A Research Program in the History of Ideas and in Cognition

By Ubiratan D'Ambrosio

This research program, with clear pedagogical implications, has its origins in our first attempts to teach a course in "History of Mathematics" in the so-called third world. A frequent option is to follow the practice of teaching History of Mathematics as a mere collection of results dispasosed in a chronological order and of names associated with them plus some anecdotal remarks, which is indeed a History of European Mathematics.

The mere identification of native practitioners of Mathematics in the academic registers and in publications--local or from Europe, in colonial times and in the early years of independence up to current days--does not change the eurocentric character of what is called Mathematics. Without the need of any adjective, by Mathematics it is understood to be the mode of thought which took shape in Greece some 2500 years ago and which was shaped through medieval and renaissance Europe into its current forms.

The overall objectives of this mode of thought are, as an etymological analysis would reveal, an art or technique (techne = tics) of understanding, explaining, learning about, coping with and managing the natural, social and political environment. The divinatory, hence mystical, nature of these objectives are undeniable, and other of such arts or techniques were well developed in the same Greece, in the civilizations of Egypt and Africa, in the Near East and the Far East, and in the trans Adantic and trans Pacific civilizations. Other cultural systems were
also looking for their own art or technique of understanding, explaining, learning about,coping with and managing the natural, social and political environment, and the dinivatory nature, hence mysticism, associated with these objectives are again undeniable.

In particular, many of these techniques rely on processes like counting, measuring, sorting, ordering and inferring. The search, which continued throughout History, has been, and continues to be, the essential motivation of well-identified cultural groups for building up corpora of knowledge which came to be called Religion, Art, Philosophy and Science. When we say well- identified cultural groups we mean a group of people who share common and distinctive civilization characteristics, such as jargon, codes of behavior, hopes and fears, or summing up, language and culture in their broad sense. We might say ethnic group in the broad acception of the root ethnos, which has been abusively associated, in the colonial minds, exclusively with race.

We call Ethnomathematics the art or technique of understanding, explaining, learning about, coping with and managing the natural, social and political environment, relying on processes like counting, measuring, sorting, ordering and inferring which result from well-identified cultural groups. In the case of the Greeks the divinatory nature of these objectives are undeniable, and this was done through techniques, learned from Egypt, Babylonia and elsewhere, of counting, ordering, measuring, inferring, among others, which were in competition with oracle practices. These arts or techniques were called by different names, among them geometry, arithmetic, ars magna. The same is true with the advances of this form of thought in Islam, and among them one was called al-iabr, the other al-mucabala.

The same is with the development of sacred geometries and number mysticism in Medieval Christian Europe. No one would then use the word mathematics, and even less Ethnomathematics, to describe such practices. It is also clear that the exposure of different cultural groups one to another brings about inevitable cultural changes. These cultural dynamics result in intense and frequent modifications of arts, techniques and the full range of manifestations of intellectual behavior, obviously including Ethnomathematics.

The successful European enterprise of bringing "civilization" to the entire globe, successfully undertaken in the more span of the 16th century, brought with it the mode of thought then beginning to be called Mathematics, carrying with it the sense of rationality, precision, efficiency and truth. This form of thought has since then been claimed as the essence of man's rationality, disregarding any of the resulting modifications which result from cultural dynamics. More than religion, art, philosophy and the sciences in general, which have been subjected to the basic principles of cultural dynamics, Mathematics has imposed itself as an eminently eurocentric mode of thought, originating from the Mediterranean and incorporating Islamic traditions, absolute in its codes and paradigms. So absolute was this imposition that most of its codes were kept, imposing themselves (through a mechanism of insertion) into languages of non-European origin. Some histories of mathematics bring examples of developments in China, India, Japan and even the Andean civilizations and try to match a few of their results and practices with European similars. The general tone has been "see how good they were! They knew the zero and they even knew a form of Pythagoras as theorem!"

Even the references of Egyptian mathematics are limited to the showing that they were able to solve a few problems which resemble manipulation of fractions. The very essence of the art or technique of understanding, explaining, learning about, coping with and managing the natural, social and political environment, relying on processes like counting, measuring, sorting, ordering, inferring or their equlvalent among the Egyptians, or the Chinese, or the Aztecs, or the Bambaras, has never been mentioned in current histories of mathematics. Indeed, what is usually called history of mathematics should be called History of European Mathematics.

Much research is needed to increase scholarship in Ethnomathematics. We need some categorization of this research in order to draw from several projects going on all over the world, under different names, but which satisfy our conceptualization of Ethnomathematics and, consequently, contribute by adding to the as yet limited knowledge of it. The categories which we use to synthesize relevant research in Ethnomathematics are:

I.Research in culturally diversified environments.

II.Curriculum development projects and classroom applications.

III.Out-of-school applications.

IV.Conceptual and theoretical foundations.

Closely related to this is the research program in the history of mathematics, which can be identified with the very conceptuaization of Ethnomathematics as described above, and takes into account cultural dynamics, which undeniably underlines the evolution of cognitive processes and places history of mathematics in the broader framework of the history of ideas and the even broader vision of general history. Clearly, all these stages of historical analysis must be faced by both the vision of the winners--in the case of Mathematics, it is European (or Academic) Mathematics--and of the losers.

In the case of Mathematics, this means investigating precolonial practices, as identified through monuments, artifacts, documents and preserved practices among communities with strong cultural roots. The program ends with a critical analysis of the transfer, as seen in the institutionalization and in the academic productivity both quantitative and qualitative, of Mathematics to peripherical nations.

In Latin America, since the mid-1970s we have been stressing a research program in the following general direction:

1. Epistemological foundations: Ethnomathe-matics.

2. Socio-cultural bases of European Mathematics: an historical approach.

3. Specificities of Iberic Science in the Middle Ages: Math of the discoveries and of early colonial period.

4. Pre-Columbian Mathematics: a historical approach.

5. Late colonial period: efforts towards introduction of Modern Mathematics in Spain and Portugal and reflection in the colonies.

6. Independence movements, modern ideas and European Mathematics in Latin America in 19th century: institutional aspects.

7. History of Native, Popular and Professional Mathematics (Mathematics in everyday use, Rural Mathematics, Commercial Mathematics, Mathematics of Engineers and Scientists): a socio-cultural approach.

8. Late 19th and 20th century introduction and production of Mathematics in Latin America: quantitative and qualitative analysis.

This is the table of contents of a book in preparation, and most of these topics have been partially presented in a series of papers:

(1) Ubiratan D'Ambrosio; History of Ibero-American Mathematics, Historica Mathematica vol.6,1980, pp.452-453.

(2) Ubiratan D'Ambrosio: L'adaptation de la structure de l'enseignement aux besoins des pays en voic de developement, Impact of Science in Society, vol.25, n. 1, 1975, pp.100-101.

(3) Ubiratan D'Ambrosio: Objectives and Goals of Mathematics Education, Proceedings of the 3rd International Congress of Mathematics Education, Karlsruhe, 1976 (LINES CO, Paris, 1979).

(4) Ubiratan D'Ambrosio: Science and Technology in Latin America during its discovery in Impact of Science on Society, vol.27, n. 3.1977, pp.267-274.

(5) Ubiratan D'Ambrosio: Knowledge Transfer and the Universities: A Policy Dilemma Impact of Science on Society, vol.29, n. 3, 1979, pp.233-240.

(6) Ubiratan D'Ambrosio: Mathematics and Society: Some Historical and Pedagogical Implications, International Journal of Mathematics Education in Science and Tech- nology, vol.11, n. 4, 1980, pp.479-488.

(7) Ubiratan D'Ambrosio: Mathematical Education in a Cultural Setting, International Journal of Mathematics Education in Science and Technolo~v, vol.16, n. 4,1985, pp. 469-477.

(8)Ubiratan D'Ambrosio: Socio-Cultural Bases for Mathematics Education, UNICAP, Campinas, 1955.

(9) Ubiratan D'Ambrosio; Da Realidades a Acao: reflexoes sobre Educacao C Matematica, Summus Editorial, Sao Paulo, 1986 (2a. edicao 1988).

(10) Ubiratan D'Ambrosio: A Methodology for Ethnoscience: The need for Alternative Epistemologies THEORIA Segunda Epoca, n. 2, 1986.

(11) Ubiratan D'Ambrosio: Socio-Cultural Influences in the Transmission of Scientific Knowledge and Alternative Methodologies, in Cross Cultural Diffusion of Science. Latin America ed. Juan Jose Saldana, Cuadernos de Quipu n.2, Sociedad Latinoamericana de Historiade las Ciencias y la Tecnologia, Mexico, 1988; pp.125-133.

With some modifications, mainly affecting chapters 4 through 8, the same program may be adapted to other regions of the world.