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Remarks on Two Papers of Paul Bernays

From the book Philosophy of Logic and Mathematics

  • Charles Parsons


The paper comments on two papers in French that Paul Bernays derived from lectures at a conference on mathematical logic held in Geneva in June 1934. The first, the well-known “On Platonism in mathematics,” sets forth a methodological version of Platonism and observes that it can be implemented in some branches of mathematics and not others. He notes that by his definition Brouwer’s intuitionism rejects all Platonism,while HermannWeyl’s reconstruction of analysis retains it for generalizations about natural numbers but not for generalizations about real numbers or higher-type objects. He rejects the then widespread idea of a crisis of foundations and argues that the questions raised are philosophical. Bernays’ second paper, “Some observations on metamathematics,” is a technical sequel to “On Platonism.” It describes some basic points in the Hilbert school’s proof theory and sketches some results, such as a simple application of Herbrand’s theorem, Gödel’s proof that if intuitionistic first-order arithmetic is consistent, then so is classical, and Gödel’s second incompleteness theorem. A full proof of the latter and a correct proof of Herbrand’s theorem only appeared in 1939, in volume II of Hilbert and Bernays, Grundlagen der Mathematik.

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