The Battle for Cantorian Set Theory

Start: 10/26/2016 - 4:15pm
End  : 10/26/2016 - 4:15pm


Joseph Dauben (CUNY)


The substance of Georg Cantor’s revolutionary mathematics of the infinite is well-known: in developing what he called the arithmetic of transfinite numbers, he gave mathematical content to the idea of actual infinity. In so doing he laid the groundwork for abstract set theory and made significant contributions to the foundations of the calculus and to the analysis of the continuum of real numbers. Cantor’s most remarkable achievement was to show, in a mathematically rigorous way, that the concept of infinity is not an undifferentiated one. Not all infinite sets are the same size, and consequently, infinite sets can be compared with one another. But so shocking and counter-intuitive were Cantor’s ideas at first that the eminent French mathematician, Henri Poincaré, once characterized Cantor’s theory of transfinite numbers as a “disease” from which he was certain mathematics would one day be cured. Leopold Kronecker, one of Cantor’s teachers and among the most prominent members of the German mathematics establishment, even attacked Cantor personally, and in the words of Arthur Schoenflies, called him a “scientific charlatan,” a “renegade” and a “corrupter of youth.” It is also well-known that Cantor suffered throughout his life from a series of “nervous breakdowns” which became increasingly frequent and debilitating as he got older. Some have linked this to his dangerous flirtations with the infinite, and Cantor has often been portrayed as the hapless victim of the infinite, due to his increasingly long periods of mental breakdown that began in the 1880s, all of which were exacerbated by the persecutions of his contemporaries. But such accounts distort the truth by trivializing the genuine intellectual concerns that motivated some of the most thoughtful contemporary opposition to Cantor’s theory. They also fail to credit the power and scope of the defense Cantor offered for his ideas in the battle to win acceptance for transfinite set theory.

Kravis Lower Court, Claremont McKenna College

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