Peano is a key figure in the contemporary search for axiom systems in mathematics. He is also the link between the development from the 1860s of rigorous mathematical analysis as conceived by Weierstrass and the emergence in the 1890s of 'mathematical logic' (his name in this sense). He realized that the language of analysis needed refinement in order to make explicit not only logical relationships between terms and propositions, but also the quantification ('for all' and 'there exists') over both individuals and predicates. In addition, he recognized the merits of Cantor's set theory within analysis (from which in fact it had arisen in the early 1870s), and brought it under his logical umbrella. Indeed much of the notation that we still use for both logic and set theory comes from him or his many followers who became known as 'the Peanists'.
Peano also imitated Weierstrass as a leader, building up around him in Turin a remarkable coterie of followers and disciples. He gave them two organs in which to publish: the Rivista for papers and the various editions of the Formulario in which their formulations of branches of mathematics were codified (often with valuable historical references). The three principal followers were C. Burali-Forti (1861-1931), A. Padoa (1868-1937) and M. Pieri (1860-1913).
The presentation in the Formulario started out from arithmetic (for which Peano had produced his axiomatization in 1889) and proceeded to set theory; the subsequent coverage of mathematics included basic algebra and geometry, real and complex numbers, the calculus, the theory of curves, and vector algebra. But the Peanists were always chary of absorbing mathematics into this logic, and distinguished logical from mathematical notions in their explanations.
This was not the reaction of a young listener to Peano and his chief trio when they had papers presented on 3 August 1900 at the International Congress of Philosophy in Paris. Russell saw Peano's structure as The System which he had been seeking, and under its impiration it he announced the 'logicist' thesis which they had eschewed.
Unlike Russell, Peano was wary of exploring philosophical issues in his logico-mathematics, but he was uncommonly sharp on several matters. For example, during the concern with the paradoxes of set theory in the 1900s he glimpsed the distinction between mathematical and semantic ones. His forte was definitions in mathematics (the subject of his Paris lecture); for example, the essentials of Russell's theory of denoting are in one of his pieces in the second edition (1897) of the Formulario.
The later editions of this work, and many of Peano's papers from the 1900s, were written in 'Latin without inflection', and from that decade on he became increasingly concerned with advocating it as an international language. But he maintained his encouragement of logic and logically minded philosophers. One of his last students was Ludovico Geymonat.
British, b: 8 August 1921, Bedfont, Middlesex. Cat: Philosopher of mind. Ints: Hume; Russell; Wittgenstein. Educ: Balliol College, Oxford. Infls: Literary influencew include Aristotle, Hume, Russell and Wittgenstein. Appts: 1948-50, Research Lecturer, Christ Church, Oxford; 1950-60, Fellow, Corpus Christi College, Oxford; 1960-88, Tutor in Philosophy, Christ Church; 1972-85, Reader in Philosophy, Christ Church; 1985-8, Professor of Philosophy, then Professor Emeritus, University of Oxford.