Georg Cantor (1845–1918)
Georg Cantor was the founder of modern set theory, which over the course of the early twentieth century came to serve as the most widely accepted foundational system for mathematics as a whole. Born in Saint Petersburg to a German family, Cantor displayed early talents as a violinist and moved with his family back to Germany in 1911. He pursued studies in mathematics at universities in Darmstadt, Zurich, and Berlin, eventually receiving his doctorate in Berlin with a thesis in number theory. Cantor quickly landed a position at the University of Halle, where he would teach for most of his professional life, failing to win appointments at more prestigious universities due to the hostility to his ideas on the part of many in the profession. In 1879, he was nonetheless promoted to full professor at a relatively young age due to his prodigious output. Motivated by correspondence with Richard Dedekind and later with Gösta Mittag-Lefler, Cantor worked at a furious pace in the years between 1874 and 1884, a decade generally recognized as the moment when set theory was founded.
Cantor’s importance for the Cahiers pour l’Analyse, and the later work of Alain Badiou is difficult to overstate. The excerpt translated by Jean-Claude Milner and published in volume nine of the Cahiers dates from 1882 and is one of the clearest statements of Cantor’s ideas concerning the transfinite and the distinction between ordinal and cardinal concepts of number. A controversial figure in his own day, whose ideas evoked fierce resistance on the part of his contemporaries, Leopold Kronecker and Henri Poincaré chief among them, Cantor suffered from posthumously diagnosed bipolar disorder, which resulted in severe bouts of depression and the curtailment of mathematical productivity for the last two decades of his life. Given to a certain mystical disposition, Cantor was aware of the philosophical import of his mathematical discovery of variably-sized infinite sets and its theological implications as well. In the last decade of his life, he was aware of Russell and Whitehead’s reliance on his work in the Principia Mathematica and the efforts of Ernst Zermelo on set theory’s behalf, but he did not leave to see his discovery established as the foundational paradigm it would become for twentieth-century mathematics. In 1926, seven years after Cantor’s death, the mathematician David Hilbert observed: ’No one shall expel us from the Paradise that Cantor has created’.
In the Cahiers pour l’Analyse
|Georg Cantor, ‘Fondements d’une théorie générale des ensembles’, CpA 10.3||[HTML]||[PDF]||[SYN]|
|Kurt Gödel, ‘La Logique mathématique de Russell’, CpA 10.5||[HTML]||[PDF]||[SYN]|
|Alain Badiou, ‘Marque et manque: à propos du zéro’, CpA 10.8||[HTML]||[PDF]||[SYN]|
|Jacques Bouveresse, ‘Philosophie des mathématiques et thérapeutique d’une maladie philosophique: Wittgenstein et la critique de l’apparence “ontologique” dans les mathématiques’, CpA 10.9||[HTML]||[PDF]||[SYN]|
- ‘Foundations of a General Theory of Manifolds: A Mathematico-Philosophical Investigation into the Theory of the Infinite’, trans. William Ewald, in Ewald, ed., From Kant to Hilbert: A Source Book in the Foundations of Mathematics, Volume II. Oxford University Press, 1996, pp. 878-920.
- Contributions to the Founding of the Theory of Transfinite Numbers, trans. Philip Jourdain. London: Dover, 1915 (translation of ‘Beiträge zur Begründung der transfiniten Mengenlehre’, parts 1 and 2 [1895, 1897]). Online at http://www.archive.org/details/contributionstot003626mbp.