Giuseppe Peano

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Giuseppe Peano (; ; 27 August 1858 – 20 April 1932) was an Italian
mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History ...
and glottologist. The author of over 200 books and papers, he was a founder of
mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal ...
and
set theory Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concer ...
, to which he contributed much
notation In linguistics and semiotics, a notation is a system of graphics or symbols, characters and abbreviated expressions, used (for example) in artistic and scientific disciplines to represent technical facts and quantities by convention. Therefore, ...
. The standard axiomatization of the
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called '' cardinal ...
s is named the
Peano axioms In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano. These axioms have been used nearly ...
in his honor. As part of this effort, he made key contributions to the modern rigorous and systematic treatment of the method of
mathematical induction Mathematical induction is a method for proving that a statement ''P''(''n'') is true for every natural number ''n'', that is, that the infinitely many cases ''P''(0), ''P''(1), ''P''(2), ''P''(3), ...  all hold. Informal metaphors help ...
. He spent most of his career teaching mathematics at the University of Turin. He also wrote an international auxiliary language, Latino sine flexione ("Latin without inflections"), which is a simplified version of Classical Latin. Most of his books and papers are in Latino sine flexione, others are in Italian.

Biography

Peano was born and raised on a farm at Spinetta, a hamlet now belonging to Cuneo, Piedmont,
Italy Italy ( it, Italia ), officially the Italian Republic, ) or the Republic of Italy, is a country in Southern Europe. It is located in the middle of the Mediterranean Sea, and its territory largely coincides with the homonymous geographical re ...
. He attended the Liceo classico Cavour in
Turin Turin ( , Piedmontese: ; it, Torino ) is a city and an important business and cultural centre in Northern Italy. It is the capital city of Piedmont and of the Metropolitan City of Turin, and was the first Italian capital from 1861 to 1865. Th ...
, and enrolled at the University of Turin in 1876, graduating in 1880 with high honors, after which the University employed him to assist first Enrico D'Ovidio, and then Angelo Genocchi, the Chair of
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", 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 arith ...
. Due to Genocchi's poor health, Peano took over the teaching of calculus course within two years. His first major work, a textbook on calculus, was published in 1884 and was credited to Genocchi. A few years later, Peano published his first book dealing with mathematical logic. Here the modern symbols for the union and
intersection In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, their ...
of sets appeared for the first time. In 1887, Peano married Carola Crosio, the daughter of the Turin-based painter Luigi Crosio, known for painting the '' Refugium Peccatorum
Madonna Madonna Louise Ciccone (; ; born August 16, 1958) is an American singer-songwriter and actress. Widely dubbed the " Queen of Pop", Madonna has been noted for her continual reinvention and versatility in music production, songwriting, a ...
''. In 1886, he began teaching concurrently at the Royal Military Academy, and was promoted to Professor First Class in 1889. In that year he published the
Peano axioms In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano. These axioms have been used nearly ...
, a formal foundation for the collection of
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called '' cardinal ...
s. The next year, the University of Turin also granted him his full professorship. The
Peano curve In geometry, the Peano curve is the first example of a space-filling curve to be discovered, by Giuseppe Peano in 1890. Peano's curve is a surjective, continuous function from the unit interval onto the unit square, however it is not injective. ...
was published in 1890 as the first example of a
space-filling curve In mathematical analysis, a space-filling curve is a curve whose range contains the entire 2-dimensional unit square (or more generally an ''n''-dimensional unit hypercube). Because Giuseppe Peano (1858–1932) was the first to discover one, spa ...
which demonstrated that the unit interval and the unit square have the same
cardinality In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...
. Today it is understood to be an early example of what is known as a
fractal In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as illu ...
. In 1890 Peano founded the journal ''Rivista di Matematica'', which published its first issue in January 1891. In 1891 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 The International Congress of Mathematicians (ICM) is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union (IMU). The Fields Medals, the Nevanlinna Prize (to be rename ...
was held in
Zürich , neighboring_municipalities = Adliswil, Dübendorf, Fällanden, Kilchberg, Maur, Oberengstringen, Opfikon, Regensdorf, Rümlang, Schlieren, Stallikon, Uitikon, Urdorf, Wallisellen, Zollikon , twintowns = Kunming, San Francisco ...
. Peano was a key participant, presenting a paper on mathematical logic. He also started to become increasingly occupied with ''Formulario'' 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 Paris () is the Capital city, capital and List of communes in France with over 20,000 inhabitants, most populous city of France, with an estimated population of 2,165,423 residents in 2019 in an area of more than 105 km² (41 sq mi), ma ...
was the venue for the Second
International Congress of Mathematicians The International Congress of Mathematicians (ICM) is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union (IMU). The Fields Medals, the Nevanlinna Prize (to be rename ...
in 1900. The conference was preceded by the First International Conference of Philosophy 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.'' "how do 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 Bertrand Arthur William Russell, 3rd Earl Russell, (18 May 1872 – 2 February 1970) was a British mathematician, philosopher, logician, and public intellectual. He had a considerable influence on mathematics, logic, set theory, linguistics, ar ...
and gave him a copy of ''Formulario''. Russell was struck by Peano's innovative logical symbols and after the conference he retired in the country "to study quietly every word written by him or his disciples." Peano's students
Mario Pieri Mario Pieri (22 June 1860 – 1 March 1913) was an Italian mathematician who is known for his work on foundations of geometry. Biography Pieri was born in Lucca, Italy, the son of Pellegrino Pieri and Ermina Luporini. Pellegrino was a lawyer. Pi ...
and Alessandro Padoa had papers presented at the philosophy congress also. For the mathematical congress, Peano did not speak, but Padoa's memorable presentation has been frequently recalled. 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 Analysis ( : analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (384 ...
, foundations and logic, made many contributions to the teaching of calculus and also contributed to the fields of
differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and ...
s and vector 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 University. In 1903 Peano announced his work on an international auxiliary language called '' Latino sine flexione'' ("
Latin Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through the power of ...
without inflexion," later called Interlingua, and the precursor of the
Interlingua Interlingua (; ISO 639 language codes ia, ina) is an international auxiliary language (IAL) developed between 1937 and 1951 by the American International Auxiliary Language Association (IALA). It ranks among the most widely used IALs and is th ...
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. On 3 January 1908, he read a paper to the ''Academia delle Scienze di Torino'' in which 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.Bodmer, Frederick (1944), ''The Loom of Language'', London: George Allen & Unwin Ltd, p.468 The year 1908 was important for Peano. The fifth and final edition of the ''Formulario'' project, titled '' Formulario mathematico'', 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 pro Interlingua''. Having previously created
Idiom Neutral Idiom Neutral is an international auxiliary language, published in 1902 by the International Academy of the Universal Language () under the leadership of Waldemar Rosenberger, a St. Petersburg engineer. History The Academy had its origin a ...
, 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 The Accademia dei Lincei (; literally the " Academy of the Lynx-Eyed", but anglicised as the Lincean Academy) is one of the oldest and most prestigious European scientific institutions, located at the Palazzo Corsini on the Via della Lungara in R ...
'' to present papers written by friends and colleagues who were not members (the Accademia recorded and published all presented papers given in sessions). During the years 1913–1918, Peano published several papers that dealt with the remainder term for various numerical quadrature formulas, and introduced the Peano kernel. 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 A myocardial infarction (MI), commonly known as a heart attack, occurs when blood flow decreases or stops to the coronary artery of the heart, causing damage to the heart muscle. The most common symptom is chest pain or discomfort which may t ...
.

* 1881: Published first paper. * 1884
''Calcolo Differenziale e Principii di Calcolo Integrale''
* 1887
''Applicazioni Geometriche del Calcolo Infinitesimale''
* 1889: Appointed Professor First Class at the Royal Military Academy. * 1889
''Arithmetices principia: nova methodo exposita''
* 1890: Appointed Extraordinary Professor of
infinitesimal calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", 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 arith ...
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 Lazzaro. * 1903: Announces '' Latino sine flexione''. * 1905: Made Knight of the
Order of the Crown of Italy The Order of the Crown of Italy ( it, Ordine della Corona d'Italia, italic=no or OCI) was founded as a national order in 1868 by King Vittorio Emanuele II, to commemorate the unification of Italy in 1861. It was awarded in five degrees for civ ...
. Elected a corresponding member of the
Accademia dei Lincei The Accademia dei Lincei (; literally the " Academy of the Lynx-Eyed", but anglicised as the Lincean Academy) is one of the oldest and most prestigious European scientific institutions, located at the Palazzo Corsini on the Via della Lungara in R ...
in
Rome , established_title = Founded , established_date = 753 BC , founder = King Romulus ( legendary) , image_map = Map of comune of Rome (metropolitan city of Capital Rome, region Lazio, Italy).svg , map_caption ...
, 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.

Bibliography

;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.

* ''
Arithmetices principia, nova methodo exposita The 1889 treatise ''Arithmetices principia, nova methodo exposita'' (''The principles of arithmetic, presented by a new method''; 1889) by Giuseppe Peano is a seminal document in mathematical logic and set theory, introducing what is now the ...
'' * Foundations of geometry