(Calcolo Geometrico, G. Peano's first publication in mathe...)

Calcolo Geometrico, G. Peano's first publication in mathematical logic, is a model of expository writing, with a significant impact on 20th century mathematics.

Giuseppe Peano was an Italian mathematician, logician, educator and author. He was the founder of symbolic logic whose interests centred on the foundations of mathematics and on the development of a formal logical language.

Background

Giuseppe Peano was born in Spinetta, near the city of Cuneo, Italy, on August 27, 1858. He was the second of five children born to Bartolomeo Peano and the former Rosa Cavallo. At the time of Peano’s birth, his family lived on a farm about three miles from Cuneo, a distance that he and his brother Michele walked each day to and from school. Sometime later, the family moved to Cuneo to reduce the boys’ travel time.

Education

At the age of twelve or thirteen, Peano moved to Turin, some fifty miles south of Cuneo, to study with his uncle, Michele Cavallo. Three years later he passed the entrance examination to the Cavour School in Turin, graduating in 1876. He then enrolled at the University of Turin and began an intensive study of mathematics. On July 16, 1880, he passed his final examinations with high honours.

In 1880, Peano passed his final examinations with high honours and was offered a job as assistant to Enrico D’Ovidio, professor of mathematics at Turin. A year later he began an eight-year apprenticeship with another mathematics professor, Angelo Genocchi.

Peano’s relationship with Genocchi involved one somewhat unusual feature. In 1883 the publishing firm of Bocca Brothers expressed an interest in having a new calculus text written by the famous Genocchi. They expressed this wish to Peano, who passed it on to his master in a letter of June 7, 1883. Peano noted to Genocchi that he would understand if the great man were not interested in writing the book himself and, should that be the case, Peano would complete the work for him using Genocchi’s own lecture notes and listing Genocchi as author.

In fact, that was just Genocchi’s wish. A little more than a year later, the book was published, written by Peano but carrying Genocchi’s name as author. Until the full story was known, however, many of Genocchi’s colleagues were convinced that Peano had used his master’s name to advance his own reputation. As others became aware of Peano’s contribution to the book, his own fame began to rise.

Peano’s first original publications in 1881 and 1882 included an important work on the integrability of functions. He showed that any first-order differential equation of the form y’ = f (x, y) can be solved provided only that f is continuous. Some of these early works also included examples of a type of problem of which Peano was to become particularly fond, examples that contradicted widely accepted and fundamental mathematical statements. The most famous of these, published in 1890, was his work on the space-filling curve.

At the time, it was commonly believed that a curve defined by a parametric function would always be limited to an arbitrarily small region. Peano showed, however, that the two continuous parametric functions x = x (t) and y = y (t) could be written in such a way that as t varies through a given interval, the graph of the curve covers every point within a given area. Peano’s biographer Hubert Kennedy points out that Peano “was so proud of this discovery that he had one of the curves in the sequence put on the terrace of his home, in black tiles on white.”

Peano’s first paper on symbolic logic was an article published in 1888 in which he continued and extended the work of George Boole, the founder of the subject, and other pioneers such as Ernst Schroeder, H. McColl, and Charles S. Peirce. His magnum opus on logic Arithmetices principia, nova methodoexposita (The Principles of Arithmetic, Presented by a New Method), was written about a year later. In it, Peano suggested a number of new notations, including the familiar symbol to represent the members of a set. He wrote in the preface to this work that progress in mathematics was hampered by the “ambiguity of ordinary language.” It would be his goal, he said, to indicate “by signs all the ideas which occur in the fundamentals of arithmetic, so that every proposition is stated with just these signs.” Also included in this book were Peano’s postulates for the natural numbers, an accomplishment that Kennedy calls “perhaps the best known of all his creations.”

In 1891 Peano founded the journal Rivista di matemática (Review of Mathematics) as an outlet for his own work and that of others; he edited the journal until its demise in 1906. He also announced in 1892 the publication of a journal called Formulario with the ambitious goal of bringing together all known theorems in all fields of mathematics. Five editions of Formulario listing a total of 4,200 theorems were published between 1895 and 1908.

By 1900 Peano had become interested in quite another topic, the development of an international language. He saw the need for the creation of an “interlingua” through which people of all nations— especially scientists—would be able to communicate. He conceived of the new language as being the successor of the classical Latin in which pre-Renaissance scholars had corresponded, a latino sine flexione, or “Latin without grammar.”

While still working as Genocchi’s assistant, Peano was appointed a professor of mathematics at the Turin Military Academy in 1886. Four years later he was also chosen to be an extraordinary professor of infinitesimal calculus at the University of Turin. In 1895 he was promoted to ordinary professor. In 1901 he resigned his post at the Military Academy, but continued to hold his chair at the university until his death of a heart attack on April 20, 1932.

Peano married Carla Crosio on July 21, 1887. She was the daughter of the painter Luigi Crosio and was particularly fond of the opera. The couple had no children.