Kurt Gödel, 28 April 1906 - 14 January 1978

Abstract Kurt Gödel did not invent mathematical logic; his famous work in the thirties settled questions which had been clearly formulated in the preceding quarter of this century. Despite sensational presentations by crackpots, philosophers and journalists (or even in poems, for example, by H. M. Enzensberger, set to music by H. W. Henze), Gödel’s results have not revolutionized the silent majority’s conception of mathematics, let alone its practice; much less so than the internal development of the subject since then. Certainly, those results refuted most elegantly each of the grand foundational ‘theories’ current at the time, of which Hilbert’s, on the place of formal rules in mathematical reasoning, and those associated with Frege and Russell, on its reduction to universal systems like set theory, were most popular. (Gödel’s own and related results also deflate the particular ‘anti-formalist’ foundations of the time, Poincaré’s and Brouwer’s constructivist and Zermelo’s infinitistic schemes being extreme examples; they are taken up in the last sections of parts II-IV.) For obvious reasons, in his original publications Gödel made a point of formulating his work in terms acceptable to the theories mentioned, and to stress its bearing on them. But it is fair to say that they were suspect anyway, and—less trivially—that they can be refuted more convincingly by simple constatations rather than by (his) mathematical theorems as explained in more detail in part II. Further, as so often with very grand schemes, the refutations put nothing comparable in the place of the discredited foundational views which are, quite properly, simply ignored in current practice.