the process by which those results were generated. Turing and Church took a first step toward including the process by formalizing the treatment of the algorithms by which proofs were derived. But this did not seem to them (or others) to be sufficient, because it did not capture the deeply creative nature of mathematics as seen in the continual introduction of new concepts and procedures that established whole new areas of mathematics. We might interpret the journey from Turing algorithm to learning as an implicit recognition of the necessity of natural time in mathematics. On the other hand, while Turing had formalized the proof process in terms of a machine which would have to operate in natural time, the machine was defined in such a way that the results that it produced would be time free. Thus in postulating strategies like chained Turing machines to represent learning, he probably assumed that the results would also be time free. However, if mathematics is understood to include the process by which it is created, it will have to involve natural time, even if the result of that creative process is time free. While Turing spoke of learning, a more appropriate term might 