Ethnomathematics

Carneiro R. L. , “Classical evolution”, in Main currents in cultural anthropology, ed. by Naroll R. and Naroll F. (Englewood Cliffs, N. J., 1973), 57–121, p. 73; Hersovits M. J. , “A geneology of ethnological theory”, in Context and meaning in cultural anthropology, ed. by Spiro M. E. (New York, 1965), 403–15, p. 408.

Bagby P. , Culture and history (Berkeley, 1963), 18; Beals R. L. and Hoijer H. , An introduction to anthropology (New York, 1965), 711.

Tylor E. B. , Primitive culture (Boston, 1874), i, 240–72. Tylor says that counting began with gestures using fingers and toes. At that stage, humans could count no further than four. Later, with the realization that words existed for hands and feet, words for numbers came into being. Still later, it was found that a system based on five (one hand) was scanty, one based on twenty (hands and toes) was cumbersome, and so the base ten was adopted, and so on. At every juncture in the story, Tylor points to a culture, from native Australians and native Americans through Polynesians and native Africans and, of course, up to – and ending – with us.

Conant L. L. , The number concept: Its origin and development (New York, 1896).

E.g., Boyer C. B. , A history of mathematics (New York, 1968); Dubisch R. , The nature of number (New York, 1952); Eves H. W. , An introduction to the history of mathematics (New York, 1st edn, 1953; 5th edn, 1982); Ifrah G. , Histoire universelle des chiffres (Paris, 1981).

An exception to the classical evolutionary paradigm is found in Seidenberg A. , “The diffusion of counting practices”. University of California publications in mathematics, iii (Berkeley, 1960), 215–300; idem, “The ritual origin of counting”, Archive for history of exact sciences, ii (1962), 1–40. Seidenberg draws on diffusionism, the major contending nineteenth and early twentieth century scheme. According to him, nonliterate peoples invented nothing whatsoever in the way of number, let alone in other areas of mathematical thought.

Some or all of these ideas are in: Aleksandrov A. D. , “A general view of mathematics”, in Mathematics: Its content, methods, and meaning, ed. by Aleksandrov A. D. Kologorov A. N. Lavrent'ev M. A. (Cambridge, Mass., 2nd edn, 1969; original Russian edn, 1956), i, 1–64, pp. 7–10; Beckmann P. , A history of π (pi) (New York, 2nd edn, 1971), 10–12; Boyer C. B. , op. cit. (ref. 5), 3–5; Brett W. F. Feldman E. B. and Sentlowitz M. , An introduction to the history of mathematics, number theory and operations research (New York, 1974), 11–13; Bunt L. N. H. Jones P. S. and Bedient J. D. , The historical roots of elementary mathematics (Englewood Cliffs, N.J., 1976), 2–3; Court N. A. , Mathematics in fun and earnest (New York, 1961; original edn, 1935), 64–66; Dantzig T. , Number (New York, 4th edn, 1967; original edn, 1930), 1–21; de Villiers M. , The numeral-words: Their origin, meaning, history, and lesson (Cape Town, 1923), 16, 46–49; Dubisch R. , op. cit. (ref. 5), 4–9; Eves H. W. , op. cit. (ref. 5, 1982), 2–5, 16; Flegg G. , Numbers, their history and meaning (New York, 1983), 8–46; Ifrah G. , op. cit. (ref. 5), 9–107; Kline M. , Mathematicsin Western culture (New York, 1972; original edn, 1953), 13–14; Kramer E. E. , The nature and growth of modern mathematics (Princeton, N.J., 1982), 5; Menninger K. , Number words and number symbols (Cambridge, Mass., 1969; original edn, 1957), 9–12, 33–40, 196–7, 223–56, 297; Miller C. D. and Heeren V. E. , Mathematical ideas (Glenview, Ill., 2nd edn, 1973), 1–4; Miller C. D. and Heeren V. E. , Mathematics an everyday experience (Glenview, Ill., 1976), 1; Myers N. , The math book (New York, 1975), 1–4; Smith D. E. , History of mathematics (New York, 1958; original edn, 1923), i, 6–14; Struik D.J. , A concise history of mathematics (New York, 2nd rev. edn, 1948), 2–5; van der Waerden B. L. and Flegg G. , History of mathematics, counting (Milton Keynes, Bucks, 1975), pt 1, 12, 40; pt 11, 42–45.

Hockett C. F. , Man's place in nature (New York, 1973), 138.

Hale K. , “Gaps in grammar and culture”, in Linguistics and anthropology: In honor of C. F. Voegelin, ed. by Kinkade M. D. Hale K. L. Werner O. (Lisse, The Netherlands, 1975), 295–316; Salzmann Z. , “A method for analyzing numerical systems”, Word, v (1950), 78–83.

Crick M. R. , “Anthropology of knowledge”, Annual review of anthropology, xi (1982), 287–313, pp. 288–98.

Dubisch R. , op. cit. (ref. 5).

Burling R. , “How to choose a Burmese number classifier”, in Context and meaning in cultural anthropology, ed. by Spiro M. E. (New York, 1965), 243–64.

E.g., Seidenberg A. , op. cit. (ref. 6, 1960), 275.

Ascher M. and Ascher R. , Code of the Quipu: A study in media, mathematics, and culture (Ann Arbor, 1981).

Conant L. L. , op. cit. (ref. 4), 4; Court N. A. , op. cit. (ref. 7), 65–66; de Villiers M. , op. cit. (ref. 7), 48–49; Eves H. W. , In mathematical circles, reprinted as part of The other side of the equation (Boston, 1973; original, 1969), 5; Flegg G. , op. cit. (ref. 7), 19; Kline M. , op. cit. (ref. 7), 14; Menninger K. , op. cit. (ref. 7), 34.

Kline M. , Mathematics, a cultural approach (Reading, 1962), 581; Wittgenstein L. , Remarkson the foundations of mathematics, ed. by von Wright G. H. Rhees R. and Anscombe G. E. M. , trans. by Anscombe G. E. M. (Oxford, 1956), 14e.

Wertheimer M. , “Numbers and numerical concepts in primitive peoples”, in A source book of gestalt psychology, ed. by Ellis W. E. (London, 1955), 265–73.

E.g., Restivo S. , “Mathematics and world view: Otto Spengler's analysis of numbers and culture”, presented at International Society for the Comparative Study of Civilizations Conference, Bloomington, Indiana, May 1981.

For a somewhat different insight into the persistence of this story of the Demara, we go to its source, Galton F. , The narrative of an explorer in tropical South Africa (London, 1853). A cousin of Darwin and a peer and friend of Taylor E. B. , Galton was considered a genius by many because of the breadth and originality of his thinking. He is included among the earliest of modern statisticians. His great interest in numbers and in measuring mental differences in people were related to an idea he advocated to which he gave the name ‘eugenics’. Discussions of Galton can be found in: Gould S. J. , The mismeasure of man (New York, 1981), 75–77; and Newman J. R. , “Commentary on Sir Francis Galton”, in The world of mathematics, ed. by Newman J. R. (New York, 1956), 1167–72. Galton's 1853 African travelogue provides a fascinating glimpse into European reactions upon just meeting others so different from themselves. Now, 125 years later, those reactions can hardly be used as scientific evidence about nonliterate peoples.

E.g., Alexsandrov A. D. , op. cit. (ref. 7), 7; Brunschvicg L. , Les étapes de le philosophie mathématique (Paris, 1972; original edn, 1912), 7; Dantzig T. , op. cit. (ref. 7), 4–6; de Villiers M. , op. cit. (ref. 7), 46–47; Dubish R. , op. cit. (ref. 7), 3–4, 7, 11; Eels W. C. , “Number systems of the North American Indians”, American mathematical monthly, xx (1913), 263–72, 293–9; Eves H. W. , op. cit. (ref. 15), 5–6; Ifrah G. , op. cit. (ref. 5), 11–12; Keyser C. J. , Mathematics as a culture clue and other essays (New York, 1947), 33–34; Kline M. , op. cit. (ref. 7), 13; Marks R. W. , “Introduction”, in The growth of mathematics, ed. by Marks R. W. (New York, 1964), 1–4, p. 1; Menninger K. , op. cit. (ref. 7), 9–10; Smith D. E. , op. cit. (ref. 7), 7; Wertheimer M. , op. cit. (ref. 17), 267; Whitehead A. N. , “Mathematics as an element in the history of thought”, in The growth of mathematics, ed. by Marks R. W. (New York, 1964), 7–24, pp. 7–8.

The exact formulation of psychic unity varies from text to text, but usually within the first ten pages there is a statement to the effect that the mental capacities of human beings throughout the world are the same. For example, contrast Morgan L. H. , Ancient society (New York, 1877), 8, 17–18, with Taylor E. B. , op. cit. (ref. 3), 6, 33.

For a discussion of the influence of Lévy-Brühl , see Cazeneuve J. , Lucien Lévy-Brühl (New York, 1972).

Lévy-Brühl L. , How natives think (London, 1926; original French edn, 1910).

Carneiro R. L. , op. cit. (ref. 1), 90–93.

Tylor E. B. , op. cit. (ref. 3), 31, 246, 272; Lubbock J. , Prehistoric times (London, 1865), 569; and idem, The origin of civilization and the primitive condition of man: Mental and social condition of savages (New York, 1870), 4–5.

Through the years Lévy-Brühl modified his opinions, although his posthumously published work shows that his position on number remained the same. See Lévy-Brühl L. , The notebooks on primitive mentality (New York, 1975), 143–4. Others followed Lévy-Brühl dressing his ideas in new clothes. See, for example, the substitution of ‘incomplete logic’ for ‘prelogical’ and ‘pre-operative’ for ‘childlike’ in Hallpike C. R. , Foundations of primitive thought (Oxford, 1979).

See Brunschvicg L. , op. cit. (ref. 20).

Menninger K. , Zahlwort und Ziffer (Göttingen, 1957); idem, op. cit. (ref. 7); and others, e.g., Ifrah , op. cit. (ref. 5).

Durkheim E. , The elementary forms of religious life (New York, 1915), 270–1; Lévi-Strauss C. , The savage mind (Chicago, 1966).

Lévy-Brühl L. , op. cit. (ref. 23), 221.

Barnes R. H. , “Number and number use in Kédang, Indonesia”, Man, xvii (1982), 1–22.

Hamill J. F. , “Trans-cultural logic: Testing hypotheses in three languages”, in Discourse and inference in cognitive anthropology, ed. by Loflin M. D. and Silverberg J. (The Hague, 1978), 19–43, p. 31; Loflin M. D. , “Discourse and inference in cognitive anthropology”, ibid., 3–16, p. 4.

Silverberg J. , “The scientific discovery of logic: The anthropological significance of empirical research on psychic unity (inference-making)”, ibid., 281–95.

Barnes S. B. , “Natural rationality: A neglected concept in the social sciences”, Philosophy of the social sciences, vi (1976), 115–26.

Hutchins E. , “Reasoning in Triobriand discourse”, Quarterly newsletter of the Laboratory of Comparative Human Cognition, i (1979), 13–17.

Cooper D. E. , “Alternative logic in ‘primitive thought’”, Man, x (1975), 238–56.

McCarthy J. , “Circumscription – a form of non-monotonic reasoning”, Artificial intelligence, xiii (1980), 27–39.

Zadeh L. A. , “Coping with the imprecision of the real world: An interview”, Communications of the Association for Computing Machinery, xxvii (1984), 304–11.

In the early 1930s syllogisms were used to investigate the reasoning abilities of a nonliterate people in Uzbekistan, Central Asia. For a discussion of this see Luria A. R. , Language and cognition , ed. by Wertsch J. V. (Washington, D.C., 1981). Luria believed syllogisms could “serve as a model for those techniques of language that make logical thinking possible” (ibid., 201). Frequently, responses were in terms of direct personal experience instead of the premises stated or there was a refusal to answer when the respondent had no knowledge of the subject. Comparing this with other responses, it was concluded that logical thinking is a concomitant of the more developed socioeconomic systems.

Scribner S. , “Modes of thinking and ways of speaking: Culture and logic reconsidered”, in Thinking – readings in cognitive science, ed. by Johnson-Laird P. N. and Watson P. C. (New York, 1977), 483–500, p. 490.

ibid., 487.

ibid., 486.

Cole M. , “An ethnographic psychology of cognition”, in Johnson-Laird P. N. and Watson P. C. , op. cit. (ref. 40), 468–82; Cole M. Gay J. Glick J. A. and Sharp D. W. , Cultural context of learning and thinking (New York, 1971); Cole M. and Griffin P. , “Cultural amplifiers reconsidered”, in The social foundations of language and thought, ed. by Olsen D. R. (New York, 1980), 343–64; Laboratory of Comparative Human Cognition, “Cognition as a residual category in anthropology”, Annual review of anthropology, vii (1978), 51–69.

Johnson-Laird P. N. and Watson P. C. , “A theoretical analysis of insight into a reasoning task”, in Johnson-Laird P. N. and Watson P. C. , op. cit. (ref. 40), 143–58; Piaget J. , “Intellectual evolution from adolescence to adulthood”, ibid., 158–65.

Cole M. Gay J. Glick J. A. Sharp D. W. , op. cit. (ref. 43), 233. For a similar view stated in a different way, see Lévi-Strauss C. , “The structural study of myth”, Journal of American folklore, lxviii (1955), 428–44, p. 444.

Marcus G. E. and Cushman D. , “Ethnographies as texts”, Annual review of anthropology, xi (1982), 25–69, p. 42.

Focus on number was difficult to broaden as the questions put to nonliterate peoples by ethnologists remained essentially the same. For example, the 1951 edition of Notes and queries, British Association for the Advancement of Science, London, contains essentially the same questions as the arithmetic section written for the 1874 edition of the same work. This book is a general guide to anthropological fieldwork.

Ascher M. and Ascher R. , Code of the Quipu databook (Ann Arbor, 1978); idem, op. cit. (ref. 14).

E.g., Kelley D. H. , Deciphering the Maya script (Austin, 1976).

Urton G. , At the crossroads of the earth and sky: An Andean cosmology (Austin, 1981).

Davis P. J. and Hersh R. , The mathematical experience (Boston, 1981), 158–9.

Neihardt J. G. , Black Elk speaks (Lincoln, Nebr., 1961), 198–200.

Carpenter E. , “Eskimo space concepts”, Explorations, v (1955), 131–45.

ibid., 143.

Oatley K. G. , “Mental maps for navigation”, New scientist, lxiv (1974), 863–6; Oatley K. G. , “Inference, navigation, and cognitive maps”, in Johnson-Laird P. N. and Watson P. C. , op. cit. (ref. 40), 537–47.

Goodenough W. H. , “Native astronomy in Micronesia: A rudimentary science”, The scientific monthly, lxxiii (1951), 105–10; Alkire W. H. , “Systems of measurement on Woleai Atoll, Caroline Islands”, Anthropos, lxv (1970), 1–73. Figure 2 follows the diagrams in ibid., 53 and in Oatley K. G. , op. cit. (ref. 55, 1974), 866.

It is usually difficult to sail in the direction one wishes to go. For example, it is impossible to sail directly into a wind. Tacking is sailing back and forth at angles to the wind. Changing from one tack to another across the wind requires changing the setting of the sails from one side of the boat to the other. Because of their design, this is harder to do on the boats of the Pacific navigators, as their entire mast with sails must be moved to the other end of the boat. The tacks by the Pacific navigators are sometimes, therefore, as long as 120 miles. For a fuller discussion see Gladwin T. , East is a big bird – navigation and logic on Pulwat Atoll (Cambridge, Mass., 1970).

Oatley K. G. , “Inference …”, op. cit. (ref. 55).

Keyser C. J. , op. cit. (ref. 20), 223.

Alpher B. , personal communication 1979.

Stanner W. E. H. , “The dreaming”, in Australian signposts, ed. by Hungerford T. A. G. (Melbourne, 1956), 51–65.

Figure adapted from Strehlow T. G. H. , Aranda traditions (Melbourne, 1947), 174.

Courrège P. , “Un modèle mathématique des structures élémentaires de parenté”, in Anthropologie et calcul, ed. by Richard P. and Jaulin R. (Paris, 1971), 126–81.

Layard J. , Stone men of Malekula: Vao (London, 1942), 98.

Deacon A. B. , Malekula, a vanishing people in the New Hebrides, ed. by Wedgewood C. H. (London, 1934), pp. xxii–iii.

ibid., p. xxiii.

Deacon A. B. , “The regulation of marriage in Ambrym”, Journal of the Royal Anthropological Institute, lvii (1927), 325–42.

ibid., 331.

ibid., 332.

ibid., 329.

Ascher M. and Ascher R. , op. cit. (ref. 14), 160–4. We plan to discuss these figures at greater length in another paper.

Wilder R. L. , Mathematics as a cultural system (New York, 1981).

Zaslavsky C. , Africa counts (Westport, Conn., 1979; original edn, 1973).

Keyser C. J. , op. cit. (ref. 20); Kline M. , op. cit. (ref. 16); Kline M. , op. cit. (ref. 7); Spengler O. , The decline of the West (New York, 1926), i; Wittgenstein L. , op. cit. (ref. 16).

MacLane S. , “Mathematical models: A sketch for the philosophy of mathematics”, American mathematical monthly, lxxxviii (1981), 462–72.