Giuseppe Peano

Giuseppe Peano: The Man Who Made Mathematics Say What It Meant

In August 1900, at the International Congress of Philosophy in Paris, Bertrand Russell watched a small Italian professor demolish opponent after opponent. "I observed that he was always more precise than anyone else," Russell wrote later, "and that he invariably got the better of any argument on which he embarked." Russell went home with a stack of Peano's papers and retired to the country to study quietly every word written by him or his disciples. Modern analytic philosophy begins roughly there.

The Farm at Spinetta

Peano was born on 27 August 1858 in a farmhouse near Cuneo in Piedmont, then the Kingdom of Sardinia. His parents were agricultural workers; he attended the village school at Spinetta and then walked five kilometres a day to school in Cuneo. The escape came through family. His maternal uncle, a priest and lawyer in Turin, saw what the boy could do and brought him to the city in 1870 for a proper secondary education at the Liceo Cavour.

He entered the University of Turin in 1876. D'Ovidio taught him analytic geometry and algebra in the first year, Angelo Genocchi taught him calculus in the second, and by the third year he was the only student in the faculty still reading pure mathematics — everyone else had transferred to Engineering. He graduated with a doctorate on 29 September 1880, and the university kept him on to assist first D'Ovidio and then Genocchi, who held the chair of calculus.

The Man Who Found the Mistakes

Peano's defining trait announced itself almost immediately. In 1882, working through the standard calculus text by Serret, he found that its definition of the area of a curved surface was wrong. He took it to Genocchi and learned that Schwarz had already caught it — but the instinct was established, and it never left him. He would spend his life reading other mathematicians' work and finding the place where the reasoning quietly failed.

This was not a hobby. When Corrado Segre defended loose statements on the grounds that the moment of discovery mattered more than rigorous formulation, Peano's reply was withering: "I believe it new in the history of mathematics that authors knowingly use in their research propositions for which exceptions are known, or for which they have no proof." He dismantled Hermann Laurent's proofs, and reviewing a book by Veronese concluded that its errors and lack of rigour took all value away from it. He made enemies at a steady rate.

When Genocchi's health failed, Peano took over the calculus lectures, and the 1884 *Calcolo differenziale e principii di calcolo integrale* appeared under Genocchi's name. Genocchi, honourably, issued a public disclaimer: everything, he said, was due to "that outstanding young man Dr Giuseppe Peano."

Two Bombshells, 1889 and 1890

In 1889 Peano published *Arithmetices principia, nova methodo exposita* — Principles of Arithmetic, Presented by a New Method. In it he set out the axioms that still carry his name, the standard axiomatization of the natural numbers, deriving all of arithmetic from a starting point, a successor operation and induction. It is at once a landmark in the history of mathematical logic and in the foundations of mathematics. He wrote it in Latin, an act one historian calls sheer romanticism.

A year later came the space-filling curve. Cantor had shown that the unit interval and the unit square have the same cardinality; Netto had shown that no such correspondence could be one-to-one and continuous at the same time. Peano constructed a continuous curve that passes through every point of the square — abandoning injectivity to keep continuity, and in the process destroying every comfortable intuition about what a curve is and what dimension means. Hausdorff called it one of the most remarkable facts of set theory. Alongside these sit the Peano existence theorem for differential equations, the Peano–Jordan measure and the Peano kernel theorem.

He was appointed Professor First Class at the Royal Military Academy in 1889, extraordinary professor of infinitesimal calculus at Turin in 1890, and ordinary professor in 1895.

A Language for Mathematics

From 1891 Peano pursued the project that would consume the rest of his working life: the *Formulario Mathematico*, an encyclopedia intended to contain all known formulae and theorems of mathematics, each stated in a standard symbolic notation of his own devising. "Such a collection," he wrote, "which would be long and difficult in ordinary language, is made noticeably easier by using the notation of mathematical logic."

The notation was the real achievement. The symbols working mathematicians use every day for membership, union, intersection and existence descend from Peano. The final edition of 1908 contained 4,200 formulae and theorems, all fully stated and most of them proved. Almost nobody read it.

Why Giuseppe Is Called a Genius

The specific faculty is a kind of ruthless semantic clarity: an ability to look at a mathematical sentence everybody accepted and see precisely which word was doing unexamined work — and then either fix it with a new symbol or blow it up with a counterexample. The space-filling curve and the axioms for arithmetic are the same talent pointed in opposite directions, one destroying a false intuition, the other rebuilding a foundation so that no intuition is needed at all.

The word has been used by people whose judgement counts. Russell called Peano's notation "an instrument of logical analysis such as I had been seeking for years." Hausdorff called the curve one of the most remarkable facts of set theory. His biographer Kennedy judged him one of the great scientists of his century.

The counter-case is unusually clear-cut, and Peano's own admirers make it. His originality is concentrated in the years before 1900; after that he poured enormous energy into two projects — the *Formulario* and an artificial language — that the mathematical community simply did not use. He was so determined to teach his symbols that he neglected the calculus he was paid to teach, and the Military Academy sacked him in 1901; only Turin's institutional autonomy saved his university post. And though Peano founded mathematical logic as a working practice, it is Gottlob Frege who is generally called its father. Peano's genius was real, narrow, and partly wasted by its owner.

Latin Without Grammar

In 1903 he announced *Latino sine flexione* — Latin stripped of inflection, with vocabulary drawn from Latin, English, French and German, intended as a universal auxiliary language for science. On 3 January 1908 he read a paper to the Academy of Sciences of Turin that began in ordinary Latin and, simplification by simplification, ended in the new tongue. He was elected director of the Academia pro Interlingua, which abandoned Idiom Neutral in his favour. He wrote the final *Formulario* in it, which did nothing to help its readership.

The Last Lecture

He married Carola Crosio, daughter of the Turin painter Luigi Crosio, in 1887. Honours accumulated steadily — the Academy of Science of Turin in 1891, Knight of the Order of Saints Maurice and Lazarus in 1901, corresponding membership of the Accademia dei Lincei in 1905, Commendatore of the Crown of Italy in 1921. He taught at Turin until the day before he died, suffering a fatal heart attack on 20 April 1932, aged seventy-three. Every student who writes that an element belongs to a set is using his handwriting.

Achievements

• Knight of the Order of the Crown of Italy‎ — 1905
• knight of the Order of Saints Maurice and Lazarus — 1901
• Notable work: Peano curve
• Affiliated with University of Turin and Royal Academy of Turin
• Educated at University of Turin and Liceo Classico e Musicale "C. Cavour"
• Worked as mathematician, philosopher and university teacher