## Guiseppe Peano

# Guiseppe Peano

**1858-1932**

**Italian Mathematician**

Guiseppe Peano was one of the outstanding mathematicians of the nineteenth century. He produced original and important work in the areas of calculus and set theory, and his work on solving systems of linear equations developed simultaneously with that of Emile Picard (1856-1941) and Hermann Schwarz (1843-1921). He spent the last part of his career, however, working on large projects of minor importance to mathematics.

Peano was born into a farming family in Italy. Recognizing his talents, his uncle, a priest and lawyer, took him to Turin where he attended secondary school. He was admitted to the University of Turin at the age of 18. There, he studied under some of the premier Italian mathematicians of the day.

Although initially planning on pursuing studies in engineering, by his third year, Peano had decided on a career in mathematics. He graduated with a doctorate in mathematics in 1880, at the age of 22, and began teaching at Turin that same year.

Peano's first work of importance was editing a textbook written by his mentor, Angelo Gennocchi in 1884. That same year he was made a professor of mathematics at Turin, taking over much of Gennocchi's teaching load. Peano's first important contribution to mathematics came in 1886 when he expanded on some work by Augustin Louis Cauchy (1789-1857) and Rudolf O. S. Lipschitz (1832-1903) in differential equations, later expanding this work even further. This was quickly followed by his work on solving systems of linear equations by the method of successive approximations, although this work turned out to have been anticipated by others. His next major accomplishment was the publishing of his book, *Geometrical Calculus*, in 1888, in which he introduced Hermann Grassmann's (1809-1877) important work in a much more approachable manner than Grassmann had done. This work also marked the beginning of Peano's work in mathematical logic, an area in which he was to make many more important contributions.

Yet another achievement of this early period was Peano's work in set theory. This began in 1889 with his publications of the Peano axioms, which defined the natural numbers in terms of sets. A year later, he introduced the concept of "space-filling curves," previously thought not to exist and whose proof was called one of the most remarkable facts in set theory.

Peano was well-known for his clarity of thought and his ability to spot logical flaws in arguments presented by others. This gave him an almost unique ability to find exceptions to mathematical proofs or theorems presented to him as well as to find small flaws in arguments. This trait also served to irritate many of his colleagues. Of this ability, Bertrand Russell commented: "In discussions... I observed that he was always more precise than anyone else, and that he invariably got the better of any argument on which he embarked. As the days went by, I decided that this must be owing to his mathematical logic."

In 1896, at the age of 38, Peano effectively ended the outstandingly creative period of his career to pursue work on a new project, the *Formulario Mathematico*. This was to be a compendium of all known mathematical theorems, organized by subject area and described solely with mathematical notation. In fact, after about 1900 Peano made virtually no additional discoveries of importance in mathematics, although work on the *Formulario* continued.

In addition to working on the *Formulario* , Peano began work in 1903 on a universal language that would be based on Latin but from which all grammar would be removed. This language, which he called *Latino sine flexione *(later called Interlingua), was to incorporate vocabulary from many of the world's languages, and Peano hoped it would eventually become widespread. Peano continued work on these and similar projects until his death in 1932.

**P. ANDREW KARAM**

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## Guiseppe Peano

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**Guiseppe Peano**