The Foundations of Mathematics

409 Pages·2015·2.84 MB·English

by  Ian Stewart

the foundations of mathematics

THE FOUNDATIONS
OF MATHEMATICS
SecondEdition
ian stewart and david tall
3
3
GreatClarendonStreet,Oxford,ox26dp,
UnitedKingdom
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©JoatEnterprisesandDavidTall2015
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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