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Introduction Chapter 1 Introduction For a long time, historians and philosophers of mathematics left out societies of oral tradition from their field of study. One may discern in this neglect the influence of philosophers such as Lucien Lévy-Bruhl, who claimed that the members of these societies are characterized by their “primitive mentality”, described by him as “prelogical”. Allegedly, this mentality made them hardly capable of dealing with abstraction or of reasoning in a rational way. Lévy-Bruhl wrote: “In defining it [primitive mentality] as prelogical, I just want to say that it does not compel itself first and foremost to refrain from contradiction, as our way of thinking does”1 (Lévy-Bruhl 1910, p. 76). As sociologist and anthropologist Jean Cazeneuve commented, “in fact, this mentality obeys a principle that is not radically opposed to the principle of non-contradiction, but is merely indifferent to it. It is this principle that Lévy-Bruhl calls the principle of participation. Under this principle, beings can be both themselves and something else, and they can be linked in a way that has nothing to do with our logic. Because logical operations and prelogical ones are intertwined in the mind of primitive men, they are much less capable of abstracting and generalizing than we are”2—my translation (Cazeneuve 1963, pp. 25–26).3 1Original text: “En l’appelant prélogique, je veux seulement dire qu’elle ne s’astreint pas avant tout, comme notre pensée, à s’abstenir de la contradiction.” 2Original text: “En vérité, cette mentalité obéit à un principe qui n’est pas en opposition radicale avec celui de la non-contradiction, mais lui est simplement indifférent. Et c’est ce principe que Lévy-Bruhl appelle le principe de participation. En vertu de cette loi, les êtres peuvent être à la fois eux-mêmes et autre chose qu’eux, et ils peuvent être unis par des rapports n’ayant rien à voir avec ceux de notre logique. Du fait que les opérations logiques et prélogiques sont, en son esprit, étroitement mêlées, il résulte que le primitif est beaucoup moins apte que nous à abstraire et à généraliser.” 3See also Keck (2008). About the use of “logic” in social sciences during the twentieth century, particularly as a means of assessing the rationality of individuals in a given society, see the article by sociologist Claude Rosental entitled “What is the logic for which rationality? Representations and uses of social logic” (Rosental 2002). © Springer International Publishing Switzerland 2015 1 E. Vandendriessche, String Figures as Mathematics?, Studies in History and Philosophy of Science 36, DOI 10.1007/978-3-319-11994-6_1 2 1 Introduction Although this view has long been abandoned by scholars, it did have a profound influence throughout the twentieth century. However, even by the 1920s, some anthropologists had made observations that were in contradiction with Lévy-Bruhl’s theory. For instance, after having spent many months in a Melanesian society in the New Hebrides (renamed Vanuatu in 1980 when this South Pacific island group became independent), Bernard Arthur Deacon underlined: The older men explained the system [kinship system] to me perfectly lucidly, I could not explain it to anymore better myself. It is perfectly clear that the natives (the intelligent ones) do conceive of the system as a connected mechanism which they can represent by diagrams4::: It is extraordinary that a native should be able to represent completely by a diagram a complex system of matrimonial classes. The way they could reason about relationships from their diagrams was absolutely on a par with a good scientific exposition in a lecture room. I have collected in Malekula, too, some cases of a mathematical ability. I hope, when I get my material together, to be able to prove that the native is capable of pretty abstract thought (Deacon 1934, p. xxiii). Deacon died tragically shortly after writing these words,5 and the questions he raised about New Hebrides natives’ mathematical ability were to be left unanswered for some decades.6 Deacon’s observations were nevertheless predictive of the epistemological turning-point which occurred on this issue in the second half of the twentieth century, particularly through the work of anthropologist Claude Lévi- Strauss, as it can be perceived in the following extract of The Savage Mind: This appetite for objective knowledge is one of the most neglected aspects of the way those we call primitive think. Although it is rarely directed to realities at the same level as those associated with modern science, it involves comparable intellectual approaches and methods of observation. In both cases, the universe is an object of thought, as well as a means to satisfy needs (Lévi-Strauss 1962, p. 13)- My translation.7 In the first societies studied by social anthropologists, mathematics did not generally appear as an autonomous category of indigenous knowledge. As it has been suggested by Deacon’s testimony, and later confirmed by a large number of ethnographies from various “traditional societies” (Deacon and Wedgwood 1934; Lévi-Strauss 1947; Gladwin 1986;Austern1939), a form of rationality seems how- ever to occur within different practices, such as navigation, calendars, ornaments, 4These diagrams were drawn for Deacon on the sand. 5Bernard Arthur Deacon (1903–1927) was a student of Cambridge anthropologist Alfred Cort Haddon. He died of malaria on Malekula Island, New Hebrides, where he had carried out fieldwork in the years 1926–1927. His ethnographical field notes were published in 1927 and 1934 by another student of Haddon, anthropologist Camilla H. Wedgwood (1901–1955) (Deacon 1927, 1934; Deacon and Wedgwood 1934). 6Mathematician Marcia Ascher has analysed some of Deacon’s materials in an ethnomathematical perspective (Ascher 1988, 1991). See below. 7Original text: “Cet appétit de connaissance objective constitue l’un des aspects les plus négligés de la pensée de ceux que nous nommons primitifs. S’il est rarement dirigé vers des réalités de même niveau que celles auxquelles s’attache la science moderne, il implique des démarches intellectuelles et des méthodes d’observation comparables. Dans les deux cas, l’univers est objet de pensée, autant que moyen de satisfaire des besoins.” 1 Introduction 3 games, kinship systems, etc. The epistemological issue is then to determine whether some of these activities relate to mathematical practices, and how. The following question will be central throughout this volume: How is an activity recognized as “mathematical” when it is not identified as such by those who practise it? What criteria should we use? This is the major question I try to answer in this book. Philosophy in general, and the philosophy of mathematics in particular, does not yet offer efficient conceptual tools to tackle this issue, as it often takes for granted all that is mathematical. According to the philosopher of mathematics Jean-Jacques Szczeciniarz, a lot remains to be done in that perspective, even though a few recent works were carried out in an attempt to better characterize mathematical objects8 (Caveing 2004). However, regarding the issue of determining whether or not an activity is related to mathematics, it can be noticed that there is a significant difference between activities that involve numbering and/or measuring, and those dealing with geometrical forms. The counting of yams with a basket as it is practised in Melanesia—in the Trobriand islands (Papua New Guinea), for instance—is quite readily recognized by academia as mathematical, as it involves the use of a particular counting method. By contrast, other practices that require “geometrical” abilities (suchasweaving,basketry,ornamentalfriezemaking...)areusuallynot—ornot so readily—regarded and analysed as mathematical by scholars. The last decades have seen the development of a new interdisciplinary field of research, called ethnomathematics, which lies between the history of science and anthropology, and aims to study cultural variations in mathematics (D’Ambrosio 1985; Ascher and Ascher 1986; Gerdes 1994; Ascher and D’Ambrosio 1994; Eglash 2000). In the 1970s and 1980s, the first ethnomathematical studies were conducted by mathematicians. In most cases, these seminal works were not carried out on the basis of first hand ethnographical data, but rather on secondary sources drawn from ethnographical literature and often (but not always) from a didactic point of view (Zaslavsky 1973;Crowe1975; Ascher and Ascher 1981; Lancy 1983; Moore 1988). In the latter case, the main idea was—and remains in more recent studies in ethnomathematics—to promote the teaching of mathematics in connection to indigenous knowledge (Nunes et al. 1993; Gerdes 1995, 1999; Knijnik 2007; Trouré and Bednarz 2010). Mathematician Marcia Ascher is one of the founders of ethnomathematics whose approach has largely inspired the first steps of my own research in this field. Instead of focusing on didactic issues, her seminal books (Ascher 1991, 2002) analyse the mathematical aspects of certain activities carried out within small- scale indigenous societies. “To avoid being constrained by Western connotations of the word mathematics”, inherited from various definitions of what mathematics is, and mainly based on Western experiences of historians and philosophers of mathematics, Ascher introduced the concept of “mathematical ideas”. She defined as a mathematical idea any idea “involving numbers, logic or spatial configurations, and even more significant, combinations or organization of these into systems 8Personal communication
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