Neil Sims

It has often been argued that Gödel’s first incompleteness theorem has major implications for our understanding of the human mind. Gödel himself hoped that the results of his theorem, combined with Turning’s work on computers and phenomenological analysis, would establish that the human mind contains an element totally different from a finite combinatorial mechanism. Decades of attempts to establish this by reasoning about Gödel’s theorem and Turing’s work are now widely taken to be unsuccessful. The present article, in accord with Gödel’s suggestion, adds extended phenomenological analysis to the discussion. It also focuses on the “going outside the system” step central to Gödel’s method of proof, rather than on the implications of the theorem itself. Analysis of the “I-it” intentional structure, held by phenomenology to underlie all ordinary experience, yields a simple model that (i) resolves long-standing conceptual problems associated with the “I-it”, the most basic structure of phenomenology, (ii) clarifies the “going outside” step crucial to Gödel’s method of proof, (iii) avoids conceptual problems associated with this step, (iv) identifies the step as an instance of a natural pre-mathematical operation of ordinary thought, and (v) suggests that the step itself is intrinsically non-algorithmic. Logical analysis of role of this step in Gödel’s proof then shows, independently of phenomenological considerations, that anything (human or not) that can prove Gödel’s theorem soundly by his method cannot be entirely algorithmic. Further implications for the nature of the mind are then suggested.

It has often been argued that Gödel’s first incompleteness theorem has major implications for our understanding of the human mind. Gödel himself hoped that the results of his theorem, combined with Turning’s work on computers and phenomenological analysis, would establish that the human mind contains an element totally different from a finite combinatorial mechanism. Decades of attempts to establish this by reasoning about Gödel’s theorem and Turing’s work are now widely taken to be unsuccessful. The present article, in accord with Gödel’s suggestion, adds extended phenomenological analysis to the discussion. It also focuses on the “going outside the system” step central to Gödel’s method of proof, rather than on the implications of the theorem itself. Analysis of the “I-it” intentional structure, held by phenomenology to underlie all ordinary experience, yields a simple model that (i) resolves long-standing conceptual problems associated with the “I-it”, the most basic structure of phenomenology, (ii) clarifies the “going outside” step crucial to Gödel’s method of proof, (iii) avoids conceptual problems associated with this step, (iv) identifies the step as an instance of a natural pre-mathematical operation of ordinary thought, and (v) suggests that the step itself is intrinsically non-algorithmic. Logical analysis of role of this step in Gödel’s proof then shows, independently of phenomenological considerations, that anything (human or not) that can prove Gödel’s theorem soundly by his method cannot be entirely algorithmic. Further implications for the nature of the mind are then suggested.