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Turing's programme, the idea that intelligence can be modelled computationally, is set in the context of a parallel between certain elements from metamathematics and Popper's schema for the evolution of knowledge. The parallel is developed at both the formal level, where it hinges on the recursive structuring of Popper's schema, and at the contentual level, where a few key issues common to both epistemology and metamathematics are briefly discussed. In light of this connection Popper's principle of transference, akin to Turing's belief in the relevance of the theory of computation for modelling psychological functions, is widened into the extended principle of transference. Thus Turing's programme gains a solid epistemological footing.