in this enlarged system there would again appear true statements that were not provable within it. This result highlighted the degree to which mathematics as it existed depended not only on axioms and logical deduction — that is, syntax — but also on the products of what was called mathematical intuition — the source of new concepts and procedures introduced to deal with new problems or problems that were otherwise intractable. In other words, mathematics was progressively extended by constructions based on these intuitions. The intuitions came with semantic content, which G[ö]{}del’s result implicitly suggested could not be eliminated, even in principle.

But there was another problem. G[ö]{}del, like Hilbert, thought of deduction as a mechanical procedure; thus the idea of certainty was closely linked to the idea of a machine operating in a completely predictable manner. Of course at the time the idea was metaphorical; the deductions would actually be carried out by a person (the person was invariably referred to as a computer), but for the results to be reliable, the person would have to follow without error a precise sequence of operations that would yield the desired result; in other words, for each deduction the person would 