The Essential role of consciousness in mathematical cognition
Robert Hadley, Simon Fraser University
Journal of Consciousness Studies, 17, No. 1-2, 2010, pp. 27-46
Hadley puts forward alternative possibilities to Penrose’s argument from the Godel theorem, in order to reach a Penrose-type conclusion about brains and computers. He argues that a system that lacked consciousness would be incapable of certain concepts and certain proofs. Hadley refers to Kant’s argument that the perception of an object requires the unity of consciousness. In modern terms, the difficulty of seeing how the unity of consciousness is achieved by the brain is referred to as the binding problem, and is not the same as, but is closely intertwined with the question of consciousness. The concept of objects is claimed to require certain assumptions about space and time, and also the categorisation of the objects themselves. Conscious experience may also be needed to understand the relationship of one object to another. In terms of mathematics, the natural numbers are an even set, which is conceived of as existing simultaneously. It is possible for human students of mathematics to think of an unbounded set of objects existing simultaneously, but this concept produces a circularity for computers.
There is also the question of understanding geometrically-based proofs, where to understand the proof, it is necessary to conceive a geometric design, as a whole or unit. This involves an argument concerning the situation where human perception is able to immediately see that an arrangement of dots comprises a hexagon, which is seen as a unit, whole or gestalt, although all that exists is a few printed dots, and there is no continuous hexagon printed on the paper. A computer analysis of the dots could generate the angles of relationship between them, but not by itself generate the idea of a geometrical objects such as a hexagon as a single cohesive whole. There needs to be a realisation that the dots at the corners of the hexagon (the only thing actually printed on the paper) belong together, and although something might be programmed in for particular dots, there is no way to generate this for arrangements of dots in general, from present forms of computation. It requires human conceptions about the parts of cohesive wholes belonging together to achieve this. Complex diagrams need to be perceived as integrated gestalt patterns. Therefore the author argues that it is not necessary to accept Penrose’s argument from the Godel theorem, in order to agree with his main conclusion that brains and existing forms of computer are different, and consciousness not possessed by computers is required for some human brain activities.