Enrico Betti Biography

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World of Mathematics on Enrico Betti

Enrico Betti, an Italian mathematician, was born near Pistoia, Italy on October 21, 1823. He died at Pisa, Italy on August 11, 1892.

With the loss of his father while still very young, Betti received his early education from his mother. He earned a degree in physical and mathematical sciences at the University of Pisa. While a university student, he championed ideas that would later propel him into the Italian war for independence.

In 1865, after a stint teaching mathematics at a high school in Pistoia, he was offered and accepted a professorship at the University of Pisa, where he remained until his death. In Pisa, he also served as rector of the university, and director of the teacher's college. He served as a member of parliament beginning in 1862, and as a senator starting in 1884.

For several months in 1874, Betti served as secretary of state for public education. His preference, however, was for academic life, with time alone or spent with close friends, such as G. Riemann (1826-1866), the person for whom Riemannian geometry and Riemann surfaces are named. Betti had met Riemann in Italy, where the latter had gone to improve his health.

In mathematics, Betti is remembered for showing the relevance of the ideas of the Frenchman Évariste Galois (1812-1832) to the research of Paolo Ruffini (1765-1822) and the Norwegian Niels Henrik Abel (1802-1829). He made contributions to the solution of algebraic equations using a specific type of operation. He demonstrated, based on the theory of substitutions, the necessary and sufficient conditions for the solution of algebraic equations at a time when many believed that research related to Galois' was unproductive. Papers published in 1852 and 1855 make fundamental contributions to the development of abstract from classical algebra.

Betti also contributed to the theory of functions, and in particular, elliptic functions. He published his work in this area between 1860 and 1861. Fifteen years later, these ideas were further developed by Karl Wilhem Theodor Weirstrass (1815-1897).

By 1870, Betti had become interested in problems associated with the connectivity of higher-dimensional figures. Betti is especially remembered for his introduction of connectivity numbers. In one dimension, the connectivity number is the number of closed curves that can be drawn that do not divide a surface into separate regions. A two-dimensional connectivity number is the number of closed surfaces in a figure that do not, taken together, bound any three-dimensional regions. Higher-dimensional connectivity numbers are analogously defined. Betti succeeded in proving that the one-dimensional connectivity number is the same as the three-dimensional connectivity number. Betti's research in this area inspired the French mathematician Henri Poincaré (1854-1912) in the latter's own work, and resulted in the naming of "Betti numbers" in his honor.

Betti's mathematical ideas were either algebraic (related to Galois' research), or physicomathematic (influenced by the work of Riemann). Betti had a special fondness for theoretical physics, and devoted part of his research to further work on methods that had already been applied to electricity. In physicomathematics, Betti made contributions to the mathematical theories of elasticity and heat.

Betti was also devoted to the preservation of classical culture. He urged the return to the teaching of Euclid (whose work he felt exemplified the ideals of discipline and beauty) in secondary schools. A gifted teacher, Betti counted among his students individuals of later distinction, including U. Dini, L. Bianchi, and V. Volterra.