MATTHIAS BAAZ, CHRISTOS H. PAPADIMITRIOU, HILARY W. PUTNAM, DANA S. SCOTT, and CHARLES L. HARPER, Jr, eds. Kurt Godel and the Foundations of Mathematics: Horizons of Truth. New York: Cambridge University Press, 2011. ISBN 978-0-521-76144-4. Pp. xxiii + 515

Abstract

Blockbuster centenary conferences are usually very unsatisfactory affairs. Too many of the great and the good will have nothing very new to say. And despite a notional common focus, speakers interpret their brief in wildly different ways. Unless editors take a stern line, the resulting conference volumes are equally unsatisfactory. This book, comprising 21 papers arising from the 2006 Vienna ‘Horizons of Truth’ symposium marking Gödel's anniversary, is even more of a mixed bag than usual. Which papers are worth looking at more than five years after the event? Some pieces can be rapidly passed over by most readers of this journal. Ivor Grattan-Guinness writes with his usual care about ‘The reception of Gödel's 1931 incompletability theorems by mathematicians, and some logicians, to the early 1960s’. There are one or two interesting anecdotes (e.g., Saunders Mac Lane studied under Bernays in Hilbert's Göttingen in 1931 to 1933, but writes that that he was not made aware of Gödel's result). But the general theme that logicians got to know about incompleteness early (with some surprising little delays), while the word spread among the wider mathematical community much more slowly, is no news. Karl Sigmund discusses ‘Gödel's troubled relationship with the University of Vienna based on material from the archives as well as on private letters’. This is also a nicely readable paper about the facts of the case, but there is nothing here that sheds new light on Gödel's intellectual development. By contrast, Georg Kreisel's contribution is not readable at all, but is like a cruel parody of late Kreiseleriana, rambling and allusive. The paper's inclusion does no kindness at all to the author though the endnotes, more than twice as long as the paper itself, have moments of interest. I was struck by one where Kreisel reports that he at one point ‘saw a good deal of Bernays, who liked to remember Hilbert …. According to Bernays … Hilbert was asked (before his stroke) if his claims for the ideal of consistency should be taken literally. In his (then) usual style, he laughed and quipped that the claims served only to attract the attention of mathematicians to the potential of proof theory.’ And Kreisel goes on to say something about Hilbert wanting to use consistency proofs to bypass ‘then popular (dramatized) foundational problems and get on with the job of doing mathematics’. This chimes with the ‘naturalistic’ reading of Hilbert, most recently defended by Curtis Franks.