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On mind & Turing's machines WILFRIED SIEG

Summary: On mind & Turing's machines
Carnegie Mellon University, Pittsburgh, PA, USA
E-mail: ws15@andrew.cmu.edu
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 GoĻ del's perspective on mechanical com-
putability 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; GoĻ del
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


Source: Andrews, Peter B. - Department of Mathematical Sciences, Carnegie Mellon University


Collections: Mathematics