## Abstract

Turing’s notion of human computability is exactly right not only for obtaining a negative solution of Hilbert’s *Entscheidungsproblem* that is conclusive, but also for achieving a precise characterization of formal systems that is needed for the general formulation of the incompleteness theorems. The broad intellectual context reaches back to Leibniz and requires a focus on *mechanical* procedures; these procedures are to be carried out by human computers without invoking higher cognitive capacities. The question whether there are strictly broader notions of effectiveness has of course been asked for both cognitive and physical processes. I address this question not in any general way, but rather by focusing on aspects of mathematical reasoning that transcend mechanical procedures.

Section 1 discusses Gödel’s perspective on mechanical computability as articulated in his [193?], where he drew a dramatic conclusion from the undecidability of certain Diophantine propositions, namely, that mathematicians cannot be replaced by machines. That theme is taken up in the Gibbs Lecture of 1951; Gödel argues there in greater detail that the human mind infinitely surpasses the powers of any finite machine. An analysis of the argument is presented in Section 2 under the heading *Beyond calculation*. Section 3 is entitled *Beyond discipline* and gives Turing’s view of intelligent machinery; it is devoted to the seemingly sharp conflict between Gödel’s and Turing’s views on mind. Their deeper disagreement really concerns the nature of machines, and I’ll end with some brief remarks on (supra-) mechanical devices in Section 4.

## Keywords

absolutely unsolvable (undecidable) axiom of infinity Church’s Thesis Diophantine problem finite machine general recursive function mechanical computability objective mathematics subjective mathematics Turing machine## Preview

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