Burali-Forti, Cesare (1861-1931)

Italian mathematician who worked in the field of vector analysis, especially on the linear transformations of vectors. He also framed the Burali-Forti paradox in 1897, which contradicted the notion that mathematics (or at least its foundations) could be adequately expressed in purely logical terms.

Burali-Forti was born in Arezzo, Tuscany, studied at Pisa, and became a professor at the Academia Militare di Artiglieria e Genio in Turin.

Burali-Forti's paradox states: ‘To every class of ordinal numbers there corresponds an ordinal number which is greater than any element of the class.’ In 1902 English philosopher Bertrand Russell demonstrated that this contradiction was of a fundamental logical character and could not be overcome by minor changes in the theory of infinite ordinal numbers.

Much of Burali-Forti's work in the field of vector analysis was done in collaboration with Roberto Marcolongo. In 1904 they published a comprehensive analysis of existing systems of vector notation, producing in 1909 their own proposals for a unified system. Burali-Forti simplified the foundations of vector analysis by the introduction of the notion of the derivative of a vector with respect to a point.

In 1912-13 Burali-Forti published more volumes on linear transformations and demonstrated their application to such things as the theory of mechanics of continuous bodies, hydrodynamics, optics, and some problems of mechanics. His last contribution, a paper on differential projective geometry, was finished in 1930.