Science Fair Project Encyclopedia
Minds, Machines and Gödel
Lucas presented the paper in 1959 to the Oxford Philosophical Society . It was first printed in Philosophy, XXXVI, 1961, then reprinted in The Modeling of Mind, Kenneth M. Sayre and Frederick J. Crosson, eds., Notre Dame Press, 1963, and in Minds and Machines, ed. Alan Ross Anderson, Prentice-Hall, 1964, ISBN 0135833930.
Lucas argues that a human mathematician cannot be accurately represented by an algorithmic automaton. Appealing to Gödel's incompleteness theorem, he argues that, for any such automaton, there would be some mathematical formula which it could not prove, but which the human mathematician could both see, and show, to be true. Other philosophers, notably Roger Penrose, have made similar arguments.
Implications: The Church-Turing Thesis
If we accept Lucas' Godelian argument, then not only is a specific form of mechanism false, but so is the Church-Turing thesis, which states that any effectively computable algorithm is Turing-computable. This thesis is generally taken to be true but unprovable; unprovable because it asserts a relationship between algorithm as an intuitive, non-mathematical notion and algorithm as a rigorous, mathematical definition.
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