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pragmatic dilemma

Between theorems and railways

Debate on positivist mathematics led to pure science in Brazil

Larissa RibeiroAccording to many historians, the foundation of the University of São Paulo, in 1934, was the beginning of modern science in Brazil. In the words of Sergio Milliet, this event “is the intellectual and scientific revolution that will change Brazilians’ economic and social concepts.” Historians add that up until that time, Brazil had been “scientifically isolated,” this being the fault of the “authoritarian and anti-science group” that had imposed “order and progress” on the Brazilians and added this slogan to the Brazilian flag. This unusual distortion resulted in “demonizing” of positivism – whose creed was faith in science to leverage progress and civilization – and led to blaming positivism as the major obstacle to national scientific development.

The accusation gains controversial contours when it falls on mathematics, the discipline viewed by France’s Auguste Comte (1798-1857) – the creator of positivism – as the base of education. “The Brazilian-style positivism during the First Republic (1889-1930) was, and still is, analyzed in a simplistic and general manner because of its “scientific-like” vision that advocated pragmatic science and mathematics as the practical instruments to solve national problems and provide material progress and social modernization. Superficial, interested interpretations accuse positivism of over-rating applied science, thus creating constraints for scientific progress, whose main driver is pure, uninterested science, “ explains mathematician Rogério Monteiro de Siqueira, a professor at the Postgraduate Program in Cultural Studies at the School of Arts, Sciences, and Humanities (EACH/USP) of the University of São Paulo. He is the author of the study Modernismo, modernidade e modernização nas ciências matemáticas brasileiras [Modernism, modernity and modernization in the Brazilian mathematical sciences}, sponsored by FAPESP.

“Of course, before 1930s, what was going on in mathematics in Brazil was nothing like what was going on in Europe. But we can’t simplify matters and say that we lacked any kind of mathematical development before USP was established. Individual mathematicians were regularly publishing papers on original themes in international journals. Therefore, the belief that the positivists prevented the attempt to develop pure science – as some of their early and current detractors state – is misleading,” he says. “Nonetheless, many people continue to insist that progress was achieved only because of “the escapes from positivism.” This throws a veil on the past, reducing mathematical progress to a restricted pantheon of ‘modern’ anti-positivists, such as Otto de Alencar (1874-1912), Manuel de Amoroso Costa (1885-1928), Theodoro Ramos (1895-1935) and Lélio Gama (1892-1981),” Rogério adds.

The battle between pure science and applied science is very complex and seldom researched, as the researcher found out when analyzing articles published in specialized journals. “Some ‘positivist’ mathematicians criticized Comte. In Brazil, positivism was not a single, radical movement; it was split into various factions with different levels of orthodoxy,” he says. The leader of the positivist movement in Brazil, Benjamin Constant de Magalhães (1833-1891), a staunch republican and professor of mathematics at military academies, was openly against the Comte interpretation of mathematics. “One must also be familiar with the entire production of the ‘moderns.’ Nowadays, we have an incomplete depiction of the debates, from which political variables and interests were extracted. Only the ‘modern’ articles they wrote are selected, and many other articles on issues related to applied science are disregarded. People also forget that the so-called ‘pioneers of pure mathematics’ were ‘hybrids,’ because, besides developing theorems, they accepted government jobs and wrote about the practice of engineering, without limiting themselves to ‘uninterested science,’” he adds.


Even Amoroso Costa, who held positivism responsible for the sorry state of the exact sciences in Brazil, was obliged to admit that “our terrain is still not ready to cultivate pure science, this contemplative, uninterested, and supreme flower of the spirit.” “This battle was a symptom of the realignment of political forces with national sciences, as exemplified by a group of engineers. These engineers, who had invested in mathematics that was foreign to their applications, following a pattern that was in vogue at that time in Europe, gradually became aware that they were no longer valuable and there was no room for them. This happened in an environment in which mathematics was increasingly being seen as an instrument of practical work for the country’s progress,” Rogério analyzes. Having been discarded, this group began to advocate the creation of an institutional venue for “uncommitted” science; more specifically, this venue was the university, which they eventually dominated.

Actually, a small group of radical positivists was against creating this space, as it was highly aware of the arm-wrestling that was under way. Many other factions disagreed with this “censorship,” and took on a non-dogmatic attitude towards Comte’s texts. “We mustn’t forget that the influence of positivism on the First Republic didn’t last for very long and the generation of 1870 – more specifically, the military summit influenced by Comte’s social reform ideas – was jettisoned from power by the oligarchies,” explains Angela Alonso, a professor at the Sociology Department of the University of São Paulo and author of Ideias em movimento: a geração 1870 na crise do Brasil-Império [Ideas in motion: the 1870 generation in the Brazilian Empire crisis] (2002). This group wanted to see a split between the military and the civilians; they felt contempt for the “bachelor’s degree community” and its liberal views and romantic attitude towards a monarchy in Brazil. To this counter-elite comprised of military personnel, engineers and physicians, all of whom had a technical-scientific background, positivism was the confirmation of their awareness of the huge gap between the country and “civilization.”

“Brazilian positivists had a unique characteristic: instead of approaching the doctrine in religious terms, they used it to discuss political issues on a social terrain. Thus, science emerges as a source of solutions,” says the professor. These positivists, fans of “Brazilian illustration,” viewed education as the panacea and saw themselves as having a “mission,” which was to become familiar with Brazil’s social reality and nature, overcoming obstacles by resorting to science and practical solutions, and thus revealing the potential of the territory. “The point was not to enhance applied science to the detriment of pure science; the point was to practice scientific knowledge with a social objective, associated with the fundamental role attributed to scientists in the new positivist Brazil,” explains Luiz Otávio Ferreira, a researcher at the Casa de Oswaldo Cruz center of Fiocruz, and coordinator of the research study O ‘ethos’ positivista e a institucionalização da ciência no Brasil [The positivist ‘ethos’ and the institutionalization of science in Brazil ](2007).

“Thus, there was no room for ‘pure mathematics’ in this urgent clearing of territories. However, there were diverging opinions after the Central School of Engineering was established in 1858. This School split the teaching of engineering between civilians and the military. The military embraced positivism at the military academies,” says Ferreira. The “pure” mathematicians aligned themselves with the civil engineers. The conflict began in 1896, when Benjamin Constant de Magalhães, then minister of Public Instruction, closed down the physical, mathematical, and natural sciences courses at the Polytechnic School of Rio de Janeiro. “Even though closing down these courses could be attributed to the fact that only 67 students had enrolled in these courses since 1874, some professors felt that the objective of this measure, was the imposition by the positivists that dominated the institution of the utilitarian view of the sciences,” Ferreira explains.

In the opinion of the “scientistic engineers,” this was a coup designed to take over their territory. The reaction came in 1898. Otto de Alencar, a former positivist, published the article “Alguns erros de mathematica na Synthese subjectiva de A. Comte,” the first “cannon ball” set off in the battle between “pure” and “ applied” mathematics. This was low-caliber ammunition, but it acted as the “fodder” for the cannon set off in 1918, at the conferences of Amoroso Costa. “The new countries are fanatic about material progress; they disregard the existence of a scientific ideal that is far superior to the man who manufactures one thousand automobiles a day or removes an appendix in 10 minutes. There is a unanimous opinion: science is useful, because engineers, doctors and military personnel need it. It is not worthwhile doing it in Brazil, as it is cheaper and more convenient importing it from Europe. This is the predominant mentality among educators and government authorities,” the mathematician stated.

Ten years later, Lélio Gama, in Rio, and Theodoro Ramos, in São Paulo, joined the battle to “retrieve” the territory of “uninterested” science. This ultimately led to the creation of USP and Rio de Janeiro’s University of Brazil, in 1939. However, was there any room for “contemplative science” prior to the 1930s? “In Europe, mathematics, physics, and engineering were soon separated; but in Brazil this didn’t happen. The separation was possible in Europe because of its rapid industrialization in the nineteenth century. In Brazil, there was no demand for technical knowledge in all fields, as was the case of medicine, for example,” points out Rogério. The anti-positivist criticisms were not “pure” either.

Reproduction from the book "O Brasil" by Marc FerrezMilitary Academy, Urca, Rio de Janeiro, 1885, a well-known center of positivistsReproduction from the book "O Brasil" by Marc Ferrez

“The taints of ´inaccuracy’ and ‘lack of scientific rigor’ thrown on the positivists are debatable. Italian mathematicians, for example, were referred to as ‘poets’ because of their alleged inaccuracy yet nobody put the blame on positivism for this. The requested ‘rigor’ was not exercised in the writings of the ‘modern’ Brazilians, whose work was not on a par with the work being done in Europe,” says the researcher. “The desire was to create ‘distinction’: the thesis of Theodoro Ramos, for example, used set theory less in favor of ‘pure mathematics’ than as a fighting strategy,” says Rogério.

Nevertheless, what was the motivation of the “purists?” “They had a feeling that something was mismatched, that ‘ideas were out of place.’ Many of them had traveled abroad and had returned with knowledge on new scientific concepts being used in Europe,” says Rogério. In the researcher’s opinion, one cannot deny the constant existence of a political component in the battle among groups that excluded each other and that wanted their places under the sun. “This is evidenced by Ramos’ link to the Revolution of 1930. It was no coincidence that he brought Italian mathematicians – many of whom were Fascists – to USP, to please Vargas, who admired Mussolini. This action was also in line with the demands of the enormous Italian community,” he adds.

Thus, the Germans and the Italians were left in charge of the exact sciences, while the French were in charge of the humanities. The first generation of mathematicians from the 1950s were the “heirs” of this decision, which included a disregard for didactics, instilled by such Italian professors as Luigi Fantappié. National mathematics, which gained international prominence in the early 1960s, was based on an “intellectual imbroglio.” “The fans of ‘pure science’ appropriated articles that came from abroad without being familiar with the context and the discussions surrounding these articles. They were appropriated directly and this resulted in a kind of ‘Brazilian-type’ mathematics,” says Rogério.

Thus, the “demonizing” of positivism deserves to be revised. “Criticism of positivist scientific ideas was not only an undertaking of young, innovative mathematicians that wanted to break the cycle of the archaic conservatism of science in Brazil. This interpretation disregards the fact that the frontier between the archaic and the modern results from the processes of social construction,” Ferreira points out. Positivism was the basis for the development of a class of scientists that was against it. “The positivists provided the ideological content necessary to shape the ‘scientist’ category. The model of the positivist intellectual, who was objective and precise, whether he was a social reformer or not, educated those who wished to be regarded as scientists.”