Giuseppe Peano ( ; 27 August 1858 – 20 April 1932) was an Italian mathematician, whose work was of exceptional
of over 200 books and papers, he was a founder of mathematical logic
and set theory
, to which he contributed much
notation. The standard axiomatization
of the natural numbers
is named in
his honor. As part of this axiomatization
effort, he made key
contributions to the modern rigorous and systematic treatment of
the method of mathematical
. He spent most of his career teaching mathematics at
the University of Turin
born and raised on a farm at Spinetta, a hamlet near Cuneo, Piedmont, Italy.
enrolled at the University of
in 1876, graduating in 1880 with high honours, after
which the University employed him to assist first Enrico D'Ovidio
, and then Angelo Genocchi
, the Chair of Infinitesimal calculus
. Due to
Genocchi's poor health, Peano took over the teaching of the
infinitesimal calculus course within 2 years. His first major work,
a textbook on calculus, was published in 1884 and was credited to
Genocchi. Three years later, Peano published his first book dealing
with mathematical logic. Here the modern symbols for the union
of sets appeared for
the first time.
In 1887, Peano married Carola Crosio, the daughter of the
Turin-based painter Luigi Crosio
for painting the Refugium
. In 1886,
he began teaching concurrently at the Royal Military Academy
was promoted to Professor First Class in 1889. The next year, the
University of Turin also granted him his full professorship.
Peano's famous space-filling
appeared in 1890 as a counterexample
. He used it to show that a
continuous curve cannot always be enclosed in an arbitrarily small
region. This was an early example of what came to be known as a
The following year Peano started the Formulario Project
. It was to be an
"Encyclopedia of Mathematics", containing all known formulae and
theorems of mathematical science using a standard notation invented
by Peano. In 1897, the first International Congress
of Mathematicians was held in Zürich.
Peano was a key participant, presenting a paper on mathematical
logic. He also started to become increasingly occupied with
to the detriment of his other work.
In 1898 he presented a note to the Academy about binary numeration
and its ability to
be used to represent the sounds of languages. He also became so
frustrated with publishing delays (due to his demand that formulae
be printed on one line) that he purchased a printing press.
Paris was the
venue for the Second International Congress
of Mathematicians in 1900.
The conference was preceded
by the First International Conference
where Peano was a member of the patronage
committee. He presented a paper which posed the question of
correctly formed definitions in mathematics, i.e.
you define a definition?". This became one of Peano's main
philosophical interests for the rest of his life. At the conference
Peano met Bertrand Russell
him a copy of Formulario
. Russell was so struck by Peano's
innovative logical symbols that he left the conference and returned
home to study Peano's text.
Peano's followers presented papers (using Peano's teachings) at the
mathematics conference, but Peano did not. A resolution calling for
the formation of an "international auxiliary language" to
facilitate the spread of mathematical (and commercial) ideas, was
proposed; Peano fully supported it.
By 1901, Peano was at the peak of his mathematical career. He had
made advances in the areas of analysis
, foundations and logic, made
many contributions to the teaching of calculus and also contributed
to the fields of differential
analysis. Peano played a key role in the axiomatization
of mathematics and was a
leading pioneer in the development of mathematical logic. Peano had
by this stage become heavily involved with the Formulario
project and his teaching began to suffer. In fact, he became so
determined to teach his new mathematical symbols that the calculus
in his course was neglected. As a result he was dismissed from the
Royal Military Academy but retained his post at Turin
In 1903 Peano announced his work on an international auxiliary
language called Latino sine
later called Interlingua, but which should not be confused with the
of the IALA
). This was
an important project for him (along with finding contributors for
'Formulario'). The idea was to use Latin vocabulary, since this was
widely known, but simplify the grammar as much as possible and
remove all irregular and anomalous forms to make it easier to
learn. In one speech, he started speaking in Latin and, as he
described each simplification, introduced it into his speech so
that by the end he was talking in his new language.
The year 1908 was big for Peano. The fifth and final edition of the
project, titled Formulario
, was published. It contained 4200 formulae and
theorems, all completely stated and most of them proved. The book
received little attention since much of the content was dated by
this time. However, it remains a significant contribution to
mathematical literature. The comments and examples were written in
Latino sine flexione
Also in 1908, Peano took over the chair of higher analysis at Turin
(this appointment was to last for only two years). He was elected
the director of Academia
. Having previously created Idiom Neutral
, the Academy effectively chose
to abandon it in favor of Peano's Latino sine flexione
After his mother died in 1910, Peano divided his time between
teaching, working on texts aimed for secondary schooling including
a dictionary of mathematics, and developing and promoting his and
other auxiliary languages
becoming a revered member of the international auxiliary language
movement. He used his membership of the Accademia dei Lincei
papers written by friends and colleagues who were not members (the
Accademia recorded and published all presented papers given in
In 1925 Peano switched Chairs unofficially from Infinitesimal
Calculus to Complementary Mathematics, a field which better suited
his current style of mathematics. This move became official in
1931. Giuseppe Peano continued teaching at Turin University until
the day before he died, when he suffered a fatal heart attack
Milestones and honors received
- 1881: Published first paper.
- 1884: Calcolo Differenziale e Principii di Calcolo
- 1887: Applicazioni Geometriche del Calcolo
- 1889: Appointed Professor First Class at the Royal Military
- 1890: Appointed Extraordinary Professor of infinitesimal calculus at the
University of Turin.
- 1891: Made a member of the Academy of Science, Torino.
- 1893: Lezioni di Analisi Infinitesimale, 2 vols.
- 1895: Promoted to Ordinary Professor.
- 1901: Made Knight of the Order of Saints Maurizio and
- 1903: Announces Latino sine
- 1905: Made Knight of the Order of the Crown of Italy.
corresponding member of the Accademia dei Lincei in Rome, the highest
Italian honour for scientists.
- 1908: Fifth and final edition of the Formulario mathematico.
- 1917: Made an Officer of the Crown of Italy.
- 1921: Promoted to Commendatore of the Crown of Italy.
- Peano's writings in English translation
- 1889. "The principles of arithmetic, presented by a new method"
in Jean van Heijenoort, 1967.
A Source Book in Mathematical Logic, 1879-1931. Harvard
Univ. Press: 83-97.
- 1973. Selected works of Giuseppe Peano. Kennedy,
Hubert C., ed. and transl. With a biographical sketch and
bibliography. London: Allen & Unwin.
- Secondary literature
- Gillies, Douglas A., 1982. Frege, Dedekind, and Peano on
the foundations of arithmetic. Assen, Netherlands: Van
- Ivor Grattan-Guinness,
2000. The Search for Mathematical Roots 1870-1940.
Princeton University Press.
- Kennedy, Hubert C., 1980. Peano: Life and Works of Giuseppe Peano.
Reidel. Biography with complete bibliography (p. 195-209).
- Collection of articles on life and mathematics of Peano (1960s