The Problem of Modelling the Mathematical Mind

Following Alan Turing’s ground-breaking 1937 paper, which introduced his notion of the Universal Turing machine, he suggested, in 1939, generalizations based on ordinal logic and oracle machines, these being apparently motivated by attempts to model the mathematical mind in a way that could evade the apparent limitations presented by Gödel’s incompleteness theorems. In this talk, I introduce the idea of a “cautious oracle” as a more human version of Turing’s oracles. Nevertheless, I show that even this fails to capture the essence of the full capabilities of our understanding. I raise the issue of possible physical processes that would appear to be needed in order to circumvent these Gödel-type restrictions. At the end of the talk, I report on some startling new experiments, which appear to point to new insights into the possible physical processes underlying conscious brain activity, and I speculate on how this might relate to the power of human understanding.