Computation, hypercomputation, and physical science

Copeland and others have argued that the Church-Turing thesis (CTT) has been widely misunderstood by philosophers and cognitive scientists. In particular, they have claimed that CTT is in principle compatible with the existence of machines that compute functions above the “Turing limit,” and that empirical investigation is needed to determine the “exact membership” of the set of functions that are physically computable. I argue for the following points: (a) It is highly doubtful that philosophers and cognitive scientists have widely misunderstood CTT as alleged.1 In fact, by and large, computability theorists and mathematical logicians understand CTT in the exact same way. (b) That understanding most likely coincides with what Turing and Church had in mind. Even if it does not, an accurate exegesis of Turing and Church need not dictate how today’s working scientists understand the thesis. (c) Even if we grant Copeland’s reading of CTT, an orthodox stronger version of it which he rejects (Gandy’s thesis) follows readily if we only accept a highly plausible necessary condition for what constitutes a deterministic digital computer. Finally, (d) regardless of whether we accept this condition, the prospects for a scientific theory of hypercomputation are exceedingly poor because physical science does not have the wherewithal to investigate computability or to discover its ultimate “limit.”