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C H A P T E R T W O The Crisis in Mathematics THE spiritual crisis in German society prior to the First World War, which, subsequent to it, became translated into political terms, has been the subject of considerable study.1 However, in mathematics as a subject matter, there was a contemporaneous crisis—or rather, crises—as alluded to earlier, in which Germans played a prominent role. The term “crisis in mathematics” as used, for example, by Hermann Weyl2 is usually taken to refer to the logical and foundational dispute aroused by the settheoretic antinomies and Zermelo’s pinpointing of the so-called Axiom of Choice, to be discussed below. However, while these questions still reverberate today, they reflect a changing view of mathematics and reaction to that change, which need to be elucidated first. This was the development of abstract conceptual notions and the increasing reliance on more general methods of proof and so-called existence theorems, in which the existence of a mathematical object with certain prescribed properties was logically demonstrated without that object ever being explicitly exhibited. As a setting of the “mathematical stage,” this chapter is devoted to a description of these crises in a manner that I hope will be accessible to the reader with little or no mathematical background. In addition , this chapter provides more than just a setting. Issues involving the mode of intellectual discovery in mathematics became critical for a number of mathematicians aligned with the Nazis. For them, this was an issue of axiomatics versus valid intuition. In their terms, it was an issue of logic-chopping axiomjuggling against true insight into the mathematics naturally displayed. The set theory created by Georg Cantor and the axiomatization of it by Ernst Zermelo, together with his highly counterinituitive Well-Ordering Theorem, were particular bêtes noires. This may have partly been because Cantor was erroneously believed to have been Jewish.3 Similarly, the dislike for abstract algebra may 1 Among other works, for example, on the Youth Movement as a response to this crisis, see Walter Laqueur, Young Germany (1962); on the development of German ideology, George Mosse, The Crisis in German Ideology (1964), and Peter Gay, Weimar Culture: The Outsider as Insider (1968); on three “important outsiders” whose thought became “inside” as the result of the crisis, see Fritz Stern, The Politics of Cultural Despair (1961). A somewhat overrated but excellent study of a particular manipulation of German ideology in crisis is Siegfried Kracauer, From Caligari to Hitler (Princeton : Princeton University Press, 1947). Among the multitude of novels that reflect one aspect or another of the development of this crisis and its ultimate resolution in Naziism are Heinrich Mann’s Der Untertan (Man of straw) and, as retrospect, Thomas Mann’s Dr. Faustus. A reader who is a peruser of novels no doubt has his or her own favorites as well. 2 Hermann Weyl, “Mathematics and Logic,” American Mathematical Monthly 53 (1946): 2–13. 3 See J. W. Dauben, Georg Cantor, His Mathematics and Philosophy of the Infinite (1979), esp. 271– 299 and 140–148. Cantor’s antecedents and religious education are discussed on 273–280. Can- T H E C R I S I S I N M AT H E M AT I C S 15 have been partly because one of its principal founders, Emmy Noether, was Jewish, female, and left-wing. Such reasons, however, are superficial. The mathematics of Cantor, Zermelo, and Noether would not have been favored by the “ideological Nazi” mathematicians in any case. At the same time, there were mathematicians who, while not Nazis in Weltanschauung, were spokespersons for the Nazi government, either by position or by choice. Even among these people, there were a variety of mathematical and political views. Ludwig Bieberbach, as will be seen in chapter 7, became a leading protagonist of Nazi attitudes among mathematicians after 1933, and had, in the 1920s, converted from his earlier beliefs on one side of the crisis in the foundation of mathematics to the other. The leading exponent of the position to which Bieberbach converted was the famous Dutch mathematician L.E.J. Brouwer, who was a noncombatant in World War I and aggressively pro-German prior to 1933. Brouwer’s ideas about mathematics being rooted in its human creators clearly would have been attractive in the Nazi period to someone like Bieberbach , who in 1940 would speak of “the rootedness of science in the people.” Yet an af...


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