Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences

Volume One

Ivor Grattan-Guiness, Ivor Grattan-Guiness

• Publisher: Routledge
• Year: 2004
• ISBN:
  9781134887552
• Edition: 1
• Topic: History
• Subtopic: World History
• Index: History

First published in 2004. This book examines the history and philosophy of the mathematical sciences in a cultural context, tracing their evolution from ancient times up to the twentieth century. Includes 176 articles contributed by authors of 18 nationalities. With a chronological table of main events in the development of mathematics. Has a fully integrated index of people, events and topics; as well as annotated bibliographies of both classic and contemporary sources and provide unique coverage of Ancient and non-Western traditions of mathematics. Presented in Two Volumes.

Part 1

Ancient and non-Western traditions

1.0

Introduction

1.1 Babylonian mathematics Jens Høyrup

1.2 Egyptian mathematics C. S. Roero

1.3 Greek mathematics to AD 300 Alexander Jones

1.4 Greek applied mathematics Alexander Jones

1.5 Later Greek and Byzantine mathematics Alexander Jones

1.6 Pure mathematics in Islamic civilization J. P. Hogendijk

1.7 Mathematics applied to aspects of religious ritual in Islam David A. King

1.8 Mathematics in Africa: Explicit and implicit C. Zaslavsky

1.9 Chinese mathematics J. C. Martzloff

1.10 Indigenous Japanese mathematics, Wasan Tamotsu Murata

1.11 Korean mathematics Yong Woon Kim

1.12 Indian mathematics Takao Hayashi

1.13 Tibetan astronomy and mathematics George Gheverghese Joseph

1.14 Mathematics in medieval Hebrew Literature Y. T. Langermann

1.15 Maya mathematics M. P. Closs

1.16 The beginnings of counting and number John N. Crossley

1.17 Some ancient solutions to the problem of fractioning numbers Karine Chemla

Although the West has dominated the development of mathematics for around five centuries, the subject has far more ancient roots in some non-Western cultures (using this adjective as explained near the beginning of §0), and many of its basic theories owe their origins to these lines. This point is a principal concern of this Part, which concentrates largely on non-Western mathematics.

The order of the articles is partly determined by chronology, beginning with some of the most ancient cultures and proceeding to those such as the Mayan and the Jewish (§1.14–1.15), in which the principal achievements were during the Middle Ages (as Westerners conceived it) and the Renaissance. In modern geographical terms, the countries of the Middle East, Greece and Africa are taken first, and then the Far East (Chinese characters are transcribed according to the Pinyin system). The two articles on number systems and fractions (§1.16–1.17) are placed at the end as they treat cross-national and cross-cultural topics; thus they review earlier articles to some extent. Some articles in Part 12 are also pertinent.

It has not been possible to treat every culture here. In some cases the history is too obscure to be summarized (as, for example, with Celtic methods; see Bain 1951).

From the modern point of view, the mathematics itself is usually arithmetic and geometry, with early algebra and/or trigonometry and their uses in mechanics and astronomy. Astronomy was particularly important, with consequences for cosmology (often religiously conceived); no attempt has been made to treat these features in detail, as they were only secondarily mathematical. Within arithmetic, many notations were introduced; Cajori 1928 gives a good selection of them.

In recent decades there has been considerable growth in the study of non-Western science in general, and some specialist journals are devoted to it (for example, the Journal of the History of Arabic Science). In addition, Archaeoastronomy considers ancient mathematics of all cultures when appropriate.

Although note is taken of the arrival of Western mathematics in non-Western countries, no account is given of the more modern developments in those countries, and indeed in the modern period none of them became a major mathematical centre (the closest candidate would be Japan). Thus there are no successor articles in Part 11, which deals with institutional developments since the seventeenth century. However, contributions from individual mathematicians of these countries – at any period – are noted in the later Parts as they arise.

BIBLIOGRAPHY

Bain, G. 1951, Celtic Art: The Methods of Construction, Glasgow: MacLellan. [Many reprints, most recently London: Constable.]

Cajori, F. 1929, A History of Mathematical Notations, Vol. 1, Chap. 2, Chicago, IL: Open Court.

1.1

Babylonian mathematics

JENS HØYRUP

1 INTRODUCTION

Babylonian mathematics was, in its origin, an offspring of early state organization. It was transmitted by scribes and basically used for practical computation. Yet Babylonian mathematics was more than a set of practitioners’ recipes. First, Babylonian calculators knew what they were doing and why they were doing it. Second, they produced a level of complex, ‘pure’ (i.e. not practically relevant) problems and pertinent techniques, especially in the field of algebra.

For lack of sources, only the mathematics of the Old Babylonian and the Seleucid periods were discussed in the literature until about 1970. Since then, however, a number of texts have been discovered which permit one to outline tentatively the development of Babylonian mathematics from the beginnings until the Late Babylonian and Seleucid periods.

2 PROTO-LITERATE AND SUMERIAN BEGINNINGS

From the eighth millennium BC, a system of arithmetical recording or accounting based on small clay tokens was used in the Near and Middle East. In the late fourth millennium, this system appears to have inspired the development both of writing and of numerical and metrological notations. Furthermore, a trend towards the harmonization of the various metrological systems emerged: the system of area measures (originally based on unconnected natural units) was keyed to the linear system, sub-unit metrologies were created, and so on.

Proto-literate mathematics was created for the purposes of practical administration in a ‘redistributive economy’ directed by a Temple institution; the replacement of ‘natural’ units by mathematically coherent metrologies corresponds to the needs of the planning and accounting official rather than to those of the immediate producer. But the complexity of the system appears to go beyond bureaucratic needs. The immediate cause of the reorganization of a bundle of arithmetical techniques as coherent mathematics seems to have been the teaching in the Temple school.

The early administration seems not to have distinguished bureaucratic from other priestly functions. Only from around the mid-third millennium is the term for ‘scribe’ found in the sources. At this time we also encounter non-bureaucratic uses of the professional tools of the scribes: literary texts and mathematical exercises beyond the context of daily administration, the exercises dealing with, for example, the division of extremely large round numbers by ‘irregular’ divisors like 7 and 33. Even though such problems would have played no significant role in practical administration, they were evidently a central concern to a scribal profession testing its own intellectual abilities.

The trend towards increasing regularization culminated under the ‘third dynasty of Ur’ (‘Ur III’, the twenty-first century BC). This regime made extensive use of systematic and extremely meticulous book-keeping. The sexagesimal place-value system was probably created for use in this context. Mathematical school exercises pointing beyond the administrative domain have not been found; parallels in other cultural domains suggest, furthermore, that the centralized state had drained away the resources for scribal intellectual autonomy and thus blocked further development of non-utilitarian mathematics.

The first survey of third-millennium mathematics was by Powell 1976; recent discoveries concerning the earliest period are presented by Nissen et al. 1991; the mathematical texts from the peripheral Ebla area are discussed by Friberg 1986.

3 MATURITY AND DECLINE

While Ur III mathematics appears to have been strictly utilitarian in orientation, non-utilitarian mathematics was central to Old Babylonian (OB) mathematics. This is the phase in the development of Babylonian mathematics which is best documented in the sources (1900 to 1600 BC, mainly the second part of this period). In this period, which was characterized by a highly individualized economy (compared to other Bronze Age cultures) and by an ideology emphasizing the individual as a private person, the scribal school developed a curriculum which stressed virtuosity beyond what was practically necessary; the triumphs of Babylonian ‘pure’ mathematics, not least the ‘algebra’, appear to be a product of precisely this OB scribal school and scribal culture.

Until Ur III, all mathematical texts had been in Sumerian; even in Semitic-speaking Ebla, Sumerian mathematics was taken over in the original language. On the contrary, OB mathematics was written in Akkadian – supplementary evidence that it represents a new genre and a break with the (plausibly more purely utilitarian) Ur III tradition. Many texts, it is true, were written predominantly by means of word-signs of Sumerian descent; all but a handful of these word-signs, however, are simply elliptic representations of Akkadian words and sentences.

Many mathematical tablets from the OB period onwards are compilations, containing a variety of problems. Often, utilitarian and ‘pure’ problems are found together; but mathematical and non-mathematical matters are not treated in the same texts. Obviously OB mathematics was not divided into fully distinct disciplines yet mathematics as a whole was an autonomous concern, perhaps even – in the form of engineering, surveying or accounting, or as a teacher’s speciality – a distinct vocation.

In 1600 BC, conquest by a warrior people put an end to the OB social order, to the age-old scribal school, to the characteristic OB scribal ideology – and at the same time to the characteristic form of OB mathematics. Scribal training was from now on based on apprenticeship within the scribal ‘family’;...