# Giuseppe Peano

Giuseppe Peano ( born August 27, 1858 in Spinetta, now part of Cuneo, Piedmont, † April 20, 1932 in Turin ) was an Italian mathematician. He worked in Turin and dealt with mathematical logic with the axioms of the natural numbers ( development of Peano axioms ) and with first order differential equations.

## Life

Peano was the son of farmers. He attended school in Cuneo and when his talent was recognized, from 1870, the high school ( Liceo ) in Turin, where an uncle was a priest and lawyer. From 1876 he studied mathematics at the University of Turin, among others, with Enrico D' Ovidio, Angelo Genocchi, Faa di Bruno Francesco and Francesco Siacci. In 1880 he received his doctorate and became an assistant to D' Ovidio and then at Genocchi. At the same time appeared in 1880 his first mathematical work. He held the lectures Analysis of Genocchi (which also came out in 1884 as a book, edited, written and with additions of Peano ). In 1884 he habilitated. Except at the university, he also lectured at the Military Academy in Turin. In 1890, he was the successor of Genocchi as a professor at the university.

In 1891 he founded the journal Rivista di matematica, mainly devoted to the foundations of mathematics and logic. In 1892 he began a project to formulate the well-known theorems of mathematics in logical rigor, the Formulario Matematico ( completed 1908), which he later used for his lectures, which was an educational failure. 1901 was therefore terminated his teaching at the Military Academy. At the University could not talk him into it. 1900 was Peano recognition at the International Congress of Philosophy in Paris.

## Peano as a mathematician

Peano's mathematical work is marked by great logical rigor. He has repeatedly found exceptions in published theorems ( for example, works by Corrado Segre and Hermann Laurent ). Also named after him Peano curve is an example of this. It is a continuous, onto mapping of the unit interval in the unit square, which is a space-filling curve, which is defined as the limit of a sequence of turns that can be constructed stepwise. Before Peano had not counted on the possibility of the existence of such a curve. Peano curves found in 1890, a little later, David Hilbert gave more examples.

In the field of analysis and differential equations Peano has done important. He found the remainder of the Simpson rule for the approximate calculation of integrals and proved the existence theorem of Peano for ordinary differential equations ( 1886). He also found independently by Emile Picard whose approximation method for solving systems of ordinary differential equations ( 1887).

Peano had a formative influence on modern logic, set theory and mathematics through several works in which he pursued a systematic formalization of mathematical facts. Peano created in his book Calcolo Geometrico 1888 for the first time an axiom system for the vector space (where he took up unnoticed ideas of Hermann Grassmann ) and there also formulated the modern axiom system for Boolean algebra, where he introduced the symbols and. In his arithmetic of 1889, he presented - regardless of Dedekind arithmetic - the first formal axioms for the natural numbers, which became famous as the Peano axioms. As a foundation for its arithmetic, he created the first formalized logic of classes, in which he, among other things, the element characters and ordered pairs (a, b ) introduced. The formalization of important logical and mathematical areas he built later in formularies further out; from them sprang, among others, the Existenzquantorsymbol.

In 1897 he gave a plenary lecture at the First International Congress of Mathematicians in Zurich ( Logica Matematica ).

## Peano as a linguist

In the field of linguistics Peano made a name when he created the plan language Latino sine flexione ( = Latin without diffraction ). This was an attempt to revive the former world language Latin by the widely known vocabulary was respected, the difficulties of the Latin language but were largely eradicated. This Latino sine flexione later went on in Interlingua.

Also parts of his book project Formulario Matematico he wrote in this language.

## Works (selection)

- Peano, Giuseppe: Calcolo geometrico, Torino, 1888.
- Peano, Giuseppe: Geometric Calculus, translated by LC Kannenberg, Boston 2000.
- Peano, Giuseppe: Arithmetices principia nova methodo exposita, 1889, in: G. Peano, Opere scelte II, Rome, 1958, 20-55
- Peano, Giuseppe: Logique mathematique, 1897, in: G. Peano, Opere scelte II, Rome 1958, 218-281
- Peano, Giuseppe De latino sine flexione, 1903, in: G. Peano, Opere scelte II, Rome 1958, 439-447