The German Genius

problems identified by David Hilbert at the Paris conference in 1900 (see Chapter 25) had for the most part been settled, and mathematicians looked out on the world with optimism. Their confidence was more than just a technical matter; mathematics involved logic and therefore had philosophical implications. If mathematics was complete, and internally consistent, as it appeared to be, that said something fundamental about the world. 13
    But then, in September 1931, philosophers and mathematicians convened in Königsberg for a conference on the “Theory of Knowledge in the Exact Sciences,” attended by, among others, Ludwig Wittgenstein, Rudolf Carnap, and Moritz Schlick. All were overshadowed, however, by a twenty-five-year-old mathematician from Brno (Brünn) whose revolutionary arguments were later published in a German scientific journal in an article titled “Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme” (On the Formally Undecidable Propositions of Principia Mathematica and Related Systems). The author was Kurt Gödel, and this paper is now regarded as a milestone in the history of logic and mathematics. Gödel was an intermittent member of Schlick’s Vienna Circle, which had stimulated his interest in the philosophical aspects of science. In his 1931 paper he demolished Frege’s, Russell’s, and Hilbert’s aim of putting all mathematics on irrefutably sound foundations, with his theorem that tells us, no less firmly than Heisenberg’s uncertainty principle, that there are some things we cannot know. As John Dawson Jr. has written, Gödel’s work raises “the spectre of unsolvability.” 14
    His theorem is difficult. The simplest way to explain his idea is by analogy and makes use of the so-called Richard paradox, first put forward by the French mathematician Jules Richard in 1905. In this system integers are given to a variety of definitions about mathematics. For example, the definition “not divisible by any number except one and itself” (i.e., a prime number), might be given one integer, say 17. Another definition might be “being equal to the product of an integer multiplied by that integer” (i.e., a perfect square), and given the integer 20. Now assume that these definitions are laid out in a list with the two above inserted as 17th and 20th. Notice two things: 17, attached to the first statement, is itself a prime number, but 20, attached to the second statement, is not a perfect square. In Richardian mathematics, the above statement about prime numbers is not Richardian, whereas the statement about perfect squares is. Formally, the property of being Richardian involves “not having the property designated by the defining expression with which an integer is correlated in the serially ordered set of definitions.” But of course this last statement is itself a mathematical definition and therefore belongs to the series and has its own integer, n . The question may now be put: Is n itself Richardian? Immediately the contradiction appears. “For n is Richardian if, and only if, it does not possess the property designated by the definition with which n is correlated; and it is easy to see that therefore it is Richardian if, and only if, n is not Richardian.”
    No analogy can do full justice to Gödel’s theorem, but this at least conveys the paradox. It was, for some mathematicians, a profoundly depressing conclusion, for Gödel had effectively established that there were limits to mathematics and to logic—and it changed mathematics for all time. 15
    One place where such questions were frequently discussed was among a group in Vienna who, in 1924, began to meet every Thursday. Originally organized as the Ernst Mach Society, in 1928 they changed their name to the Wiener Kreis, the Vienna Circle. 16 Under this title they became what is arguably the most important philosophical movement of the last century. The guiding spirit was Moritz Schlick (1882–1936) , Berlin-born who, like many members of the Kreis, had trained as a scientist, in his case as a physicist under Max Planck, from 1900 to 1904. The twenty-odd members of the circle that Schlick put together included Otto Neurath from Vienna, a remarkable Jewish polymath; Rudolf Carnap, a mathematician who had been a pupil of Gottlob Frege at Jena; Philipp Frank, another physicist; Heinz Hartmann, a psychoanalyst; Kurt Gödel, the mathematician we have just met; and at times Karl Popper, who became