Wittgenstein's Reception of Gödel's Incompleteness Theorems

Abstract

Wittgenstein’s remarks on Gödel’s (first) incompleteness theorem were initially dismissed by scholars who thought Wittgenstein misunderstood the syntactic nature of Gödel’s proof. Following the release of the 1998 Bergen Electronic Edition of Wittgenstein’s Nachlass, remarks on Gödel that were excluded in the two editions of the Remarks on the Foundations of Mathematics were brought to light, reviving an interest in the topic. As part of the same efforts of reinterpretation, this thesis delineates an alternative approach to Wittgenstein’s reception of Gödel. Rather than simply comparing Wittgenstein’s comments on Gödel to Gödel’s seminal paper on Undecidability, the present work does not assume that “Gödel’s theorem,” as an object of philosophical investigation, is identical to “Gödel’s theorem” as conceived within the discipline of mathematics. Hence, the object of Wittgenstein’s discussions must be determined first. To do this, I give an account of late Wittgenstein’s philosophical methods in general, and interpret his remarks on the aims of his discussions of the foundations of mathematics. Wittgenstein’s philosophy is a dialectical exposition of our tendencies in thinking which avoids issuing any apodictic claims. He proposes a kind of philosophical investigation which describes rather than explains. Thus, his intention is not to advance competing theories or interpretations of mathematical theories, but to describe the use of words which appear in mathematics but originate in ordinary language. Given these strictures, a more charitable reading of Wittgenstein’s reception of Gödel is available. For Wittgenstein, Gödel’s theorem is a case study of our tendencies to imagine metaphysical theories, being misled by pictures associated with words that occur in the prose which accompanies the proof. Wittgenstein shows us how we are tempted to interpret words like “truth” and “proof” and thus to impose unwarranted expectations on Gödel’s proof. This is further demonstrated by appealing to passages from the Nachlass (MS 163, 43r-47v) that are for the first time shown to concern Gödel’s paper, passages in which Wittgenstein problematizes the notion of “interpreting content.” As a result, this thesis shows that by a close study of Wittgenstein’s method and stated intentions, the common conclusion that Wittgenstein misunderstood Gödel can be avoided.