Realism and the debate on impredicativity, 1917-1944*

Summary

It is fair to say that the acceptability of impredicative definitions and reasoning in mathematics is not now, and hasn't been in recent years, a matter of major controversy. Solomon Feferman may regret that state of affairs, even though his ownwork contributed a great deal to bringing it about. I see thework of Feferman andKurt Schütte on the analysis of predicative provability in the 1960's as bringing to closure one aspect of the discussion of predicativity that began with Poincaré ‘s protests against “non-predicative definitions” in the first decade of the twentieth century and with Russell's making a “vicious circle principle” a major principle by which constructions in logic should be assessed. Although in the 1950's Paul Lorenzen and Hao Wang had undertaken to reconstruct mathematics in such a way that impredicativitywould be avoided, insistence on this (to which evenWang did not subscribe) was very much a minority view, and Feferman in particular sought principally to analyze what predicativity is, with the understanding that some aspects of this enterprise would require impredicative methods. The picture has changed since then by work to which he has also contributed, which has brought to light how much of classical analysis in particular can be done by methods that are logically very weak, in particular predicative.

The pioneer of this latter effort, as Feferman has analyzed in detail, was Hermann Weyl. Weyl also brought about the most dramatic episode in the early history by claiming that there is a “vicious circle” of the kind pointed to by Poincaré and Russell in some basic reasonings in analysis. Curiously, his promising beginning for a predicative reconstruction of analysis was not pursued further at the time either by him or by others. It may seem that Weyl's charge of a vicious circle found few adherents, but this appearance may be misleading because intuitionist analysis, which was just then being developed seriously by Brouwer, was not thought to be subject to the same difficulty. For some time thereafter, however, workers in foundations who accepted classical mathematics thought it necessary to reply to Weyl.