José Sebastião E Silva
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José Sebastião e Silva (12 December 1914 in
Mértola Mértola (), officially the Town of Mértola (), is a town and municipality in southeastern Portuguese Alentejo near the Spanish border. In 2011, the population was 7,274, in an area of approximately : it is the sixth-largest municipality in Portu ...
– 25 May 1972 in
Lisbon Lisbon ( ; ) is the capital and largest city of Portugal, with an estimated population of 567,131, as of 2023, within its administrative limits and 3,028,000 within the Lisbon Metropolitan Area, metropolis, as of 2025. Lisbon is mainlan ...
) was a Portuguese mathematician who made contributions to
functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (for example, Inner product space#Definition, inner product, Norm (mathematics ...
, distribution theory, and mathematical education. After graduating from the
University of Lisbon The University of Lisbon (ULisboa; ) is a public university, public research university in Lisbon, and Portugal's largest university. It was founded in 1911, but the university's present structure dates to the 2013 merger of the former Universit ...
in 1937 and earning his doctorate in 1949, he taught at the
Instituto Superior de Agronomia Instituto Superior de Agronomía (ISA), School of Agronomy – University of Lisbon, is a faculty for graduate and post-graduate studies in Agronomy, Forestry, Food Science, Landscape Architecture, Environment, Animal Production, Plant Protectio ...
before becoming Director of the Centre for Mathematical Studies at the University of Lisbon. Silva is particularly remembered for his influential approach to teaching calculus, which combined intuitive understanding with mathematical rigour, and for his co-authored textbook ''Compêndio de Álgebra'' that shaped mathematics education in Portugal for decades. His pedagogical methods emphasised introducing students to mathematical concepts through concrete examples before progressing to formal definitions.


Early life and education

Born in Mértola, Silva graduated in mathematics from the
University of Lisbon The University of Lisbon (ULisboa; ) is a public university, public research university in Lisbon, and Portugal's largest university. It was founded in 1911, but the university's present structure dates to the 2013 merger of the former Universit ...
in 1937. In 1942 he received a grant from the to study in
Rome Rome (Italian language, Italian and , ) is the capital city and most populated (municipality) of Italy. It is also the administrative centre of the Lazio Regions of Italy, region and of the Metropolitan City of Rome. A special named with 2, ...
, where he worked with members of the
Italian school of algebraic geometry In relation to the history of mathematics, the Italian school of algebraic geometry refers to mathematicians and their work in birational geometry, particularly on algebraic surfaces, centered around Rome roughly from 1885 to 1935. There were 30 ...
. His first doctoral thesis on
geometric transformation In mathematics, a geometric transformation is any bijection of a set to itself (or to another such set) with some salient geometrical underpinning, such as preserving distances, angles, or ratios (scale). More specifically, it is a function wh ...
s was rejected by
Federigo Enriques Abramo Giulio Umberto Federigo Enriques (5 January 1871 – 14 June 1946) was an Italian mathematician, now known principally as the first to give a classification of algebraic surfaces in birational geometry, and other contributions in algebrai ...
, so he composed a second on
functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (for example, Inner product space#Definition, inner product, Norm (mathematics ...
, earning his doctorate in 1949 from the University of Lisbon.


Academic career

From 1951 to 1961, Silva was professor of mathematics at the
Instituto Superior de Agronomia Instituto Superior de Agronomía (ISA), School of Agronomy – University of Lisbon, is a faculty for graduate and post-graduate studies in Agronomy, Forestry, Food Science, Landscape Architecture, Environment, Animal Production, Plant Protectio ...
. He then returned to the University of Lisbon as Director of the Centre for Mathematical Studies, a post he held for twenty years. His research spanned analytic functionals, the theory of distributions (including vector‑valued distributions and ultradistributions), the
operational calculus Operational calculus, also known as operational analysis, is a technique by which problems in Mathematical Analysis, analysis, in particular differential equations, are transformed into algebraic problems, usually the problem of solving a polynomia ...
, and
differential calculus In mathematics, differential calculus is a subfield of calculus that studies the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus—the study of the area beneath a curve. ...
in locally convex spaces.


Educational contributions

In 1951, Silva argued in the journal ''Gazeta de Matemática'' that
infinitesimal calculus Calculus is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations. Originally called infinitesimal calculus or "the calculus of ...
ought to be reintroduced into
secondary education Secondary education is the education level following primary education and preceding tertiary education. Level 2 or ''lower secondary education'' (less commonly ''junior secondary education'') is considered the second and final phase of basic e ...
, warning that "when nothing is sown, what can be harvested?" and stressing that first exposure to the ideas of
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-orie ...
and limit should occur in
lyceum The lyceum is a category of educational institution defined within the education system of many countries, mainly in Europe. The definition varies among countries; usually it is a type of secondary school. Basic science and some introduction to ...
classrooms. Two years later, in partnership with José da Silva Paulo, he co‑authored the (first ed. 1956; second ed. 1957), which was selected by national competition as the official algebra textbook for Portugal’s third‑cycle lyceums until 1968. Silva's
pedagogical Pedagogy (), most commonly understood as the approach to teaching, is the theory and practice of learning, and how this process influences, and is influenced by, the social, political, and psychological development of learners. Pedagogy, taken ...
design unfolds as follows: # Introduction of the concept of infinitesimal as a more intuitive precursor to limits. # Successive treatment of limits of sequences followed by limits of functions. # Continuity of a function. # Definition of the derivative via the limit: f'(x_0) =\lim_\frac =\lim_\frac where f'(x_0) denotes the rate of change of f at x_0. To avoid
abstraction Abstraction is a process where general rules and concepts are derived from the use and classifying of specific examples, literal (reality, real or Abstract and concrete, concrete) signifiers, first principles, or other methods. "An abstraction" ...
, Silva began with the mechanical notion of
velocity Velocity is a measurement of speed in a certain direction of motion. It is a fundamental concept in kinematics, the branch of classical mechanics that describes the motion of physical objects. Velocity is a vector (geometry), vector Physical q ...
—interpreting the
slope In mathematics, the slope or gradient of a Line (mathematics), line is a number that describes the direction (geometry), direction of the line on a plane (geometry), plane. Often denoted by the letter ''m'', slope is calculated as the ratio of t ...
of a distance–time graph as speed—and the geometric notion of slope (
coefficient In mathematics, a coefficient is a Factor (arithmetic), multiplicative factor involved in some Summand, term of a polynomial, a series (mathematics), series, or any other type of expression (mathematics), expression. It may be a Dimensionless qu ...
of a
secant line In geometry, a secant is a line (geometry), line that intersects a curve at a minimum of two distinct Point (geometry), points.. The word ''secant'' comes from the Latin word ''secare'', meaning ''to cut''. In the case of a circle, a secant inter ...
approaching the
tangent In geometry, the tangent line (or simply tangent) to a plane curve at a given point is, intuitively, the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points o ...
on a curve). Only after these concrete and symbolic motivations did he present the formal ε–δ definition, reserving rigorous
mathematical proof A mathematical proof is a deductive reasoning, deductive Argument-deduction-proof distinctions, argument for a Proposition, mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use othe ...
s for
higher education Tertiary education (higher education, or post-secondary education) is the educational level following the completion of secondary education. The World Bank defines tertiary education as including universities, colleges, and vocational schools ...
. This sequence closely mirrors
David Tall David Orme Tall (1941-2024) was an Emeritus Professor in Mathematical Thinking at the University of Warwick. One of his early influential works is the joint paper with Vinner "Concept image and concept definition, Concept image and concept defi ...
's "three worlds" of mathematics—corporeal (sensory intuition), symbolic (manipulation of
algebraic expression In mathematics, an algebraic expression is an expression built up from constants (usually, algebraic numbers), variables, and the basic algebraic operations: addition (+), subtraction (-), multiplication (×), division (÷), whole number pow ...
s) and formal ( axiomatic proof)—by weaving together concrete examples, algebraic computation and formal definitions in a single narrative. Analyses of lyceum students' notebooks from 1978 confirm that Silva's examples—such as computing the derivative of x^2 at a given point—were reproduced verbatim in class, and that subsequent textbooks built on his approach by adding lateral derivative concepts and further exercises while preserving his intuitive framework.


Legacy

Silva's integration of intuition and rigour in teaching the derivative shaped Portuguese mathematics education for decades. The ''Compêndio de Álgebra'' remained a pedagogical benchmark long after its official adoption, and successor texts continued to draw on his method of motivating abstract definitions through concrete phenomena.


References

{{DEFAULTSORT:Silva, Jose Sebastiao E 1914 births 1972 deaths People from Mértola 20th-century Portuguese mathematicians Mathematical analysts University of Lisbon alumni