Gödel’s Theorems and the Synthetic-Analytic Distinction
Vojtěch Kolman2006, Miscellanea Logica VI, UK FF, Praha 2006, pp. 45–64
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Abstract
When posing the old question ‘What are arithmetical truths about?’ (‘What is their epistemic status?’ or ‘How are they possible?’) we find ourselves standing in the shadow of Gödel, just as our predecessors stood in the shadow of Kant. Of course, this observation may be a bit misleading if only for the reason that Gödel’s famous incompleteness theorems are not of a philosophical nature, at least not in the first place. There are plenty of texts, however, explaining them as philosophically relevant, i.e. as having some philosophical implications. In this article I am not aiming to add a new interpretation to the old ones. Rather, I am proposing to see the incompleteness as a link in the chain of certain great (positive or negative) foundational results such as Frege’s calculization of logic, Russell’s paradox, G¨odel’s completeness theorem, Gentzen’s proof of consistency etc. The foundational line described in this way can then be critically examined as relatively successful with respect to some of its leading ideas and as unsuccessful with respect to others. What I have particularly in mind here is the idea of reducing arithmetic to logic, with its decisive influence on the rebirth and subsequent development of modern (mathematical) logic. Hence, the key issue of this article may be formulated as follows: ‘What do Gödel’s theorems tell us about the alleged analyticity or syntheticity of arithmetic?’.
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