Gödel’s conclusions bear on the question whether a calculating machine can be constructed that would match the human brain in mathematical intelligence. Today’s calculating machines have a fixed set of directives built into them: these directives correspond to the fixed rules of inference of formalized axiomatic procedure. [But] the resources of the human intellect have not been, and cannot be, fully formalized, and…new principles of demonstration forever await invention and discovery. We have seen that mathematical propositions which cannot be established by formal deduction from a given set of axioms may, nevertheless. be established by “informal” meta-mathematical reasoning. It would be irresponsible to claim that these formally indemonstrable truths established by meta-mathematical arguments are based on nothing better than bare appeals to intuition.

Nor do the inherent limitations of calculating machines imply that we cannot hope to explain living matter and human reason in physical and chemical terms. The possibility of such explanations is neither precluded nor affirmed by Gödel’s incompleteness theorem. The theorem does indicate that the structure and power of the human mind are far more complex and subtle than any non-living machine yet envisaged. Gödel’s own work is a remarkable example of such complexity and subtlety. It is an occasion, not for dejection, but for a renewed appreciation of the powers of creative reason.