CORINNA ROSSI. Architecture and mathematics in ancient Egypt. xxii+280 pages, 102 illustrations, 9 tables. 2003. Cambridge: Cambridge University Press; 978-0-521-69053-9 paperback £ 19.99 & $ 39.99; 978-0-521-82954-0 hardback £ 65 & $110.
ELEANOR ROBSON. Mathematics in ancient Iraq: a social history. xxx+442 pages, 75 illustrations, 63 tables. 2008. Princeton (NJ) & Oxford: Princeton University Press; 978-0-691-09182-2 hardback £ 20.95.
The history and prehistory of the exact sciences has traditionally been the province of mathematicians, physicists and astronomers, using approaches that owe little to either history or archaeology (much less anthropology or linguistics). Such modes of analysis held as axiomatic the Platonic notion that mathematics was discovered rather than invented, and treated cultural variability in mathematics in terms of deviation from the ideal, which was held to reside in Western practice from Euclid onward. At best, the mathematics of ancient Near Eastern societies could be compared to the 'Greek miracle' by envisioning them as proto-Western, limited and tentative steps towards Truth, which could be treated in a handful of pages in history of mathematics textbooks.
Two recent publications, Corinna Rossi's Architecture and mathematics in ancient Egypt, and Eleanor Robson's Mathematics in ancient Iraq, challenge these assumptions forcefully. Without falling into the trap of a muddled relativism that would deny any mathematical truths, each author makes a conscious effort to avoid ethnocentrically relating ancient Near Eastern concepts and practices to Greek or modern ones. In so doing, they locate the mathematics of each society within its own technical practices, social institutions (and tensions), and textual traditions, bringing the history of ancient mathematics into the forefront of contemporary discussions in social history and the history of science. The integration of archaeological and textual data is fundamental to each, and archaeologists with an interest in science, mathematics, and architecture will find much of interest and relevance to their own work.
Rossi's challenge is perhaps the greater one, because the study of the mathematics of Egyptian architecture inevitably brings her into conflict with the numerologically inclined, who see the Golden Ratio (phi = 1.618) in temples and proto-Pythagorean plans in pyramids. Cautioning against pattern-seeking based on the presumptions of Western scholarship, Rossi argues that the lack of an emic concept of phi in Egyptian knowledge systems makes it inappropriate as an analytical tool for modern scholars. She thus issues a significant challenge, not only to scholars of the exact sciences in Egypt, but to archaeologists interested in prehistoric architecture, astronomy and mathematics for which emic concepts are far less available, such as Neolithic Britain. Bringing architecture and iconography together, she shows that, in some periods, Egyptian aesthetic canons for representing the human body parallel architectural ratios. This is no Greek mathematics - but neither does she wish to deny the long-evident influences of Egyptian mathematics on other Mediterranean civilisations.
Perhaps Rossi's most significant accomplishment is her detailed demonstration that, far from being characterised by uncanny precision, Egyptian architecture was not the result of complex mathematical pre-planning. While drawings (though not scale drawings), three-dimensional models and textual descriptions exist, actual practice involved 'incomplete preliminary planning' (p. 174) that could be adjusted during construction as circumstances arose. Ptolemaic assertions of extreme precision are rightly discounted as ideological claims of ancient wisdom, and in demolishing them Rossi puts to rest the assertions of the 'pyramidiots' while at the same time making the case for a flexible and clever Egyptian architectural tradition. The tension between the desire for the maximum pyramid height possible and the technical and physical limitations of construction can neither be resolved to a set of fixed ratios nor to a set of practices divorced completely from abstract concerns.
Rossi therefore contributes to the archaeological literature on monumental architecture which seeks to understand building in terms of conspicuous consumption (e.g. Trigger 1990), within a system of technical capacities and constraints. One wishes, however, that more attention had been paid to the social over the technical - to illuminate how particular techniques relate to elite social interests, to changing political contexts, and in particular to the ways in which architects and scribes were educated and integrated within Egyptian society. Rossi grants us remarkable insight, for instance, into the way in which two-dimensional architectural plans relate to three-dimensional constructions, but far less into the way those plans relate to other textual genres, or how continuities and breaks in architectural practice reflected changing Egyptian sociopolitical circumstances.
Robson's scope is far greater than Rossi's - her book is a synthetic and diachronic examination of four millennia of mathematics in Iraq from Uruk to the Seleucids. Her title is an intentional indicator of the break with previous work: the term 'Mesopotamia' is virtually excised in an effort not to regard Iraq through the lens of Hellenisation. Robson sets out an unconventional but linear periodisation (though occasionally breaking with this chronology when dealing with specific topics such as tabular organisation in cuneiform tablets). Arguing that 'cuneiform culture and mathematical culture were more or less coextensive' (p. 2), she interweaves cuneiform mathematics with scribal and institutional activities in each period. She thus strikes a careful balance - recognising temporal continuities, for example, the long history of sexagesimal place-value notation, while at the same time noting differences rooted in social, rather than technical, developments.
Using three general modes of mathematical practice - 'numerate apprenticeship', 'metrological justice' and 'divine quantification' (p. 263) - Robson relates each to particular constellations of elite interest and scribal practice. Thus she links the token systems of the protoliterate period to the cuneiform mathematics of the domestic economy of production in the third millennium BC. For the second millennium and most of the first, she links literary concerns (the use of round and non-round numbers for literary effect in the epic of Gilgamesh) with metrological ones (by far the most common mathematical text in the cuneiform tradition is the metrological table). Both genres reflect an interest in rectification, balancing, exactness and accurate conversion. Seleucid Babylonian mathematics is put in the context of a relatively closed group of families for whom mathematical astronomy was closely linked to religious practice. While some of this knowledge was eventually transmitted to Greece, the Greek mathematical tradition must be considered as essentially independent from the Babylonian one.
The identification of cuneiform mathematics as a genre is non-trivial. Nowhere is this more apparent than in Robson's efforts to distinguish mathematical scribal exercises from applied arithmetic; the former can contain numbers that are spuriously round or suspiciously difficult, or either implausibly large or small (p. 29). Resolving these issues can never be formulaic, but rather requires paying attention to what is known of social practices from a particular period. A focus on the construction of mathematical texts as part of scribal practice leads to a substantially richer view of the genre. In preference to treating the sexagesimal place-value system as the unique Babylonian contribution to world mathematics, Robson insists that we understand it as a tool for temporarily converting numbers to a form allowing them to be used for computation, while results were written using decimal numerals that were never used for calculation.
In propounding an 'alienating' linguistic turn in cuneiform mathematics, Robson, like Rossi, seeks to understand ancient mathematical traditions in the way they were understood by their originators, rather than as exemplars of a universalising mathematical tendency. Her work is informed both by the ethnomathematical work of cultural anthropologists such as Urton (1997) and Ascher (1991), and by Said's (1978) critique of ethnocentric scholarship on the Middle East. The most striking chapter of the entire book is the Epilogue, an extended historiography of 'Mesopotamian' mathematics in which Robson pays tribute to her predecessors such as Neugebauer (1957) while at the same time utterly rejecting the Western-centred and Greek-oriented translations and interpretations of cuneiform texts.
Given the similarities between these two books and the approaches to ancient knowledge that they describe, one is forced to ask, if the comparison with ancient Greece is not to be forced, whether comparison to one another is nonetheless warranted. Robson notes that Old Babylonian mathematics shares many discursive features with those of ancient India, Egypt and China (p. 288). If this is so, why is it so? Despite the differences between Babylonian and Egyptian practice, are there interrelations between the two traditions worthy of note (e.g. Friberg 2005)? More broadly, in a socially-informed, linguistically-nuanced approach to ancient exact sciences, is there room for the concept of 'ancient mathematics'? The answers provided by Rossi and Robson to these questions are tentative at best, and demand a comparative approach that examines similarities and differences without prioritising either, and without viewing ancient mathematics as a necessary step or mere precursor.
For archaeologists and archaeologically-minded historians, both Rossi and Robson provide significant new insights into the mathematics of ancient civilisations, while challenging us to consider how language, material culture, and socio-technical practices are integrated, not only in mathematics, but in many domains. To treat a tablet as a tablet, not just as a text, or to treat a floor-plan as a visual representation with a point of view, is to insist on the relevance of archaeological insights in the understanding of a most abstract phenomenon.
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