Peano , Giuseppe

(1858–1932) Italian mathematician and logician

Peano, who was born at Spinetta near Cuneo, in Italy, studied at the University of Turin and was an assistant there from 1880. He became extraordinary professor of infinitesimal calculus in 1890 and was full professor from 1895 until his death. He was also professor of the military academy in Turin from 1886 to 1901.

Peano began his mathematical career as an analyst and, like Richard Dedekind before him, his interest in philosophical and logical matters was awakened by the lack of rigor in some presentations of the subject. Peano was particularly keen to avoid all illegitimate reliance on intuition in analysis. His discovery in 1890 of a curve that was continuous but filled space went against intuition. A similar discovery was Karl Weierstrass's famous function that was everywhere continuous but nowhere differentiable. As with Weierstrass's function, Peano's curve shows that the concept of a continuous function cannot be identified with that of a graph.

His interest in rigorous and logical presentation of mathematics led Peano naturally to an interest in the mathematical development of logic. In this field he was one of the great pioneers along with Georg Boole, Gottlob Frege, and Bertrand Russell. Peano's achievement was twofold. First he devised, in his Notations de logique mathématique (1894; Notations in Mathematical Logic), a clear and efficient notation for mathematical logic which, as modified by Bertrand Russell, is still widely used. Secondly, he showed how arithmetic can be derived from a purely logical basis. To do this he formulated, in his Nova methodo exposita(1889; New Explanation of Method), nine axioms, four dealing with equality, and the remaining five, listed below, characterizing the numbers series:

1 is a number

The successor of any number is a number

No two numbers have the same successor

1 is not the successor of any number

Any property that belongs to 1 and the successor of any number that also has that property, belongs to all numbers (mathematical induction).

Peano's axioms had been proposed, in a more complicated form, by Dedekind a year earlier.

Peano also did notable work in geometry and on the error terms in numerical calculation. Among his extramathematical interests he was a keen propagandist for a proposed international language, Interlingua, which he had developed from Volapük.