VI.62 Giuseppe Peano
b. Spinetta, Italy, 1858; d. Turin, 1932
Analysis; mathematical logic; foundations of mathematics
Known above all for his (and DEDEKIND’S [VI.50]) axiom system for the natural numbers, Peano made important contributions to analysis, logic, and the axiomatization of mathematics. He was born in Spinetta (Piedmont, Italy) as the son of a peasant, and from 1876 studied at the University of Turin, taking his doctoral degree in 1880. He remained there until his death in 1932, becoming full professor in 1895.
During the 1880s Peano worked in analysis, achieving what are generally considered to be his most important results. Particularly noteworthy are the continuous space-filling Peano curve (1890), the notion of content (a precedent of MEASURE THEORY [III.55]) developed independently by JORDAN [VI.52], and his theorems on the existence of solutions for differential equations of the first order (1886, 1890). The textbook he published in 1884, Calcolo Differentiale e Principii di Calcolo Integrale, partly based on lectures by his teacher Angelo Genocchi, was noteworthy for its rigor and critical style, and is counted among the very best nineteenth-century treatises.
The years 1889-1908 saw Peano dedicating himself intensively to symbolic logic, axiomatization, and producing the encyclopedic Formulaire de Mathématiques (1895-1908, five volumes). This ambitious assembly of mathematical results, compactly presented in the symbols of mathematical logic, was given completely without proofs. This was by no means standard at the time, but it shows what Peano expected from logic: it was supposed to bring precision of language and brevity, but not a greater level of rigor (something that was, by contrast, crucial for FREGE [VI.56]). In 1891, together with some colleagues, he founded the journal Rivista di Matematica, gathering around him an important group of followers.
Peano was an accessible man, and the way he mingled with students was regarded as “scandalous” in Turin. He was a socialist in politics, and a tolerant universalist in all matters of life and culture. In the late 1890s Peano became increasingly interested in elaborating a universal spoken language, “Latino sine flexione”; the last edition of the Formulario (1905-8) appeared in this language.
Peano followed closely the work of German mathematicians such as Hermann Grassmann, Ernst Schröder, and Richard Dedekind; for example, the 1884 textbook defined the real numbers by Dedekind cuts, and in 1888 he published Calcolo Geometrico Secondo I’Ausdehnungslehre di H. Grassmann. In 1889 there appeared (notably in Latin) a first version of the famous PEANO AXIOMS [III.67] for the set of natural numbers, which he refined in volume 2 of the Formulaire (1898). It aimed at filling the most significant gap in the foundations of mathematics at a time when the arithmetization of analysis had essentially been completed. It is no coincidence that other mathematicians (Frege, Charles S. Peirce, and Dedekind) published similar work in the same decade. Peano’s attempt is better rounded than Peirce’s, but simpler and framed in more familiar terms than those of Frege and Dedekind; because of this, it has been more popular.
Peano’s work on the natural numbers was at the crossroads of his diverse mathematical contributions, linking naturally his previous research in analysis with his later work on logical foundations, and being a necessary prerequisite for the Formulaire project. Actually, Arithmetices Principia can be regarded as a simplification, refinement, and translation into logical language (the “nova methodo” in its title) of Grass-mann’s Lehrbuch der Arithmetik (1861). Grassmann had striven to elaborate a stern deductive structure, stressing proofs by mathematical induction and recursive definitions. But curiously, unlike Peano, he did not postulate an axiom of induction; thus, Peano presented the basic assumptions much more clearly, bringing induction to center stage as the key defining property of the natural numbers.
Further Reading
Borga, M., P. Freguglia, and D. Palladino. 1985. I Contributi Fondazionali della Scuola di Peano. Milan: Franco Angeli.
Ferreirós, J. 2005. Richard Dedekind (1888) and Giuseppe Peano (1889), booklets on the foundations of arithmetic. In Landmark Writings in Western Mathematics 1640-1940, edited by I. Grattan-Guinness, pp. 613-26. Amsterdam: Elsevier.
Peano, G. 1973. Selected Works of Giuseppe Peano, with a biographical sketch and bibliography by H. C. Kennedy. Toronto: University of Toronto Press.