Constructivism in logic and mathematics. (Wittgenstein v. Godel: some selected contemporary readings)

In recent years, Wittgenstein's remarks on Godel have produced many disputes and publications. The discussion overcame the framework of historical and philosophical research, and directly touches upon series of problems adjacent to the proof of the first incompleteness theorem. After the publication in 1956 of Remarks on the Foundations of Mathematics, Kraisel and Bernays attributed a total misunderstanding of Godel to Wittgenstein. There were also apologetic readings. To date, almost all friendly or hostile interpretations consider Wittgenstein's remarks irrespective of the mathematical (purely syntactic) proof of the first incompleteness theorem (Wittgenstein's notes must allegedly be understood in the context of the semantic and philosophical implications of the theorem). Among such interpretations it is important to point out the historical and philosophical studies of Floyd (proofs of un-provability are considered here in the context of Wittgenstein's general position regarding the proof of the impossibility of certain geometric constructions). Or there are attempts to see Wittgenstein as the pioneer of paraconsistent logic. Undoubtedly, the semantic and general philosophical consequences of the theorem are of great importance today, and can be better understood in the context of Wittgenstein's remarks. In the article, the author considers the friendly reading of Wittgenstein by Putnam and Floyd (also considering Bays' criticism) (Parts I and II of the article) and the ground-breaking reading by Lampert (Part III). Lampert is notable for the following: he was the first to propose an apologetic reading of Wittgenstein under the condition of correlating the remarks directly with the proof of the first incompleteness theorem (Wittgenstein's approach to algorithmic and metamathematical (constructive) kinds of proof is also considered). In the final part, the author briefly dwells on the difference between two selected interpretations of Wittgenstein's remarks and on the general context of Wittgenstein's attitude toward Godel.

Keywords

Authors

References