Peano, Giuseppe(1858–1932)
Giuseppe Peano, an Italian mathematician and logician, was a professor of mathematics at the University of Turin from 1890 to 1932 and also taught at the military academy in Turin from 1886 to 1901. In 1891 he founded the Rivista di matematica, which was later also published in French (Revue de mathématique) and in Interlingua (an international language developed from Latino sine flexione, an auxiliary language based on Latin), which Peano propounded in 1903. In 1898 Peano acquired a small printing establishment in Turin, and he soon became an accomplished printer; his skill seems to have been of help to him in the process of simplifying logico-mathematical symbolism.
Peano's contributions to mathematics include the first statement of vector calculus (Elementi di calcolo geometrico, Turin, 1891) and the first example of integration by successive approximations within the theory of ordinary differential equations; with the single hypothesis that the data were continuous he proved the existence of the integrals of such equations. He submitted to rigorous criticism the foundations of arithmetic, of projective geometry, and of the general theory of sets. Peano's postulates (1899) were a set of five postulates for the arithmetic of natural numbers that allowed arithmetic to be constructed as a hypothetical-deductive system. In 1882 Peano first arrived at the principle that rigorous language can be separated from ordinary language both within and without mathematics. As Bertrand Russell wrote, Peano's method "extended the region of mathematical precision backwards towards regions which had been given over to philosophical disagreement" ("My Mental Development," in The Philosophy of Bertrand Russell, Paul A. Schilpp, ed., Evanston, IL, 1951, p. 11).
In 1890 Peano introduced the use of iota and inverted iota to distinguish a one-member class from its member, which permitted him to overcome previous confusion between ∈ ("being a member of"), ⊃ ("contained in"), and = ("equal to"). In general, Peano showed the importance of distinguishing the properties of a class from those of the individuals of that class, a need shown, for example, by his "sophism" (actually, a paralogism): "Peter and Paul are apostles; the apostles are twelve; therefore Peter and Paul are twelve."
Peano's work in mathematical logic is to be found in his "Formulario completo," which includes, among other items, the well-known Formulaire de mathématiques, a compendium of mathematics derived from a set of postulates by means of a new notation. The "Formulario," in its encyclopedic, high-level approach, anticipated the thorough expositions of Bourbakism. In using a notation at least as rigorous as those of C. S. Peirce and Gottlob Frege, and more comprehensive and expedient than theirs, Peano's work marked a transition from the old algebra of logic to contemporary methods. His notation is still partially in use, mainly through its adoption by Russell and A. N. Whitehead in Principia Mathematica.
After 1913 Peano ceased to follow developments in symbolic logic. He regarded as artificial Russell's interpretation of numbers as classes of classes. Peano made several hints concerning the need for analyzing the relation of formal language to ordinary language, but he was not himself interested in undertaking such analysis. A philosophical interpretation of some of Peano's techniques is to be found in the work of his pupil Giovanni Vailati, who pointed out the general importance of Peano's discoveries concerning recursiveness, implicit definitions, and the theory of postulates. The "Formulario completo," however, still offers suggestions for research.
Computability Theory; Frege, Gottlob; Logic, History Of: Modern Logic; Mathematics, Foundations Of; Peirce, Charles Sanders; Russell, Bertrand Arthur William; Vailati, Giovanni; Whitehead, Alfred North.
Bibliography
Works by Peano
Opere scelte. 3 vols, edited by Ugo Cassina. Rome: Edizioni Cremonese, 1957–1959. Includes much of what is referred to as the "Formulario completo": The 5 vols. below, plus minor publications of 1888–1913.
Formulaire de mathématiques. Vol. I, Turin: Bocca Frères, Ch. Clausen, 1894; Vol. II, in 3 parts, Turin, 1897–1899; Vol. III, Paris, 1901.
Formulaire mathématique. Vol. IV. Turin, 1903.
Formulario mathematico. Vol. V. Turin, 1905–1908. In Latino sine flexione.
Works on Peano
Cassina, Ugo. "L'oeuvre philosophique de Giuseppe Peano." Revue de métaphysique et de morale 40 (4) (1933): 481–491.
Cassina, Ugo. "Vita et opera de Giuseppe Peano." Schola et Vita 7 (3) (1932): 117–148. In Interlingua.
Couturat, Louis. "La logique mathématique de M. Peano." Revue de métaphysique et de morale 7 (4) (1899): 616–646.
Jourdain, P. E. B. "Giuseppe Peano." Quarterly Journal of Pure and Applied Mathematics 43 (1912): 270–314.
Russell, Bertrand. The Principles of Mathematics. Cambridge, U.K.: Cambridge University Press, 1903.
Stamm, E. "Józef Peano." Wiadomości Matematyczne 36 (1933): 1–56.
Terracini, Alessandro, ed. In memoria di Giuseppe Peano. Cuneo, 1955. Essays by various authors.
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