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Concepção etnoantropológica de matemática

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AN ADEQUATE HISTORIOGRAPHY FOR NON-WESTERN MATHEMATICS *

Ubiratan D’Ambrosio

Mathematics, and the same is true for Science, as generally understood nowadays, emerged in a distinctive form in Europe . But every culture generates something equivalent to Mathematics, and Science, that works satisfactorily in its context. These are corpora of knowledge that haven generated in a particular context, with specific motivations, and that have been and are subject to unsatisfaction, criticism and the changes resulting from exposition to other cultures. which The major challenge is the human species itself. It is moved by the drives of survival and transcendence. To survive through the encounter with the other and with nature as a whole, and to transcend the moment, searching into the past and probing into the future. Knowledge is the response to these drives. How is it generated, organized intellectually and socially, and diffused? Particularly, language and mathematics offer the major challenges. Both have grown differently in different cultures. And both have been affected by cultural encounters. Particularly important for our analysis are the encounters which occurred after the 15^th century between European and non-European cultures.

I agree with Urs Bitterli when he reduces the encounter of European and non-European cultures to three basic phases: contacts, collisions and relationship. He shows that they do not occur necessarily in this order, that they are not mutually excludent and it is necessary the occurrence of the three types. In some cases the contacts lead directly to relationship.[0]

These types of encounter are convenient for understanding cultural dynamics of the encounters from the beginning of European overseas conquest through early industrial relations.

At the moment of the conquest, Mathematics was beginning to establish itself as the field of knowledge which would be central for the development of the industrial civilization.

History and Philosophy of Mathematics focuses on the Mathematics which synthesizes centuries of ideas developed in the Mediterranean basin enriched by contacts with Africa and the Far East. Historiography is largely based on written sources and relies on names, epochs, dates and places proposed by early historians.

Although the conquered civilizations had mathematical knowledge, that historiography is absolute inadequate to recognize the mathematics in the contact. Its nature and history are practically unknown. The collision phase obviously asked for the denial of the forms of knowledge of the conquered. The relationship is marked by an effort to transfer Mathematics from the European tradition to the colonies.

The condition of consumers of knowledge produced in Europe continued in the colonies until the transition from the 19^th through the 20^th century, when local production of Mathematics, as originated in Europe, start to be delineated.

This paper focus on the social, political and cultural factors in the dynamics of the transfer and production of mathematical knowledge in the colonies and on the recognition of non-European forms of mathematics, extant or buried in the colonial process. The proposal is an historiography which allows to put together scraps of information and recognize extant mathematical practices and relying in the memory of people and events which survived in a literate era.

Introductory remarks

The great navigations since the 16^th century mutually exposed forms of scientific knowledge from different cultural environments. The several ethnosciences involved in the encounters, which obviously include European Science, have been subjected to great changes as a result. In this paper I will examine some of the consequences of this mutual exposure of cultures.

By ethnosciences I mean the corpora of knowledge established as systems of explanations and ways of doing accumulated through generations in distinct cultural environments.

Particularly important for us is ethnomathematics as the corpora of knowledge derived from quantitative and qualitative practices, such as counting, weighing and measuring, sorting and classifying. As with academic Western Science and Mathematics, the two have a symbiotic relation.

Both are not new disciplines. Rather they are part of a research program on history and epistemology. The pedagogical implications are obvious. Both research and educational programs take into account all the forces that shape a mode of thought, in the sense of looking into the generation, organization (both intellectual and social) and diffusion of knowledge.

FIGURE: The cycle of knowledge

The research program, typically interdisciplinarian, brings together and interrelates, results from the cognitive sciences, epistemology, history, sociology and education. An essential component is the recognition that mathematics and science are intellectual constructs of mankind in response to needs of survival and transcendence.

The need for an intellectual framework to organize the corresponding systems of codes, norms and practices gave rise to many aspects of science and mathematics.[1]

In the research program particular attention is given to those dimensions of knowledge which bear some relation to what became known as the several discipline of science and mathematics in European civilization after the 15th century.

Ethnoscience, both as corpora of knowledge and as pedagogical practices, is supported by the history of science and reflect the dynamics of cultural acquisition. Some examples illustrate this.

All over the World, much of the weather explanations and predictions, agriculture practices, processes of cure, dressing and institutional codes, culinary, and commerce, came from the European tradition developed in the Middle Ages and the Renaissance. But we see, all over the World, practices performed in a very distinctive. These practices, which have their origins in native communities, are significantly modified as a result of mutual exposition of cultural forms since colonial times. For example, it is common to see indigenous peoples in the Americas using Indo-Arabic numerals, but performing the operations from bottom to top, explaining that this is the way trees grow. But it is also common to identify, in the more advanced notions, the influence of this mutual exposition in everyday life and practices.

Practices of daily life which are scientifically based are easily recognized. This is evident by looking into professions that require some scientific knowledge and mathematical abilities.

Practices and perceptions of learners are the substratum upon which new knowledge is built. Thus new knowledge has to be based on the individual and cultural history of the learner and it has to be recognized the diversity of extant cultures, present in specific communities, all over the world. This is the essence of a new educational posture called Multicultural Education.

A new educational posture depends on a new historical attitude which recognizes the contribution of past cultures in building up the modern world and modern thought, and which avoids omissions and errors of the past treatment of cultural differences.

We easily identify two categories of scientific knowledge: Scholarly (or "formal" or "academic") science, supported by a convenient epistemology, and whose practice is restricted to professionals with specialties; Cultural (or "practical" or "popular" or "street") science.[2] These categories are closely related and their main distinction refers to criteria of rigor, to the nature, domain and breadth of its pursuits, that is to what and how much one can do with them.

For example, pre-Columbian cultures had different styles of doing their measurements and computations and these practices are still prevalent in some native communities. Most Amazonian tribes have counting systems that goes as "one, two, three, four, many". And that is all, since with these numbers they can satisfy all their needs.[3] We also see important ways of dealing with pottery, tapestry and everyday knowledge with strong mathematics characteristics in several cultures.[4] The same with African cultures.[5] The people from these cultures have no problems at all in assimilating the current European number system and deal perfectly well with counting, measurement and money when trading with individuals of European culture. Land measurement, as practiced by peasants in Latin America, comes from ancient geometry transmitted to medieval surveyors since land property and measurement (geo-metry) is strange to Pre-Columbian cultures. Another example comes from Africa, where the people deal with numbers and counting according to their specific cultural background.[6]

The high prestige of science comes mainly from its recognition as the basic intellectual instrument of progress. It is recognized that modern technology depends on science and that the instruments of validation in social, economic and political affairs, mainly through storing and handling data, are based on science and mathematics. Particularly important in this respect is statistics. This evidently brings to science an aura of essentiality in modern society. There is a general feeling that there are practically no limits to what can be explained by science. Many of the applications which give science such a prestigious position are part of various forms of cultural conflict.

Studies of ethnoscience and ethnomathematics are motivated by the demands of the natural and cultural environment and are present everywhere. It is a fact that, even without recognizing it, just about everybody deals with mathematical practices, incorporated in daily routines. When walking or driving, people memorize routes, in most cases optimizing trajectories, which is a practice of a mathematical nature. Also when dealing with money, with measurements and quantifications in general, we recognize an intrinsic mathematical component. The same with the capability of classifying, ordering, selecting and memorizing routines.

These practices are generated, organized and transmitted informally, the same as language, to satisfy immediate needs of a population. They are incorporated in the pool of common knowledge which keeps a group of individuals, a community, a society together and operational, and this is what is called culture. Culture thus manifests itself in different, obviously interrelated, forms and domains. Cultural forms, such as language, mathematical practices, religious feelings, family structure, dressing and behavior patterns, are thus diversified. They are of course associated with the history of the groups of individuals, communities and societies where they are developed. A larger community is partitioned into several distinct cultural variants, each owing to its own history and responsive to differentiated cultural forms.

Some remarks on Historiography

History, as a major academic discipline, carries with it an intrinsic bias which makes it difficult to explain the ever present process of cultural dynamics which permeates the evolution of mankind. This paves the way for paternalism and arrogance, for intolerance and intransigence. And clearly interferes with the understanding, for different cultural groups, of each other processes of building up their cultural realities when trying to satisfy their needs of survival and transcendence.

These biases have been methodological as well as ideological, particularly in the History of Science. Helge Kragh says that "History of Science has its own 'imperialism' that partly reflects the fact that viewed historically and socially science is almost purely a western phenomenon, concentrated on a few, rich countries. While science may be international, history of science is not."[7]

This seems to be almost unavoidable in the framework of historiographies which rely on reductionist approaches, such as it is the case of the various supposedly autonomous histories, in particular in the History of Sciences. The mere fact that to pursue historical analyses one talks about the Sciences, such as Physics, Chemistry, Mathematics, as distinct from Religion, from Art, from Politics, obviously impedes the understanding of the processes of evolution of ideas and methods, of reflection and action, which underlies man's struggle to find explanations, to understand and cope with its environment, and of conviviality with nature.

The reductionism which characterizes several of the so-called autonomous histories and also histories based on facts and names, on places and dates, naturally derive from the prevailing ideology and justify current actions. Even when we move a step further than narrative history and go to historiography, the facts get immersed in the processes and we may be led to be satisfied with the false impression of having approached the past because we have data verified and facts described and explained. I agree with Armando Saitta in saying that historiography should be focused on a problem, never losing the view of all the forces which play in the historical reality, and avoiding the unilateral approach of the specialist and the reduction of the historical flow to a few elements. Saitta asks for the historian to look into "What today isn't but tomorrow will be"[8]. He clearly proposes a global history. When he refuses the history of the "if", he opens the way to an evaluation of all the alternatives which were present in the process and he claims that the one alternative which have succeeded should not imply the rejection of he others. E. H. Carr has the same opinion when he says that the historical moment in which several alternatives were open does not imply abandoning those which did not succeed, but rather looking into the reason for which some did not succeed and what was the cost of these decision.[9]

Paraphrasing Miguel León-Portilla, it is a matter of listening also to the looser.[10] History has been mostly the history of the winners. This is particularly true in the History of Science.

For obvious reasons, the vision of the looser has been marginalized, and this is more noticeable in the chapters which deal with the origins of Modern Science. We use the term Modern Science as the set of ideas which have supported in paradigms established in the XVII and XVIII centuries, mainly through the works of R. Descartes, I. Newton, G. W. Leibniz and followers.

The dawn of Modern Science is identified with the modern geography of the world, and the appearance of privileges for those capable of mastering Modern Science and Technology. How did this privileged role came into being? Why conquered and colonized still have problems in mastering Science and Technology? Why have Science and Technology progressed so rapidly and in this progress has left aside, indeed eliminated, social and above all ethical concerns, thus paving the way for enormous social, political and environmental distortions? These questions are germane to the concept of knowledge itself.

Building-up scientific knowledge

We see knowledge as emanating from the people, essentially a products of man's drive towards explaining, understanding and coping with his immediate environment and with reality in general, reality understood in its broadest sense and in permanent change as a result of man's own action. This drive, obviously holistic, is dynamically subjected to a process of exposure to other members of society -- people -- and thanks to communication, both immediate and remote in time and space, goes through a process of codification, intertwined by an associated underlying logic, inherent to the people as a form of knowledge – some call wisdom. The modes of communication and the underlying logic are recognized as the result of the prevailing cognitive processes. Cognitive evolution, related to environmental specificity, gives rise to different modes of thought and different underlying logic, communication and codification. Hence knowledge is structured and formalized subjected to specificity of a cultural nature. Power structure, which itself rises from society as a form of political knowledge, appropriates, indeed expropriates, structured knowledge and organize them in institutions. In this form and under the control of the establishment and the power structure, which mutually support each other, knowledge is given back to the people, who in the first instance generated it, through systems and filters which are designed to keep the established power structure.

The generation, transmission, institutionalization and diffusion of knowledge is clearly an holistic approach to knowledge and to the dynamics of change. This is the essence of the research program on the History of Science which I call ”Ethnomathematics”.[11]

The disciplinary approach to knowledge focus on cognition, epistemology, history and sociology. This clearly makes it difficult to understand the dynamics of change. Mutual exposure of distinct approaches to knowledge, resulting from distinct environmental realities, is global, embracing the entire cycle from the generation through the diffusion of knowledge.

The process of cultural dynamics which takes place in the exposure is based on mechanisms which balance the process of change, which I call acquiescence -- that is, the capability of consciously accepting change (modernity) -- and the cultural ethos -- which acts as a sort of protective mechanism against change that produces new cultural forms.

This behavior can be traced back throughout the entire history of mankind. These conceptual tools are close to the ethos and schismogenesis introduced by Gregory Bateson in dealing with cultural contact and enculturation.[12]

In the encounter of the two worlds (Europe and America) this was violated in many instances. The origin of these violations may be related to distinct views of nature. A scientific conceptualization, which resulted from an intertwining of medieval Judeo, Christian and Greco-Arabic thought, and developed in Europe, lead man to look at nature and at the universe as an inexhaustible source of richness and to exploit these resources with a mandatory drive towards power and possession.

This behavior towards nature and life has lead man to favor a single model of development, hence to ignore the cultural, economical, spiritual and social diversities which constitute the essence of our species.

These reflections question the set of current concepts and models, and calls for the acceptance of the idea that survival depends of a global and holistic view of reality. This demands a radical change which applies to all levels of knowing and doing. Thus we are lead to look for radical changes in our models of development, of education and of civilization, based in the recognition of a plurality of models, of cultures, of spirituality and of social and economical diversity, with full respect for each one of the distinct options.

Visions of the World

The European navigators of the end of the 15^th and early 16^th centuries reached all of America, Africa, India and China. In the case of Africa and in Asia, previous contacts with civilizations which had shared, before, many encounters among themselves and with Europeans. Thus the encounters of the 15^th and early 16^th centuries were, indeed, an amplifications and deeper contacts. But meeting the “new”, the unknown, the unexpected, was experienced by Columbus and the Spaniards, in 1492 and the subsequent voyages.

Although earlier contacts with the Americas are known. But the motivations and behavior of earlier navigators was completely different from the Spanish and Portuguese, and afterwards the English, French and Dutch.[13]

The influence of the navigators and chroniclers, particularly Portuguese, in building up the mode of thought which underlies modern European science is noticeable. In the words of Joaquim Barradas de Carvalho "the authors of the Portuguese literature of the navigation made it possible the Galileos and the Descartes"[14] essentially through the development of "objective and serene curiosity, rigorous observations and creative experimentation"[15].

The low recognition of Portuguese science in the 15^th and 16^th centuries illustrates the observations above about biased historiography. Indeed, the important Tractatus de sphera (early thirteenth century) written by Johannes de Sacrobosco, was recognized as "the clearest, most elementary, and most used textbook in astronomy and cosmography from the thirteenth to the seventeenth century"[16], received two important translations with commentaries in Portugal. By Pedro Nunes, in 1537 and by João de Castro, possibly in 1546. The translation with comments by Pedro Nunes, an important mathematicians of the 16^th century, incorporates much of the observational and experimental science which had been pursued by Portuguese navigators since early fifteenth century and registered in their writings. Curiously enough, neither are recognized in the most important study of Sacrobosco, written by L. Thorndike.

Particularly important as chronicles are the Crónica dos feitos de Guiné of Gomes Eanes de Zurara (1453) and the Esmeraldo de situ orbis, by Duarte Pacheco Pereira, written between 1505 and 1508, probably the first major scientific work reporting on what was observed and experimented in the newly "discovered" environments. In fact, we have to understand the sense of the word "discovery" among the Portuguese authors of that period to better realize the role of the navigations in paving the way for modern science. In his important historiographical contribution, Joaquim Barradas de Carvalho (see Note 15) gives both an exhaustive study of the Esmeraldo de situ orbis and the discussion of the meaning of the word ”discovery”.

The voyages themselves allowed a broader view of the world. Mainly venturing to the Southern Hemisphere demanded two major enterprises, the construction of the caravel, an extremely versatile ship built by the Portuguese in the fifteenth century as the result of a remarkable engineering project,[17] and novel navigation techniques, relying on tables constructed from systematic recorded observation carried on by the commanders of those ships. Themselves with commanding function they were also responsible for recording the "different skies" which they were the first Europeans to look at. The contributions of Gil Eanes crossing the Bojador Cape in 1434, Nuno Tristão reaching in 1443 the coast of Mauritania, and the major achievement of Diogo Cão crossing the Equator line in 1483, all paved the way for Bartolomeo Dias to cross the Cape of Hope in 1488 and for Vasco da Gama to reach Calicut in India, in 1498. Together with Columbus reaching the Western lands in 1492, the vision of the World changed. All lands and peoples were within the reach of the navigators. It is the beginning of a new phase in the History of Mankind.

The “new Sciences” seen in the encounter

As said above, America and to some extent Africa, were more surprising to Europeans than what was seen in lands which had been reached before by land routes. Particularly, America showed peoples with new forms of explanation, of rituals and of societal arrangement. Reflections on the so-called Natural Philosophy or the Physical Sciences, particularly Astronomy, were part of the overall cosmovision of the pre-columbian civilizations. In other words, the scientific establishment and scientists, surely present in the society of the conquered cultures, have not been recognized as such by the conquerors. One of the earliest registers of these cultures, Fray Bernardino de Sahagún writes, in the 16^th century, that "The reader will rightfully be bored in reading this Book Seven [Which treats Astrology and Natural Philosophy which the naturals of this New Spain have reached],...trying only to know and to write what they understood in the matter of astrology and natural philosophy, what is very little and very low".[18] The important report of Sahagún explains much of the flora and fauna, as well as of medicinal properties of herbs of Nueva España. But he does not give any credit to indigineous formal structured knowledge. This is typical of what might be called an epistemological obstacle of the encounter.

Another important book is the Sumario compendioso ... con algunas reglas tocantes al Aritmética by Juan Diaz Freyle, printed in Mexico in 1556, the first arithmetic book printed in the New World. It has a description of the number system of the Aztecs. But this book soon disappeared of circulation and the Aztec arithmetic was replaced by the Spanish system.

Much research is needed on the Science of the encounter. But this needs a new historiography, since names and facts, on which current history of science heavily rely, have not been a concern in the registry of these cultures. A history "from below", which might throw some lights in the modes of explanation and of understanding reality in these cultures, have not been common in the History of Science.

There is some more availability of sources for the history of the natural and health sciences.

The Basin Metaphor and a Sociology of Mathematics

There is no way to deny that [Western] Mathematics is essential in the modern world. Public opinion is ready to support investment in mathematical research in spite of being absolutely unable to guess what kind of research is being supported, professionally successful parents invest in the mathematical education of their children and even accept that a child does an entire year again if he/she fails in the final exam - in spite of him/her, successful parent, declaring that while they were in school and up to nowadays never understood mathematics. "Miraculously" they graduated in spite of successive failures in Mathematics and "miraculously" they became very successful. Their children have to proceed - suffering and struggling - so they will not depend of miracles! Less successful parents, which did not have an opportunity of schooling and nave not the slightly idea of Mathematics punish their children if they don't show good marks in Mathematics! And peers and society in general regard those that get good grades in Mathematics as potential geniuses, while those that do not do well in Mathematics are regarded as stupid. Socially, this has been instrumental in the selection of elite, as it has been well studied by Pierre Samuel in his classic paper on this theme. On the other hand, the evidence from research showing that both individual and social creativity is enhanced by self-esteem is not taken into account for those that do beautifully in the Arts or in Sports but fail in Mathematics. Let us introduce at this moment some concepts and reflections that result from what is now called Social Studies of Science or Science Policy. This is basically the study of the politics of scientific development, the backbone of funding agencies. It is very interesting to analyze the substitution of the colonial discourse by the discourse of aid - both multilateral, like UNESCO, and bilateral, like ORSTOM, the British Council and similar. The nature of the deprived populations did not change in the span of less than ten years. The strategies to keep them as faithful consumers had to change.[19] But let us not deviate from the main objective of this paper, which is the production of scientific, in particular mathematical, knowledge.

When deciding on investments in Science and Technology, it is natural to expect social benefits. These investments have been substantial, both through funding agencies, either governmental or through aiding agencies, either bi- or multilateral. The outcomes in the so-called Third World have not been encouraging, as recently mentioned by the Director-General of UNESCO. The gap between central nations and peripheral nations in the production of scientific knowledge is enlarging. Over 80% of the benefits of scientific and technological research benefits the First World. "The gap between rich and poor countries is a gap of knowledge" as says Federico Mayor[20]. It is clear that scientific productivity is related to the cultural atmosphere and self-esteem. Self-esteem can hardly prevail among a population deprived of its history.

Referring to what was discussed above, the main instrument in the colonial period was to deprive the conquered peoples of their history or to produce a history "favorable" to the conqueror. There is no need to elaborate on the vision of slavery passed on by official history nor to question why Zumbi (1655-1695) is practically unheard of by Brazilian students while Cardinal Richelieu and, of course, D'Artagnan, are so familiar.

We may consider, as it is frequent in discussions of policy and specially in the United Nations and other national and international agencies, the production of scientific and technological, particularly mathematical, knowledge as measurable. Scientometrics relies on several indicators and the studies of quantitative history allow us to speak of central nations, those who produce new knowledge, and peripheral nations, those who absorb new knowledge. Production and absorption of knowledge are clearly distinguishable. The sad situation is that the peripheral nations have been slow in absorbing new knowledge. The lack of infrastructure acts as a barrier for this process.[21] The basin metaphor helps to understand the process. The picture speaks for itself. The main producers of knowledge (central nations) are represented by the main stream. The water fertilizes their margins. They will produce their effect in the margins of the tributaries (peripheral nations) much later, when the waters have already flown along the stream (thus producing the gap or obsolescence of knowledge). The Water (knowledge) does not flow up stream of the tributaries.

FIGURE: The basin metaphor

The water of the tributaries surely fertilizes their margins and will add and contribute, as the brain drain, to the volume of water of the main stream. This is manifest in the emigration of academics and, worst, on the subordination of laboratories and research institutions of the peripheral nations to the priorities of their major homologous in the central nations.[22] As an example of this, we see the efforts to entice research institutions in the peripheral nations to join major biotechnology research plans.

The co-optation of scientists in the periphery is normally done by the attractive of sending experts, in many cases scientists with a high reputation, to the periphery for short visits and conferences, by offering fellowships, by giving stipends higher than the current national salaries, by sending equipment, in many cases obsolete or already heavily used equipment, and offering international travel to seminars and congresses. This is true in academics and, in the more developed peripheral nations, also in industry. Particularly in mathematics, we have numerous examples of such practices in the post-war period. The presence of monies of the USA Army, Navy and Air Force research agencies, as well as of the NSF, of the CNRS, of the British Council, of the DAAD and other agencies, following the pattern mentioned above, is noticeable.

These cases have not been studied in detail as yet. Both have the common feature of producing human resources and results without any analysis of the capability of the peripheral countries to absorb and to make these resources and results useful for their priority needs. Normally this is the result of a lack of qualitative directives in Science Policy of the peripheral nations. Practically every scientific development plan in the periphery is a program entirely based in quantitative goals.

Perversely, World Bank, UNDP and other financing agencies rely on, indeed stimulate, plans based on quantitative goals. Clearly, they are easier to check. But the benefits for the poor populations of the peripheral nations are practically nil.

In the basin metaphor, the sources of the rivers, both the main stream and the tributaries, correspond to ethnomathematical knowledge. Ethnomathematical knowledge, like the waters, flow fertilizing the margins of the tributary in their way and eventually mixing in a major stream, contributing to this flow. Waters of the main stream do not go up-stream through the tributaries.

The notion of progress carried on by the main stream will benefit the margins of the tributary after a long way through difficult land paths - which correspond to the acquisition of knowledge from other socio-cultural and environmental sources. The need of the margins - peripheral cultures - are met by the water of the tributaries and only later receive the benefits coming from the main stream. These are useful only in fertile grounds.

An alternative to main stream and tributaries would be a large lake, were all the sources contribute equally to the main body of water. Each source producing according to its environmental history and all the waters of the lake fertilizing all the margins.

Erosion of the basin in favor of the creation of a great lake - the deterioration of the current world order - hopefully will lead to a new planetary order.

Final remarks

The conquest and colonization had as a consequence an enormous influence in the course of development of the civilization. The chroniclers of the conquest tell of absolutely different ways of explaining the cosmos and the creation and of dealing with the surrounding environment. Religious systems, political structures, architecture and urban arrangements, sciences and values were, in a few decades, suppressed and replaced by those of the conqueror. A few remnants of the original behavior of these cultures were and still are outlawed or treated as folklore. But they surely integrate the cultural memory of the peoples descending from the conquered. Much of these behaviors are easily recognized in everyday life.

Mathematics, as an human endeavor, is not different. This is one focal point of the research program known as Ethnomathematics, which deals with the generation, the intellectual and social organization and the diffusion of different ways, styles, modes (tics) of explanation, understanding, learning, coping with and probing beyond (mathema) the immediate natural and socio-cultural environment (ethno). This clearly results from the mutual exposition of different cultures and the dynamics of this process is a major problem we face in doing the history of ideas in every region of this world.

The conquest paved the way to colonization. In the Americas, the early colonizers, the Spanish and the Portuguese, paved the way for the French, the English and the Dutch colonizer and later on for Africans, Europeans and Asiatic immigrants. With them came new forms of coping with the environment, of dealing with daily life, and new ways of explanations and learnings. The result was the emergence of a synthesis of different forms of knowing and explaining which were generated by and available to the different communities, to workers and to the people. We recognize the emergence very soon of new religions of new cuisine, new music, new arts and new languages, particularly in the Americas -- the Creoles. All of these absolutely interrelated as a synthesis of the cultural forms of the ancestors.

Mathematics, as a cultural form, is not different. The emergence of new cultural behavior, particularly of Mathematical behavior, is another focal point of the research program known as Ethnomathematics.

Particularly in the Americas, the variety and peculiarity of the expositions of cultures and the specificity of the population migrations reveal an effort of the colonizer to transfer, with minor adaptations, the forms of social, economical and political organization and administration prevailing in the metropolises, including schooling and scholarship (academies, universities, monasteries). The new institutions in the Americas were based on the styles prevailing in the metropolises, mostly under influence, and even control, of religious orders.

All this, which took place during most of the 16th, 17th and 18th centuries, occurred while new philosophical ideas, new sciences, new ways of production and new political arrangements were flourishing in Europe. The cultural facts produced in Europe were assimilated in the Americas under specific, mostly precarious, conditions. Indeed the Americas were the consumer of some of these new cultural facts. There is a clear co-existence of cultural goods, particularly knowledge, produced in the Americas and produced abroad. The former consumed mostly by the lower strata of society, the people and workers, and the later by the domineering classes. These boundaries are not clearly defined and the mutual influence of the resulting intellectual productions are evident.

This poses the following

BASIC QUESTION

What are the relations between the producers and consumers of cultural goods?

This guides my proposal for a historiography of Mathematics and what I have called "the basin metaphor". Although this is a question affecting the relations between academia and society in general, hence between the ruling elites and the population as a whole, it is particularly important for understanding the role of intellectuality in the colonial era. Thus Ethnomathematics comes as a fundamental instrument of historical analysis.[23] These views own much to the Annales proposal.

Curiously enough, the factors influencing the consumption of what we might call Academic Mathematics produced in an alien cultural environment, and what "outsiders" of the profession -- that is, non-mathematicians -- have to say about Mathematics, have not been given attention in the prevailing historiographies.[24] My proposal incorporates to the History of Mathematics, in an essential way, the views of aliens, in both senses, about Mathematics. This broader look, suggested by new historical scholarship, comes under severe attack, in what became to be known as the Science Wars.[25]

THE CYCLE OF KNOWLEDGE