# Giuseppe Peano

Giuseppe Peano

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

## Life

Giuseppe Peano and his wife Carola Crosio 1887

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 with Enrico D'Ovidio , Angelo Genocchi , Francesco Faà di Bruno and Francesco Siacci, among others . In 1880 he received his doctorate and became an assistant to D'Ovidio and then to Genocchi. At the same time, his first mathematical work appeared in 1880. He gave the analysis lectures by Genocchi (which came out as a book in 1884, edited, written and supplemented by Peano). In 1884 he completed his habilitation. Besides the university he also gave lectures at the Military Academy in Turin. In 1890 he succeeded Genocchi as a professor at the university.

In 1891 he founded the journal Rivista di matematica , which was mainly devoted to the basics of mathematics and logic. In 1892 he began a project to formulate the well-known mathematical theorems with logical rigor, the Formulio Matematico (finished in 1908), which he later also used for his lectures, which became an educational failure. In 1901, therefore, his teaching at the military academy ended. At the university, however, you couldn't talk him into it. In 1900 Peano found recognition at the International Congress of Philosophy in Paris.

## Peano as a mathematician

Four levels of a Peano curve

Peano's mathematical work is characterized by great logical rigor. He has repeatedly found exceptional cases in published theorems (for example, the work of Corrado Segre and Hermann Laurent ). The Peano curve named after him is an example of this. It is a continuous , surjective mapping of the unit interval into the unit square, i.e. a space-filling curve that is defined as the limit value of a sequence of curves that can be constructed step by step. Before Peano, the possibility of such a curve was not anticipated. Peano found the curves in 1890, and David Hilbert gave further examples a little later . He was also a pioneer of measure theory with Camille Jordan (and with a similar concept as Jordan) with his book from 1887 on the geometric applications of calculus, but only Émile Borel found a satisfactory concept of measure .

Peano has also achieved important things in the field of analysis and differential equations. He found the remainder of Simpson's rule for the approximate calculation of integrals and proved Peano's existence theorem for ordinary differential equations (1886). Independently of Émile Picard, he also found his approximation method for the solution of systems of ordinary differential equations (1887).

Peano had a formative influence on modern logic, set theory and mathematics through some works in which he pursued a consistent formalization of mathematical facts. In his book Calcolo Geometrico from 1888, Peano first created a system of axioms for vector space (taking up neglected ideas from Hermann Grassmann ) and there he also formulated the modern system of axioms for Boolean algebra , where he introduced the symbols and . In his Arithmetik of 1889 he established - independently of Dedekind's arithmetic - the first formal axioms for the natural numbers, which became famous as Peano axioms . As a foundation for his arithmetic, he created the first formalized class logic , in which he also introduced the element sign and ordered pairs (a, b). He later expanded the formalization of important logical and mathematical areas in formulas; among other things, the existential quantifier symbol comes from them . ${\ displaystyle \ cap}$${\ displaystyle \ cup}$ ${\ displaystyle \ in}$ ${\ displaystyle \ exists}$

In 1906 he gave a new proof of Cantor-Bernstein's theorem , which led to a dispute with Ernst Zermelo , who published a similar proof (published only in 1908). Richard Dedekind (1899 in a letter) had anticipated both , but did not publish it.

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

In 1899 Peano gave a counterexample , the Peano surface , to a conjecture about the existence of a local extremum of a function of two variables.

Aritmetica generale e algebra elementare , 1902

## Peano as a linguist

Peano made a name for himself in the field of linguistics when he created the planned language Latino sine flexione (= Latin without inflection). This was an attempt to revive the former world language Latin by preserving the largely known vocabulary, but largely eliminating the difficulties of the Latin language. This Latino sine flexione later became part of Interlingua .

The Formulario Mathematico V Peano (1905/1908) wrote in Latino sine flexione.

## Works (selection)

• Sulla integrabilità delle funzione, Atti Accad. Sci. Torino, Volume 18, 1882/83, pp. 439-446 (English translation in Kennedy, Peano, Selected Works, 1973, pp. 37-43)
• with Angelo Genocchi : Calcolo differenziale e principii di calcolo integrale, Turin 1884, Archives
• German translation (by G. Bohlmann, A. Schepp): Angelo Genocchi: Differentialrechnung und Grundzüge der Integralrechner, Teubner 1899, in the appendix with the translation of the work of Peano from Sulla definizione di integrale, Ann. mat. pura appl., Vol. 23, 1895, pp. 153-157, by Studii di logica matematica , Atti Accad. Sci. Torino, Volume 32, 1897, pp. 565-583 and Sulla formula di Taylor, Atti Accad. Sci. Torino, Volume 27, 1891, pp. 40-46, Archives
• Sull'integrabilità delle equazioni differentenziali del primo ordine, Atti Accad. Sci. Torino, Volume 21, 1886, pp. 677-685 (English translation: On the integrability of first order differential equations, in Kennedy, Peano, Selected Works, 1973, pp. 51-57)
• Applicazioni geometriche del calcolo infinitesimale, Turin 1887, Archives
• Integration par séries des equations differentielles linéaires, Mathematische Annalen, Volume 32, 1888, pp. 450–456, SUB Göttingen
• Calcolo geometrico secondo l'extension theory di H. Grassmann, preceduto dalle operazioni della logica deduttiva, Turin 1888, digitized
• English translation: Geometric Calculus, translated by LC Kannenberg, Boston 2000.
• Arithmetices principia: nova methodo, Turin: Bocca 1889, Archive (also in Opera Scelte, Volume 2, 1958, pp. 20–55)
• English translation: The principles of arithmetic, presented by a new method, in Jan van Heijenoort, From Frege to Goedel, Harvard University Press 1967, pp. 83-97, another English translation is in Kennedy, Peano, Selected Works, 1973, Pp. 101-134
• I principii di geometria logicamente esposti, Turin 1889, digitized , archive
• Démonstration de l´intégrabilité des equations differentielles ordinaires, Mathematische Annalen, Volume 37, 1890, pp. 182–228, SUB Göttingen
• Sur une courbe qui remplit tout une aire plane, Mathematische Annalen, Volume 36, 1890, pp. 157–160, SUB Göttingen
• Sopra alcune curvi singolari, Atti della Reale Accademia delle Scienze di Torino, Classe di Scienze Fisiche, Matematiche e Naturali, Volume 26, 1890/91, pp. 221–224, Biodiversity Heritage Library (English translation: On some singular curves in Kennedy, Peano, Selected Works 1973)
• Principii di logica matematica, Rivista di Matematica, Volume 1, 1891, pp. 1–10 (English translation in Kennedy, Peano, Selected Works 1973, pp. 153–161)
• Sul concetto di numero, Rivista di Matematica, Volume 1, 1891, pp. 87-102, 256-267
• Gli elementi di calcolo geometrico, Turin 1891, Archives
• German translation: The basics of the geometric calculus, Teubner 1891, Archives
• Generallizazione della formula de Simpson, Atti Accad. Sci. Torino, Volume 27, 1892, pp. 68-612 (English translation in Kennedy, Peano, Selected Works 1973)
• Lezioni di analisi infintesimale, 2 volumes, Turin 1893, Archives, volume 1
• Sur la définition de la limite d'une fonction, exercise de logique mathèmatique, American J. Math., Volume 17, 1894, pp. 27-68
• Notations de logique mathèmatique, Turin 1894, digitized
• Saggio di calcolo geometrico, Atti Accad. Sci. Torino, Volume 31, 1895/96, pp. 852-957 (English translation: Essay on geometrical calculus in Kennedy, Peano, Selected Works 1973)
• Studii in logica matematica, Atti Accad. Sci. Torino, Volume 32, 1896/97, pp. 565-583 (English translation in Kennedy, Peano, Selected Works 1973)
• Formules de logique mathématique, Rivista di Matematica, Volume 7, 1900, pp. 1-41, Archives
• Les définitions mathématiques, Congrès Int. de Philosophy, Paris 1900, Volume 3, pp. 279–288
• Aritmetica generale e algebra elementare, Turin 1902, Archives
• De Latino Sine Flexione. Lingua Auxiliare Internationale, Rivista di Matematica, Volume 8, Turin, 1903, pp. 74-83, reprinted in: G. Peano, Opere scelte II, Rome 1958, pp. 439-447, Project Gutenberg
• Vocabulario de Latino internationale comparato cum Anglo, Franco, Germano, Hispano, Italo, Russo, Græco et Sanscrito, Turin 1904
• Super Theorema de Cantor-Bernstein, Rendiconti del Circolo Mat. Di Palermo, Volume 21, 1906, pp. 360–366 (reprinted in Rivista di Mathematica, Volume 8, 1906, pp. 136–157 with addendum)
• Super teorema de Cantor-Bernstein et additione, Rivista di Matematica, Volume 8, 1902–1906, pp. 136–157 (English translation: Supplement to On the Cantor-Bernstein theorem , in Kennedy, Peano, Selected Works, 1973)
• Vocabulario Commune ad linguas de Europa, Turin 1909
• Formulaire des mathématiques, 5 volumes, Turin 1895, 1897, 1901, 1903, 1908 (Volume 5 in Peano's simplified Latin as Formulio matematico), Volume 1, digitized, Gallica , Archives, Volume 1 , Volume 2 , Volume 3 , Volume 4 , Volume 5
• Sui fondamenti dell´analisi, Bolletino Mathesis Societa Italiana di Mat., Volume 2, 1910, pp. 31-37 (English translation: On the foundations of analysis, in Kennedy, Peano, Selected Works 1937)
• L´importanze dei simboli in matematica, Scientio, Volume 18, 1915, pp. 165–173 (English translation in Kennedy, Peano, Selected Works 1973)
• La definizioni in matematica, Periodico di matematiche, Volume 1, 1921, pp. 175-189 (English translation in Kennedy, Peano, Selected Works 1973)
• Giochi di aritmetica e problemiinteresting, Turin 1925

## expenditure

• Peano: Opere Scelte, 3 volumes, Rome: Cremonese 1957 to 1959 (editor Ugo Cassina )
• Hubert C. Kennedy (Ed.): Selected Works of Giuseppe Peano, Allen and Unwin and University of Toronto Press 1973
• G. Peano: Work on Analysis and Mathematical Logic, ed. G. Asser, Leipzig (Teubner) 1990.

## Web links

Commons : Giuseppe Peano  - collection of images, videos and audio files

## Individual evidence

1. Pierre Dugac, foundations of analysis, in J. Dieudonné, history of mathematics, Vieweg 1985, p 409
2. Hubert Kennedy: The origins of modern axiomatics , in: American Mathematical monthly, 79 (1972), 133-136. Also in: Kennedy: Giuseppe Peano , San Francisco, 2002, p. 15
3. Hubert Kennedy, Peano, 2006, p. 164. This is also analyzed in Hinkis, Proofs of the Cantor-Bernstein theorem, a mathematical excursion, Birkhäuser 2013, and presented in a historical context.