id: 05258172
dt: j
an: 2009c.00016
au: Pacheco, José M.; Pérez-Fernández, Francisco J.; Suárez, Carlos
ti: On the reception and spread of metaphysical explanations of imaginary
numbers in spain.
so: Rev. Acad. Canar. Cienc. 18(2006), No. 1-2, 101-132 (2007).
py: 2007
pu: Academia Canaria de Ciencias, La Laguna, Tenerife
la: EN
cc: A30
ut: imaginary quantities; complex numbers; Kant; Spain; 19th century
ci:
li:
ab: This is a welcome paper. It is related to a number of important papers of
the authors dealing with the 19th century developments of mathematics
in Spain. In the 19th century, Spain was experiencing a turbulent
political and intellectual scenario, after the Napoleonic invasion and
having lost her colonial empire. While there was a vigorous development
of rigorous mathematics and philosophy in Europe, in Spain a few
isolated individuals, mainly in seminaries, military schools and in the
secondary educational system, were aware of what was going on the rest
of Europe. This paper focuses specially one of such individuals, José
Maria Rey Heredia (1818-1861). Heredia was preparing for an
ecclesiastical career, but did not receive the orders and decided for a
career as a teacher. He took degrees in Laws and in Philosophy and
Letters, and became a Professor of Logic at a High School in Madrid. In
1853, Rey wrote a textbook, called Elementos de Lógica, which is a
classical treatment of logics. This book was sort of a springboard for
his most important book, the Teoría Transcendental de las Cantidades
Imaginarias, which he started to write in 1855. The book was
posthumously published in 1865, referred to by the authors as a
“beautifully printed book". The authors give a good overall
description of the Teoría. They discuss the important question of
sources and contacts of Rey with other European scholars. Rey went
through successive exams in order to obtain his academic degrees and,
in preparation for them, updated his philosophical and mathematical
knowledge. It seems that he had good access to English sources. The
authors analyze the Teoría by books. Book I is very important, since
it is where Rey’s conception of imaginary numbers is discussed. He
recognizes the influence of A.-Q. Buée (1748-1826), a Frenchman
leaving in England and a very interesting example of someone with
original ideas, but not well recognized in current historiography. Rey
is indebted to Buée for his conception of imaginary num bers as
quantities modified by a quality, which express directions in a plane.
Commenting on Rey’s approach, the authors give a very interesting
explanation of quantity and quality as mathematical categories. Books
II, III and IV treat algebraic manipulations with complex numbers.
According to the authors, Books II and III are commonplace. Book IV is
more advanced. It deals with graduation for raising to powers. This
book treats more advanced themes and reveal the mathematical background
of Rey. The authors reveal disappointment with Book IV, mainly because
Rey reveals rigor in the metaphysics and logics, but lacks mathematical
rigor. Last part of the paper is about the reception of Rey’s ideas
in later 19th century and early 20th century. The authors claim that
“the theories of Rey enjoyed a certain popularity in the following
years, and several mathematicians incorporated some of his ideas in
their texts". This is illustrated with a brief analysis of five
authors. The Conclusion applies to historiography in general. There is,
justly, a dominant presence of scholars who have, effectively,
contributed to the advancement of mathematics. But others, whose
contribution are not brilliant and determinant for the advances of
mathematics, like Rey, are, unjustly, ignored. It is important to
recognize their importance in influencing new generations of
mathematicians which will have effective academic contributions. This
interesting and carefully written paper exemplifies this.
rv: Ubiratan D’Ambrosio (São Paulo)