Making up Numbers: A History of Invention in Mathematics

280 Pages·2020·10.046 MB·English

by  Ekkehard Kopp

Making up Numbers

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A History of Invention in Mathematics
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KKEHARD OPP H
Making up Numbers
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Making up Numbers: A History of Inventi on in Mathemati cs off ers a detailed but

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accessible account of a wide range of mathema� cal ideas. Star� ng with elementary
O A History of Invention in Mathematics
concepts, it leads the reader towards aspects of current mathema� cal research.
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Ekkehard Kopp adopts a chronological framework to demonstrate that changes

in our understanding of numbers have o� en relied on the breaking of long-held

conven� ons, making way for new inven� ons that provide greater clarity and widen

mathema� cal horizons. Viewed from this historical perspec� ve, mathema� cal

abstrac� on emerges as neither mysterious nor immutable, but as a con� ngent,

developing human ac� vity.

Chapters are organised thema� cally to cover: wri� ng and solving equa� ons,

geometric construc� on, coordinates and complex numbers, a� tudes to the use

of ‘infi nity’ in mathema� cs, number systems, and evolving views of the role of

axioms. The narra� ve moves from Pythagorean insistence on posi� ve mul� ples to

gradual acceptance of nega� ve, irra� onal and complex numbers as essen� al tools
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in quan� ta� ve analysis.
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Making up Numbers will be of great interest to undergraduate and A-level students N
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of mathema� cs, as well as secondary school teachers of the subject. By virtue of
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its detailed treatment of mathema� cal ideas, it will be of value to anyone seeking P

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to learn more about the development of the subject.
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As with all Open Book publica� ons, this en� re book is available to read for free on B
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the publisher’s website. Printed and digital edi� ons, together with supplementary R
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digital material, can also be found at www.openbookpublishers.com
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Cover images from Wikimedia Commons. For image details see capti ons in the book. KKEHARD OPP
Cover Design by Anna Gatti .
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book
ebook and OA edi� ons
also available
www.openbookpublishers.com
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MAKING UP
NUMBERS

Making up Numbers
AHistoryofInventioninMathematics
Ekkehard Kopp
Contents
Preface vii
Prologue: NamingNumbers 1
1. Naminglargenumbers 2
2. Verylargenumbers 4
3. Archimedes’Sand-Reckoner 5
4. Alonghistory 10
Chapter1. ArithmeticinAntiquity 13
Summary 13
1. Babylon: sexagesimals,quadraticequations 14
2. Pythagoras: allisnumber 19
3. Incommensurables 34
4. DiophantusofAlexandria 41
Chapter2. WritingandSolvingEquations 45
Summary 45
1. TheHindu-Arabicnumbersystem 45
2. ReceptioninmediaevalEurope 50
3. Solvingequations: cubicsandbeyond 58
Chapter3. ConstructionandCalculation 67
Summary 67
1. ConstructionsinGreekgeometry 67
2. ‘Famousproblems’ofantiquity 70
3. Decimalsandlogarithms 76
Chapter4. CoordinatesandComplexNumbers 85
Summary 85
1. Descartes’analyticgeometry 86
2. Pavingtheway 93
3. Imaginaryrootsandcomplexnumbers 98
4. Thefundamentaltheoremofalgebra 103
Chapter5. StruggleswiththeInfinite 107
Summary 107
1. ZenoandAristotle 108
2. Archimedes’‘Method’ 111
iv CONTENTS
3. Infinitesimalsinthecalculus 115
4. Critiqueofthecalculus 128
Chapter6. FromCalculustoAnalysis 131
Summary 131
1. D’AlembertandLagrange 131
2. Cauchy’s‘Coursd’Analyse’ 136
3. Continuousfunctions 142
4. Derivativeandintegral 146
Chapter7. NumberSystems 151
Summary 151
1. Setsofnumbers 152
2. Naturalnumbers 155
3. Integersandrationals 162
4. Dedekindcuts 170
5. Cantor’sconstructionofthereals 176
6. Decimalexpansions 180
7. Algebraicandconstructiblenumbers 184
8. Transcendentalnumbers 186
Chapter8. Axiomsfornumbersystems 193
Summary 193
1. Theaxiomaticmethod 193
2. ThePeanoaxioms 200
3. Axiomsfortherealnumbersystem 205
4. Appendix: arithmeticandorderinC 208
Chapter9. Countingbeyondthefinite 211
Summary 211
1. Cantor’scontinuum 211
2. Cantor’stransfinitenumbers 217
3. Comparisonofcardinals 223
Chapter10. SolidFoundations? 233
Summary 233
1. Avoidingparadoxes: theZFaxioms 234
2. Theaxiomofchoice 236
3. Tribalconflict 240
4. Gödel’sincompletenesstheorems 244
5. Alogician’srevenge? 251
Epilogue 257
Bibliography 259
NameIndex 261
Index 263
ForMarianne

Preface
Humanbeingshaveaninnateneedtomakethingsup.Peoplemakeupsto-
ries,nationsmakeuphistories,scientistsmakeuptheoriestoexplainhow
the world works and philosophers ponder how we know things and how
we should live and behave. These made-up tales often conflict with each
other, but perhaps there is one thing on which we can all agree: that it is
necessarytomakeupnumberstohelpuscopewithlifeandwitheachother,
fromtimeswhen‘one,two,many’seemedtobeenough,rightdowntothe
modern concepts of number used by scientists and mathematicians today.
We might not always agree, nor even think about, what numbers are, but
no-oneislikelytodenythatweneedthem.
Numberscropupeverywhereinmodernlife:onclocks,calendars,coins
andincashdispensers,forexample. Atprimaryschoolweallspentmuch
timelearningtomanipulatenumbers:weaddedandsubtracted,learntmul-
tiplication tables by rote, practised long division—some of us even learnt
howtocomputesquareroots. Muchofthisisnowdoneroutinelywithcal-
culatorsandcomputersandweforgettheeffortspentinacquiringthebasics
whenwewereyoung—perhapsweevenforgethowtousethem.
If you have ever wondered how all this came about, how our concept
ofnumbershasdevelopedoverthecenturies,andhowvariouspuzzlesand
conceptual problems encountered along the way were resolved, then this
bookshouldbeofinteresttoyou. Youmightbeacurrentorintendingmath-
ematicsundergraduate,orakeenstudentofA-levelmathematics,orindeed
beteachingthesubjectatsecondaryschool. Oryoumightsimplybeinter-
estedinmathematicsandseektolearnmoreaboutitsdevelopment.
Thetraditionalmathematicssyllabus,atschool,collegeoruniversity,at
bestmakespassingreferencetothefascinatinghistoryofoursubject. Stu-
dentsseekingtotracethedevelopmentofmathematicalideasoftenfindthat
therearerelativelyfewdetailedbutaccessiblesourcestoguidethem; and
while texts presenting ‘popular mathematics’ can provide much fun with
examplesandinterestinganecdotes,thethreadofconceptualdevelopment
sometimessuffersintheprocess.
Thisbook makesnopretence tobean academictreatise inthehistory
ofmathematics,norisitamathematicstextbook. Itseekstotellastory,one
viii PREFACE
thatIhopemayinformreaderswhosepriorexperienceofabstractmathe-
maticalargumentsisnotextensive.
To understand what mathematics does and how it has developed, it
is essential to do some mathematics. In presenting problems whose solu-
tions led to ever wider classes of number, as well as discussing concep-
tual obstacles that were overcome, I make use of mathematical notation,
basicmanipulationofequationsandstep-by-stepmathematicalreasoning.
Some of this has been placed in shaded sections that readers in a hurry
maydecidetoskip, hopefullywithoutlossofcontinuity. Toassistreaders
seeking more detail on particular points, an online resource—available at
https://www.openbookpublishers.com/product/1279#resources—entitled
Mathematical Miscellany (abbreviated to MM in the text) accompanies this
book. Its purpose is to remind the reader of basic mathematical concepts,
provide simple technical details, as well as some longer proofs, that are
omittedinthetext, andprovidemorebackground, mathematicalandhis-
torical,ontopicsaddressedinthebook.
It may seem that nothing more needs to be said about numbers. So
itmaysurprisesomereadersofthefinalchaptersthatmathematiciansto-
dayarenotimmunetodoubtsaboutthefoundationsoftheirsubject. After
all,therigourofmathematicalproofandthetimelessnessofmathematical
truthshavebeenhallmarksofthedisciplineeversinceAncientGreece,more
than 2000 years ago. Until quite recently, countless generations of school
pupilsspentyearswrestlingwiththeinexorablelogicofthegeometriccon-
structionsandtheoremsinEuclid’sElements. Todaytheyalsoencounterthe
abstractionofalgebraicsymbolsinsolvingequationsand(somewhatlater)
marvelattheapparentlymiraculoussuccessoftheCalculusinthequanti-
tativeanalysisofmotionandforcesinourphysicaluniverse,whichled,in
turn,totechnologicalrevolutionsthatnowgovernoureverydaylives. Why,
indeed,shouldanyofthisbesubjecttodoubt?
Naturally,IamnotclaimingthatIambesetwithdoubt. Rather,Iregard
mathematicsasahumanactivity,whosehistoricaldevelopmentreflectsthe
continuing refinement and abstraction of its concepts—including the con-
cept of number, and even that of proof—as a process of evolution. This
process is conducted collectively and is stimulated by careful observation
ofourenvironment,creativeuseoftheimagination,andintellectualrigour.
From that perspective it does not seem so different from other human en-
deavours. Itisnotinfallible,norareitspreceptsbeyondquestion,however
well-hiddenorabstrusetheymaybe. Inthefinalchapterofthisbookthis
is illustrated, in a graphic account of disputes over the foundations of the
subject,bytheeminentmathematicianJohnvonNeumann,who,oversev-
entyyearsago,explainedtheconundrumposedtheremorevividlythanI
cantoday.