To the Editor:

It was a pleasure to read Jeremy Bernstein’s graceful account of Kurt Gödel’s life and profound accomplishment [“Gödel’s Universe,” September].

I do wonder, though, that Mr. Bernstein did not mention Gödel’s contributions to set theory, which together with Paul Cohen’s work comprise a magnificent contribution to 20th-century intellectual life. In addition, logicians might wish that Mr. Bernstein had placed somewhat greater stress on the distinction between Gödel’s first and second incompleteness theorems. Gödel provided a full proof for the first, which established the incompleteness of arithmetic, but only a sketch of the second, which established the impossibility of demonstrating the consistency of arithmetic from within arithmetic itself. It remained for the American logician Solomon Feferman to complete the story in the 1960’s.

Mr. Bernstein mentions antecedents to Gödel’s great incompleteness theory, but without going into detail. The story is exceedingly poignant. Emil Post was an American mathematician of Polish birth, a member of the faculty at City College for his entire professional life. He saw the essential facts of incompleteness a full ten years before Gödel published his epochal paper, but was unable to organize his insights into a coherent and accessible proof. In 1941, he asked Hermann Weyl, then the editor of the American Journal of Mathematics, to publish a long retrospective he had written in which his own prophetic work would be brought to the attention of the mathematical community (“Absolutely Unsolvable Problems and Relatively Undecidable Propositions, Account of an Anticipation”).

Weyl declined, remarking fairly enough that “Gödel, [Alonzo] Church, and others have done what they have done, and the American Journal is no place for historical accounts.” But Weyl did add that “[p]ersonally you may be comforted by the certainty that most of the leading logicians . . . know in a general way of your anticipation.” The exchange (reprinted in Martin Davis’s introductory essay to a 1994 volume of Emil Post’s collected works) gives one an idea of the insanely high intellectual standards prevailing in mathematical logic during the first half of this century, standards that would be impossible to implement today, even if they could be attained.

There is, finally, the curious fact that Gödel, “the greatest logician since Aristotle”—the description is J. Robert Oppenheimer’s—failed to win a full professorship at the Institute for Advanced Study until 1953. No doubt, as Mr. Bernstein speculates, academic politics played the role it habitually plays, which is to ensure the suppression of genius. There is, as well, the fact that many mathematicians of the 1930’s and 1940’s regarded mathematical logic with frank hostility. Their attitude is still current: when I began teaching mathematics at the University of Paris in the late 1970’s, French colleagues assured me that my own interest in logic was a form of intellectual pathology. And there is also the fact that until at least the mid-1950’s, mathematical logic was more or less an esoteric discipline, occupying only a handful of scholars. Alonzo Church published his magisterial Introduction to Mathematical Logic in 1956, but Volume I did not include the Gödel incompleteness theorems and ten years later, when I found myself Church’s student, the material on incompleteness still existed only in mimeographed form.

This has now changed, of course, but in assessing the sluggish failure of the Institute properly to appreciate the force of Gödel’s work, it should be remembered that it required more than 30 years for mathematical logicians to establish the social apparatus necessary for the communication of their ideas.

David Berlinski
San Francisco, California

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