1 October 23, 2017 Where Do Numbers Come From? DRAFT T. W. Korner¨ October 23, 2017 Senseless as beasts I gave men sense, possessed them Of mind. I speak not in contempt of man; I do but tell of good gifts I conferred. In the beginning, seeing they saw amiss, And hearing heard not, but, like phantoms huddled In dreams, the perplexed story of their days Confounded; knowing neither timber-work Nor brick-built dwellings basking in the light, But dug for themselves holes, wherein like ants, That hardly may contend against a breath, They dwelt in burrows of their unsunned caves. Neither of winter’s cold had they fixed sign, Nor of the spring when she comes decked with flowers, Nor yet of summer’s heat with melting fruits Sure token: but utterly without knowledge Moiled, until I the rising of the stars Showed them, and when they set, though much obscure. Moreover, number, the most excellent Of all inventions, I for them devised, And gave them writing that retaineth all, The serviceable mother of the Muse. Aeschylus, Prometheus Bound, translation by G. M. Cookson What would life be without arithmetic but a scene of horrors. Sydney Smith, letter to Miss Lucie Austin God made the integers, all else is the work of man. Attributed to Kronecker ‘When I use a word,’ Humpty Dumpty said in rather a scornful tone, ‘it means just what I choose it to mean — neither more nor less.’ ‘The question is,’ said Alice, ‘whether you can make words mean so many different things.’ ‘The question is,’ said Humpty Dumpty, ‘which is to be master — that is all.’ Lewis Carroll, Through the Looking Glass We should never forget that the functions, like all mathematical constructions, are only our own creations, and that when the definition, from which one begins, ceases to make sense, one should not ask: what is it, but what is it convenient to assume so that I can always remain consistent. Thus for example, the product of minus by minus. Gauss, letter to Bessel, 1811, Volume 10 of his collected works I have learnt one thing from my Arab masters, with reason as guide, but you another [from your teachers in Paris]: you follow a halter, being enthralled by the picture of authority. For what else can authority be called other than a halter? As brute animals are led wherever one pleases by a halter, but do not know where or why they are being led, and only follow the rope by which they are pulled along, so the authority of written words leads many people into danger, since they just accept what they are told, without question. So what is the point of having a brain, if one does not think for oneself? Adelard of Bath Conversations with his nephew (Adelard was one of those who introduced the Indian system of writing numbers to Europe) Now you may ask, ‘What is mathematics doing in a physics lecture?’ We have several possible excuses: first, of course, mathematics is an important tool, but that would only excuse us for giving the formula in two minutes. On the other hand, in theoretical physics we discover that all our laws can be written in mathematical form; and that this has a certain simplicity and beauty about it. So, ultimately, in order to understand nature it may be necessary to have a deeper understanding of mathematical relationships. But the real reason is that the subject is enjoyable, and although we humans cut nature up in different ways, ...we should take our intellectual pleasures where we find them. Feynman, Addition and multiplication Section 22-1 of the Feynman Lectures of Physics Volume 1 There is no excellent beauty that hath not some strangeness in the proportion. Bacon, Essays Have nothing in your houses that you do not know to be useful, or believe to be beautiful William Morris, Hopes and Fears for Art v Mathematical rigour is very simple. It consists in affirming true statements and in not affirming what is not true. It does not consist in affirming every truth possible. Peano, quoted in [16] There are still people who live in the presence of a perpetual miracle and are not astonished by it. Poincar´e, The Value of Science It seems to me, that the only objects of the abstract sciences or of demonstra- tion are quantity and number, and that all attempts to extend this more perfect species of knowledge beyond these bounds are mere sophistry and illusion. As the component parts of quantity and number are entirely similar, their relations become intricate and involved; and nothing can be more curious, as well as use- ful, than to trace, by a variety of mediums, their equality or inequality, through their different appearances. Hume, An Enquiry into Human Understanding I wouldbe MOST GRATEFUL to be sentlistsof errors. Send to [email protected]. I would be very interested by any other comments. Contents Introduction xi I The rationals 1 1 Counting sheep 3 1.1 Afoundationmyth ......................... 3 1.2 Whatwerenumbersusedfor? . 8 1.3 AGreekmyth............................ 11 2 The positive rationals 17 2.1 AnIndianlegend .......................... 17 2.2 Equivalenceclasses . 20 2.3 Propertiesofthepositiverationals . 25 2.4 Whathaveweactuallydone? . 29 3 The rational numbers 31 3.1 Negativenumbers.......................... 31 3.2 Definingtherationalnumbers . 35 3.3 Whatdoesnaturesay? . .. .. .. .. .. .. 41 3.4 Whenaretwothingsthesame? . 42 II The natural numbers 47 4 The golden key 49 4.1 Theleastmember .......................... 49 4.2 Inductivedefinition . 53 4.3 Applications............................. 55 4.4 Primenumbers ........................... 61 vii viii CONTENTS 5 Modular arithmetic 67 5.1 Finitefields ............................. 67 5.2 Somenicetheorems. .. .. .. .. .. .. .. 70 5.3 Anewuseforoldnumbers . 74 5.4 Moremodulararithmetic . 80 5.5 Problems of equal difficulty..................... 83 6 Axioms for the natural numbers 89 6.1 ThePeanoaxioms.......................... 89 6.2 Order ................................ 93 6.3 Conclusionoftheconstruction . 97 6.4 Ordernumberscanbeusedascountingnumbers . 100 6.5 Objections..............................105 III The real numbers (and the complex numbers) 111 7 What is the problem? 113 7.1 Mathematicsbecomesaprofession . 113 7.2 Roguenumbers ...........................114 7.3 Howshouldwefoundcalculus? . 120 7.4 Thefundamentalaxiomofanalysis. 123 7.5 Equivalentformsofthefundamentalaxiom . 129 8 And what is its solution? 135 8.1 Aconstructionoftherealnumbers . 135 8.2 Someconsequences. 145 8.3 Aretherealnumbersreal? . 149 9 The complex numbers 153 9.1 Constructingthecomplexnumbers . 153 9.2 Analysis for C ............................157 9.3 Continuous functions from C ....................160 10 A plethora of polynomials 163 10.1 Preliminaries . .. .. .. .. .. .. .. .. 163 10.2 Thefundamentaltheoremofalgebra . 168 10.3 Liouvillenumbers. 173 10.4 Anon-Archimedeanorderedfield . 176 CONTENTS ix 11 Can we go further? 183 11.1 Thequaternions. 183 11.2 Whathappenednext . 187 11.3 Valedictory .............................191 A Products of many elements 193 B nth complex roots 199 C How do quaternions represent rotations? 203 D Why are the quaternions so special? 207 Introduction Faced with questions like ‘What is truth?’ or ‘What is justice?’, practical people dismiss them as useless speculation and intellectuals enjoy the vague contem- plation of matters beyond the reach of practical people. The best philosophers provide answers, which may not be final, but which illuminate the paths that we follow. The question ‘What are numbers?’ is clearly less important, but has inter- ested several important philosophers and mathematicians. The answer given is this book is essentially that given by Dedekind in two essays ‘Stetigkeit und ir- rationale Zahlen’ (‘Continuity and irrational numbers’) and ‘Was sind und was sollen die Zahlen?’ (‘What are numbers and what should they be?’) [7]. Starting with the natural numbers N+ (that is to say, the strictly positive in- tegers), we shall construct the strictly positive rational numbers Q+ and then use these to construct the rational numbers Q. We then use the rational numbers to construct the real numbers R. Once R has been constructed, we construct the complex numbers C and we have all the numbers required by modern analysis (that is to say, calculus). Of course, we still have to say where N+ comes from. Dedekind showed that all the properties of the positive integers can be derived from a very small number of very plausible rules. The question whether to accept these rules can now be left to the individual mathematician. In the real world, we build the foundations before the building. Historically, mathematicians have tended to install foundations when the building is half com- pleted. Pedagogically there are good reasons for only studying the construction of the various number systems after the student has acquired facility in using them. However, there will always be apprentice mathematicians who ignore the ad- vice of their more experienced teachers and demand a look behind the curtain. This book is written for them and, for that reason, I have tried to follow an arc going from the relatively easy to the relatively difficult. After a general discussion setting out some of the properties of the natural numbers, I show how to construct the strictly positive rational numbers (possibly better known to the reader as the positive fractions). By modifying the ideas xi xii INTRODUCTION involved, I then show how the construct negative numbers so that we get all the rationals. We now take a long detour during which I discuss induction and modular arith- metic. The ideas are, I think, interesting in themselves and provide a background to the rather abstract arguments we then use to construct the natural numbers. The final part of the book deals with the construction of the reals from the rationals and involves a substantial increase in the level of difficulty. Both the question of why we need the real numbers and the techniques used to construct them require ideas and techniques which only emerged slowly and painfully in the course of the 19th century.