The German Genius

the Theory of Trigonometric Series.” In this paper, with its jaw-breaking title, Cantor made the concept of “set” one of the most interesting terms in both mathematics and philosophy (he is generally regarded as the founder of set theory). 37 But it was his next step that took mathematicians by surprise (though in truth it was also a surprise that no one had noticed this before). The series, 1, 2, 3… n , was an infinite set and so was 2, 4, 6… n . But it followed from this that some infinite sets were larger than others—there are more integers in the infinite series, 1, 2, 3… n than in 2, 4, 6… n . Next came Cantor’s proof that the infinite number of points on a line segment is equal to the infinite number of points in a plane figure. “I see it, but I don’t believe it!” he wrote to Dedekind on June 29. It is as well to remember this sentence.
    Not everyone thought that the revolution was anything of the kind. From Berlin, Cantor’s former professor, Leopold Kronecker, attacked the new ideas, as did Hermann von Helmholtz and even Friedrich Nietzsche, who agreed that numbers, though necessary, were “fiction.” By now both Gottlob Frege at Jena, upriver from Cantor at Halle, and the Italian Giuseppe Peano, were also grappling with the nature of number. Frege’s answer was less complicated than Peano’s (and Dedekind’s) but used a special notation he had himself devised. 38
    Born in 1848, Frege is now known for two fundamental works, the Begriffsschrift of 1879 and Die Grundlagen der Arithmetik ( Foundations of Arithmetic ) of 1884, in which his basic idea was that language described logic much as mathematics does and that by comparing them, the essential elements of logic would become clear. This was an approach that interested another of Weierstrass’s students, Edmund Husserl, of Halle, whose doctoral dissertation, Über den Begriff der Zahl ( On the Concept of Number ), was followed in 1891 with his ambitiously titled Philosophie der Arithmetik ( Philosophy of Arithmetic ). Having used Frege’s Foundations of Arithmetic in his own work, Husserl sent the Philosophy to Frege as a mark of respect. 39 Instead of trying to define a set mathematically, Husserl asked how the mind forms generalities—to convert multiplications into units in the first place. In other words, it was a philosophical or epistemological problem before it was a mathematical one. And he gave a Kantian answer. The continuum of real numbers, Husserl said, could never be made present to consciousness. Continuity was like space or time, or infinity, a creation of our minds. This was too much for Frege, who dismissed Philosophy of Arithmetic as a “devastation.” 40
    The youngest of the second great generation of German mathematicians, David Hilbert (1862–1943), was in the Frege/Husserl mold in that he viewed mathematics philosophically. But he was also in the Cantor/ Dedekind mold, being equally interested in the mathematics of sets.
    Born in Königsberg, East Prussia, now Kaliningrad, he attended the Collegium Friedrichianum, the school Kant had himself attended 140 years before. Hilbert was a professor at Königsberg until 1895, when Felix Klein lured him to Göttingen. There he became a mentor to many other subsequently famous mathematicians, including Hermann Weyl, Richard Courant, and John von Neumann. 41
    Hilbert was interested in number and in the difference between intuition and logic. He thought that certain aspects of number (for example, order and some sets) were intuitive, and he wanted to define where logic took over from intuition. He became best known for his “exceptional” identification of twenty-three unsolved problems of mathematics, which he presented at the International Congress of Mathematicians in Paris in 1900, although these were, he said, just a sample of problems that remained to be discovered. 42 Later he became interested in what he called “infinite dimensional Euclidean space,” later called a “Hilbert space,” and he worked with Einstein on the final form of General Relativity, the so-called Einstein-Hilbert action.
    Physics and mathematics had, in a sense, a conceptual overlap, both being concerned with the nature of continuity and particularity. That concern would help produce dividends in the digital revolution—but decades in the future.

Sensibility and Sensuality in Vienna
     
    I n early September 1887, Arthur Schnitzler—physician, writer, amateur pianist—was out