The Foundations of Mathematics PDF
Preview The Foundations of Mathematics
the foundations of mathematics THE FOUNDATIONS OF MATHEMATICS SecondEdition ian stewart and david tall 3 3 GreatClarendonStreet,Oxford,ox26dp, UnitedKingdom OxfordUniversityPressisadepartmentoftheUniversityofOxford. ItfurtherstheUniversity’sobjectiveofexcellenceinresearch,scholarship, andeducationbypublishingworldwide.Oxfordisaregisteredtrademarkof OxfordUniversityPressintheUKandincertainothercountries ©JoatEnterprisesandDavidTall2015 Themoralrightsoftheauthorshavebeenasserted FirstEditionpublishedin1977 SecondEditionpublishedin2015 Impression:1 Allrightsreserved.Nopartofthispublicationmaybereproduced,storedin aretrievalsystem,ortransmitted,inanyformorbyanymeans,withoutthe priorpermissioninwritingofOxfordUniversityPress,orasexpresslypermitted bylaw,bylicenceorundertermsagreedwiththeappropriatereprographics rightsorganization.Enquiriesconcerningreproductionoutsidethescopeofthe aboveshouldbesenttotheRightsDepartment,OxfordUniversityPress,atthe addressabove Youmustnotcirculatethisworkinanyotherform andyoumustimposethissameconditiononanyacquirer PublishedintheUnitedStatesofAmericabyOxfordUniversityPress 198MadisonAvenue,NewYork,NY10016,UnitedStatesofAmerica BritishLibraryCataloguinginPublicationData Dataavailable LibraryofCongressControlNumber:2014946122 ISBN978–0–19–870644–1(hbk.) ISBN978–0–19–870643–4(pbk.) Printedandboundby CPIGroup(UK)Ltd,Croydon,CR04YY LinkstothirdpartywebsitesareprovidedbyOxfordingoodfaithand forinformationonly.Oxforddisclaimsanyresponsibilityforthematerials containedinanythirdpartywebsitereferencedinthiswork. TO PROFESSORRICHARDSKEMP whosetheoriesonthelearningofmathematicshavebeen aconstantsourceofinspiration PREFACE TO THE SECOND EDITION Theworldhasmovedonsincethefirsteditionofthisbookwaswritten ontypewritersin1976.Forastart,thedefaultuseofmalepronouns is quite rightly frowned upon. Educationally, research has revealed newinsightsintohowindividualslearntothinkmathematicallyastheybuild 1 ontheirpreviousexperience(see[3]). Wehaveusedtheseinsightstoadd commentsthatencouragethereadertoreflectontheirownunderstanding, thereby making more sense of the subtleties of the formal definitions. We have also added an appendix on self-explanation (written by Lara Alcock, Mark Hodds, and Matthew Inglis of the Mathematics Education Centre, LoughboroughUniversity)whichhasbeendemonstratedtoimprovelong- term performance in making sense of mathematical proof. We thank the authorsfortheirpermissiontoreproducetheiradviceinthistext. Thesecondeditionhasmuchincommonwiththefirst,sothatteachers familiarwiththefirsteditionwillfindthatmostoftheoriginalcontentand exercises remain. However, we have taken a significant step forward. The firsteditionintroducedideasofsettheory,logic,andproofandusedthem tostartwiththreesimpleaxiomsforthenaturalnumberstoconstructthe realnumbersasacompleteorderedfield.Wegeneralisedcountingtocon- siderinfinitesetsandintroducedinfinitecardinalnumbers.Butwedidnot generalisetheideasofmeasuringwhereunitscouldbesubdividedtogivean orderedfield. InthiseditionweredressthebalancebyintroducinganewpartIVthat retainsthechapteroninfinitecardinalnumberswhileaddinganewchapter onhowtherealnumbersasacompleteorderedfieldcanbeextendedtoa largerorderedfield. This is part of a broader vision of formal mathematics in which certain theoremscalledstructuretheoremsprovethatformalstructureshavenatural interpretationsthatmaybeinterpretedusingvisualimaginationandsym- bolicmanipulation.Forinstance,wealreadyknowthattheformalconceptof acompleteorderedfieldmayberepresentedvisuallyaspointsonanumber lineorsymbolicallyasinfinitedecimalstoperformcalculations. 1 NumbersinsquarebracketsrefertoentriesintheReferencesandFurtherReading sectionsonpage383. PREFACETOTHESECONDEDITION | vii Structuretheoremsofferanewvisionofformalmathematicsinwhichfor- maldefinedconceptsmayberepresentedinvisualandsymbolicwaysthat appeal to our human imagination. This will allow us to picture new ideas and operate with them symbolically to imagine new possibilities. We may thenseektoprovideformalproofofthesepossibilitiestoextendourtheory tocombineformal,visual,andsymbolicmodesofoperation. In Part IV, chapter 12 opens with a survey of the broader vision. Chap- ter13introducesgrouptheory,wheretheformalideaofagroup—asetwith anoperationthatsatisfiesaparticularlistofaxioms—isdevelopedtoprove astructuretheoremshowingthatelementsofthegroupoperatebypermut- ingtheelementsoftheunderlyingset.Thisstructuretheoremenablesusto interpret the formal definition of a group in a natural way using algebraic symbolismandgeometricvisualisation. Followingchapter14oninfinitecardinalnumbersfromthefirstedition, chapter15usesthecompletenessaxiomfortherealnumberstoproveasim- plestructuretheoremforanyorderedfieldextensionKoftherealnumbers. This shows that K must contain elements k that satisfy k > r for all real numbers r, which we may call ‘infinite elements’, and these have inverses h = 1/kthatsatisfy0 < h < rforallpositiverealnumbersr,whichmaybe called‘infinitesimals’.(Therearecorrespondingnotionsofnegativeinfinite numbersksatisfyingk < r forallnegativerealnumbersr.)Thestructure theoremalsoprovesthatanyfiniteelementkinK (meaninga < k < bfor realnumbersa,b)mustbeoftheforma+hwhereaisarealnumberandhis zerooraninfinitesimal.Thisallowsustovisualisetheelementsofthelarger fieldK aspointsonanumberline.Theclueliesinusingthemagnification m : K → K givenbym(x) = (x–a)/hwhichmapsato0anda+hto1, scalingupinfinitesimaldetailaroundatobeabletoseeitatanormalscale. This possibility often comes as a surprise to mathematicians who have workedonlywithintherealnumberswheretherearenoinfinitesimals.How- ever,inthelargerorderedfieldwecannowseeinfinitesimalquantitiesina largerorderedfieldaspointsonanextendednumberlinebymagnifyingthe picture. Thisrevealstwoentirelydifferentwaysofgeneralisingnumberconcepts, one generalising counting, the other generalising the full arithmetic of the realnumbers.Itoffersanewvisioninwhichaxiomaticsystemsmaybede- fined to have consistent structures within their own context yet differing systems may be extended to give larger systems with different properties. Whyshouldwebesurprised?Thesystemofwholenumbersdoesnothave multiplicativeinverses,butthefieldofrealnumbersdoeshavemultiplica- tiveinversesforallnon-zeroelements.Eachextendedsystemhasproperties that are relevant to its own particular context. This releases us from the viii | PREFACETOTHESECONDEDITION limitationsofourreal-worldexperiencetouseourimaginationtodevelop powerfulnewtheories. Thefirsteditionofthebooktookstudentsfromtheirfamiliarexperience in school mathematics to the more precise mathematical thinking in pure mathematicsatuniversity.Thissecondeditionallowsafurthervisionofthe widerworldofmathematicalthinkinginwhichformaldefinitionsandproof leadtoamazingnewwaysofdefining,proving,visualising,andsymbolising mathematicsbeyondourpreviousexpectations. IanStewartandDavidTall Coventry2015 PREFACETOTHESECONDEDITION | ix
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