On understanding understanding

Abstract

It is argued, by use of specific examples, that mathematical understanding is something which cannot be modelled in terms of entirely computational procedures. Our conception of a natural number (a non‐negative integer: 0, 1, 2, 3,…) is something which goes beyond any formulation in terms of computational rules. Our ability to perceive the properties of natural numbers depends upon our awareness, and represents just one of the many ways in which awareness provides an essential ingredient to our ability to understand. There is no bar to the quality of understanding being the result of natural selection, but only so long as the physical laws contain a non‐computational ingredient.