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The History of Mathematics: A Very Short Introduction

Jacqueline Stedall OXFORD UNIVERSITY PRESS 2012, 144 PAGES PRICE (PAPERBACK) £7.99 ISBN 978-0-19-959968-4

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In The History of Mathematics: A Very Short Introduction, Jacqueline Stedall considers a variety of themes in seven chapters. Mathematics is clearly portrayed as a human endeavour evolving from dead ends, failure and the capacity of a mathematician to use, refine and finesse the work of another. The book is a manual on how to approach a historical study, detailing the constraints and possible motivations. Stedall rejects the linearity of progress stepping stone approach right from the start. She takes a wider view that also includes the most written about period of post-15th century Western European mathematics, considering the input from earlier times across the globe.
Rarity of source material and their format is considered with increasing inference taken from the cultural practice of the time and artefacts of the period. An amazing illustration of mathematics literally entwined in cultural development and source material is given by the building made of clay tablets, many of them showing mathematical calculations from a school that stood in ancient Babylonian times and giving the chance to derive the mathematical curriculum. This curriculum grows in complexity, in terms of sexagesimal arithmetic progressing to complicated tables of multiplicative inverses.
Stedall is particularly keen to emphasise the role of the historian in assessing work in its original form. Giving text a contemporary meaning in translation has been a problem. She refers to a commentator on Thomas Heath’s late 19th century edition of Apollonius that ‘thanks to skilful compression and the substitution of modern notation for literary proofs, [it] occupied less than half the space of the original’. Stedall explains the inherent danger in this approach as ‘…. one comes to regard unfamiliar ideas as no more than archaic renderings of what we can now do, as we see it, more efficiently’.
An interesting insight is given into change in mathematical representation from geometry to algebra in the 17th century. This contrasts with the original intention being of a visual interpretation, an example of which is Pythagoras’ theorem in Euclid’s Elements.
The Euclidean axiomatic approach to mathematical construction is contrasted with the succeeding haphazard one, prevalent from the 2nd century BC to the 19th century AD. Stedall refers to the development of calculus, where the theory was not axiomatically sound, although functional. Ironically, branches of mathematics were being axiomatically defined in the late 19th and early 20th centuries just as debate raged in the mathematical community on the impact of the rigidity of Euclidean geometry in education.
There is a strong contrast between everyday usage of mathematics and elite mathematicians who were valued in ancient times for their skill in setting dates for religious ceremonies, sought out and given patronage by important figures of the time. Stedall comments on the term mathematician, practitioner awareness of it and adoption on a wider scale.
The establishment of professional societies, journals and eventually international conferences that display the characteristics of any defined profession based on the 19th century German approach of academic structure, practice and research is profiled in the context of the close collegiality evidenced in European mathematics in the 17th and 18th centuries. An unbroken link between mathematicians and their working together in an established tradition of social cohesion is a practice that exists today. The book gets to the heart of major issues in the history of mathematics and is readily accessible to the non-specialist. A Very Short Introduction that makes a bigger impact.
Wallace A Ferguson CMath MIMA CSci
Chatham and Clarendon Grammar School, Ramsgate
Book review published directly onto IMA website (June 2013)