METHODOLOGICAL QUESTIONS IN STUDYING

THE HISTORY OF MATHEMATICS

IN COLONIAL LATIN AMERICA

Acta historiae rerum naturalium necnon technicarum, vol. 3, 1999; pp. 139-151.

For Lubos Nový on his 70^th birthday

It is impossible to give in a short paper an account of the developments in Mathematics in Latin America in colonial times. The History of Mathematics in Latin America is a field open for research. But research requires some specific methodology. Current historiography seems to be inappropriate for such studies. In this paper I will propose some methodological directions and will also mention a few sources which give serve as basic references for the period. Particularly interesting is the fact that the first mathematician recognized in Brazilian history is a Czech. Among the most relevant early mathematical works in Latin America is the research of Valentim Stancel, S.J. who, after his studies in Ormuz and Prague, spent most of his life in Brazil, in the 17^th century.

THEORETICAL FRAMEWORK

This paper is situated in a discussion of the nature of scientific knowledge and a search for alternative research methodologies in the history and philosophy of science.

I discuss the problem of the generation, exchange, and interaction of mathematics in cultures confronting themselves during the colonial period in the Americas, particularly Latin America.

Although much attention has been given to the dynamics of scientific creation, little has been reserved for a consideration of the process of the confrontation of different ways of approaching science, such as in the case of the conquest and colonization in what is today Latin America.

I will look at the process from the point of view of the analysis of the cultural dynamics seen in the colonial process. We see the basic power structure manifested in legitimization schemes based on certain forms of authority that combine magic (knowledge, language, values, science) and technique (ability, art, arms, production, kinds of work, barter, and money) as the essence of a synthesis of Western culture. In fact, this synthesis is often heavily reflected in the paradigms that permeate both science and art, paradigms that serve as the basis for establishing a conceptual structure for creativity.

Although it is such a broad concept that it is difficult to define, creativity is understood to have many meanings, always converging on the production of something that is outside the ordinary, or the expected, and that brings a new dimension to an undertaking. Creativity is manifested in various ways and is recognized by its results: a creative poem, a fabulous goal in a soccer game, a particularly good joke, a clever proof of a mathematical theorem. All these manifestations presume that there is something new, appropriate for what already exists and legitimized by the rules and conventions of society. It is basically an "action" that results in the conscious immersion into a reality and the unconscious release of some form of energy. The enigma has to do precisely with the intensity with which that energy is related to existing codes. This is the process to be understood. Certainly, this relationship - to break or to conform with the codes - brings to the surface the dialectical process of unconscious thought entwined with conscious thought, that, at the same time, can be conceived of as a dialectical process of the limits and extension and transgression of what is accepted.

With relation to creativity and with relation to the dialectical process of extension-transgression of limits, reflections on reality become a decisive step through creative action, that will have entered into the same reality that was the initial step.

The cultural dynamics implicit in the scheme ... reality - individual - action - reality ... finds there its total dimension. In fact, as gears in a mechanism of individual action, be they individuals in the usual sense, be they established cultural patterns, or social behaviors depends on something that adapts correctly to individual behavior. In a discussion on cultural anthropology we indicate kinds of behavior that clarify the process of the basic confrontations of different cultural patterns. Recent works on ethnoscience and attempts to understand the basic structure of such manifestations are elements of great worth in an understanding of this process.

Of all the manifestations of creativity, scientific creativity deserves a position of distinction. The controversial matters raised when someone asks if creativity would be considered to be the transgression of limits versus the extension of limits, or if its origins are socio-cultural versus individual-psychological, find fertile ground for debate in the study of science as a human force. Science has played a pioneering role in the revolt against transcendental taboos such as magic, authority, myth, and fear of the unknown, transgressing limits. Science is also cumulative by its very nature, building upon previous results, whose acceptance supports, probably, the most elaborated and hermetic legitimization process. In fact, criteria of acceptance that permeate creativity are in some sense a mystery of science, and without a doubt depend on an organized body of practitioners [1].

Constructed on a system of thought originated in Greece and developed by a complicated cultural complex, modern western science was carried along with the conquest, by western invaders, to non-western civilizations. Although the invaders had made various concessions with respect to many fundamental matters such as religion, life styles, economics, and politics, no concession of universal scope was permitted with respect to a commitment to forms of explanation other than those offered by so-called western rationalism, nor with its most successful realization which is science and technology. This makes the above mentioned concessions illusory. It would be enough to raise the problem of western invasions and conquest of non-western civilizations, but the process repeats itself in an internal pattern seen in the so-called modern world that affects countries, communities, families, and individuals. The prevailing image of modern science is the result of advanced technology and of miraculous conquests and even of a vague idea on the way those marvels have appeared. The set of knowledge and methods, universal, and "ideologically neutral" that constitute modern science, reign above the understanding and challenge of society as a whole. We raise again the problem of the controversy of the socio-cultural versus individual-psychological nature of creativity. In this case, individual behavior is substituted by the behavior of a minority identified by a set of deceits, common background, and interests. More than a controversy, this is in fact an internal conflict of Western civilization. It is very important in educational and scientific policies, especially in developing countries. Certainly, considerations of this nature are of fundamental importance for the questioning of the relationship between the educational system and the intensification of creativity, understood as a manifestation of a synthesis of natural elements.

I return now to the basic question of what constitutes scientific creativity. In other words, what causes science to advance? We can examine folklore and the accounts of Kakulé, Hadamard, Poincaré, and others to suggest the role of the unconscious in the process in scientific creation [2]. In all the accounts we find a clearly defined structure that Morazé puts as informare – cogitare -- intelegere [3]. There is a sense of being immersed in reality, in a global reality that includes the socio-cultural and natural medium.

This basically coincides with the position of Paul Feyerabend in his interesting and polemic sketch of an anarchistic theory of knowledge. He says that scientists do not solve problems using a magic wand of methodology or a theory of rationality. Instead it is because they study a problem for a long time, because they know the situation perfectly, because they are not foolish, and because the excesses of one scientific school are almost always compensated for by the excesses of others. Furthermore, scientists rarely solve their problems, they make a lot of mistakes, and many of their solutions are useless. Basically there is little difference between the process that leads to the pronouncement of a new scientific law and that which leads to a law of society: someone informs all the citizens, or at least those who are interested, others collect facts and precedents, the matter is discussed and finally they vote. While democracy strives to explain the process so that all can understand it, scientists hide it or distort it to conform to their sectarian interests. The problem is then one of legitimization of a process. The responsibility resides essentially in the pact between society and scientists, implicit since the parallel appearance of modern science, technology, and the concept of political interest, in the 17^th century. This pact has led to the point of an almost total control by the stratocracy which resulted from interweaving modern science and technology with national, social and public interest that has been increasing since the 17^th century, even becoming an incontestable force today. The renaissance of the pact between society and scientists and the efforts to bring this pact to new dimensions, focuses on more direct and immediate social interest. This brings back the commitment of Isaac Newton in saying that "Scientia ex usu conciliatur gratia". It suggests a reexamination of the scientific process itself and also on an examination of the strategies for developing societies. Certainly, these strategies are implicit in educational systems, which leads us to an analysis of educational practices and schooling.

I will now discuss the link between knowledge and power. The relationships among these categories of human behavior in Western culture are implicit in the conflict between the Creator and his creation in the Book of Genesis, which has permeated western thought. In fact, the concept of knowledge, primitively confused with wisdom, has become identified with the acquisition of facts, experiences, codes, and symbols accumulated by the human species. In this process of accumulation, abstract thought has been preponderant, opening the way to become what is today western thought. This sustains an ideology with profound implications in the social structure of the modern world. An analysis of the patterns of work production and consequently of forms of trade are of great importance in understanding the predominant social structure. An analysis of the forms of trade has to consider the question of how to connect, in the history of ideas, the concrete and material concepts of utility with the essentially abstract concept of value. In this connection one finds the appearance of money and the origins of monetary society, that marks western society and the distinction between intellectual and manual labor. In fact, there is the key to the privileged status of knowledge in the invasion, by Western civilization, of its rivals. An analysis of the relationship between manual and intellectual labor as a critique of epistemology made by Alfred Sohn-Rethel suggests the inherent nature of the search for power in the construction of the preponderant role of knowledge [4]. More specific examples of this are shown in the substitution of the concept of wisdom, in traditional African cultures, by the concept of knowledge, as has been discussed quite well by Kwasi Wiredu [5]. Also, examining the philosophical basis of the conquest and colonization of America, we recall the synthesis of the jurist Juan Lopez de Palacios Rubios at the beginning of the 16^th century: "the infidels should, because of their ignorance, serve those who have wisdom, as servants to masters."

This leads us to the question of what kind of knowledge we are talking about. Everyone accepts that a global knowledge exists, general and structured in a certain way, following a special logic, a knowledge that is above the domination of a certain cultural group, and that we call science. Therefore, we encounter the necessity for alternative epistemologies when we want to explain alternative forms of knowledge. Although derived from the same natural reality, this knowledge is structured in a different way.

Our focus remains in between the history of science and cultural anthropology. Ethnoscience can be conceptualized as the study of scientific, and, by extension, technological phenomena in a direct relationship with their cultural, economic, and social formation. Particularly interesting is the case of mathematical ideas understood in this context. That is, of ethnomathematics.

Much has already been said about the universality of science. This concept of universality does not seem to be easy to defend, as has been shown by recent research carried out by anthropologists that provides evidence of typically scientific activities such as observing, counting, ordering, choosing, measuring, and weighing, being performed in more radical ways than is commonly taught in school systems. These observations have given the incentive for studies on the evolution of scientific concepts and mathematical practices in an anthropological and cultural structure. I feel that this has been done only to a limited and even timid extent. On the other hand, there is a reasonable quantity of literature on this matter, written by anthropologists. Linking the research of anthropologists and cultural historians to science and mathematics is an important step in the recognition of the different modes of thought that lead to different kinds of science, or to that which I call the Programme Ethnomathematics.

Almost nothing is known about the logic used in the Incan treatment of number, but it is known that the "quipus" represented a language that was mystical, quantitative, and qualitative. The concept of experience or experimental method is something that can be discussed. When we follow the strong argument of René Thom in favor of a Heraclitian position and his challenge in favor of theoretical reflection on what can be called "the experimental base of scientific knowledge" we have to admit the possibility of a new concept of experience. These observations invite us to an analysis of the history of science in a broader context, thus adding to it other forms of knowledge of natural phenomena. Let's go even farther in our considerations by saying that this is more than an academic exercise, once the pedagogical implications are clear, principally due to the recent advances in the theory of cognition, that show how culture and cognition are intimately related. Despite the indications of the strong link between the mechanisms of cognition and the cultural environment, a reductionist tendency that has its origins in Descartes, and that up to a certain point grew parallel to the development of science, attempted to dominate education until recently, recognizing and encouraging models of cognition that are free of culture. The recent recognition of the interpenetration of biology and culture opens a fertile field of research on culture and scientific cognition.

In order to do research effectively in this area it is necessary to develop a research method that assimilates and understands ethnoscience. This demands a mix of research methods from anthropology with research from the other sciences, a field of study that has been little cultivated. The social history of science, that aims to understand the influence of many of the socio-cultural, economic, and political factors in the development of science, is a theme that we judge essential, not as an exercise in itself, as some universities have been interested, but, yes, as the basis upon which we can understand the evolution of scientific knowledge.

The history of mathematics requires also a more global, holistic approach, not only through the consideration of methods, objectives, and contents of mathematical knowledge in solidarity, but principally through the incorporation of results of the anthropological discoveries. This is very different from what has been done, which is to analyze mathematical contents disregarding the context.

There are many implications for the research priorities in the history of mathematics, and even a counterpart in the development of mathematics itself. The distinction between science and mathematics and technology has to be interpreted in a different way. What was labeled as mathematics is the natural result of the evolution of the discipline with an economic, cultural, and social model, which cannot be separated from the main expectation of a certain socio-cultural group in a historical moment. Equally important are considerations about ideology, which is implicit in dress, dwellings, titles, and, of course, in ways of thinking, including the inherent logic of structured knowledge. Mathematics in intrinsically related to logics, which sustains the ideological roots of western civilization.

We assume, throughout the Programme Ethnomathematics, a broad concept of mathematics, that permits an analysis of common practices that are apparently unestructured forms of knowledge. This arises from a concept of culture that is the result of the hierarchization of behavior, proceeding from individual to social and then to cultural, and is based on the behavior model referred to in the cycle ... reality - individual - action - reality .. The concept of mathematics that results from this model permits the inclusion of what can be considered as marginal practices of a mathematical nature. Naturally, these common practices are imbued with ideological connotations rooted in the cultural texture of the group who uses them.

A complete understanding of these ideological connotations is what establishes the research program on alternative methodologies.

This is the theoretical framework which I propose to study the history of mathematics in Latin America. It is important in this study the adoption of a convenient chronology and the identification of sources.

MATHEMATICS IN COLONIAL TIMES

IN LATIN AMERICAA BRIEF SURVEY

In 1492 Christopher Columbus reached new lands navigating under the sponsorship of the Catholic Kings, Isabella of Castile and Fernando of Aragon. In 1500 the Portuguese Pedro Alvares Cabral reached the coast of Brazil. By the end of the first quarter of the 16^th century, most of the territory known as the Americas had been conquered in the name of the Kings of Spain and Portugal. Immediately the establishment of colonies in the conquered lands took place.

We may say that the origins of mathematical knowledge is as yet an unresolved issue. It is difficult to disagree that the search for explanations (religions, arts and sciences), systems of values and behavior styles (communal and societal life), the psycho-emotional and the imaginary and models of production and of property are related to mathematical thinking. There is growing scholarship in the search of different styles and modes of building up knowledge in different natural and cultural environments, where the development of mathematical ideas is recognized. In the civilizations conquered and colonized by Europeans the development of mathematics ideas followed different paths from Europe, Asia and Africa. Reading the chroniclers of the conquest we easily recognize 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.

To study of the development of European mathematics introduced in the conquered lands, we need a specific chronology.

My proposal is a chronology based on five major periods:

1. Pre-columbian;

2. conquest and early colonial times (roughly 16^th and 17^th centuries);

3. the established colonies (18^th century);

4. independent countries (19^th century);

5. the 20^th century.

This division is justified when we look into the most relevant turning points in the development of the region. Of course, mathematical development is subordinated to the overall scenario of society.

We have also to take into account geographic divisions. For the pre-Columbian period, sources are available mainly for the Aztec, Maya and Inca civilizations. An enlarged concept of sources, mainly drawn from anthropologists, is needed to look into other civilizations, such as for example, those of the prairies and of the Amazon basin. Much finer divisions, taking into account both political and cultural specificities, are needed for a special study of pre-Columbian mathematics. A similar situation occur in studying traditional African cultures.

After the conquest of the Americas, the most appropriate is to follow the administrative organization in Viceroyalties: New Spain (roughly what is today Mexico and upper Central America), New Granada (southern Central America, approximately Costa Rica, Colombia, Venezuela, Ecuador), Peru (roughly Peru and Bolivia), La Plata (roughly what is now Chile, Paraguay, Argentina and Uruguay) and the Viceroyalty of Brazil, which was a Portuguese conquest.

The current political division of the regions in countries is practically the same as resulted from the independence movements since the 19^th century.

In this paper I will not cover the developments in this period. For a brief introduction see my paper "Science and Technlogy in Latin America During the Discovery"" in Impact of Science on Society, vol. 27, n° 3, 1977, pp. 267-274.

In the early colonial times, the Spanish and the Portuguese tried to establish schools, mostly run by Catholic religious orders. The demand for mathematics in these schools were essentially for economic purposes related to trade, but there was also an interest on mathematics related to astronomical observations. Reliance on indigineous knowledge was limited, but there was some interest in the nature of native knowledge.

An important source justifying this assertion is the first non-religious book published in the Americas is an aritmethics book related to mining, the Sumario compendioso de las quentas de plata y oro que en los reinos del Pirú son necessarias a los mercaderes y todo genero de tratantes. Con algunas reglas tocantes al arithmética, by Juan Diez freyle, printed in New Spain in 1556. It is a book on arithmetics as practiced by the natives, to which the author adds some questions on the resolution of quadratics.

Other important source is the Historia del Nuevo Mundo [1653], by Bernabé Cobo [6]. The archives of the Jesuit missionaires, as well as of other religious orders, are rich in historic material, but they are as yet to be explored.

Already in the first century after the conquest, we have practical books published in Mexico, such as the Arte menor de arithmetica, by Pedro de Paz, in 1623, and Arte menor de arithmética y modo de formar campos, de Atanasio Reaton, in 1649. It is also to be noticed the book Nuevas proposiciones geométricas, written by Juan de Porres Osorio, in Mexico.

Astronomy was a major area of interest in Latin America in the 17^th century. There are important discussions on the meaning of comets. Many of the interpretations related to their puorpose of conveying divine messages and messages to mankind. In another words, these were search for scientific explanations. The figure of Don Carlos de Sigüenza y Góngora, of Mexico, towers. His works focus on astronomical observations and calculations. His Libra astronómica y filosófica, written in 1690, is considered one of the most important works of Latin American Science. In it Sigüenza y Góngora refutes prevailing astrological arguments about comets.

In Brazil, research on comets was of major importance. The same tone of the reflections of Sigüenza y Góngora we see in the work of Valentin Stancel (1621-1705), a Jesuit mathematician from Prague who lived in Brazil from 1663 until his death [7]. His astronomical measurements are mentioned in Newton’s Principia [8]. Several polemical exchanges of letters and papers from these times reveal interesting epistemological arguments. Particularly revealing of the internal tensions in such established school of thought as the jesuit order is the reports on Stancel by his superior, Antonio Vieira (1608-1697), commenting on his views about the nature of comets [9].

Also in the Viceroyalty of Peru we have the same concerns. The first to be recognized as a mathematician in Peru is Francisco Ruiz Lozano (1607-1677), who wrote Tratado de los Cometas, essentially a treatise of medieval mathematics explaining the phenomenon.

In late colonial times, since the middle of the 18^th century, a good number of expatriates and criollos played an important role in creating a scientific atmosphere in the colonies. This happened under the influence of the Ilustración [Enlightenment], the important intellectual revival that began in Spain under Charles III and in Portugal under José I and his strong minister, the Marquis of Pombal [10].

A number of intellectuals well versed in a variety of areas of knowledge were responsible for introducing mathematics to the colonies. These include Juan Alsina and Pedro Cerviño in Buenos Aires, who lectured on Infinitesimal Calculus, Mechanics and Trigonometry. In Peru, Cosme Bueno (1711-1798), Gabriel Moreno (1735-1809) and Joaquín Gregorio Paredes (1778-1839) are best known. I do not have access to details of their life and work.

In Brazil, José Fernandes Pinto Alpoim (1695-1765) wrote two books, Exame de Artilheiros (1744) and Exame de Bombeiros (1748), both focused on what we might call Military Mathematics, and both were written in the form of questions and answers. Not much is known of his life and mathematical work.

Where and with whom did they study? These are questions open for research.

In South America.

Colombia had a privileged situation in the pre-independence days, which reflected in the developments in the 19^th century. A rather distinguished figure was José Celestino Mutis (1732-1808), who was the author of an unpublished translation of Newton, but was also responsible for bringing to Colombia new ideas of mathematics in Colombia, mainly relying on the books by Christian Wolff. He was the founder of the Observatorio de Bogotá, in 1803. His most distinguished disciple, Francisco José de Caldas (1771-1816), became the director of the Observatory. Caldas was deeply involved in the Independence War and was shot by the Spaniards.

In Venezuela, a Real y Pontificia Universidad de Caracas was founded in 1721, with no Mathematics contemplated in its plan of studies.

In Chile, the Universidad Real de San Felipe, which was inaugurated in 1747 in Santiago, was provided with a ‘catedra’ of Mathematics. Fray Ignacio León de Garavito, a self-instructed criollo mathematician, was responsible for this chair.

The most significant developments of mathematics in Latin America in these days took place in Mexico. The preponderance of New Spain in the Americas in that period was responsible for effects of these developments in the rest of the Spanish colonies.

It is important to recognize that much of the development of Mathematics in Europe in the Renaissance was due to the the incipient industrialization and to the emergence of new metropolises. The same is true in the colonies. Practical problems related to building up the economy of the colonies and establishing the centers of administrative power resulted in the development of special kinds of applied mathematics. In all the colonies in the Americas urbanization was a major challenge. The transfer of the idea of an European city to the new world posed quite interesting problems, which influenced mathematical development. A sort of mathematics applied the most immediate questions posed by the economy of the region, such as mines and urban development [11].

The influence of Mexico is particularly felt in Guatemala, which included Costa Rica. The most renowned scholar in the period is José Antonio Liendo y Goicoechea (1735-1814), who had a Mexican background. He taught at the Universidad de San Carlos de Guatemala, which had already become an important academic center after a new plan of studies was published in 1785. This plan was written in Latin in the form of 25 theses, under the title Temas de Filosofia Racional y de Filosofia Mecánica de los sentidos, de acuerdo con los usos de la Física; y de otros tópicos físico-teológicos según el pensamiento de los modernos para ser defendidos en esta Real y Pontificia Academia Guatemalteca de San Carlos ..." [12]. This was essentially a medieval proposition. Goicoechea was responsible for modernizing this plan of studies, incorporating experimental physics to the project. He introduced mathematics incorporating newtonian ideas, based on the texts of Christian Wolff.

The Viceroyalties of Nueva España, Nueva Granada, Peru, La Plata and Brazil achieved their independence in the first quarter of the 19^th century. This period goes beyond the scope of this paper.

REFERENCES

[1] Michel Polányi: Personal knowledge. London: Routledge & Kegan Paul, 1964.

[2] Arthur Koestler: The act of creation. London: Picador, 1975.

[3] Charles Morazé: Literacy invention, in R. Macksey & E. Donato, eds., The structuralist controversy, Baltimore: The Johns Hopkins University Press, 1972; pp. 22-55.

[4] Alfred Sohn-Rethel: Intellectual and manual labor: A critique of epistemology. New York: Humanities Press, 1979.

[5] Kwasi Wiredu: Philosophy and an African culture. Cambridge: Cambridge University Press, 1980.

[6] Bernabé Cobo: Historia del Nuevo Mundo [1653], Madrid: Atlas, 1964.

[7] See the paper by Juan Casanovas S.J. and Philip C. Keenan: The Observation of Comets by Valentine Stansel, a seventeenth century missionary in Brasil, Archivum Historicum Societaatis Iesu, LXII, 1993; p.319-330.

[8] Isaac Newton: The Principia. Mathematical Principles of Natural Philosophy. A new Translation, by I. Bernhard Cohen and Anne Whitman, Berkeley: University of California Press, 1999; p.927.

[9] For details see the paper by Carlos Ziller Camenietzki: O Cometa, o Pregador e o Cientista. Antônio Vieira e Valentin Stancel observam o céu da Bahia no século XVII, 1° Seminário Nacional de História da Ciência e da Tecnologia, Ouro Preto, 1995 [to appear].

[10] A recent book on the Marquis of Pombal brings new elements to understand Science and Mathematics in this period. See Kenneth Maxwell: Pombal, paradox of the enlightment, Cambridge: Cambridge University Press, 1995.

[11] See the book by José Sala Catalá Ciencia y Técnica en la Metropolización de América, Madrid: Theatrum Machinae, 1994. The author studies the problems posed by mining and the urban development of three cities: Mexico, Lima and Recife.

[12] "Themes of Rational Philosophy and of Mechanical Philosophy of the senses, according to the uses of Physics; and of other physical-philosophical topics following the thought of moderns to be defended in this Royal and Pontifical Guatemalan Academy of San Carlos..."