What Platonism ? Reflections on the thought of Kurt Godel

Por • 2 ago, 2021 • Sección: Filosofía

Jaakko Hintikka

Godel is routinely called a Platonist in his philosophy of logic and mathematics. What does that mean ? From the point of view of a working mathematician or logician such labels make little difference. For instance, take the idea of antirealism. A typical manifestation of antirealism in the foundations of mathematics is the claim that in mathematical contexts quantifiers do not really express existence and universality with respect to some subject matter, as they do in everyday applications. (Cf. here Benacerraf 1973.) But what is really meant by «not really» here ? What difference does it make for one’s actual work in logic ? It is almost certain that when a soi-disant antirealist logician practices model theory, he or she will treat quantifiers as ranging over a class of values, which is tantamount to taking them to express existence and universality in that class of values. And this is likely to be the case no matter whether the logician in question pays homage in his or her philosophical Sunday prayers to Plato or to Michael Dummett.

2I am of course keenly aware that there are approaches to semantics that are not referential and in which the meaning of quantifiers is supposed to be explained in terms other than existence and universality. I am equally painfully aware that I cannot refute them in one single paper. All I can say is therefore that I find such treatments of semantics to be on a par with attempts to stage Hamlet without the Prince of Denmark. In any case it is eminently clear that Godel wanted to interpret quantifiers in a mathematical context realistically, no matter whether this is thought of as a reason for calling him a Platonist or not.

3To take another example of this vacuity of current philosophical terminology, almost all philosophers of mathematics call David Hilbert a «formalist», even though on closer examination he turns out to be an axiomatist and even a proto-model-theorist. (Cf. here Hintikka, forfhcoming(a).) Sigue en…

Revue internationale de philosophie 2005/4 (n° 234), pages 535 à 552

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