The claim that machines are
intelligent is denoted by AI, which stands for Artificial
Intelligence. “The objectives of AI
are to imitate by means of machines, normally electronic ones, as much of human
mental activity as possible, and perhaps eventually to improve upon human
abilities in these respects.” (Penrose [1989] p.14) The claim that machines can “think” as we
humans do, are “intelligent” and “conscious” just as we are, is
denoted by strong AI. “According
to strong AI,” as Penrose explains, “... mental qualities of a sort can be
attributed to the logical functioning of any computation device, even
the very simplest mechanical ones, such as a thermostat.” (Penrose [1989] p.21 By effectively
computable or recursive is meant any process that is performed by a
digital machine. The aim of this paper
is to refute the doctrine of strong AI.
The methodology of this paper is to achieve this by developing further
mathematical insight into the nature of proof to show that no effectively
computable process can match it.
In proofs we encounter axioms and rules. Traditionally, the axioms were regarded as
primitive propositions apprehended by “the mind” by means of “intuition”. Such a philosophy of mathematics does not
cohere with Strong AI, which is consistent with a view of mathematics known as formalism. Formalism maintains that axioms may be
mechanically formed in a language whose syntax can be recursively
enumerated as a list; the rules are transformations of expressions of
the formal language into other expressions.
It is assumed that the operations of all rules and axioms thus given can
be physically simulated in a machine.
According to
formalism the central concept in mathematics is that of a formal system. Such a system is defined by a set of conventions
... we start with a list of elementary propositions, called axioms,
which are true by definition, and then give rules of procedure by means
of which further elementary theorems are derived. The proof of an elementary proposition then
consists simply in showing that it satisfies the recursive definition of
elementary theorem. (Curry [1954], p. 203)
I shall take this quotation as a definition of formalism. Sometimes with formalist theories reference is made to semantics and in particular to model theory. Yet here there is no primitive notion deployed of meaning, intension or concept. The relation explored in effective formal semantics between a language and its model is conceived as another effective relation between one structure and another. In the formalist conception of semantics we don’t break out of the formalism into anything other than more formalism. In the mathematics of formal systems there is only syntax: semantics is given by syntax. Curry, a nominalist, regards interpretation of the language of mathematics as irrelevant: “Why not abolish the object language altogether and understand that the tokens are objects which we can take as symbols if we want to?” (Curry [1954], p. 204) Such nominalism is an essential aspect of formalism and proponents of strong AI must adopt it.
This is part of a longer thesis advancing a refutation of
strong AI. To download the thesis visit:
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For an
introduction to the work as a whole visit: -
Introduction to Poincare’s
thesis by Peter Fekete