From Shadows of the Mind, pp. 64-65.
“What did Goedel’s theorem achieve? It was in 1930 that the brilliant young mathematician Kurt Goedel startled a group of the world’s leading mathematicians and logicians, at a meeting in Koenigsberg, with what was to become his famous theorem. It rapidly became accepted as being a fundamental contribution to the foundations of mathematics – probably the most fundamental ever found – but I shall be arguing that in establishing his theorem, he also initiated a major step forward in the philosophy of mind.
Among the things that Goedel indisputably established was that no formal system of sound mathematical rules of proof can ever suffice, even in principle, to establish all the true propositions of ordinary arithmetic. This is certainly remarkable enough. But a powerful case can also be made that his results showed something more than this, and established that human understanding and insight cannot be reduced to any set of computational rules. For what he appears to have shown is that no such system of rules can ever be sufficient to prove even those propositions of arithmetic whose truth is accessible, in principle to human intuition and insight – whence human intuition and insight cannot be reduced to any set of rules. It will be part of my purpose here to try to convince the reader that Goedel’s theorem indeed shows this, and provides the foundation of my argument that there must be more to human thinking than can ever be achieved by a computer, in the sense that we understand the term ‘computer’ today.”