E Making up Numbers A History of Invention in Mathematics KKEHARD EKKEHARD KOPP Making up Numbers: A History of Inventi on in Mathemati cs off ers a detailed but K Making up Numbers accessible account of a wide range of mathema� cal ideas. Star� ng with elementary OPP concepts, it leads the reader towards aspects of current mathema� cal research. A History of Invention in Mathematics 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 M 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 in quan� ta� ve analysis. AKING Making up Numbers will be of great interest to undergraduate and A-level students of mathema� cs, as well as secondary school teachers of the subject. By virtue of UP its detailed treatment of mathema� cal ideas, it will be of value to anyone seeking N to learn more about the development of the subject. UMBERS As with all Open Book publica� ons, this en� re book is available to read for free on the publisher’s website. Printed and digital edi� ons, together with supplementary digital material, can also be found at www.openbookpublishers.com Cover images from Wikimedia Commons. For image details see capti ons in the book. KKEHARD OPP Cover Design by Anna Gatti . E K book eebook and OA edi� ons also available www.openbookpublishers.com OBP https://www.openbookpublishers.com c 2020 Ekkehard Kopp This work is licensed under a Creative Commons Attribution 4.0 Interna- tional license (CC BY 4.0). This license allows you to share, copy, distribute and transmit the text; to adapt the text and to make commercial use of the text providing attribution is made to the authors (but not in any way that suggests that they endorse you or your use of the work). Attribution should include the following information: Ekkehard Kopp, Making Up Numbers: A History of Invention in Mathematics Cambridge, UK: Open Book Publishers, 2020, https://doi.org/10.11647/OBP.0236 In order to access detailed and updated information on the license, please visit https://www.openbookpublishers.com/product/1279#copyright Further details about CC BY licenses are available at https://creativecommons.org/licenses/by/4.0/ All external links were active at the time of publication unless otherwise stated and have been archived via the Internet Archive Wayback Machine at https://archive.org/web Updated digital material and resources associated with this volume are avail- able at https://www.openbookpublishers.com/product/1279#resources Every effort has been made to identify and contact copyright holders and any omission or error will be corrected if notification is made to the pub- lisher. For image details see captions in the book. ISBN Paperback: 978-1-80064-095-5 ISBN Hardback: 978-1-80064-096-2 ISBN Digital (PDF): 978-1-80064-097-9 DOI: 10.11647/OBP.0236 Cover images from Wikimedia Commons. For image details see captions in the book. Cover design by Anna Gatti. CHAPTER 10 Solid Foundations? Democritus said: ’That truth did lie in profound pits, and when it was got it need much refining.’ Sir Francis Bacon, in A Collection of Apophthegms, New and Old, 1625 Summary In this final chapter we observe how various paradoxes led to the effec- tive abandonment of the logicist programme in favour of the establishment of an axiom system for set theory—initiated by Ernst Zermelo in 1908—that avoids these paradoxes and that has (to date) not been shown to be inconsis- tent. We then focus on debates about the Axiom of Choice, which was not included in Zermelo’s system, but first made explicit in his proof of Cantor’s Well-Ordering Principle. It created lively debates about permissible proof methods in mathematics, particularly in France. Differences in perception between the two giants of mathematics around the turn of the twentieth century, Poincaré in France and Hilbert in Germany, later developed—via the injection of the intuitionist philosophy of the Dutch mathematician L.E.J. Brouwer into this debate—into open conflict and a cri- sis of confidence in the foundations of the subject. A technical discussion of these arguments is beyond the scope of this book and we present only a brief outline. Hilbert’s hopes of founding mathematics on a formalist viewpoint, avoid- ing all discussion of the ‘nature’ of mathematical objects, was dealt a lethal blow in 1930 by the incompleteness theorems of Kurt Gödel. Thus, advances in mathematical logic, a subject that owes its early development to the ques- tions raised by the work of Cantor and Dedekind, ultimately forced math- ematicians to adopt a more cautious attitude to the nature of mathematical truth. I cannot hope to improve on the succinct, non-technical, description given to these events by John von Neumann, written in 1947 and quoted here at some length. The final section of this chapter, however, is devoted to a c Ekkehard Kopp CC BY 4.0 https://doi.org/10.11647/OBP.0236.10 234 10. SOLID FOUNDATIONS? brief description of how infinitesimals, having been banished in the late nine- teenth century, have made a modest comeback since 1960, notably through the efforts of the logician Abraham Robinson. 1. Avoiding paradoxes: the ZF axioms Although held in high regard by specialists, Russell and Whitehead’s Principia did not have a decisive influence on the directions of mathematical research in the early decades of the twentieth century. In part, this was due to the formulation, by Ernst Zermelo in 1908, of an apparently consistent system of axioms for set theory within mathematics, using the notion of set as an undefined term (much as Euclid does with point and line, despite appear- ances). Zermelo’s axiom system was later added to and completed by Abra- ham Fraenkel (1891-1965) and Thoralf Skolem (1881-1963) and is today known simply as ZF. The relationship of the ZF axiom system to key questions that Cantor’s groundbreaking work had left unresolved, such as whether every set can be well-ordered and the Continuum Hypothesis, became a signifi- cant topic of research over the next half-century. Any precise specification of the axioms of ZF requires a background in formal mathematical logic—which was not flagged as a pre-requisite for reading his book! We must therefore be content with a brief informal de- scription of the restrictions on set formation that Zermelo and his successors considered necessary to avoid antinomies such as the ones indicated at the end of the previous chapter. Cantor had already expressed the hope that avoiding the use of notions such as ‘the set of all sets’ of a specific kind might suffice to banish contra- dictions, but he never specified how this might be done without also inval- idating proofs that appeared to produce useful correct results. Russell also suggested early on that avoiding notions such as the ‘class of all entities’ of a particular type would lead ‘naturally’ to the view that the set of objects satisfying a functional proposition should be required to be equipotent to some initial segment of the ordinal numbers. These ideas, however, did not distinguish successfully between all types of antinomies. In particular, a ‘semantic’ paradox first described by the French math- ematics teacher Jules Richard (1862-1956) in 1905, applied Cantor’s second diagonal argument to the set E of all decimal expansions between 0 and 1 that can be defined (in English, say) in a finite number of words. This set must be countable, and thus can be presented as a sequence (rn)n of real num- bers: start with all two-letter combinations in alphabetical order, then all three-letter combinations similarly, and so on. Delete all those that do not define a real number between 0 and 1 (represented here by an infinite dec- imal expansion). Thus the set E of all real numbers that can be defined in a finite number of words can be written as a sequence, that is, as a well- ordered denumerable set fr1; r2; :::; rn; :::g: Now, by using this sequence (in 1. AVOIDING PARADOXES: THE ZF AXIOMS 235 other words, by making reference to E), Richard describes how to define a decimal expansion (hence a real number) that is not equal to any of the rn: Call this decimal expansion x: If the decimal expansion defining rn has digit p in the nth place and p is neither 8 nor 9; then the nth digit of x will be th th defined as p + 1: If the n digit of rn is 8 or 9 we take the n digit of x as 1: But now we have used a finite number of (English) words to define x and x is not in E; which contradicts the definition of E: Richard’s paradox was picked up by Henri Poincaré (1854-1912), then the undisputed doyen of French mathematics, who had remained critical of var- ious aspects of research into the foundations of mathematics.