We show that the name “Lucas-Penrose thesis” encompasses several different theses. All these theses refer to extremely vague concepts, and so are either practically meaningless, or obviously false. The arguments for the various theses, in turn, are based on confusions with regard to the meaning of these vague notions, and on unjustified hidden assumptions concerning them. All these observations are true also for all interesting versions of the much weaker thesis known as “Gö- del disjunction”. Our main conclusions are that (...) pure mathematical theorems cannot decide alone any question which is not purely mathematical, and that an argument that cannot be fully formalized cannot be taken as a mathematical proof. (shrink)

The alleged proof of the non-mechanical, or non-computational, character of the human mind based on Gödel’s incompleteness theorem is revisited. Its history is reviewed. The proof, also known as the Lucas argument and the Penrose argument, is refuted. It is claimed, following Gödel himself and other leading logicians, that antimechanism is not implied by Gödel’s theorems alone. The present paper sets out this refutation in its strongest form, demonstrating general theorems implying the inconsistency of Lucas’s arithmetic and the semantic inadequacy (...) of Penrose’s arithmetic. On the other hand, the limitations to our capacity for mechanizing or programming the mind are also indicated, together with two other corollaries of Gödel’s theorems: that we cannot prove that we are consistent, and that we cannot fully describe our notion of a natural number. (shrink)

This is a paper for a special issue of Semiotic Studies devoted to Stanislaw Krajewski’s paper. This paper gives some supplementary notes to Krajewski’s on the Anti-Mechanist Arguments based on Gödel’s incompleteness theorem. In Section 3, we give some additional explanations to Section 4–6 in Krajewski’s and classify some misunderstandings of Gödel’s incompleteness theorem related to AntiMechanist Arguments. In Section 4 and 5, we give a more detailed discussion of Gödel’s Disjunctive Thesis, Gödel’s Undemonstrability of Consistency Thesis and the definability (...) of natural numbers as in Section 7–8 in Krajewski’s, describing how recent advances bear on these issues. (shrink)

The aim of this paper is to challenge Hao Wang's presentation of Gödel's views on the language of mathematics. Hao Wang claimed that the language of mathematics is for Gödel nothing but a sensory tool that helps humans to focus their attention on some abstract objects. According to an alternative interpretation presented here, Gödel believed that the language of mathematics has an important role in acquiring knowledge of the abstract mathematical world. One possible explanation of that role is proposed.