Physical Computability Theses

The Church-Turing thesis asserts that every effectively computable function is Turing computable. On the other hand, the physical Church-Turing Thesis (PCTT) concerns the computational power of physical systems, regardless of whether these perform effective computations. We distinguish three variants of PCTT – modest, bold and super-bold – and examine some objections to each. We highlight Itamar Pitowsky’s contributions to the formulation of these three variants of PCTT, and discuss his insightful remarks regarding their validity. The distinction between the modest and bold variants was originally advanced by Piccinini (Br J Philos Sci 62:733–769, 2011). The modest variant concerns the behavior of physical computing systems, while the bold variant is about the behavior of physical systems more generally. Both say that this behavior, when formulated in terms of some mathematical function, is Turing computable. We distinguish these two variants from a third – the super-bold variant – concerning decidability questions about the behavior of physical systems. This says, roughly, that every physical aspect of the behavior of physical systems – e.g., stability, periodicity – is decidable (i.e. Turing computable). We then examine some potential challenges to these three variants, drawn from relativity theory, quantum mechanics, and elsewhere. We conclude that all three variants are best viewed as open empirical hypotheses.