Giuseppe Peano
Peano held that arithmetic can be built on a few axioms for the natural numbers and that mathematics and logic can be written in an exact symbolic notation, which Russell and Whitehead later adopted.
Placement
Italian mathematician who gave the natural numbers a short axiomatic form and designed a symbolic notation for logic that Russell and Whitehead adopted; Frege defended his own concept-script against it in 1897.
Key ideas
- Axioms for the natural numbers
- A symbolic notation for logic and mathematics
Influence
Influenced 1
Bertrand Russell
influenced ✓
Russell met Peano at the Paris congress of 1900, and Principia Mathematica adopts Peano’s logical notation. SEP “Bertrand Russell” (A. D. Irvine), chronology; SEP “The Algebra of Logic Tradition” (S. Burris, J. Legris)
Evidence: also in Wikidata · Wikidata
Bertrand Russell › 1872–1970
Criticism
Mutual critique 1
Gottlob Frege ⇄ mutual
criticised disputed
Peano’s 1895 review of Frege’s Grundgesetze I in the Rivista di matematica prompted Frege’s replies of 1896–97; what exactly Peano objected to was not checked. Bibliographic data from the Frege–Peano literature (G. Besler; Rivista di matematica 5, 1895, 122–128 and 6, 1896–99, 53–61); review text not read
Evidence: AI draft
criticised by
In 1897 Frege compared Peano’s logical notation with his own and defended the superiority of his concept-script. Frege, “Über die Begriffsschrift des Herrn Peano und meine eigene” (1897), listed in SEP “Frege’s Logic” (R. Cook, 2023)
Evidence: AI draft
Gottlob Frege › 1848–1925
Works
Short Latin treatise that states the axioms for the natural numbers in symbolic notation, the form now known as the Peano axioms.
Collection of mathematical theorems written in Peano’s symbolic notation, begun in 1891; the fifth and last edition appeared in 1908. Russell was struck by its logical symbols when he met Peano in 1900.
Position
Peano is on the map for two things that shaped logic and the foundations of mathematics: a short axiomatisation of the natural numbers, set out in Latin in Arithmetic Principles (1889) and now called the Peano axioms, and a symbolic notation for logic and mathematics, developed in the Rivista and in the Formulario, a collection of mathematical theorems written in it, of which the fifth and last edition appeared in 1908. At the Paris congress of 1900 he met Bertrand Russell, who was struck by his logical symbols and went on to study his work and that of his school; the Principia Mathematica adopts Peano’s notation. Frege compared Peano’s notation with his own concept-script in a paper of 1897 and defended the superiority of his own; Peano’s review of Frege’s Basic Laws of Arithmetic, vol. 1 (1895), which prompted Frege’s replies of 1896–97, has not been checked in the sources, and the line from Frege to Peano is marked as disputed.
Life & context
Giuseppe Peano was born on 27 August 1858 in Spinetta near Cuneo in Piedmont. He studied at the University of Turin from 1876, graduated in 1880 with high honours and worked first as assistant to Enrico D’Ovidio and then to Angelo Genocchi. From 1886 he also taught at the Royal Military Academy, which dismissed him later because he neglected calculus in favour of his new symbols, and from 1890 he was a professor at Turin. In 1890 he founded the Rivista di matematica, whose first issue appeared in 1891. In 1903 he announced an auxiliary language based on Latin vocabulary with simplified grammar, Latino sine flexione, in which he wrote most of his later work. He died on 20 April 1932 in Turin.
Further main works
- Rivista di matematica (journal, founded 1890)
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