Church turing thesis and non-computability

Notice that the Turing-Church thesis does not entail thesis M; the truth of the Turing-Church thesis is consistent with the falsity of Thesis M (in both its wide and narrow forms). A thesis concerning effective methods - which is to say, concerning procedures of a certain sort that a human being unaided by machinery can carry out - carries no implication concerning the extent of the procedures that machines are capable of carrying out (since, for example, there might be, among a machine's repertoire of atomic operations, operations that no human being who is working effectively is able to perform). The above-mentioned evidence for the Turing-Church thesis is not also evidence for Thesis M.

An interesting question is whether the computation model represented by concrete programming languages is Turing equivalent. While the computation of a real computer is based on finite states and thus not capable to simulate a Turing machine, programming languages themselves do not necessarily have this limitation. Kirner et al., 2009 have shown that among the general-purpose programming languages some are Turing complete while others are not. For example, ANSI C is not Turing-equivalent, as all instantiations of ANSI C (different instantiations are possible as the standard deliberately leaves certain behaviour undefined for legacy reasons) imply a finite-space memory. This is because the size of memory reference data types is accessible inside the language. However, other programming languages like Pascal do not have this feature, which allows them to be Turing complete in principle. It is just Turing complete in principle, as memory allocation in a programming language is allowed to fail, which means the programming language can be Turing complete when ignoring failed memory allocations, but the compiled programs executable on a real computer cannot.

Thesis M is not the only problematic thesis that is linked to the Church-Turing thesis. An error which, unfortunately, is common in modern writing on computability and the brain is to hold that Turing's results somehow entail that the brain, and indeed any biological or physical system whatever, can be simulated by a Turing machine. For example, the entry on Turing in the recent A Companion to the Philosophy of Mind contains the following claims: "we can depend on there being a Turing machine that captures the functional relations of the brain", for so long as "these relations between input and output are functionally well-behaved enough to be describable by ... mathematical relationships ... we know that some specific version of a Turing machine will be able to mimic them" (Guttenplan 1994: 595). Searle writes in a similar fashion:

Church turing thesis and non-computability

church turing thesis and non-computability

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