaves.Hereisaquantitythatturnsarithmeticonitshead,makingitseementirelyfeasiblethat1=0.Hereisaquantitythatenablesustocramasmanyextraguestsaswelikeintoanalreadyfullhotel.Mostbizarrelyofall,itisquiteeasytoshowthattheremustbesomethingthatisbiggerthaninfinitywhichsurelyshouldbethebiggestthingtherecouldpossiblybe.

Althoughthereisnosciencemoreabstractthanmathematics,whenitcomestoinfinity,ithasprovedhardtokeepspiritualconsiderationsoutoftheequation.Whenhumanbeingscontemplatetheinfinite,itisalmostimpossibletoavoidthingstheological,whetherinanattempttodisproveorprovetheexistenceofsomethingmore,somethinggreaterthanthephysicaluniverse.Infinityhasthisstrangeabilitytobemanythingsatonce.Itisbothpracticalandmysterious.Scientistsandengineersuseitquitehappilybecauseitworksbuttheyconsideritablackbox,havingthesamerelationshipwithitthatmostofusdowithacomputeroramobilephone,somethingthatdoesthejobeventhoughwedontquiteunderstandhow.

Thepositionofmathematiciansisratherdifferent.Forthem,modernconsiderationsofinfinityshakeup

the comfortable, traditional world in the same way that physicists suffered after quantum mechanicsshattered the neat, classical view of the way the world operated. Reluctant scientists have foundthemselves having to handle such concepts as particles travelling backwards in time, or being in twooppositestatesatthesametime.Ashumanbeings,theydontunderstandwhythingsshouldbelikethis,butasscientists theyknowthat if theyaccept thepicture ithelpspredictwhatactuallyhappens.Asthegreattwentieth-centuryphysicistRichardFeynmansaidinalecturetoanon-technicalaudience:

Itismytasktoconvinceyounottoturnawaybecauseyoudontunderstandit.Yousee,myphysicsstudentsdontunderstanditeither.ThatisbecauseIdontunderstandit.Nobodydoes.2

Infinityprovidesasimilartantalizingmixofthenormalandthecounter-intuitive.Allofthismakesinfinityafascinating,elusivetopic.Itcanbelikeadeer,spottedinthedepthsofa

thickwood.Youwillcatchaglimpseofbeautythatstopsyouinyourtracks,butmomentslateryouarenotsureifyousawanythingatall.Then,quiteunexpectedly,themagnificentanimalstalksoutintofullviewforafew,fleetingseconds.

Arealproblemwith infinityhasalwaysbeengetting through thedenseundergrowthofsymbolsandjargon thatmathematicians throw up. The jargon is there for a very good reason. Its not practical tohandle thesubjectwithoutsomeuseof thesenear-magical incantations.But it isverypossible tomakethem transparent enough that theydontget in theway.Wemay thenopenupclearviewson thismostremarkable of mathematical creatures a concept that goes far beyond sheer numbers, forcing us toquestionourunderstandingofreality.

Welcometotheworldofinfinity.

2COUNTINGONYOURFINGERS

AlexanderweptwhenheheardfromAnaxarchusthattherewasaninfinitenumberofworlds;andhisfriendsaskinghimifanyaccidenthadbefallenhim,hereturnsthisanswer:

Do you not think it a matter worthy of lamentation that when there is such a vast multitude of them, we have not yetconqueredone?

Plutarch,OntheTranquillityoftheMind

COUNTING A SEQUENCE OF NUMBERS, one after the other, is a practice that is ingrained in us fromchildhood.Thesimple,step-by-stepprogressionof thenumerals issostrong that itcanbesurprisinglydifficult to break out of the sequence. Try counting from one to ten as quickly as you can out loud inFrench(oranotherlanguagewhereyouknowthebasics,butarentparticularlyfluent).Nowtrytokeepupthesamespeedcountingbackdownfromtentoone.Theresultisusuallyhesitation;therhythmbreaksas we fumble around for the next number. We are tripped up trying to untangle that deep-seatedprogression.

Numbersequenceshavebecomeembeddedinourculture,oftenasthefocusofchildhoodrhymes.Themostbasicofthesearesimplememoryaidsdatingbacktoourfirstattemptstocount:

One,two,bucklemyshoe,Three,four,knockonthedoor,Five,six,pickupsticks,Seven,eight,laythemstraight,Nine,ten,abig,fathen,Eleven,twelve,diganddelve,Thirteen,fourteen,maidsacourting,Fifteen,sixteen,maidsinthekitchen,Seventeen,eighteen,maidsinwaiting,Nineteen,twenty,myplatesempty.

Theres a degree of desperation in some of the later rhymes in the sequence, but also a fascinatingreminder of a world that has disappeared in its imagery of shoe buckles andmaids. The repetitious,hypnoticrhythmhelpsdrivethenumbervaluesintoplace.

Otherdoggerel ismoreorientedtosingingthanthebasicchantedrepetitionofone, two,bucklemyshoe,atypicalexamplebeing

One,two,three,four,five,OnceIcaughtafishalive,Six,seven,eight,nine,ten,ThenIletitgoagain.

Equally valuable for practicewith counting are songs likeTenGreen Bottles, running downwards tomakeachildsgraspofthenumbersmoreflexible.

Butnumberrhymesarentlimitedtohelpinguslearnthebasicsofcounting.Moresophisticatedversesaddsymbolismtosequence.Itsdifficultnottofeelthepowerofnumberscomingthroughinarhymelikethemagpieaugury.Thistraditionalverseform,familiartoUKchildrensTVviewersofthe1970sfromits use in the theme song of the magazine programme Magpie, links the number of magpies (oroccasionallycrows)seenatonetimewithapredictionofthefuture.Itsnotsomuchaboutcountingasaboutfortune-telling.

TheTVshowusedthefirstpartofacommonsanitizedversion:

Oneforsorrow,twoforjoy,Threeforagirlandfourforaboy,Fiveforsilver,sixforgold,Sevenforasecret,nevertobetold,Eightforawishandnineforakiss,Tenforamarriagenevertobeold.3

ButtheresmoreearthyrealisminthisearlyLancashirevariant:

Oneforanger,twoformirth,Threeforaweddingandfourforabirth,Fiveforrich,sixforpoor,Sevenforabitch[orwitch],eightforawhoreNineforaburying,tenforadance,ElevenforEngland,twelveforFrance.4

Manychildrendevelopafascinationwiththebasicsequenceofcountingnumbers.Onceyoungstershavetakenonboardtherulesfornamingthenumbers,itsnotuncommonfortheirparentstohavetobegthemtostopastheyspendaninordinateamountoftimecountingupandup.Perhapstheirintentionistogettotheend,tonamethebiggestnumber.Butthisisataskwhereasenseofcompletenessisnevergoingtobeachieved.Achildcouldcountfortherestofhisorherlifeandtherewouldstillbeasmanynumberstogo. It seems that children are fascinated by the order, the simple pattern of such a basic, step-by-stepnumberstring.

Itispartofhumannaturetolikeorder,toseepatternsevenwherenopatternsexist.Whenwelookatthestars,weimaginethattheyformconstellationsshapesthatlinkthesepointsoflightintoaskeletalpicture where in reality there is no link between them. You only have to consider the Centaurusconstellationinthesouthernsky.Itsbrighteststar,AlphaCentauri,isthenearesttoours,amerefourlightyearsdistant;thenextbrightestintheconstellation,BetaCentauri(orAgena),lies190lightyearsaway,more than45 timesmoredistant.Wearemistakenly linking together twoobjects that are separatedbyaround1,797,552,000,000,000kilometres.

OurownSunismuchclosertoAlphaCentaurithanAgenais,yetwewouldhardlythinkoftheSunandAlphaCentauriasformingapattern.AlphaandBetaCentauriarenomoreconnectedthanHoustonandCairoaresimplybecausetheylienearthesamelatitude.Oureyesandbrains,lookingforstructureinthemyriadofwinkingpointsinthesky,deceiveusintofindingpatterns.

Welookforpatternsprimarilytoaidrecognition.Ourbrainssimplifythecomplexshapesofapredatororofanotherhumanfaceintopatternstoenableustocopewithseeingthemfromdifferentdirectionsanddistances. In the sameway thatwe look for patterns in the physical objects around us,we appreciatepatternsinnumbers,and,ofthese,fewaresimplerandmoreeasilygraspedthantheseriesofwholeorcountingnumbers1,2,3,4,5,6,...

Theellipsisat theendof thesequence, thatcollectionof threedots... isashorthandthatstretchesbeyondmathematics,thoughwehavetobealittlecarefulabouthowitisbeingusedhere.Innormalusageitsimplymeansandsoonformoreofthesame,butmathematicians,morefussythantherestofus,takeitspecificallytomeanandsoonwithoutanyend.Thereisnopointatwhichyoucansaythesequencehasstopped,itjustgoeson.Andon.Andon.

From the earliest days of pre-scientific exploration of both the naturalworld and theworld of themind, such chains of numbers were examined with fascination. They are prime inhabitants of thelandscapeofmathematics,as richanddiverseasanyfamilyofanimals in thebiological terrain.Some

sequencesarealmostassimpleasthebasiccountingnumbers,forexampledoublingthepreviousnumberinthesequencetoproduce

1,2,4,8,16,32,64,128,...

Butthingsdonthavetobesoneatlyordered.Youcanhavesequencesthatvaryindirection,buildinginadance-likemovement,alternatingtwostepsforwardandonestepback:

1,3,2,4,3,5,4,6,5,7,...

Oreachvaluecanbethesumoftheprevioustwo,theso-calledFibonaccinumbers:

0,1,1,2,3,5,8,13,21,...

Moreover,wecanmoveonfromadditionandsubtraction towilderexcursions thatmultiply, takingofflike a flock of birds surprised on a lake: the squares, the original counting numbers multiplied bythemselves,forinstance,

1,4,9,16,25,36,49,...,

orthefast-acceleratingprogressofasequencethatmultipliesthetwopreviousentries,

1,2,2,4,8,32,256,8192,...

MostofthesetypesofsequencewereknownbytheGreekphilosopherswhofirstponderedthenatureofnumbers.Butoneparticularclassseemedparticularlytohavefascinatedthem.Thesewereprogressions,notofwholenumbers,butoffractions.

Thesimplestfractionalsequencetakeseachwholenumberandmakesitthebottompartofthefraction:

Thissequenceofnumbers isnotparticularlyspecial. Ifwewere toaddeachtermto thenext, the totalwouldgrowwithoutlimit.ButtheGreekphilosophersnoticedaverydifferentabizarrelydifferentbehaviourwhentheymadeatinychange.Insteadofusingthecountingnumbersinthebottompartofthefraction,thenewsequenceisformedbydoublingthebottompartofthefraction.Theresult,

hasaverystrangeproperty,strangeenoughforthephilosopherZenotouseitasthebasisfortwoofhisstill-famousparadoxes.

WeknowverylittledirectlyofZenoswork.Allhiswritingwaslostexceptforafewhundredwords(andeventheattributionofthosetohimisquestionable).Whatremainsisthesecond-handcommentaryofthelikesofPlatoandAristotle,whowereanythingbutsympathetictoZenosideas.WeknowthatZenowasastudentofParmenides,whowasbornaround539BC.ParmenidesjoinedthesettlementatEleainsouthernItaly.TheruinsofthisPhoeniciancolonycanstillbeseenoutsidethemodern-dayItaliantownofCastellammare di Velia. It was here that the Eleatic school pursued a philosophy of permanent,unchangingonenessbelievingthateverythingintheuniverseisasitis,andallchangeandmotionisbut

illusion.For all we know, Zenomay have contributedmuch to Eleatic philosophy, but now he tends to be

rememberedasamathematicalone-hitwonder.Whathascometous,eventhoughonlyasadimreflectionin the comments of others, is his fascination with tearing apart the way we think of motion. ThisdemonstratesafundamentalbeliefoftheEleatics,thedenialoftheexistenceofchange,buteveninthisindirectformitseemspossibletodetectametaphoricalglintinZenoseyeasheputsuphisarguments.The later writers who pass on the paradoxes point out that this was a youthful effort, and though theintention of describing them this way was to be scornful, in fact there is a very positive element ofyouthfulchallengeintheseideas.

Inall,fortyofZenosreflectionsonthestaticuniversehavebeenrecorded,butitisfourofthemthatcontinuetocapturetheimagination,anditisthesefourthatparticularlyimpingeontheconsiderationofmotionandofthatstrangesequenceofnumbers

ThemoststraightforwardofthefourtellsthestoryofAchillesandthetortoise.Achilles,arguablythefastestmanofhisday,theequivalentofamodernsportsstar,takesontheponderouslyslowtortoiseinarace.ConsideringtheresultofarathersimilarraceinoneofAesopsfables(roughlycontemporarywithZenos paradoxes), its not too surprising that the tortoise wins. But unlike the outcome of the racebetween the tortoise and the hare, this unlikely result is not brought on by laziness and presumption.InsteaditisthesheermechanicsofmotionthatZenousestogivethetortoisethewinnerslaurels.

Zenoassumes thatAchilles iskindenoughtogive the tortoisean initialadvantageafterall this ishardlyaraceofequals.Heallowsthetortoisetobeginsomeconsiderabledistanceinfrontofhim.Inafrighteningly small time (Achilles is quite a runner), our athletic hero has reached the point that thetortoisestartedfrom.Bynow,though,howeverslowthetortoisewalks,ithasmovedonalittleway.Itstillhasalead.Inanevensmalleramountoftime,Achillesreachesthetortoisesnewpositionyetthatextratimehasgiventhetortoisetheopportunitytomoveon.Andsotheendlessracecarriesonwiththephysicalequivalentofthosethreedots,Achilleseternallychasingthetortoisebutneverquitecatchingit.

Anotheroftheparadoxes,calledthedichotomy,isverycloselyrelated.Thisshowsthatitshouldbeimpossibletocrossaroomandgetoutofit.Beforeyoucrosstheroom,youhavetoreachthehalf-waypoint.Butyoucantgetthereuntilyouvereachedone-quarterofthewayacrosstheroom.Andthatcantbereachedbeforeyougettotheone-eighthpointandsoon.Thesequence

givesaninsightintowhatishappening.Youareneverevengoingtoreachthe ,becauseyouhaventfirstreachedthe ,becauseyoudidntgettothe thmarker,andsoonandyoucannevergetstartedbecauseyoucantdefinetheinitialpoint.Youcanrundownthesequence,splittingthedistancefurtherandfurther,forever.Which is the first point you reach?Wecant say, and so,Zeno argues, you cannever achievemotion.

Its easy todismiss these twoparadoxesbypointingout thatAchilles stepsdonotget smaller andsmaller,so,assoonasheiswithinapaceofthetortoise,hisnextstridewilltakehimpastit.Thesamegoes for our escape from the room, but in reverse the first step you takewill encompass all of thesmaller parts of the sequence up towhatever fraction of theway across the roomyou get to.But thatmisses the point of the story. Zeno, after all, was trying to show that the whole idea of motion as acontinuousprocessthatcouldbedividedupasmuchasyoulikewasuntenable.

ItishelpfultoputZenosparadoxofAchillesandthetortoisealongsideoursequence

WhenweconsiderAchillesandthetortoise,letsmaketheratherrashassumptionthatthetortoisemoveshalfasfastastheathlete(maybeAchilleswashavingabadday,orthetortoisewasonsteroids).TheninthetimeAchillesmovedametre,thetortoisewouldhavemovedhalfametre.InthetimeAchillesmadeupthathalfmetre,thetortoisewoulddoanextraquarter.AsAchillescaughtupthequarter,thetortoisewouldstruggleonaneighth.Allofasudden, thosenumbersare lookingveryfamiliar.But theparallelisntallthatmakesitinteresting.Becauseoncewestartaddingupthenumbersinthatseries,somethingstrangehappens.Letslookatthesameseriesoffractions,butaddeachnewfiguretothepreviousonetomakearunningtotal.Nowwehave

Howeverfaryou take theseriesyouendupwith1andsomethingmore,somethingmore that isgettingcloserandclosertoanother1(making2inall),butneverquitegettingthere.Itdoesntmatterhowfaryougo,theresultremainslessthan2.Thinkofthebiggestmultipleof2youcanpossiblyimaginewellgiveitamade-upname,athrupple.Thenbythetimeyoureachityouwillhave

butnotquite2.Givingahugenumberanarbitrarynamelikethruppleisntquiteasbizarreasitseems.Thebiggest

numberaroundwithanon-compoundname,thegoogol,doesntjustsoundlikeachildishname,itactuallywasdevisedbyachild.Accordingtothestoryofitsorigin,AmericanmathematicianEdwardKasnerwasworkingonablackboardathomein1938andforsomereasonhadwrittenoutthenumberbelow.Thatlookslikeagoogol,saidhisnine-year-oldnephewMiltonSirrota.Andthenamestuck.ThatpartofthestoryseemsunlikelyitsmorecrediblethatKasnerwassimplylookingforanameforanumberbiggerthananyonecansensiblyconceiveandaskedyoungMiltonforsuggestions.Eitherway,agoogol is theentirelyarbitrarynumber:

10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000

or1with100noughtsafterit.

Backatthesequence ,whatwefind,then,isthathowevermanytimesweaddanitemtothelist,wewillneverquitereachatotalof2.AssoonasAchillesstridebreaksthisapparentbarrierthetortoisesleadisdonefor,buttheseriesitselfcantbreakoutofitscloserandcloserapproachto2.

Amodern equivalent of the translation of this sequence into a physical reality, anAchilles and thetortoiseforthetwenty-firstcentury,wouldbetoimagineaseriesofmirrorsreflectingaparticleoflight,aphoton, eachmirror half the distance from the next, set in a spiral. The obvious paradox here is thathowevermanymirrorsyoucramin,howevermanyreflectionsyouallowfor,thelightwillonlytravelalimiteddistance.Butthereisanother,moresubtleconsideration.Whathappenstothephotonattheendoftheprocess?Wheredoesitgo?Weknowthatafterthefirstmirroritbeginstospiralinwards,butlightisincapableofstopping.Thephotonmustcontinuetravellingat300,000kilometrespersecond.Sowhere

doesthisparticlegoto?

Figure2.1Thedisappearingphoton.

Inpracticethisseemstobeoneofthosequestionsthathavenomeaning,becausewewouldhavetostraybeyondtheboundsofphysicalrealitytoreachtheendresult.Evenifitwerepossibletodivideupspaceintoinfinitelysmallchunks(wewillreturntothepracticalitiesofthisinthepenultimatechapter),weknowthatphysicalmatterisntacontinuoussubstancethatcanbedividedforever,alwayscomingupwithasmallerversionofthesamething.Reflectionoflightdependsonaninteractionbetweenthephotonoflightandanelectroninthematerialthatitisbeingreflectedoff.Eventually,asthemirrorsgotsmallerandsmallertofitintotheremainingspacetheywouldhavetobecomesmallerthananatom,smallerthananelectronatwhichpointreflectioncouldnolongeroccurandthephotonwouldcontinueonitsjourneywithoutbeingfurtherreflected.

Inamomentwewillreturntothatelusivesequence ,whichseemsasifitshouldaddupto2butneverquitemakesit,butfirst,forneatness,letsfinishoffZenoandhisparadoxes.

TheothertwomentalpicturesZenopaintedattacktheconventionalviewofmotioninadifferentway.PerhapsmostfamousofallisZenosarrow.Hedescribesanarrow,flyingthroughspace.Afteracertainamountoftimehaspassed,itwillhavemovedtoanewposition.Butnowletsimagineitataparticularinstantintime.Thearrowmustbesomewhere.Youcanimagineithanginginspacelikeasingleframefromamovie.Thatswherethearrowisat,say,exactlytenminutespasttwo.

Thisiswherethevisualimageryoffilmcomesinhandy.Thereisnowavideotechniqueavailablethatseems to make time stop. An object freezes in space as the camera pans around it, showing it fromdifferentdirections.(Infactwhatishappeningisthataseriesofcamerasatdifferentanglescapturethemoment, and their images are linked together by a computer to produce the illusion that the camera ispanning.)Imaginethatwedothisforreal.Westoptimeatthatoneinstantandviewthearrow.

Nowletsdothesameforanotherarrowthatisntmovingatall.Wewontworrytoomuchabouthowthissecondarrowissuspendedinspace.Ifitsreallyaproblemforyou,wecouldworktheparadoxwithtwo trucks, onemoving andone stationary, butZenoused an arrow, so Id like to stickwith that.ThequestionZenoasksis:howdowetellthedifference?Howdoesthearrowtellthedifference?Howdoesthefirstarrowknowthatitmustchangepositionsinthenextmoment,whiletheother,seeminglyidenticalinoursnapshot,staysstill?

ItsawonderfulproblemwhereAchillesandthetortoisecanseemalmostamatterofsemantics,thisone isa realposer. Infact,arguably itwasntpossible to trulyanswerZenountilanothergreat thinkeralsoimaginedbringingsomethingtoastop.Wehavetoleapforward2,400yearstofindAlbertEinsteinlyingonagrassybank,lettingthesunlightfilterthroughhiseyelashes.

RelaxingonasummersdayataparkintheSwisscityofBerneintheearly1900s,Einsteinimaginedfreezingabeamof light,notby takinga snapshot in timebutby ridingalongside it at the samespeed.

Now,asfarasEinsteinwasconcerned,thelightwasstopped.Thisisjustthesameasifyouwereinacar,andatruckwasalongsideyougoingatexactlythesamespeedfromyourviewpointthetruckwouldnotbemoving.ButEinsteinsdaydreamofstoppinglightwasarealproblem,becausethemechanismthattheScottishphysicist JamesClerkMaxwellhadused to explain theworkingsof light some fiftyyearsbeforewouldnotallowit.

Maxwells explanation of light depended on electricity and magnetism supporting each other in acontinuousdance,aninterplaythatcouldonlyworkatonespeed,thespeedoflight.Ifitwerepossibleforlight to slow down, the delicate interaction of electricity andmagnetismwould collapse and the lightwouldceasetoexist.

If Maxwell was right, and Einstein assumed that he was, light could only continue to exist if ittravelledatthatonespeed.AndsoEinsteinmadetheremarkableleapofthinkingthatlightwouldalwaysmoveatthatoneparticularspeed,howeverfastyoumovedtowardsitoralongsideit.Wherewenormallyaddspeedstogetherwhenwemovetowardsanothermovingobject,ortakespeedsawayfromeachotherwhenwetravelinthesamedirectionassomethingelse,lightisaspecialcasethatwontplaythegame.This idea is at the heart of special relativity,which then, conveniently,makesZenos arrow less of aproblem.BecauseitsoonbecameapparenttoEinsteinthatfixinglighttoasinglespeed(around300,000kilometresasecond)changestheapparentnatureofreality.

Einsteincombinedthisfixedspeedoflightwiththebasicequationsofmotionthathadstoodunchangedsince Newtons time. He was able to show that being in motion would change the appearance of anobject.Everyday, apparently fixed, properties such as size,mass and the passage of time itselfwouldseemdifferentforanobserverandtheobjectthatwasobserved.Thiseffectisnotveryobviousuntilyougetclosetothespeedoflight,butitisalwaysthere.Einsteinwasabletoshowthatmovingatanyspeedwith respect to something else changed both how you looked to that observer and how the observerlooked to you.Being inmotion changes yourworld.And this provides amechanism for the arrow toknow that it is inmotion,because theworld looksdifferent in comparisonwith theworld seenbyastaticarrow.

RelativityalsodealswithZenosfinalandleastobviousparadox.This,calledthestadium,imaginestworowsofpeoplepassingeachotherinoppositedirections.Anathleteinoneofthesetworowswillhavepassedtwiceasmanybodywidthsintheotherrowashewouldhavedoneifhehadbeenrunningbesidestationarypeople.Eventhoughthereisaphysicallimittohisspeed,heseemstohaverunattwicethat rate. In a way, what happens here is that our better understanding of relativity vindicates Zenosviews.Hewasusingtheexampletoshowthattheideaofmovingatacertainspeedismeaninglessandhewasright.Itisonlyeverpossibletosaywearemovingatacertainspeedrelativetosomethingelse.

So with Zenos four paradoxes in place, lets return to that sequence: Is there anumber of fractions we could add together that would bring the total up to 2? It would hardly besurprising ifyousaidan infinitenumber.Butwhatdoes thatmean? Ifyouaskchildrenwhat infinity iswhentheyarefirstintroducedtoit,theyoftensayitisthebiggestnumberthattherecanbe.Yetwevealreadysaidthat,howevermanytimesyouaddononeofthesedecreasingfractions,youwillneverquitemake2.Ifinfinityisthebiggestnumberthatthereis,thensurelyweendupwithafractionthatis

stillnottheroundfigureof2.This was a problem that proved an irritation for the ancient Greek philosophers who spent time

thinkingaboutsuchseries.But toget into theframeofmindofaZenoorPlato,weneedto lookat thenumbers as the Greeks did themselves. The very wording of the sentence at the end of the previousparagraphwouldhavebeenmeaninglesstotheGreeks,becausetheydidnothavethesameconceptofafraction thatwedo. In fact,numbers ingeneralwerehandled inaverydifferentwaywhen theancientGreekcivilizationflourished.

3ADIFFERENTMATHEMATICS

Ifthedoorsofperceptionwerecleansedeverythingwouldappeartomanasitis,infinite.WilliamBlake,TheMarriageofHeavenandHell(AMemorableFancy)

INMODERNTIMESWETAKEFORGRANTEDtwoveryspecifictoolsofmathematics.Oneisbeingabletotreatnumbersaspartofanumberlineanimaginaryhorizontalrulerwith0inthemiddle,stretchingoffintothedistanceineitherdirectionwithaseriesofmarkersalongthewayrepresenting1,2,3,etc.,asyouhead to the right and the negative numbers to the left. Between each of thewhole numbers sit all thefractionsyoudcaretoimagine(andquiteafewyouwouldnt).Forchildreninschooltoday,thenumberlineisanexplicitpartoftheirearlylearningofarithmetic.Whensubtracting,forexample,theywilloftendrawanumberlineandjumpbackdownitbytheappropriatevalue.Forolderreaders,thenumberlinemighthavebeenimplicit,butitwasstillpresentinoureducation.

Thesecondtoolthatcomesintoplaywhenweworkonthenumbersfromthenumberlineistotakeanalgebraicapproachusingaseriesofsymbolsthatimplychangingandcombiningthenumbers(orplacemarkersrepresentinganundefinednumber)invariouswaystocometoaresult.If,forexample,wewanttoworkoutapricewithsalestaxfromabasicprice,wecouldhaveatablewitheverypossiblebasicpriceonitandlookuptheappropriatevalue.Alternativelywecouldsaythat

Price=basicprice(1+taxrate).

Thisisasimpleshorthand,sayingthat,toproducetheprice,wereplacebasicpricewithavalue,taxratewithanothervalueandcarryoutthecalculation.Thisis,inessence,thesecretofalgebra.Tokeepthingsshort(andunfortunatelyalsomakingthemlessreadable),mathematicianshaveapassionforusingsingle letters (if necessary straying into Greek and other alphabets) for these labels, so they wouldprobablysaysomethinglike

p=b(1+t),

buttheeffectisthesame.Similarly,insteadofhavingtoexplicitlywritedownhowmuchenergyeverymovingobjecthasasaresultofitsspeed,physicistssaythatthiskineticenergyis mv2,wheremisthemassoftheobjectandvisitsvelocity.Itdoesntmatterwhatthevaluesofmandvare,putthemintotheequationanditwillwork.Lookatanymathematicalpaperwritteninthelast300yearsanditislikelytodependuponthealgebraicmanipulationofmanysymbolsinordertoreachaconclusion.

The Greeks, it seems, at least at the height of their philosophical work, took a totally differentapproach.Theirmathematicswasnotalgebraic,butgeometric.Itwasthemathematicsofimagery.

Studies of the brain have shown that it can operate in two different modes, often labelled byassociationwith its two halves left-brain and right-brain thinking. Left-brain thinking is systematic,analytical and numerical. It is the principal approachof science andbusiness andmathematics.Right-brainthinkingisholistic,focusingoncolourandimagery.Itunderliesthethoughtprocessesthatareusedin thearts.Ever since thedevelopmentof the scientificmethod, the left-brainapproachhasdominatedmathematics.Yetallofusarecapableofbothleft-andright-brainthinking,andmanypeoplecanhandleinformationbetterfrompicturesthanfromwordsandnumbers.Ourmodernprocessestendtopushusinto

left-brainmode,butfortheancientGreeksthingswerequitedifferent.Mathematicallyatleast,left-brainthoughtseemstohavebeenalientothem.Toperformarithmetictheythoughtinpictures.

Withdiagramsitwaspossibletohandleconceptslikeequalto,largerthan,smallerthan,halfthesizeof,twicethesizeof,andsoon,withouteverresortingtoanumber.WheretheGreeksdidusenumbersmathematicallytheywouldnotproduceastringofdigits1,2,3,4,5,...aswewouldhave,butratherasetof descriptions of objects corresponding to the numbers. In his book The Mathematics of PlatosAcademy,DavidFowlerof theUniversityofWarwicksayswecould thinkof thecorrespondingGreekseriesas

duet,trio,quartet,quintet,...5

(asoloitemisomittedasitwouldhavebeentreatedseparatelyfromthelargernumbers).Similarly,thesequencewewouldwriteas

wouldeffectivelybeformulatedinwordsas

thehalfpart,thethirdpart,thequarterpart,thefifthpart,...

DavidFowlerpointsoutthatweshouldnotmentallytranslatetheseintofractionswhattheGreekswerethinkingofbythehalfpart,forinstance,wasashapethatwassmallerthanaunititembyafactorof2.WenotonlyhavetothinkoftheGreekfractionsvisibly,as(forexample)asquarewhichishalftheareaof a rectangle twice the size, but also to remember that the relationship was seen in terms of wholenumbers.Whatwas observedwas that the larger shapewas twice the size of the smaller, not that thesmallerwashalfthesizeofthelarger.

If quantitieswerewrittendownat all (diagramswereused to put acrossmuchof themathematicalinformationthatwasrequired)theresultwouldbeaclumsywordphrase,allthemoreconfusingastheconventionofthetimewastowritewithnogapsbetweenthewords.RevielNetzpointsoutinhisTheShapingofDeductioninGreekMathematicsthattowriteA+B=C+D(somethingthatwouldnotbeliterally done, as there was no algebraic concept of replacing an unknown number with a letter), theEnglishequivalentoftheGreekformulawouldbe

THEAANDTHEBTAKENTOGETHERAREEQUALTOTHECANDTHED6

This doesnt mean that the ancient Greeks didnt have symbols for numerals. It would have madeaccountingverydifficultiftheywererestrictedtowords,andinfactthereweretwoapproachestakentowritingnumbersdown.Theolderform,technicallyreferredtoasAcrophonicbutsometimescalledAttic,was avery simple approach.Thenumber1was just avertical bar |,with furtherbars added to countupwards.Fiveletterswereusedtostandinforbiggervaluestoavoidtoomanybarsbeingwritten:5waspi(thesymbolusedissometimesconfusedwithgamma(),as itwasoftenwrittenwithashorterrightdownstroke),10wasdelta(),100eta(H),1,000chi(X)and10,000mu(M).

ThetermAcrophoniccomesfromthewaytheparticularsymbolswerechosenaseachonewasthefirstletterofthenameoftherelevantnumber.Mfor10,000,forinstance,wasshortformyriad(or,moreprecisely,forwhereiswhatwewouldnowcallthelowercaseversionofM,mu).

Tocombinesymbols,theycouldeitherbestrungtogether,Romanfashion,so42wouldbe,or

theycouldhaveasuperscriptasamultiplier.Thiswouldmeanthattoget52youwouldmultiply5by10andadd2:.

However,itis,perhaps,misleadingtosaythatthiswastheearlyGreeknumbersystem,becausethereisstrongevidencethatitwashardlyeverusedforanythingbutinscriptions.Ithasntbeenfoundinworksof mathematics. It seems likely that this Acrophonic system was used in early accounting you canimagineascribetickingoffitemsonatabletbutitmayneverhavebeenatoolofthemathematicians.

Likethe(now)morefamiliarRomannumerals,theAcrophonicapproachisveryclumsywhenitcomestodealingwithlargernumbers.Amoreflexiblesystemwasadopted,andwascertainlyinwidespreaduseby the time Greek culture was centred on Alexandria. This used the letters of the Greek alphabet torepresent numbers ( = 1, = 2, = 3, and so on), but hadnt got the conveniencewe have of usingplacing to represent tens, hundreds, and so on. This meant that yet more letters were called in tosymbolizebiggernumbersupto900(,forinstance,was80).Afterthat,astrokewasplacedatthestartofthenumbertotakeitaboveathousand,andthecapitalmu(M)wasmaintainedfromtheAcrophonicsymbolfor10,000.LargernumbersstillwereachievedbywritinganumberabovetheM,whichtookthesystemuptoamyriadmyriad100,000,000.Allinall,quiteamess.Thingsdidntgetanyeasierwhendealingwith fractions.Aswehavealreadyseen, thesewere thoughtofaswholenumberparts.A fifthwasntreallyafraction,somuchasathingofwhichthewholeisfivetimesbigger.Thismeantthattoget a fraction like youwould have to have three lots of the fifth part. These parts usually used thestandardnumberwithanaccent.Athirdpart,forexample,wouldbeshownsomethinglike.However,justincasethingswereseemingtooeasy,didnotmeanahalfpart,buttwothirdparts.Bizarrely,ahalfhadseveralspecialsymbols,onelikealightningzigzag ,butnonerelatedtothealphabet.Thestandardorderinwhichpartswerelistedwas

puttingtheoddtwothirdpartscharacteratthefrontofthesequence.Towriteoutanumberthatcombinesanintegerwithoneormoreparts,theintegerwouldbewritten

first,thenacombinationofparts.So,8 wouldberepresentedby ,combining8, and .(Ifyouarefamiliarwith theGreek alphabet and arewonderingwhy is 8 and not 7, some letters that had beendroppedfromthealphabetwerestillusedasnumbers.Therewasadoublegamma,digamma ,whichwasusedas6doublethevalueofgammaat3.)

Ofcourse,producingthesecombinationsoffractionsisnotinitselftrivial.Wecanworkoutthat plusmakes byturningthehalfintotwoquartersandaddingthetoppartsofthefractions butwehave to constantly remind ourselves that theGreekswerent dealingwith fractions aswe know them.Withouttheeasyvisibilityofthepartsofourfractions,theGreeksresortedtothestandardapproachofanyonefacedwithacomplexcalculationusingamechanicalaid.

These days we would turn to a spreadsheet or a calculator; then, the solution was to consultlaboriouslyworked-outtablesprovidingtherelationshipbetweenparts.Eachfractionwouldbebrokendownintoasetofcomponentstoenablemanipulation.

WhentheGreekswerelookingataninfiniteseriesliketheonebehindZenosparadoxes,

itisquiteprobablethatthiswouldbewritteninwordsasthehalf,thequarter,theeighthpart,andsoon,but it seems equally likely that any observations on the sum of the series would have been made

visually,aspositionsmarkedonalineorrectangle,ratherthanconceivingthesumaswedotodayandthis approach is just as effective (in fact, perhaps more so) in demonstrating the tendency towards alimitingvalue.

Its easy to be dismissive of this visual approach as crude compared to the surgical precision ofalgebraic equations, but it can still be extremely enlightening. Id like to take you to a classroom inRusholmeon theoutskirts ofManchester in the early1970s.Amathematics lessonhas just finished inwhich the teacherhaddiscusseda strangeproblem. It involveda frog,hopping towardsa comfortablelily-pad for a rest. The frogwas growing tired, so each hopwas half the length of the previous one.Becauseofthis,thefrogwouldnever,evermakeitontothelily-pad.Thestudentswerenttolditatthetime,butitwasarewritingofoneofZenosparadoxes,beingusedtoillustrateoneoftheconceptsbehindthemathematicaltoolofcalculus.

Figure3.1Avisualrepresentationoftheseries

Whentheotherstudentshadlefttheroom,oneremained.Hewentuptotheteacherandconfessedthathejustcouldnotacceptthisidea.Surelyifthefroghoppedaninfinitenumberoftimes,itwouldgettoitsdestination,howeversmallthehopsbecame,wouldntit?Themathematicsteacherwasagoodteacher,andratherthandismissthiswithitdoesntmatterthatyoudontunderstandit,justuseit,hegavesomeextrathoughtsonthesubject,andnextdayturnedupwithabookthatwentintomuchmoredetailontherelationshipbetweenthefrogandlily-padexampleandsequencesoffractions.Yetdespiteallthathelp,despitethebook,itwasntuntilthestudentsawavisualrepresentation,similartoFigure3.1above,thatitallclickedintoplace.

Ihavetoconfessthatthestudentwasme.Takingavisualapproachisnotalwaysaweakness.Aswehaveseen,manyofusthinkbettervisuallythanwithnumbers.Itswhygraphscanbemuchmorepowerfulthan tables of numerals atmaking financial data comprehensible.When thephysicistRichardFeynmanwasworkingontheproblemofexplaininghowlight interactedwithmatter,hisprimary toolwasnotacomplexequationbuta(relatively)simplediagram,nowcalledtheFeynmandiagraminhishonour.Thiswasntawayofsimplifyingthetheory,quantumelectrodynamics,forignorantstudents,butratherthetoolFeynmanandhiscolleagueschosetoexploreandexplainwhatwashappening.TheancientGreekvisualapproachmaynotbeidealformuchofmathematics,butitwouldbeaseveremisunderstandingtodismissitaspurelyprimitive.

TotheancientGreeks,dealingwiththeseseriesofnumberswasonething,butinfinityitselfwasquitea differentmatter. The nearest word they had to infinitywas apeiron, but this was a termwith quitedifferentconnotationstothefeelingswenowhavefortheinfinite.Itmeantwithoutbounds,outofcontrolanduntidyapeironwasnotapleasant thing.Althoughitdidnothavethesamemeaning,apeironhadsimilarnegativeassociationstothosewenowgivetochaosinnormalEnglish(asopposedtothequitedifferent,neutralmathematicaldefinition).

ItiseasytoseewhyinfinitycausedtheGreekssuchdiscomfortwhensomeseriesaretakentotheinfiniteextremetheyproduceveryworryingresults.Mostofuscanmakethementalleapthatwillallow

tograduallyapproachasumof2whenthecomponentsareaddedtogether.Ineffect,thesumwouldreach2ifthenumberofitemsintheserieseveractuallymadeittoinfinity.Butwhatistheoutcomeofaddinguptheelementsoftheapparentlyevensimplersequence

1,1,1,1,1,1,...

Atfirstglance,thiswillobviouslyaddto0,aseachalternateentryinthesequencecancelsoutthenext.Buttheresultyougetdependsonhowyoudoyourpairingup.Itsperfectlytruethat

(11)+(11)+(11)+...

willaddto0aseachpairinbracketscanceleachotherout.Butsimplymovingthebracketsgetsus

1+(1+1)+(1+1)+(1+1)+...

Again,thebracketscancelthemselvesout,leavingnot0but1asatotal.Thesameseriesseemscapableofaddingupto0or1.

Thisparadoxicalresultissometimestransformedintothevisualexampleofturningalightonandoffaninfinitenumberoftimesdoesitendupofforon?Orisitinsomestrangeindeterminatestate,neitheroffnoron?Suchastateofbeingisnotasunlikelyasitsounds.AnexactparallelliesbehindthecuriousconditionofSchrdingerscat,ahypotheticalanimalbelovedofquantumphysicists.

Whentheworldisexaminedatthequantumlevelofindividualparticlesofmatter,normalexpectationsof behaviour go out of the window. Such particles, the electron for instance, can be in a number ofeither/orconditions,ratherlikethewaythatabitinacomputercanbe0or1,oracoincancomeupheadsortailswhenflipped.Anexampleofsuchaconditionisspinanelectroncanbeinoneoftwostates, described as spin-up and spin-down (in other words, spinning clockwise or anticlockwise,dependingonwhichwayupyouconsidertheaxisroundwhichitisspinningtobe).Technicallywedontknowiftheelectronisactuallyspinningaround,butitdoeshavethesetwoopposingstates,andthelabelseemedonethatwaseasytovisualize,soitstuck.

Whenaparticleisproduced,saybyatomicdecay,andexamined,itwillbefoundtobeinoneorotherof thesestates randomly.But thestrangenessof thequantumworldmeans that,before it isexamined,aparticle,whichcould turnout tobe ineitherstatewhenmeasured,exists inbothstatessimultaneously.Thisso-calledsuperpositionofstatesmeansthat,forexample,anelectronwouldbespinningbothwayssimultaneouslyuntil it isobserved,atwhichpoint thesuperposition issaid tocollapseand itbecomesspin-uporspin-down.

Bemused by this concept, the Austrian physicist Erwin Schrdinger imagined a cat whose veryexistencedependedonthestateofaparticle.Hishypotheticalexperimentalanimalisputintoaboxwitha lethal device that is triggered by a particle going into one of two available states. Until the box isopened,Schrdingerargued,ifsuperpositiontrulyexistsandtheparticleisyettobeobserved,thenitisinbothstatesatonceandthatmeansthatthecatshouldalsobeinbothpossiblestates.

Thispoorimaginarycatissimultaneouslydeadandaliveuntiltheboxisopenedandthesuperposedstatescollapse.Somearguethatthedevicethatchecksthestateoftheparticleisdoingtheobserving,andso the superposed states are collapsed as soon as the lethal device is given the option of triggering.Othershavesuggested that thisexampledemonstrates that thewhole ideaof superposedstates isnota

validone.Yettheconceptispartoftheonlyconsistentlyperformingmodelthathasyetbeenputforwardtoexplainwhatisactuallyobserved.

Similarly, it seems that, in the infinite seriesof switchingonandoff, the lightbulbendsup inbothstatesand in this case there isnoway toobserve it andcollapse the superposition,becausewecannever get to the end of the infinite series. (Realistswill point out that the bulbwill have blown longbeforeaninfinitenumberofswitchingsisreached,soitwilldefinitelybeoff.)

Itseems,then,thatthesequence

1,1,1,1,1,1,...

addsuptoboth0and1,andunlessweacceptthat0=1itbecomesobviousthatinfinityismorethanalittletrickytodealwith.TheGreekanswerwasnottodealwithitatall,orrathertohideitawayinafashionthatmadeitunnecessarytothinkaboutit.Aristotle,whoseviewswerealsotoshapesomuchoflaterWesternscientificthinking,madeadistinctiononthematterofinfinitythatwastoproveuseful,butalsowasafudgethatmadeitpossibletoavoidtherealissueforacoupleofthousandyears.

UnlikemanyoftheGreekphilosopherswhowouldhaveahandinthedevelopmentofourideaoftheinfinite,weknowafairamountaboutAristotle.Hewasbornin384BCinStagirusinnorthernGreeceandattheageof17joinedPlatosAcademyinAthens,wherehewastoremainfor20years.

Thewordacademyhasbecomesufficientlypartofthenormalvocabularythatitisworthpointingoutthatthiswastheoriginal.Plato,bornin427BCinAthens,hadfoundedhisschoolwhenheleftmilitaryserviceacoupleofyearsbeforeAristotlesbirth.Likemanywhohaveservedinthemilitaryhewasnonetoo impressedwith the capabilities of politicians, and set up his schoolwith the specific intention ofimprovingthequalityofthoseinpubliclife.Theschool,basedinAthens,wassituatedinagroveoftreesbelongingtoamannamedAcademos,hencethenameAcademy(andthelater,rathertwee,expressionthe groves of academe). By the time Aristotle arrived, the Academy was well established andrespected.InfactitwouldremainoperationaluntilAD529,aremarkable900yearsofexistenceeventhe Universities of Oxford and Cambridge, the oldest English-speaking universities (schools wereestablishedatOxfordfirst,butbothgottheircharterstogetherin1231),wontcatchupuntilthe2130s.

Politicalupheavalseems tohavemade itwise for the thenwell-establishedAristotle tomoveforawhiletotheislandofAssos,beforereturningtoAthenstosetuphisownschool,theLyceum.ButitwaswhilehewasstillatPlatosAcademythatAristotlewrotehisPhysicsinwhichhewastotrytotidyupthedangerousapeiron.

Aristotle introduced the idea of something being potentially infinite. So when he counted up thesequence of whole numbers he accepted that it had no limit, because there was no point at which itstoppedbutwouldnotconcedethatiteveractuallyreachedinfinity,asthiswasanunrealconcept.Theworldseemedtoneedsomethinglikeinfinitytime,afterall,hadnoobviousend,anditseemedpossibletotakeastraightlineanddivideitinhalfasmanytimesasyoulikeyetbydevisingapotentialinfinity,Aristotlecouldsafelyplacetheinfiniteoutsideofrealityandsimplyheadtowardsit.

Thisisnotanarbitrarydecision.Aristotleargueshisreasoningverycarefully.Hebeginsbymakingitclearwhytheinfiniteisnotjustamatterofcasualorevenphilosophicalcuriosity,butapracticalpieceofknowledgerequiredforanunderstandingofnature:

Thescienceofnatureisconcernedwithspatialmagnitudesandmotionandtime,andeachoftheseatleastisnecessarilyinfiniteorfinite...Henceitisincumbentonthepersonwhospecializesinphysicstodiscusstheinfiniteandtoinquirewhetherthereissuchathingornot,and,ifthereis,whatitis.7

Aristotle then begins his exploration of infinity by summing up the opinions of his predecessors andcontemporaries.BoththePythagoreansandPlato,hetellsus,considerinfinitytobeanentityinitsownright, rather than just an attribute of something else. The Pythagoreans, it seems, considered what isoutsideheaventobeinfinite,whereasPlatothoughttherewasnothingoutsidetheboundsofheavenandtheinfinitemanagedtofitwithintheuniverse.Others,thoseAristotlereferstoasthephysicists,includingAnaxagorasandDemocritus(whocameupwiththeforerunnerofatomictheory),thoughtofinfinityasanattributeofasubstance,likeitscolour,ratherthanatrueentity.

Whyshouldpeoplebelieveininfinity?Aristotlegivesusfivereasons.Thesimplest isbasedonhisassertionthat timeis infinite,whichheseemstotakeasincontrovertible.But,forAristotle,onereasonstandsoutabovetherest:

Most of all, a reason which is peculiarly appropriate and presents the difficulty that is felt by everybody not only number but alsomathematicalmagnitudesandwhatisoutsidetheheavenaresupposedtobeinfinitebecausetheynevergiveoutinourthought.8

Inotherwords,astrongpointinfavouroftheexistenceofinfinityisthatwecantconceiveofthingsliketheseriesofcountingnumbersandtheextentoftheuniverseeveractuallycomingtoastop.

Withtheseargumentsinplace,Aristotlebeginstorevealhisownviews.Theinfinitecantbeathing,anymorethannumberormagnitudecanbeathing.Nextheasksifabodycanhavethispropertyiftherecanbesuchathingasaninfinitebody.Thishecountersbyhisdefinitionofabodyassomethingthatisboundedbysurfaceasinfinityprettywellbydefinitioncannotbecompletelyboundedoritwouldntbeinfinite,thisdoesntmakesense.(Aristotlecouldntsaythatinfinityhadnoboundsatall,though.Taketheseriesofwhole,positivenumbers.It is infinite,but ithasaclearstart,soithasat leastonebound, itsstartingpoint.)

Afterarathershakydiscussionofwhyaninfinitebody,shouldtherebesuchathing,couldneitherbecompoundnorsimple(andhencecouldnotexist),Aristotlehasprovedprettyfirmlyfromhisviewpointthatyoucanthaveabody that is infinite insize.But thenhe turns theargumentbackon itself. If therewerenoinfinity,thentimemusthaveabeginningandanend,linescantbeindefinitelydivided,numbersmuststop.Aristotlepresentsuswithachallenge:

If,then,inviewoftheaboveconsiderations,neitheralternativeseemspossible,anarbitermustbecalledin;andclearlythereisasenseinwhichtheinfiniteexistsandanotherinwhichitdoesnot.9

Andnow,at last,comesthatsubtlechimera,potential infinity, infinitythatneitherisnorisnt.Aristotleremindsusthatthewordiscouldmeaneithersomethingthatisactuallyinexistence,orsomethingthathasthepotentialtoexist.Perhapstheanswertotheproblemofinfinityisthatitonlypotentiallyexists.

But thephrase potentialexistence isambiguous.Whenwespeakof thepotentialexistenceofastatuewemean that therewillbeanactualstatue.Itisnotsowiththeinfinite.Therewillnotbeanactualinfinite.Thewordishasmanysenses,andwesaythattheinfiniteisinthesenseinwhichwesayit isdayorit isthegames,becauseonethingafteranotherisalwayscomingintoexistence.Forofthesethingstoothedistinctionbetweenpotentialandactualexistenceholds.WesaythatthereareOlympicGames,bothinthesensethattheymayoccurandthattheyareactuallyoccurring.10

Its a neat distinction, particularly in that last example.Whenwe say there areOlympicGames, youcould hardly arguewith this statement. ThereareOlympicGames: they happen every four yearswithmillionsofpeoplewatchingthem.Yetunlesswehappentobeattherightpointinthefouryearcycle,ifapassing alien asked showme theseOlympicGamesofwhichyou speak,wewouldhave aproblem.Theydoexist,butuntilthenextOlympiadtheyareonlypotential.Theyarecoming,buttheyarenthere.ThiswasthenatureofAristotlespotentialinfinityexcepttheInfiniteOlympiadneverarrives.AristotleendsBook3ofthePhysicswiththesewords:

Thisconcludesmyaccountofthewayinwhichtheinfiniteexists,andofthewayinwhichitdoesnotexist,andofwhatitis.11

The finalityof thisclosingphrase, solidlyputtinga full stopon infinityandmovingon toother things,wouldkeepmathematicianshappyallthewayuptothenineteenthcentury.Whetherinfinitywasrealorunreal,comprehensiblebythehumanorthedivine,wouldoccupyphilosophers,butAristotlesmoveofinfinityintothevirtualworldofthepotentialmadethemathematicswork.

Even so, this dismissal of the ultimate did not stop others from working with larger and largernumbers,asiftheyweretryingtofindawayofdescribingtheinfiniteitself.

4THEPOWEROFNUMBER

ToseeaWorldinaGrainofSand,AndaHeaveninaWildFlower,HoldInfinityinthepalmofyourhand,AndEternityinanhour.

WilliamBlake,AuguriesofInnocence,1

WEHAVESEENHOWACHILDgaveanarbitrary(andletsfaceit,childish)nametothevalue

10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000.

Agoogolonewithahundredzerosfollowing.Althoughtherearebiggernamednumbersderivedfromitagoogolplexisonewithagoogolnoughtsafterittherereallyisnoneedtogivenamestonumbersofthisscale.Atonelevel,itsjustagame,likethechildsattemptstocounthigherandhigher.Butitsworthalsobearinginmindthatnumbersthemselveswereconsideredtohavesignificantpowerinancienttimes,andtobeabletonamealargenumbercouldbethoughtofasamarkofauthority.

For the Greek philosophers of the school of Pythagoras (he of the theorem that has doggedschoolchildrenforhundredsofyears),numberbecame identifiedwithcreation.Specificnumbersweregivenparticularmeanings.Theschoolevenhadanumericalmotto:Allisnumber.

Pythagoraswasbornaround569BContheIonianGreekislandofSamos,setintheAegeanSea.Hisfatherwasamerchant,andtheyoungPythagorastravelledfrequentlywithhim.ThismadehimmorethanusuallyreceptivetotheideaoftravellingtoEgyptwhenitwassuggestedinhisearlythirtiesthatitwouldhelphiseducationandit isprobablytherethatPythagoraswasinfluencedinthedirectionthatwouldeventuallyleadtothecreationofhisschoolatCrotoninsouthernItaly.

ThePythagoreansconsiderednumberstobeamongthebuildingblocksoftheuniverse.Infact,oneofthe most central of the beliefs of Pythagoras mathematikoi, his inner circle, was that reality wasmathematical in nature. This made numbers valuable tools, and over time even the knowledge of anumbers name came to be associatedwith power. If you could name something you had a degree ofcontroloverit,andtohavepoweroverthenumberswastohavepowerovernature.

Particular properties were ascribed to each of the numbers from one to ten. These numbers haddifferent personalities somemale, some female. Often the nature associated with each number wasreflectedintheshapeofapatternofdotsmadeupfromthatnumber.

One,forexample,stoodforthemindemphasizingitsOneness.Twowasopinion,takingastepawayfromthesingularityofmind.Threewaswholeness(awholeneedsabeginning,amiddleandanendtobemore than a one-dimensional point), and four represented the stable squareness of justice. Five wasmarriagebeingthesumofthreeandtwo,thefirstodd(male)andeven(female)numbers.(ThreewasthefirstoddnumberbecausethenumberonewasconsideredbytheGreekstobesospecialthatitcouldnotformpartofanordinarygroupingofnumbers.)

This allocation of interpretationswent on up to ten,which for thePythagoreanswas the number ofperfection.Notonlywasitthesumofthefirstfournumbers,butwhenaseriesoftendotsarearrangedinthe sequence 1, 2, 3, 4, each above the next, it forms a perfect triangle, the simplest of the two-

dimensionalshapes.SoconvincedwerethePythagoreansoftheimportanceoftenthattheyassumedtherehad to be a tenth body in the heavens on top of the known ones, an anti-Earth, never seen as it wasconstantly behind the Sun. This power of the number ten may also have linked with ancient Jewishthought, where it appears in a number of guises the ten commandments, and the ten sefirot, thecomponentsoftheJewishmysticalcabbalatradition.

SuchnumerologyascribinganaturalorsupernaturalsignificancetonumberscanalsobeseeninChristian works, and continues in some new-age traditions. In theOpus majus, written in 1266, theEnglishscientist-friarRogerBaconwrote:

Moreover,althoughamanifoldperfectionofnumberisfoundaccordingtowhichtenissaidtobeperfect,andseven,andsix,yetmostofalldoesthreeclaimitselfperfection.12

Ten,wehavealreadyseen,wasallocatedtoperfection.SevenwasthenumberofplanetsaccordingtotheancientGreeks,while thePythagoreanshaddesignated six as thenumberof theuniverse.Six alsohasmathematicalsignificance,asBaconpointsout,becauseifyoubreakitdownintothefactorsthatcanbemultipliedtogethertomakeitone,twoandthreetheyalsoadduptosix.

123=6=l+2+3.

Baconreckonedthatthreewasevenmorespecialbecauseitwastheonlynumberthatwasthesumofallthepartsofitthatdiddivideintoit(one)andthepartsthatarentafactor(two).Arguablythisisabitofjustificationafterthefact,withBacontryingtoshowwhythree,representingtheHolyTrinityofFather,SonandHolySpirit,wassosignificant.Numerologynotonlycommentedonthesignificanceofnumbersbutalsousedthemasameansofdiviningthefuture,oftenconvertingwordsandnamesintonumberformandaddingtheresultstogethertoproduceaprediction.

Yet the significanceofnumberswasnot limited to these small,building-blockvalues.According toBuddhisttradition,whenGautamaBuddhawasayoungman,hehadtonameawholeseriesofnumbers,starting in themillionsandworkingup tofigurescomparablewithagoogol,withnamesforeachstep.Buddhawasableeffortlesslytoname,forexample,

100,000,000,000,000,000

asachobya,anamewhichinisolationhasverylittlevalue(youareunlikelytogointoashopandaskforanachobyaofanything),butwhichdemonstratedhiscommandovertheimpossiblylarge.13

Archimedes,writingatleast300yearsearlierthanthestoryofBuddhasnumberingskills,alsotooktheopportunitytoplungeintotheactoftaminghugenumbers,numbersthatwereofsuchagreatsizethattheymightwellhavebeenconsideredclose to infinite.Despite thecumbersomeGreeknumbersystem,whichhadnodirectwayofreferringtoanynumbergreaterthanamyriad(10,000),Archimedesmanagedtoestimatethenumberofgrainsofsandthatwouldfilltheuniverse.

Weknow little that is factual aboutArchimedes life.Abiographyproduced by an unknownwriterHeracleideswasmentioned in otherGreek sources, but it no longer exists. (This could have been thesameHeracleidesthatArchimedesreferstoinoneofhisworks,makinghimratherlikeanancientGreekBoswell to ArchimedesJohnson). But Archimedes is thought to have died in the Roman attack onSyracuse,andif,aslegendhasit,hewasaround75atthetime,thiswouldputhisbirthdatearound287BC.

Atthetimeofhisdeath,Archimedesmaywellhavebeenbestknownforthedevicesheplannedandperhapsconstructed.Forhim,accordingtotheRomanwriterPlutarch(writingagood350yearslater),

thesewerediversionsofgeometryatplay14butthedefendersofSyracusemighthaveseenthingsverydifferently.TheyweresaidtohaveusedarangeofmechanicalenginesdevisedbyArchimedestoattacktheRomanfleet,andeventohavecontemplated(butneverbuilt)asun-powereddeathrayintheformofhugefocusingmirrorstoconcentratetheSunslightondistantRomanshipsandsetthemalight.

ButitwasnotArchimedesundoubtedmechanicalgenius,norhisdetailedstudyofshapesandthree-dimensional bodies that brings him into the story of infinity, but rather one of the strangest books inhistory. It iscalled theSand-reckoner, and seems tobeoneofArchimedes lastworks. ItwaswrittenafterhehadreturnedtohisnativeSyracusefromthetimeheisbelievedtohavespentinAlexandria,andisaddresseddirectlytothekingofSyracuse,Gelon(orGelo).

Archimedes seems to have had a close relationshipwith bothGelon and the previous kingHieron(therehasbeenasuggestion thathewasrelated to theroyalfamily15),actingasasortof interpreterofphilosophicalmysteries to thecourt.TheSand-reckoneropenswithachallenge to theking tobemoreimaginativethanmanyofhiscontemporaries:

Therearesome,KingGelon,whothink that thenumberof thesand is infinite inmultitude;andImeanby thesandnotonly thatwhichexistsaboutSyracuseandtherestofSicilybutalsothatwhichisfoundineveryregionwhetherinhabitedoruninhabited.Againtherearesomewho,withoutregardingitasinfinite,yetthinkthatnonumberhasbeennamedwhichisgreatenoughtoexceeditsmagnitude.Anditisclearthattheywhoholdthisview,iftheyimaginedamassmadeupofsandinotherrespectsaslargeasthemassoftheEarth,includinginitalltheseasandthehollowsoftheEarthfilleduptoaheightequaltothatofthehighestmountains,wouldbemanytimesfurtherstillfromrecognizingthatanynumbercouldbeexpressedwhichexceededthemultitudeofthesandsotaken.

But Iwill try toshowyoubymeansofgeometricalproofs,whichyouwillbeable to follow, that,of thenumbersnamedbymeandgivenintheworkwhichIsenttoZeuxippus,someexceednotonlythenumberofthemassofsandequalinmagnitudetotheEarthfilledupinthewaydescribed,butalsotothatofamassequalinmagnitudetotheuniverse.16

Fightingtalk.Andaseriouschallengetothelargelynon-numericalGreekmathematics.Archimedeshasfirst to definewhat hemeans by universe.Ourmodern picture is a vast expanse of space, coveringbillionsoflight-yearsandfilledwithahugecollectionofgalaxies.Atthetime,thepopularpicturewasofa(relatively)smallsphere,centredontheEarth,containingnestedspheresthatheldtheplanetsinplacewith the stars in the final outer sphere. In passing, Archimedes tantalizingly refers to the first knownsuggestion that in fact theSun rather than theEarthwas at the centre of things, the heresy forwhichGalileo would eventually be imprisoned. It is tantalizing because the work Archimedes refers to nolongerexists,soitsratherasiftheonlyviewwehadofaShakespeareplaywasadescriptionbyacritic:

Nowyouare aware that universe is thenamegivenbymost astronomers to the spherewhose centre is the centreof theEarth andwhoseradiusisequaltothestraightlinebetweenthecentreoftheSunandthecentreoftheEarth.

This is thecommonaccountasyouhaveheard fromastronomers.ButAristarchusofSamosbroughtoutabookconsistingofsomehypotheses,inwhichthepremiseleadstotheresultthattheuniverseismanytimesgreaterthanthatnowcalled.HishypothesesarethatthefixedstarsandtheSunremainunmoved,thattheearthrevolvesaroundtheSuninthecircumferenceofacircle,theSunlyinginthemiddleoftheorbit,andthatthesphereofthefixedstars,situatedaboutthesamecentreastheSun,issogreatthatthecircleinwhichhesupposestheEarthtorevolvebearssuchaproportiontothedistanceofthefixedstarsasthecentreofthespherebearstoitssurface.17

ItissometimessaidthatArchimedesdismissedthetheoryAristarchusputforward,becausehisnextlineis:

Nowitiseasytoseethatthisisimpossible...18

ButtobefairtoArchimedes,allheispointingoutisthatitisntmeaningfultosaythattheratioofthesizeoftheuniversetotheEarthsorbitisthesameastheratioofthecentreofaspheretoitsradius.Afterall,saysArchimedes,

sincethecentreofthespherehasnomagnitude,wecannotconceiveittobearanyratiowhatevertothesurfaceofthesphere.19

However,ratherthandismissAristarchusoutofhand,insteadArchimedesoffersapracticalinterpretationofwhatAristarchushadsaid:

Wemust,however,takeAristarchustomeanthis:sinceweconceivetheEarthtobe,asitwere,thecentreoftheuniverse,theratiowhichtheEarthbearstowhatwedescribeastheuniverseisthesameastheratiowhichthespherecontainingthecircleinwhichhesupposestheEarth to revolve bears to the sphere of the fixed stars. For he adapts the proofs of his results to a hypothesis of this kind, and inparticularheappearstosupposethemagnitudeofthesphereinwhichherepresentstheearthasmovingtobeequaltowhatwecalltheuniverse.20

The language is more than a little convoluted, but reasonably logical. Archimedes supposes thatAristarchusmeantthatthesizeofhisuniversewasinthesameproportiontothesizeoftheorbitoftheEarthasthesizeofthetraditionaluniverse(theorbitoftheSun)wastothesizeofthetraditionalcentreoftheuniversetheEarthitself.AnditisthisinterpretationofAristarchusuniversethatArchimedesgoesontofillwithvirtualsand:

Isaythenthat,evenifasphereweremadeupofthesand,asgreatasAristarchussupposesthesphereofthefixedstarstobe,Ishallprovethat,ofthenumbersnamedinthePrinciples,someexceedinmultitudethenumberofthesandwhichisequalinmagnitudetothespherereferredto,providedthatthefollowingassumptionsaremade...21

Like all goodmathematicians,Archimedes is about to tell us his givens, or axioms, the fundamentalassumptionsonwhichhis theory isbuilt.Thebookhe refers toas thePrinciples isone thatno longerexists,probablytheonehementionsearlierastheworkwhichIsenttoZeuxippus.Fromthecontextitclearlydealtwiththenamingofnumbers,butbeyondthatweknownothingaboutit.

To achieve an approximation of the size of the universe, Archimedes makes a total of four initialassumptions:

1.ThattheperimeteroftheEarthisabout3,000,000stadesandnotgreater;2.ThatthediameteroftheEarthisgreaterthanthediameteroftheMoonbutlessthanthediameterof

theSun;3.ThatthediameteroftheSunisabout30timesgreaterthantheMoon;and4.ThatthediameteroftheSunisgreaterthanthesideofthechiliagon(apolygonwith1,000equal

sides)inscribedinthegreatestcircleinthesphereoftheuniverse.

Letsconsiderthevalidityofthoseassumptions.Eachstadiumofmeasurementwasaround180metresinlength(westilloccasionallyrefertodimensionsintermsofstadium-likemeasurements,suchasaroundthe size of a football pitch). So his guess that the perimeter of the Earth is about 3,000,000 stadesmakestheEarthscircumferencearound54,000kilometres.Thisisntbadatall,bearinginmindthatthekilometrewasoriginallydefinedasl/10,000thofthedistancefromthepoletotheequatorthroughParismaking40,000kilometresagoodroundfigureforthis.

Archimedessecondassumption,thattheEarthwasbiggerthantheMoonbutsmallerthantheSunwas,ofcourse spotonandnot totallyobviouswhenyouconsider that theSunand theMoonhavealmostidentical apparent diameters in the sky (otherwise solar eclipses would not be anywhere near asspectacularastheyare).

ThingsgetalittlemoreshakyonthethirdassumptionthatthediameteroftheSunisabout30timesthatoftheMoon.Infact, theMoonisaround3,480kilometresindiameter(aquarter thesizeof theEarth),while theSunmeasuresupat1,392,000kilometresacross,400timesthesizeof theMoon.It ishardlysurprisingthatArchimedesgotitwrongthescaleisquiteincrediblebutevenso,itonlymadehima

factoroftenout.ThefinalassumptionwasthisratherstrangeassertionthatthediameteroftheSunisgreaterthanthe

sideofthechiliagoninscribedinthegreatestcircleinthesphereoftheuniverse.Inotherwords,itwasassumed that the whole distance around the Suns orbit (remember this was assuming a traditionaluniverseoftheSunsorbitoftheEarth)was(roughly)nomorethan1,000timesbiggerthanthediameteroftheSun.Again,thiswassurprisinglygood,aslongasyoureversetherelationshipbetweentheSunandtheEarth.

Archimedes based this assumption on one report and one experiment. Aristarchus, he tells us,discoveredthattheSunappearedtobeabout1/720thpartofthecircleofthezodiac.22ArchimedesthenimprovedonAristarchusmeasurementbytakingalongrodwithasmallcylinderordiscfixedtotheend.Justaftersunrise,whenhethoughtitwassafetolookat theSun(itneveris,butat least itreducedtherisk), he held up the rod andmoved it backwards and forwards until the disc just covered the Sunssurface.HefoundthattheangletheSunsdiscmadewasbetween1/164thand1/200thofarightangle,andfromthis,withreasonablysimplegeometry,hewasabletocomeupwithhisestimate.

In reality, theEarths orbit around theSun (the closestwe can come toArchimedes sphereof theuniverseaveragesaround939,000,000kilometres,makingtheorbitaround675timesthesizeoftheSun,comfortablyclose toArchimedesfactorof1,000.Ofcourse, thissizebearsnoresemblance to the truemagnitudeofthesolarsystem,letalonethetrueuniverse.

Withthoseassumptionsinplace,ArchimedescouldshowgeometricallythathisuniversewasnomorethanamyriadtimesthesizeoftheEarth.Hethensomewhatarbitrarilydecidedthattherearenomorethanamyriadgrainsofsandinthesizeofapoppyseed,andthatapoppyseed(aconvenientlysmallitem)isnotlessthanl/40thofafingersbreadthinsize.Nowhecangetontocountingthosesandgrains.ButjustwritingitoutintheGreekformatwouldbeanimpracticaltask.Archimedeshastodeviseanewsystemofnumbersthatcouldcopewithsuchahugequantity.

Withamyriadas thebiggestnamednumbers, itwaseasyenoughtodescribeamyriadofmyriads100,000,000. Rather than give 100 million a specific name, he described all the numbers up to100,000,000asnumbersofthefirstorder.Amyriadmyriadsthenbecametheunitforasecondorder,sothebiggestnumberinthesecondorderwasthisunitofunits,whatwewouldrepresentas100,000,0002,namely100,000,000100,000,000,or10,000,000,000,000,000.Thehighestnumberinthesecondorderbecame the unit of the next, going up to 100,000,0003, and so itwent on, all theway to 100,000,000multipliedbyitself100,000,000times.

But that wasnt enough for Archimedes. The whole set of numbers up to this enormous value, hedescribed as the first period, forming a next level of the hierarchy. The next period began with thisnumberandwentupto100,000,000timesbigger.Andsoon.Justtowriteoutthelastnumberofthefirstperiodwouldrequire1followedby800,000,000zeros this isno toynumberingsystem.Archimedesthenworkedouta relativelysimplemechanismforperformingarithmeticaloperationson theseorders.Finally,hewasreadytoworkupwards,countingthegrainsofsandinasphereofafingersbreadth(nomore than10unitsof the secondorder), and soon,up in leapsof ahundred, througha stadium to thediameterofthetraditionaluniverse.

Hehadalreadyassumedthatthisuniverse,basedontheimaginedorbitoftheSun,wasnobiggerthan10,000,000,000stades,whichleftthenumberofgrainsofsandtofilltheuniverseasnomorethan1,000unitsoftheseventhorderprettytriflinginthegrandschemeofhisnumberingsystem.IfyoupreferredAristarchus bigger universe, it was still amere 10,000,000 units of the eighth order. Towrap it up,Archimedescomments:

Iconceivethatthesethings,KingGelon,willappearincredibletothegreatmajorityofpeoplewhohavenotstudiedmathematics,butthattothosewhoareconversanttherewithandhavegiventhoughttothequestionofthedistancesandsizesoftheEarth,theSunandMoonandthewholeuniverse,theproofwillcarryconviction.AnditwasforthisreasonthatIthoughtthesubjectwouldbenotinappropriateforyourconsideration.23

ItwouldseemfromArchimedeswordsthatGelonwasmorefamiliarwithmathematicsthanmanykings.But the conclusion of the Sand-reckoner doesnt answer whyArchimedes should want to number thegrainsofsandthatwouldfilltheuniverse,ordeviseanumberingsystemthatwentsofarbeyondtheneedsofsuchaproblem.Althoughwehavenoevidencetobackthisup,it isquiteprobablethatArchimedeswasrespondingtothelooseremarksofthosewhowouldquitehappilyrefertothenumberofgrainsofsandonabeachas infinite tomake it clear that thesewereonly triflingnumberscomparedwith thenumbers that mankind (and Archimedes in particular) could name putting them nowhere near thatuncomfortableconcept,theinfinite.

ItalsogaveArchimedesachancetoplaywithastronomicalideasinawaythathadntreallybeendoneup to then.After all, his fatherwas himself an astronomer.When considering the diameter of theSun,Archimedescommentsintheonlyknownreferencetohisfather:

Eudoxus declared [the diameter of the Sun] to be nine times as great [as the diameter of theMoon], and Pheidiasmy father twelvetimes.24

ItisinterestingthatArchimedesthengoesontosaythatAristarchusthoughtthisratiotobebetween18and20times,while

IgoevenfurtherthanAristarchus,inorderthatthetruthofmypropositionmaybeestablishedbeyonddispute,andIsupposethediameteroftheSuntobeabout30timesthatoftheMoonandnotgreater.25

Perhaps part of the reason for this whole book was not so much to deny the outlandish ideas ofAristarchusbuttosupportthemtoshowthatmovingfromatraditionaluniversetothatofAristarchusdidnotrequireahugechangeinthought,merelyinvolvingamovementinsizefromtheseventhordertotheeighth.Whatever,Archimedeshadmappedoutsomeseriouslylargenumbers.

Although theSand-reckoner would remain something of a curiosity, references to it do crop up inpoeticwriting.JohnDonne, thegreatEnglishpoetof theseventeenthcentury,at least twice referred toArchimedesefforts,makingtheapparentlyhugequantityArchimedeshadnamedsmallincomparisontoaspectsofGod,andinthisparticularLentensermon,theendlessnessofsufferinginhell:

Menhavecalculatedhowmanyparticulargrainesofsand,wouldfillupallthevastspacebetweentheEarthandtheFirmament:andwefind,thatafewlinesofcipherswilldesigneandexpressthatnumber...Butifeverygrainofsandwerethatnumber,andmultipliedagainbythatnumber,yetallthat,allthatinexpressible,inconsiderablenumber,madeupnotoneminuteofthiseternity...26

ItsaverygraphicpictureofDonnesimaginingofaneternityofsuffering.Inanothersermon,this timefromEasterDay,hecomparesthelimitationsofarithmeticasheknewit(nodoubtdevoidofinfinity)tothelimitlessnessofGod:

HowbarrenathingisArithmetique!(andyetArithmetiquewilltellyou,howmanysinglegrainesofsand,willfillthishollowVaulttotheFirmament)Howempty a thing isRhetorique! (andyetRhetoriquewillmake absent and remote thingspresent to your understanding)Howweakathingispoetry!(andyetPoetryisacounterfaitCreation,andmakesthingsthatarenot,asthoughtheywere)Howinfirme,howimpotentareallassistances,iftheybeputtoexpressethisEternity.27

Donnemighthave limited sandgrains tohis sermonsbut a successorwas tobring them intopoetry.Archimedesattempts todealwithhugenumbersseemreflected in theopening linesofoneofWilliamBlakesbest-knownpoems. Itwaswritten in the firstyearsof thenineteenthcentury,butnotpublished

until1863,some36yearsafterBlakesdeath,inacollectionofhisworkbyDanteGabrielRossetti.TheauthorofTyger,TygerandJerusalemdemonstrateshisdistasteformansabilitytoabusenature,

andproducedanaugury that is significant forusnot as a counting rhyme like themagpieauguries thatappearonpage6,butratherinitsuseofasandgrainandinfinityamongtheimagesthatemphasizehowthesmallactionsofmanimpactallthevastnessofcreation.IvegonebeyondtheusualfirstfourlinesofAuguries of Innocence below to show the context of that first stanza that is so often plucked out inisolation:

ToseeaWorldinaGrainofSandAndaHeaveninaWildFlower,HoldInfinityinthepalmofyourhandAndEternityinanhour.

ArobinredbreastinacagePutsallHeaveninarage.AdovehousefilldwithdovesandpigeonsShuddersHellthroallitsregions.AdogstarvdathismastersgatePredictstheruinofthestate.AhorsemisusdupontheroadCallstoHeavenforhumanblood.EachoutcryofthehuntedhareAfibrefromthebraindoestear.Askylarkwoundedinthewing,ACherubimdoesceasetosing.ThegamecockclippdandarmdforfightDoestherisingSunaffright.EverywolfsandlionshowlRaisesfromHellahumansoul.28

ForBlakeandDonnebeforehimitwasnotonlytheattempttocountthegrainsofsandintheuniversebuttheinfiniteitselfthatwaslinkedtoGod.Therewasanalmostinevitablejuxtapositionoftheultimateinnumberandtheultimatereality,thecreatorofeverything.Thiswasnonewconcept,asweshallseeinthenextchapter.

5THEABSOLUTE

ForthepastandboundlesseternityduringwhichGodabstainedfromcreatingmanissogreat,that,compareitwithwhatvastanduntoldnumbersofagesyouplease,so longas there isadefiniteconclusionof this termof time, it isnotevenas ifyoucomparedtheminutestdropofwaterwiththeoceanthateverywhereflowsaroundtheglobe.

StAugustine,TheCityofGod,BookXII.12

BLAKESRELIGIOUSBELIEFSwereneverentirelyorthodox,butitwouldnotbesurprisingifhisconceptofinfinityembracedGodorevenifhehadequatedtheinfinitewithGod.Itisaverynaturalthingtodo.Ifyoubelieveinadivinecreatorwhoismorethantheuniverse,unboundedbytheextentoftime,itshardnottomakeaconnectionbetweenthisfigureandinfinityitself.

Therehavebeenexceptions,philosophersandtheologianswhowereunwillingtomakethis linkage.SuchwastheancientGreekdistasteforinfinitythatPlato,forexample,couldonlyconceiveofanultimateform, theGood, that was finite. Aristotle saw the practical need for infinity, but still felt the chaoticinfluenceofapeiron was too strong, and so came up, aswe have seen,with the concept of potentialinfinitynotarealthing,butadirectiontowardswhichrealnumberscouldhead.ButsuchideaslargelydiedoutwithancientGreekintellectualsupremacy.

Itishardtoattributethebreakawayfromthistraditiontooneindividual,butPlotinuswasoneofthefirstoftheGreekstomakeaspecificone-to-onecorrespondencebetweenGodandtheinfinite.BorninAD 204, Plotinus was technically Roman, but was so strongly influenced by the Greek culture ofAlexandria(hewasbornintheEgyptiantownofAsyut)thatintellectually,atleast,hecanbeconsideredaGreekphilosopher.Heincorporatedamysticalelement(largelyderivedfromJewishtradition)intotheteachingsofPlato,sparkingoffthebranchofphilosophysincecalledNeoplatonismasfarasPlotinuswas concerned, though, he was a simple interpreter of Plato with no intention of generating a newphilosophy.

Heargued thathis rather looselyconceivedgod, theOne, had tobe infinite, as to confine it to anymeasurable number would in some way reduce its oneness, introducing a form of duality. This waspresumablybecauseonceafinitelimitwasimposedonGodtherehadtobesomethingelsebeyondtheOne,andthatmeantthecollapseofunity.

The early Christian scholars followed in a similar tradition.Although theywere aware thatGreekphilosophywasdevelopedoutsideoftheChristianframework,theywereabletotakethecoreofGreekthought,particularlytheworksofAristotleandPlato,andfixit intoastructurethatmadeitcompatiblewiththeChristianityofthetime.

StAugustine,oneofthefirsttobringPlatosphilosophyintolinewiththeChristianmessage,wasnotlimitedbyPlatos thinkingon infinity. In facthewas toarguenotonly thatGodwas infinite,butcoulddealwithandcontaininfinity.

AugustineisoneofthefirstChristianwritersaftertheoriginalauthorsoftheNewTestamentwhoseworkisstillwidelyread.BorninAD354inthetownofTagaste(nowSoukAhrasinAlgeria),Augustineseemedoriginallytobesetonaglitteringcareerasascholarandorator,firstinCarthage,theninRomeandMilan.AlthoughhismotherwasChristian,hehimselfdabbledwiththedualistManicheansect,butfounditsclaimstobepoorlysupportedintellectually,andwasbaptizedaChristianin387.Heintendedatthispointtoretireintoamonasticstateofquietcontemplation,buttheChurchhierarchywasnotgoingtolet amanofhis talentsgo towaste.Hewasmadeapriest in391andbecameBishopofHippo (now

AnnabaorBona,ontheMediterraneancoast)in395.The Roman Empire that Augustine knew was the sad relic of an earlier age, already poised for

collapse. In410,Rome itself fell to theattackofAlaric theVisigoth,abarbarianwho, itwas thought,shouldnothavestoodachanceagainstthemightofRomethathadstoodunconqueredforsolong.Ratherthanacceptthedeclinethathadbeenunderwayforsomanyyears,somecommentatorsblamedthenewofficialreligion,Christianity.ItwasonlywhentheseweakChristianshadtakenover,itwasargued,thatRomewaslost.

AlthoughRomeitselfhadfallen,theRomanhierarchyinthemoreremoteareaswasinitiallyrelativelyunaffected. An imperial commissioner, Marcellinus, who was visiting the region, asked Augustine towritearesponsetothenegativefeelingsthatChristianitywasgeneratingtomakethepeoplebothmoreawareof thenatureofChristianityandhowitssuperiorityover thedisplacedRomanreligionsmade itanythingbutasuitablescapegoatforthefallofRome.

Theresultwasalong,sprawlingseriesofbooksthatcametogetheroveraround15yearsasTheCityofGod.Bythetimethiscollectedworkwascomplete,Augustinewasalreadyarespectable71(hediedfive years later). The book, published in parts as eachwaswritten, became an immediate success. Itreceived what must be one of the first extant laudatory reviews, fromMacedonius, Vicar of Africa.Macedoniuspraisesthebookforits

happycombinationoferuditionandargument,doctrineandeloquence29

AsAugustinebuiltonhisinitialtheme,TheCityofGodbecamemuchmorethanaresponsetodoubters,concernedbythefallofRome.Thefinalcollectionof22booksisratheranattempttopulltogetheratotalviewofChristianityasitwasthenunderstood.TotheviewpointofamodernChristianitmightseemoddthat Augustine should concern himself at all with numbers, but it should be remembered that he wasstronglyinfluencedbyPlatoandtookonboardPlatostheoriesonthesignificanceofcertainnumbers.

Augustinesnumericalquestbegannotwithinfinitybutwiththenumbersix.HepointedoutthatinthebookofGenesiscreationtooksixdaystocomplete,yetthisdurationcouldntpossiblyhavebeenamatterofnecessity.Thisthoughtisquitelogical,thoughnotobviouswhenwearethinkinginournormal,finitewayofthetimeittakesustocompleteatask.But,asAugustineemphasized,thereisnoparticularreasonwhyGodwouldrequireaprotractedtime.Ifwearetoconsiderthepointlogically,anall-powerfulGodshouldbeabletoundertakethewholeactofcreationinaninstant.Instead,Augustineargued:

Theseworksarerecordedtohavebeencompletedinsixdays...becausesixisaperfectnumber...becausetheperfectionoftheworkswassignifiedbythenumbersix.Forthenumbersixisthefirsttobemadeupofitsownparts,i.e.,ofitssixth,thirdandhalf,whicharerespectivelyone,twoandthree,andwhichmakeatotalofsix.30

WhatAugustinemeanthere,asRogerBaconlaterechoed,isthattheonlynumbersthatdivideintosix(itsfactors)areone,twoandthree.Multiplythesefactorstogetherandyougetsix.Addthemtogetherandyoualsogetsix.Suchnumbers,madeupof thesumof theirfactorshadbeentermedaperfectnumbereversincethetimeoftheGreeks.Unlessyouconsiderthenumberone,whichisalwayssomethingofanoddity,sixisthesmallestvalueexhibitingthisperfectquality.

Augustine,thoughhappytousesuchscienceinthefurtheranceofhiswork,wasquicktoadmitthatthiswasnotaspecialistarea,andsohehadtousewhathesaidwithcare.Themodestywithwhichhepointsthisoutistouching.Aftertalkingaboutsix,healsosaidthatsevenisaperfectnumber,butforadifferentreason.Hewenton:

Muchmoremightbesaidabouttheperfectionofthenumberseven,butthisbookisalreadytoolong,andIfearlestIshouldseemtocatchatanopportunityofairingmylittlesmatteringofsciencemorechildishlythanprofitably.31

Lateron,inthenextbookofTheCityofGod,Augustinebegantoworktowardsthingseternal.Hestartedbylookingatdifferenttheoriesoftheoriginoftheworldthatwerecurrentatthetime,someofwhichhavesurprisingly strong echoes in present-day cosmological theories, frommultiple parallel universes to acyclicallyrebornuniverse:

Therearesome,again,whothoughtheydonotsupposethatthisworldiseternal,areofopinioneitherthatthisisnottheonlyworld,butthattherearenumberlessworlds,orthatindeeditistheonlyone,butthatitdies,andisbornagainatfixedintervals,andthistimeswithoutnumber.32

From this hemovedon toputGodalongside the conceptof infinity.Somepeople, he argued, say thatalmostbydefinition,theinfinitecantbehandledbyanymodeofknowledge.EvenGod,theysuggest,hasonly finite conceptions of his creations. Again, Augustine resorted to numbers to help out with thisargument:

Asfortheirotherassertion,thatGodsknowledgecannotcomprehendthingsinfinite,itonlyremainsforthemtoaffirm,inorderthattheymaysoundthedepthsoftheirimpiety,thatGoddoesnotknowallnumbers.Foritisverycertainthattheyareinfinite;since,nomatteratwhatnumberyousupposeanendtobemade,thisnumbercanbe,Iwillnotsayincreasedbytheadditionofonemore,buthowevergreatit be, and however vast be themultitude ofwhich it is the rational and scientific expression, it can still be not only doubled, but evenmultiplied.Moreover,eachnumberissodefinedbyitsownpropertiesthatnotwonumbersareequal.Theyarethereforebothunequalanddifferentfromeachother;andwhiletheyaresimplyfinite,collectivelytheyareinfinite.33

Augustinehadassertedquiteeffectivelythatthenumbers(bywhichhemeantthecountingnumbers)goonforeverhoweverbiganumberis,youcanalwaysfindabiggerone,doubleit,orsquareit.NexthesetGodtoworkatthevirtualabacus:

DoesGod, therefore,notknownumbersonaccountof this infinity;anddoesHisknowledgeextendonlytoacertainheight innumbers,whileoftherestHeisignorant?Whoissolefttohimselfastosayso?Yettheycanhardlypretendtoputnumbersoutofthequestion,ormaintainthattheyhavenothingtodowiththeknowledgeofGod;forPlato,theirgreatauthority,representsGodasframingtheworldonnumericalprinciples;andinourbooksalsoitissaidtoGod,Thouhastorderedallthingsinnumber,andmeasure,andweight....Farbeit,then,fromustodoubtthatallnumberisknowntoHimwhoseunderstanding,accordingtothePsalmist,isinfinite.34

WhileAugustinesquotefromPsalm147fitswithamoderntranslation,itmightbealittleingenuousasanargument, in that it depends on what the psalm writer had in mind by the word that is translated asinfinity.Similarlyitwouldbeeasytodismisshisotherbiblicalreferenceasnon-literal,aliteraryturnofphraseratherthanliteraldescription.Evenso,thereisconsiderablestrengthinAugustinesargumentifGoddoes exist, it is hard indeed to envisage that such an unlimited creator could not conceive ofanythingbiggerthanacertainnumber(whateverthatnumberhappenstobe).

Augustine then turned thenumerical argument backon thosewho imagine that there is some sort ofcyclicalcreation,wheretheuniverseisbornanddiestimeandagainthroughoutcreation.Thisview,hesaid,wasbasedontheinabilitytoseehowGodcouldmanagetoworkthroughaninfinite,timelessperiodbefore the universe came into existence, somehow seeing ahead to the point when creationwould berequired:

The infinity of number, though there be no numbering of infinite numbers, is yet not incomprehensible byHimwhose understanding isinfinite.And thus, if everythingwhich is comprehended is defined ormade finite by the comprehension of himwho knows it, then allinfinityisinsomeineffablewaymadefinitetoGod,foritiscomprehensiblebyHisknowledge.35

Laterheavyweight theologianswouldpullbackalittlefromAugustinescertaintythatGodwasabletodealwiththeinfinite.WhileGodhimselfwasinsomesensesequatedwithinfinity,itwasdoubtedthathecould reallydealwith infiniteconceptsother thanHimself,notbecausehewas incapableofmanagingsuchathing,butbecausetheycouldnotexist.ThosewhorestrictedGodsimaginationinthiswaymight

arguethathesimilarlycouldnotconceiveofasquarecircle,notbecauseofsomedivinelimitation,butbecause theresimplywasnosuch thing to imagine.Agoodexample is theargumentput forwardbyStThomasAquinas.

Aquinas,bornatRoccaseccainItalyin1225,joinedthethennewlyformedDominicanorderin1243,despitehismothersattemptstopreventhim(shelockedhimupinthefamilycastleformorethanayear).HisprimeyearsofinputtophilosophyandtheteachingsoftheChurchwerethe1250sand1260s,whenhe managed to overcome the apparent conflict between Augustines dependence on spiritualinterpretation,andthenewlyre-emergingviewsofAristotle,flavouredbytheintermediaryworkoftheArabscholarAverros,whichplacedmuchmoreemphasisondeductionsmadefromthesenses.

Aquinasmanagedtobringtogetherthesetwo,apparentlyincompatibleviewsbysuggestingthat,thoughwe can only knowof things through the senses, interpretation has to come from the intellect,which isinevitablyinfluencedbythespiritual.Whenconsideringtheinfinite,Aquinasputforwardtheinterestingchallengethat

althoughGodspowerisunlimited,hestillcannotmakeanabsolutelyunlimitedthing,nomorethanhecanmakeanunmadething(forthisinvolvescontradictorystatementsbeingbothtrue).36

Sadly, Aquinass argument is not very useful, because it relies on the definition of a thing as beinginherentlylimited(echoingAristotlesargument that therecantbeaninfinitebodyasabodyhastobeboundedbyasurface,andinfinitycantbetotallybounded).Simplysayingthatathingcantbeinfinitebecausea thinghas tobe finite isacircularargument thatdoesnt take thepointanyfurther.Hedoes,however,haveanothergoatshowinghowcreationcanbe finite,even ifGod is infinite, thathasmorelogicalstrength.

InhisbookSummatheologiae,Aquinasalsoarguesthatnothingcreatedcanbeinfinite,becauseanyset of things, whatever they might be, have to be a specific set of entities, and the way entities arespecified is by numbering themoff.But there are no infinite numbers, so there can be no infinite realthings.Thiswas a point of view thatwould have a lot going for it right through to the late nineteenthcenturywhen,asweshallsee,infinite,countablesetscrashedontothemathematicalscene.

The linkage ofGodwith the infinite comes up again and again in practically all themodernworldreligions,fromthepoetryoftheHinduscripturestoJewishmysticism.IntheBhagavadGita,bytraditionspokenbyKrishnahimself,weread:

OLordof theuniverse, I seeYoueverywherewith infinite form ...Neitherdo I see thebeginningnor themiddlenor theendofYourUniversalForm.37

AndintheJewishreligion,whichgavebirthtoboththeChristianandMuslimtraditions,thereareveryspecific references to infinity in the secretive collection of Jewish mystical writings known as thecabbala.Thisterm,fromwhichwegetourwordcabbalistic(thoughthisnowhasdarkerconnotations),impliesthehandingdownofancientwisdom.

ThesupportersofthecabbalaitwouldimplytoomuchstructuretocallthemasecretsocietylikethePythagoreansbeforethemwereobsessedwiththesignificanceofnumber.Butratherthanapproachingnumber from a geometrical perspective, they were particularly interested in numerology, linking eachletterinthealphabetwithaparticularnumber,sowordscouldbeimbuedwithnumericalsignificance.

At theheartof thecabbalawere tenpropertiesorcomponents. Justas theGreekswereparticularlyfond of seven, ten had a major significance for the cabbalists after all, God had delivered tencommandments to human creatureswith ten fingers and ten toes. The ten componentswere called thesefirot(again,notethesignificanceofnumbersefirotmeanscountings).Thesecomponentswereall

consideredsubsidiarytotheGodheadandthatwasgiventhenameEinSofineffect,infinity.38MoremodernphilosophicalconsiderationshavebeennotsomuchaboutGodsgraspofinfinityasour

own. David Hume, the eighteenth-century philosopher, not only dismissed our ability to comprehendinfinity, but also drew from that limitation an assumption that the infinite and particularly theinfinitesimallysmallcouldnotexistinreality(seepage219).

At first sight, this reluctance to accept that the finite humanmind can copewith envisaging infinityseems reasonable.WhenGalileo,oneof theRenaissance thinkers to consider the infinite,wasdealingwiththesubject,heconsideredthatitwasincomprehensibletoourfiniteintellect,surpassingthecapacityofourimagination.39Yettherearewaysinwhichitseemspossibletogetahandleontheinfinite.Whenwe represent a sequence like 1, 2, 3, 4, ... , it doesnt seem so hard to envisage the idea that thisprogression goes on forever and never stops.We dont have to be able to grasp every number in thesequence,allthewaytoinfinity,toseethat,justaswedonthavetobeabletoenvisageeverymillimetreofgroundbetweenLondonandHongKongtobeabletograsptheideaoftravellingfromoneplacetotheother.

Thereis,infact,asShaughanLavine,AssociateProfessorofPhilosophyattheUniversityofArizona,points out in hisUnderstanding the Infinite,40 a very simple way that anyone can, to some degree,envisageinfinity.Aslongasyoucangraspthemeaningoffiniteandthemeaningofnot,youshouldhaveasimplisticpictureofwhatinfinityisallabout.

Theargumentsaboutwhetherwe(orGod)