Specificity of the Americas.
Recent scholarship traces pre-Columbian cultures back to 40,000 years.
It is easy to recognize 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 were developed in a completely different
way in these cultures. Although there are evidences of early contacts
with Europe, Asia and Africa, there is no indication at all of mutual
influences between the so-called Old World and the New World.
This section deals with the Americas. For this region we need a
specific chronology. The chronology adopted in current historiographies
of Science, particularly of Mathematics, does not apply to this region.
Of course, if we disregard cultural aspects of mathematical development,
doing a strict internalist history, the usual chronology is acceptable.
But if we consider Mathematics as a cultural endeavour, we have to
accept what we might call a "situated" chronology.
My proposal is a chronology based on five major periods:
1. Pre-Columbian
2. Conquest and early colonial times (roughly 16th and 17th centuries)
3. The established colonies (18th century)
4. Independent countries (19th century)
5. The 20th century
Also, geographic divisions are very important. 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.
For the period 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. Since independence, we have
roughly the current political division.
In what follows, historical periods are defined according to the
general chronology associated with the conquest and colonization
of the Americas. Beginning with the independence movements in late
18th and early 19th century, until present times, the cultural map
is roughly the same.
1. Pre-Columbian History of Mathematics.
I will not cover the developments in this period. But a few remarks
are necessary.
The imposition of the culture of the conqueror obviously depended on
the culture of the conquered. But our knowledge of the pre-Columbian
period is still very incomplete. There was a clear effort made by the
colonial regimes to ignore or obliterate any sense of the history or
historic achievement of the native cultures. Today we are faced with
the difficult task of reconstructing the histories of these cultures,
both looking into the chronology of the events and understanding the
important migratory currents that shaped their developments. Of course,
this leads us to look into the History of Mathematics in pre-Columbian
times.
The emergence of this scholarship rely much on a new reading of the
chroniclers who described the Maya stellae, reported on the Peruvian
quipus, described Aztec daily life, and indeed reported on every
aspect of the conquered people. But this views are biased and,
understandably, they failed to identify and barely recognized any form
of mathematical knowledge in these cultures. There are many references
from the period.
An important source is the first non-religious book published in the
Americas is an arithmetics book related to mining, the Sumario
compendioso de las quentas de plata y oro que en los reinos del Peru/
son necessarias a los mercaderes y todo genero de tratantes. Con
algunas reglas tocantes al arithme/tica, 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.
A basic general chronicler is Bernabe/ Cobo, who published Historia del
Nuevo Mundo [1653], Atlas, Madrid, 1964. And the archives of the Jesuit
missionaires, as well as of other religious orders, which are without
any doubt rich in historic material, are as yet to be explored. A very
important survey of pre-Columbian Mathematics is the book Native
American Mathematics ed. Michael Closs, University of Texas Press,
Austin, 1986.
2. Conquest and early colonial times.
Mexico in itself has a very rich colonial history. It goes beyond the
scope of this section to deal with the history of mathematics in the
Viceroyalty of Nuove Espan~a. Only a few references and remarks are
necessary for understanding the developments in Central and South
America, which in many cases depended on the important and strategic
position of Mexico in the New World.
In 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 indigenous knowledge was
limited, but there was some interest in the nature of native knowledge.
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 arithme/tica y modo de formar
campos, de Atanasio Reaton, in 1649. It is also to be noticed the book
Nuevas proposiciones geome/tricas, written by Juan de Porres Osorio,
in Mexico.
Astronomy was a major area of interest in Latin America in the 17th
century. There are important discussions on the meaning of comets.
Many of the interpretations related to their purpose of conveying
divine messages and messages to mankind. In another words, these were
search for scientific explanations. Several polemical exchanges of
letters and papers are known from these times, with important
epistemological arguments. The figure of Don Carlos de Sigu"enza y
Go/ngora, of Mexico, towers. His works focus on astronomical
observations and calculations. His most important book, considered
one of the most important works of Latin American Science, is Libra
astrono/mica y filoso/fica, written in 1690. In it Sigu"enza y Go/ngora
refutes prevailing astrological arguments about comets.
In Brazil, research on comets was of major importance. The same tone
of the reflections of Sigu"enza y Go/ngora we see in the work of
Valentin Stancel (1621-1705), a Jesuit mathematician from Prague who
lived in Brazil from 1663 until his death. His astronomical measurements
are mentioned in Newton's Principia. A polemic, which includes another
Jesuit, Antonio Vieira (1608-1697) reveals how important was the
discussion about the nature of comets in building up modern scientific
ideas.
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.
3. The established colonies.
In late colonial times, since the middle of the 18th 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 Ilustracio/n [Enlightement], the important intellectual
revival that began in Spain under Charles III and in Portugal under
Jose/ I and his strong minister, the Marquis of Pombal.
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 Cervin~o in Buenos Aires, who
lectured on Infinitesimal Calculus, Mechanics and Trigonometry.
In Peru, Cosme Bueno (1711-1798), Gabriel Moreno (1735-1809) and
Joaqui/n Gregorio Paredes (1778-1839) are best known. In Brazil, Jose/
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.
Among the South Americans of the pre-independence days, a rather
distinguished figure is Jose/ Celestino Mutis (1732-1808), who not only
is responsible for an unpublished translation of Newton, but was also
responsible for introducing modern mathematics in Colombia, mainly
relying on the books by Christian Wolff. He was the founder of the
Observatorio de Bogota/, in 1803. His most distinguished disciple,
Francisco Jose/ Caldas (1771-1816), became the director of the
Observatory. Caldas was deeply involved in the Independence War and
was shot by the Spaniards.
In Chile, the Universidad Real de San Felipe, which was inaugurated
in 1747 in Santiago, was provided with a "catedra" of Mathematics.
Fray Ignacio Leo/n de Garavito, a self-instructed criollo mathematician,
was responsible for this chair.
Again we have to mention Mexico, where the most important developments
of mathematics in Latin America in these days took place. In the first
half of the 18th century there were a number of textbooks on Geometry,
Arithmetics and Astronomy used in Mexico. These were not important in
the sense that they were mostly minor works. But in the second part of
the century we recognize some important contributions. Particularly
noticeable are the Lecciones matema/ticas, of Jose/ Ignacio Bartolache,
published in 1769. In 1772, an anonymous built a "calculating wheel",
capable of performing the four basic operations for numbers up to 108
digits. Benito Bails publishes, also in 1772, the Elementos de
matema/ticas, which treated infinitesimal calculus and analytic
geometry. It is remarkable the development of a special kind of
applied mathematics, stimulated by the complexity of problems related
to water and to mining. These two constitute the most important problems
in the technological development of the country. A "subterraneous
geometry" became a major theme in Mexican Science. Particularly
important was the efforts for urbanization which took place in all the
colonies. The book Comentarios a las Ordenanzas de Minas, by Francisco
Javier Gamboa, published in 1761, is most representative of these
developments.
Let us return now to the region covered in this paper. In Guatemala,
which included Costa Rica, the most renowned scholar is Jose/ Antonio
Liendo y Goicoechea (1735-1814). He taught at the Universidad de San
Carlos de Guatemala, which had already become a very important academic
center after a 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 Meca/nica de los sentidos, de acuerdo
con los usos de la Fi/sica; y de otros to/picos fi/sico-teolo/gicos
segu/n el pensamiento de los modernos para ser defendidos en esta Real
y Pontificia Academia Guatemalteca de San Carlos ...". This was
essentially a medieval proposition. Goicoechea was responsible for
modernizing this plan of studies, incorporating experimental physics
to the project. He introduced modern mathematics based on the texts of
Christian Wolff.
4. Independent countries.
The independence of the Viceroyalties of Nueva Espan~a, Nueva Granada,
Peru, La Plata and Brazil was achieved in the first quarter of the
19th century. The process of modernization in the newly independent
countries did not change the prevailing attitude towards Mathematics.
The political division in countries following the independence is
practically the same as today. The independence of Guatemala as an
independent country in 1821 lessened the influence of Mexico in Central
and South America. The establishment of new university and the renewal
of the old ones, immediately preceding and after the independence,
generated open attitudes with respect to sources of knowledge on which
to build up the newly established countries of Latin America. Formerly
restricted to an almost exclusivity of influences coming from Spain and
Portugal, the new countries attracted considerable attention from the
rest of Europe, and a number of scientific expeditions were sent to
South America. They had a great influence in creating new intellectual
climates throughout the region. This new source of intellectual
interest is seen very strongly in the building up of large and
diversified libraries, both public and private, and the acquisition of
modern literature. The influence of Auguste Comte towards the end of the
century was very important and, although impregnated by the demands of
the emerging political elites to build up the ideological framework of
the new countries, influenced a considerable development of Mathematics
and the Sciences in general.
In Costa Rica the colonial authorities established the Casa de Ensen~anza
de Santo Tomas in 1814, in which the most influential teacher was Rafael
Francisco Osejo, born in 1780. He wrote in 1830 Lecciones de aritme/tica,
written in the form of questions and answers, a common feature in that
period, as noticed above when referring to Alpoim in Brazil. In 1843 the
Casa de Ensen~anza was transformed in the Universidad de Santo Tomas,
where careers in Engineering were established. But no career in Mathematics.
Colombia soon attracted foreign mathematicians. The Frenchman Bergeron
introduced Descriptive Geometry in the country. The Italian Agusti/n
Codazzi (1793-1859) was influential in creating the Colegio Militar.
Lino Pombo (1797-1862) was particularly influential in founding the
Academia de Matematicas de Venezuela. He wrote a complete course of
Mathematics.
In Brazil, the transfer of the royal family of Portugal to escape Napoleon
invasion, in 1808, was decisive and changing cultural life in the colony.
The Portuguese court settled in Rio de Janeiro, where they have to create
an infrastructure to run, from a colonial town, the Kingdom of Portugal.
They founded a major Library and the Escola Militar [Military Academy],
the first institution of higher learning in the colony. Both were
influential in the development of Mathematics in Brazil. In the school a
doctorate in Mathematics was established and a number of theses were
submitted and defended. Translation of the textbooks of Lacroix, of
Legendre and others were quite important in generating what we might
call a mathematical style in Brazil.
Particularly interesting is the case of Joaquim Gomes de Souza (1829-
1863), known as "Souzinha", the first Brazilian mathematician with an
European visibility. He presented his results in the Academie des
Sciences de Paris and in the Royal Society. Only short notice of the
papers were given, and they were posthumously published as Me/langes
du Calcul Inte/gral, as an independent printing by Brockhaus, of
Leipzig, 1889. This work, dealing mainly with partial differential
equations, is permeated by very interesting historical and philosophical
remarks, revealing access to the most important literature then available.
This was possible due to the existence of important private collections
in Maranha~o, his home state in the Northeast. The knowledge of these
libraries is as yet an open field of research.
Argentina, independent since 1816, had a remarkable intellectual
development. In 1822 is founded the ephemorous Sociedad de Ciencias
Fi/sicas y Naturales. We soon see the emergence of private libraries
in Buenos Aires. Particularly important is the private library of
Bernardino Speluzzi (1835-1898), which listed the main works of Newton,
D'Alembert, Euler, Laplace, Carnot and several other modern classics.
Valentin Balbin (1851-1901), while Rector of the National College of
Buenos Aires proposed in 1896 a new study plan which included history
of mathematics as a distinct discipline. This is probably the first
formal interest in the History of Mathematics in South America, which
eventually led to an important school of History of Science in Argentina.
In Peru, it is to be mentioned a development in Statistics, beginning
with the book Ensayo de estadi/stica completa de los ramos econo/mico-
poli/ticos de la provincia de Aza/ngaro... by Jose/ Domingos
Choquechuanca (1789-1858), published in 1833.
In Chile, the Universidad de Chile was created in 1842, with a Faculty
of Physical and Mathematical Sciences. A most distinguished member of
the Faculty is Ramo/n Picarte, a lawyer, who had his paper La divisio/n
reducida a una adicio/n, accepted and published by the Academy of
Sciences of Paris in 1859. Much emphasis is given to teacher training.
An agreement with the government of Germany provided the pedagogical
support to reforming the education in the country. Fifteen German
mathematicians, most with a doctorate, emigrated to Chile in 1889.
Again, this is an as yet unexplored field of research.