Reviews Osmo Pekonen, Editor A Brief History of Numbers by Leo Corry OXFORD: OXFORD UNIVERSITY PRESS, 2015, 336 PP., £24.99, ISBN: 9780198702597 REVIEWED BY JESPER LU¨TZEN he concept of number has an interesting and intriguing history and this is a highly recommendable TT book about it. When were irrational numbers (or negative numbers) introduced into mathematics? Such questions seem precise and simple, but they cannot be answered with a simple year. In fact, the years 2000 BC and AD 1872 and many years Feel like writing a review for The Mathematical in between could be a valid answer to the question about Intelligencer? Contact the column editor if you wish to irrational numbers, and many years between 1 and 1870 could be defended as an answer to the question about the submit an unsolicited review, or if you would welcome introduction of negative numbers. This is not just because modern mathematicians may consider pre-1870 notions of being assigned a book to review. numbers as nonrigorous but also because the historical actors were not in agreement about the nature and extent of the concept of number. Therefore the answer to the questions raised previously must necessarily be a long and contorted story. Many treatises and articles on the history of mathematics tell aspects of this story, and Corry’s book may be considered a synthesis of these more specialized works. Still, Corry also adds his own original viewpoints and results to the standard history. The last chapters of the book dealing with the modern foundations of the number system from the end of the 19th and the beginning of the 20th cen- turies obviously draw on Corry’s own research on the establishment of modern structural thinking that has con- tributed seminally to our understanding of the development of mathematics in this period. Moreover, Corry has also shed new light on the gradual merger of quantity, ratio, and number through a study of the transmission and edition of Euclid in the Renaissance. However, the book is not primarily a research mono- graph but rather a semipopular book addressed to an audience of college students and mathematics teachers as well as high-school students and professional mathemati- cians. It is written in a pleasant, captivating style characteristic of the best popular mathematics books, but, because it is based on the most up-to-date scholarship, it paints an historically honest picture of the development of the concept of number. Many popular books on the history â Submissions should be uploaded to http://tmin.edmgr.com of mathematics try to make the presentation more readable or to be sent directly to Osmo Pekonen, by simplifying the historical account. Corry, on the con- osmo.pekonen@jyu.fi trary, does not cut corners. He presents the historical Ó 2017 Springer Science+Business Media New York, Volume 39, Number 3, 2017 87 DOI 10.1007/s00283-017-9707-0 development in all its complexity, and that is exactly why surprisingly sophisticated story. For example, I was struck by the book is so captivating. the minute differences Corry can discern in the development The main plot of the book is a story of how the ancient of the concept of a fraction. Greeks distinguished between numbers (collections of In a book about the development of a mathematical units: 2,3,4,…), magnitudes (for example, line segments), concept, one has to distinguish between explicit definitions and ratios (between numbers or magnitudes) and how this of mathematical objects and the more implicit conceptual distinction was afterward gradually blurred, leading to our ideas that one can extract from the technical use of the modern concept of the real numbers. But this story is full of object. Corry repeatedly emphasizes the discrepancy twists and turns and it has many side stories involving between the two. For example, negative and complex aspects of the history of algebra, geometry, and analysis. In numbers could be handled in a consistent manner long addition to the real numbers, Corry also deals with the before the concepts had been properly defined or philo- development of complex numbers, quaternions, and the sophically grasped. Similarly, fractions or irrational ratios infinite cardinals and ordinals. Infinitesimals, on the other were treated as numbers long before the definition of a hand, are only mentioned in passing. Number symbols and number as a collection of units had been replaced by a new mathematical notation are discussed, but mostly only definition that would allow fractions to be considered as insofar as they have had an impact on the concept of numbers. By insisting on contrasting the technical mathe- number. Corry primarily deals with Western mathematics matical practice and the more foundational or philosophical and the mathematics in cultures that have had an impact on aspects of a concept at a given time, Corry draws a richer Western mathematics, such as ancient Egypt and Babylon and more complex picture than one customarily sees in as well as Medieval Arab (Islamic) mathematics. Far Eastern popular books on the history of mathematics. and South American concepts of number are not discussed. The book begins with a simple nonhistorical chapter on Because Corry wants to display the conceptual com- ‘‘The System of Numbers’’ that presents the various sets of plexities in the history of numbers, he needs to handle a lot numbers. It continues with a chapter on the idea of a of technical mathematics. However, the technicalities are positional system, illustrating the idea with the Egyptian, kept to a minimum and they are presented in a very ped- Babylonian, and Greek number signs. agogical way, making them understandable to a good high- Chapter 3 introduces the main dichotomy that must be school student. Corry often engages in a dialog with his kept in mind throughout the book, namely the Greek dis- readers to urge them to delve into important mathematical tinction between discrete numbers and continuous arguments. For example, as an introduction to Cantor’s geometric magnitudes. The problem of incommensurability diagonal argument for the uncountability of the real num- that necessitated this distinction is explained in a simple bers, Corry writes: ‘‘I want to present Cantor’s proof now. It manner, and Corry explains how geometric magnitudes is not particularly lengthy, and understanding its details were not measured but compared in the Greek pure math- does not require any specific mathematical background. If ematical texts. He also explains the basic idea behind you find my explanation somewhat difficult to follow, I Eudoxos’s theory of proportion (details in an appendix) and urge you to carry on bravely nonetheless to the end, if only the Greek use of fractional numbers in more applied texts. for the sake of coming close to grasping the main ideas Chapter 4 deals with ‘‘Construction Problems and behind the famous argument.’’ Some of the most techni- Numerical Problems in the Greek Mathematical Tradition.’’ cally demanding arguments have been relegated to Euclid’s number theory and Diophantus’s numerical prob- appendices after each chapter. This holds, for example, for lems are discussed and are contrasted with Greek geometric p the proof of the irrationality of 2, Euclid’s discussion of problem solving. It is important to insist on this contrast the area of a circle, Khayyam’s geometric solution of the because modern students will have encountered a different cubic equation, Dedekind’s theory of cuts compared with style of geometry in which the problems mostly concern Eudoxos’s theory of proportions, and the proof that the set finding the length of certain line segments or the measure of of algebraic numbers is countable. certain angles; that is, the problems ask for numbers and the When presenting old mathematical texts and ideas, an solutions mostly involve calculation. As explained by Corry, historian of mathematics must take into account the mathe- classical Greek geometric problems ask for the construction matical background of his/her modern reader. The historian (with ruler and compass or otherwise) of geometric objects. may find it convenient to translate the old mathematics into They involve no numbers or calculations. A modern reader modern terminology and symbolism to show the similarity of will be tempted to translate many of the Greek geometric the old ideas with more familiar modern ideas. To a mathe- problems and their solutions into algebra (geometric alge- matician for whom sameness (isomorphism) is important, bra), but as pointed out by Corry one must keep in mind that such translations are appealing. For an historian of mathe- such a translation only developed gradually during the matics, however, the development of mathematical ideas subsequent two millennia. only shows itself when one focuses on the minute differ- The following chapter on the Medieval Islamic tradition ences among different texts. Corry is very much aware of this does not handle the spread of the Hindu–Arabic decimal problem. He often presents the old ideas in a modern number system but mainly the emergence of algebraic translation or notation but he insists on the conceptual dif- techniques, the treatment of fractions, and Abu Kamil’s and ferences between this translation and the old original. In this Al-Khayyam’s uneasy attempts to merge numbers with way he shows subtle conceptual differences among different geometric magnitudes. About Al-Khwarizmi’s treatment of mathematical texts, and consequently he is able to tell a fractions, Corry concludes: 88 THE MATHEMATICAL INTELLIGENCER ‘‘All of this can be seen as direct evidence of how the and Hamilton’s invention of quaternions are the main arithmetic of fractions was practiced in the early stages of subjects discussed in Chapter 9. Islamic mathematics and how fractions were conceptual- In Chapter 10, Corry has reached the late 19th century ized.