This section will be devoted to a brief consideration of Georg Cantor and Henri
Poincare, two mathematicians with life spans astride the nineteenth and twenti
eth centuried, and who exerted a considerable influence on much of the mathe
matics of present times. It is also natural to insert a few words about Leopold
Kronecker, the harsh and relentless critic of Cantor's mathematics of the in
finite. Georg Ferdinand Ludwig Philip Cantor was born of Danish parents in St.
Petersburg, Russia, in 1845, and moved with his parents to Frankfurt, Germany,
Cantor's father was a Jew converted to Protestantism and his
mother had been born a Catholic. The son took a deep interest in medieval
theology and its intricate arguments on the continuous and the infinite. As a
consequence, he gave up his father's suggestion of preparing for a career in
engineering for concentrating on philosophy, physics, and mathematics. He
studied at Zurich, Gottingen, and Berlin (where he came under the influence of
Weierstrass and where he took his doctorate in 1867). He then spent a long
teaching career at the University of Halle from 1869 until 1905. He died in a
mental hospital in Halle in 1918.
Cantor's early interests were in number theory, indeterminate equations,
and trigonometric series. The subtle theory of trigonometric series seems to
have inspired him to look into the foundations of analysis. He produced his
beautiful treatment of irrational numbers, which utilizes convergent sequences
of rational numbers and differs radically from the geometrically inspired treat
ment of Dedekind, and commenced in 1874 his revolutionary work on set
theory and the theory of the infinite. With this latter sork, Cantor created a
whole new field of mathematical research. In his papers, he developed a theory
of transfinite numbers, based on a mathematical treatment of the actual infinite,
and created an arithmetic of transfinite numbers analogous to the arithmetic of
finite numbers. Some of this matter is amplified in Section 15-4.
Cantor was deeply religious, and his work which in a sense is a continua
tion of the arguments connected with the paradoxes of Zeno, reflects his sym
pathetic respect for medieval scholastic speculation on the nature of the infi
nite. His views met considerable opposition, chiefly from Leopold Kronecker
of the University of Berlin, and it was kronecker who steadfastly opposed
Cantor's efforts toward securing a teaching post at the university of Berlin.
Today, Cantor's set theory has penetrated into almost every branch of mathe
matics, and it has proved to be of particular importance in topology and in the
foundations of real function theory.