**Giuseppe Peano** was the founder of symbolic logic and his interests centred on the foundations of mathematics and on the development of a formal logical language.

- Giuseppe took exams at Ginnasio Cavour in 1873 and then was a pupil at Liceo Cavour from where he graduated in 1876 and, in that year, he entered the University of Turin.
- Among Peano's teachers in his first year at the University of Turin was D'Ovidio who taught him analytic geometry and algebra.
- In his second year he was taught calculus by Angelo Genocchi and descriptive geometry by Giuseppe Bruno.
- Peano continued to study pure mathematics in his third year and found that he was the only student to do so.
- The others had continued their studies at the Engineering School which Peano himself had originally intended to do.
- On 29 September 1880 Peano graduated as doctor of mathematics.
- Peano joined the staff at the University of Turin in 1880, being appointed as assistant to D'Ovidio.
- Peano was appointed assistant to Genocchi for 1881-82 and it was in 1882 that Peano made a discovery that would be typical of his style for many years, he discovered an error in a standard definition.
- Genocchi was by this time quite old and in relatively poor health and Peano took over some of his teaching.
- Peano was about to teach the students about the area of a curved surface when he realised that the definition in Serret's book, which was the standard text for the course, was incorrect.
- Peano immediately told Genocchi of his discovery to be told that Genocchi already knew.
- This book Course in Infinitesimal Calculus although based on Genocchi's lectures was edited by Peano and indeed it has much in it written by Peano himself.
- published with additions by Dr Giuseppe Peano.
- Peano received his qualification to be a university professor in December 1884 and he continued to teach further courses, some for Genocchi whose health had not recovered sufficiently to allow him to return to the University.
- In addition to his teaching at the University of Turin, Peano began lecturing at the Military Academy in Turin in 1886.
- In 1888 Peano published the book Geometrical Calculus which begins with a chapter on mathematical logic.
- A more significant feature of the book is that in it Peano sets out with great clarity the ideas of Grassmann which certainly were set out in a rather obscure way by Grassmann himself.
- This book contains the first definition of a vector space given with a remarkably modern notation and style and, although it was not appreciated by many at the time, this is surely a quite remarkable achievement by Peano.
- In 1889 Peano published his famous axioms, called Peano axioms, which defined the natural numbers in terms of sets.
- Genocchi died in 1889 and Peano expected to be appointed to fill his chair.
- Before the appointment could be made Peano published another stunning result.
- Peano's continuous space-filling curves cannot be 1-1 of course, otherwise Netto's theorem would be contradicted.
- In December 1890 Peano's wait to be appointed to Genocchi's chair was over when, after the usual competition, Peano was offered the post.
- In 1891 Peano founded Rivista di matematica, a journal devoted mainly to logic and the foundations of mathematics.
- The first paper in the first part is a ten page article by Peano summarising his work on mathematical logic up to that time.
- Peano had a great skill in seeing that theorems were incorrect by spotting exceptions.
- When Corrado Segre submitted an article to Rivista di matematica Peano pointed out that some of the theorems in the article had exceptions.
- It was not only Corrado Segre who suffered from Peano's outstanding ability to spot lack of rigour.
- Of course it was the precision of his thinking, using the exactness of his mathematical logic, that gave Peano this clarity of thought.
- From around 1892, Peano embarked on a new and extremely ambitious project, namely the Formulario Mathematico.
- In many ways this grand idea marks the end of Peano's extraordinary creative work.
- Peano began trying to convert all those around him to believe in the importance of this project and this had the effect of annoying them.
- However Peano and his close associates, including his assistants, Vailati, Burali-Forti, Pieri and Fano soon became deeply involved with the work.
- When the calculus volume of the Formulario was published Peano, as he had indicated, began to use it for his teaching.
- Peano, who was a good teacher when he began his lecturing career, became unacceptable to both his students and his colleagues by the style of his teaching.
- The professor was a law unto himself in his own subject and Peano was not prepared to listen to his colleagues when they tried to encourage him to return to his old style of teaching.
- The Formulario Mathematico project was completed in 1908 and one has to admire what Peano achieved but although the work contained a mine of information it was little used.
- However, perhaps Peano's greatest triumph came in 1900.
- Peano remained in Paris for this Congress and listened to Hilbert's talk setting out ten of the 23 problems which appeared in his paper aimed at giving the agenda for the next century.
- Peano was particularly interested in the second problem which asked if the axioms of arithmetic could be proved consistent.
- Even before the Formulario Mathematico project was completed Peano was putting in place the next major project of his life.
- In 1903 Peano expressed interest in finding a universal, or international, language and proposed an artificial language "Latino sine flexione" based on Latin but stripped of all grammar.
- Peano's career was therefore rather strangely divided into two periods.
- Peano may not only be classified as a 19th century mathematician and logician, but because of his originality and influence, must be judged one of the great scientists of that century.

Born 27 August 1858, Cuneo, Piemonte, Kingdom of Sardinia (now Italy). Died 20 April 1932, Turin, Italy.

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Algebra, Origin Italy, Set Theory

**Oâ€™Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive