Giuseppe Peano

# Giuseppe Peano

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Giuseppe Peano was an Italian
Italy
Italy , officially the Italian Republic languages]] under the European Charter for Regional or Minority Languages. In each of these, Italy's official name is as follows:;;;;;;;;), is a unitary parliamentary republic in South-Central Europe. To the north it borders France, Switzerland, Austria and...

mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....

, whose work was of philosophical value. The author of over 200 books and papers, he was a founder of mathematical logic
Mathematical logic
Mathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...

and set theory
Set theory
Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...

, to which he contributed much notation. The standard axiomatization of the natural number
Natural number
In mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively...

s is named the Peano axioms
Peano axioms
In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are a set of axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano...

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
Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers...

. He spent most of his career teaching mathematics at the University of Turin
University of Turin
The University of Turin is a university in the city of Turin in the Piedmont region of north-western Italy...

.

## Biography

Peano was born and raised on a farm at Spinetta, a hamlet now belonging to Cuneo
Cuneo
Cuneo is a city and comune in Piedmont, Northern Italy, the capital of the province of Cuneo, the third largest of Italy’s provinces by area...

, Piedmont
Piedmont
Piedmont is one of the 20 regions of Italy. It has an area of 25,402 square kilometres and a population of about 4.4 million. The capital of Piedmont is Turin. The main local language is Piedmontese. Occitan is also spoken by a minority in the Occitan Valleys situated in the Provinces of...

, Italy
Italy
Italy , officially the Italian Republic languages]] under the European Charter for Regional or Minority Languages. In each of these, Italy's official name is as follows:;;;;;;;;), is a unitary parliamentary republic in South-Central Europe. To the north it borders France, Switzerland, Austria and...

. He attended the Liceo classico Cavour
Liceo classico Cavour
Liceo Ginnasio statale "Camillo Benso di Cavour" is the oldest Liceo classico in Turin and one among the oldest and most prominent high schools in Italy, for boys aged 14 to 19. It was transferred to its present location in 1931.-History:...

in Turin
Turin
Turin is a city and major business and cultural centre in northern Italy, capital of the Piedmont region, located mainly on the left bank of the Po River and surrounded by the Alpine arch. The population of the city proper is 909,193 while the population of the urban area is estimated by Eurostat...

, and enrolled at the University of Turin
University of Turin
The University of Turin is a university in the city of Turin in the Piedmont region of north-western Italy...

in 1876, graduating in 1880 with high honours, after which the University employed him to assist first Enrico D'Ovidio, and then Angelo Genocchi
Angelo Genocchi
Angelo Genocchi was an Italian mathematician who specialized in number theory. He worked with Giuseppe Peano. The Genocchi numbers are named after him.-References:...

, the Chair of Infinitesimal calculus
Infinitesimal calculus
Infinitesimal calculus is the part of mathematics concerned with finding slope of curves, areas under curves, minima and maxima, and other geometric and analytic problems. It was independently developed by Gottfried Leibniz and Isaac Newton starting in the 1660s...

. 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. A few years later, Peano published his first book dealing with mathematical logic. Here the modern symbols for the union
Union (set theory)
In set theory, the union of a collection of sets is the set of all distinct elements in the collection. The union of a collection of sets S_1, S_2, S_3, \dots , S_n\,\! gives a set S_1 \cup S_2 \cup S_3 \cup \dots \cup S_n.- Definition :...

and intersection
Intersection (set theory)
In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B , but no other elements....

of sets appeared for the first time.
In 1887, Peano married Carola Crosio, the daughter of the Turin-based painter Luigi Crosio
Luigi Crosio
Luigi Crosio was an Italian painter who lived and worked in Turin, Italy. He died in Turin and is recorded as having been born in Alba, but the town of Aqua a few miles north of Alba claims Crosio was born there....

, known for painting the Refugium Peccatorum
Refugium Peccatorum
meaning Refuge of Sinners is a Roman Catholic title for the Blessed Virgin Mary. Its use goes back to Saint Germanus of Constantinople in the 8th century....

Images of the Madonna and the Madonna and Child or Virgin and Child are pictorial or sculptured representations of Mary, Mother of Jesus, either alone, or more frequently, with the infant Jesus. These images are central icons of Roman Catholicism and Eastern Orthodox Christianity where Mary remains...

. In 1886, he began teaching concurrently at the Royal Military Academy, and 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 curve
Space-filling curve
In mathematical analysis, a space-filling curve is a curve whose range contains the entire 2-dimensional unit square...

appeared in 1890 as a counterexample
Counterexample
In logic, and especially in its applications to mathematics and philosophy, a counterexample is an exception to a proposed general rule. For example, consider the proposition "all students are lazy"....

. 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 fractal
Fractal
A fractal has been defined as "a rough or fragmented geometric shape that can be split into parts, each of which is a reduced-size copy of the whole," a property called self-similarity...

.

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
International Congress of Mathematicians
The International Congress of Mathematicians is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union ....

was held in Zürich
Zürich
Zurich is the largest city in Switzerland and the capital of the canton of Zurich. It is located in central Switzerland at the northwestern tip of Lake Zurich...

. 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
Binary numeral system
The binary numeral system, or base-2 number system, represents numeric values using two symbols, 0 and 1. More specifically, the usual base-2 system is a positional notation with a radix of 2...

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
Paris is the capital and largest city in France, situated on the river Seine, in northern France, at the heart of the Île-de-France region...

was the venue for the Second International Congress of Mathematicians
International Congress of Mathematicians
The International Congress of Mathematicians is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union ....

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 Russell
Bertrand Arthur William Russell, 3rd Earl Russell, OM, FRS was a British philosopher, logician, mathematician, historian, and social critic. At various points in his life he considered himself a liberal, a socialist, and a pacifist, but he also admitted that he had never been any of these things...

and gave 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 students Mario Pieri
Mario Pieri
Mario Pieri was an Italian mathematician who is known for his work on foundations of geometry.Pieri was born in Lucca, Italy, the son of Pellegrino Pieri and Ermina Luporini. Pellegrino was a lawyer. Pieri began his higher education at University of Bologna where he drew the attention of Salvatore...

Alessandro Padoa was an Italian mathematician and logician, a contributor to the school of Giuseppe Peano. He is remembered for a method for deciding whether, given some formal theory, a new primitive notion is truly independent of the other primitive notions...

had a 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
Mathematical analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...

, foundations and logic, made many contributions to the teaching of calculus and also contributed to the fields of differential equation
Differential equation
A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...

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
Latino sine Flexione
Latino sine flexione , or Peano’s Interlingua , is an international auxiliary language invented by the Italian mathematician Giuseppe Peano in 1903. It is a simplified version of Latin, and retains its vocabulary...

("Latin
Latin
Latin is an Italic language originally spoken in Latium and Ancient Rome. It, along with most European languages, is a descendant of the ancient Proto-Indo-European language. Although it is considered a dead language, a number of scholars and members of the Christian clergy speak it fluently, and...

without inflexion," later called Interlingua, but which should not be confused with the later Interlingua
Interlingua
Interlingua is an international auxiliary language , developed between 1937 and 1951 by the International Auxiliary Language Association...

of the IALA
International Auxiliary Language Association
The International Auxiliary Language Association was founded in 1924 to "promote widespread study, discussion and publicity of all questions involved in the establishment of an auxiliary language, together with research and experiment that may hasten such establishment in an intelligent manner and...

). 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 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
The Academia pro Interlingua was an organization dedicated to the promotion of international auxiliary languages, and is associated in particular with Prof...

. Having previously created Idiom Neutral
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...

, the Academy effectively chose to abandon it in favor of Peano's Latino sine flexione
Latino sine Flexione
Latino sine flexione , or Peano’s Interlingua , is an international auxiliary language invented by the Italian mathematician Giuseppe Peano in 1903. It is a simplified version of Latin, and retains its vocabulary...

.

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, , is an Italian science academy, located at the Palazzo Corsini on the Via della Lungara in Rome, Italy....

to present papers written by friends and colleagues who were not members (the Accademia recorded and published all presented papers given in sessions).

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
Myocardial infarction
Myocardial infarction or acute myocardial infarction , commonly known as a heart attack, results from the interruption of blood supply to a part of the heart, causing heart cells to die...

.

• 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.
• 1890: Appointed Extraordinary Professor of infinitesimal calculus
Infinitesimal calculus
Infinitesimal calculus is the part of mathematics concerned with finding slope of curves, areas under curves, minima and maxima, and other geometric and analytic problems. It was independently developed by Gottfried Leibniz and Isaac Newton starting in the 1660s...

at the University of Turin
University of Turin
The University of Turin is a university in the city of Turin in the Piedmont region of north-western Italy...

.
• 1893: Lezioni di Analisi Infinitesimale, 2 vols.
• 1895: Promoted to Ordinary Professor.
• 1901: Made Knight of the Order of Saints Maurizio and Lazzaro
Order of Saints Maurice and Lazarus
The Order of Saints Maurice and Lazarus is an order of chivalry awarded by the House of Savoy, the heads of which were formerly Kings of Italy...

.
• 1903: Announces Latino sine flexione
Latino sine Flexione
Latino sine flexione , or Peano’s Interlingua , is an international auxiliary language invented by the Italian mathematician Giuseppe Peano in 1903. It is a simplified version of Latin, and retains its vocabulary...

.
• 1905: Made Knight of the Order of the Crown of Italy
Order of the Crown of Italy
The Order of the Crown of Italy was founded as a national order in 1868 by King Vittorio Emanuele II, to commemorate the unification of Italy in 1861...

. Elected a corresponding member of the Accademia dei Lincei
The Accademia dei Lincei, , is an Italian science academy, located at the Palazzo Corsini on the Via della Lungara in Rome, Italy....

in Rome
Rome
Rome is the capital of Italy and the country's largest and most populated city and comune, with over 2.7 million residents in . The city is located in the central-western portion of the Italian Peninsula, on the Tiber River within the Lazio region of Italy.Rome's history spans two and a half...

, the highest Italian honour for scientists.
• 1908: Fifth and final edition of the Formulario mathematico
Formulario mathematico
Formulario Mathematico is a book by Giuseppe Peano which expresses fundamental theorems of mathematics in a symbolic language developed by Peano...

.
• 1917: Made an Officer of the Crown of Italy.
• 1921: Promoted to Commendatore of the Crown of Italy.