 The Church-Turing thesis is a thesis about the extent of effective methods, and therein lies its mathematical importance. Putting this another way, the thesis concerns what a human being can achieve when working by rote, with paper and pencil (ignoring contingencies such as boredom, death, or insufficiency of paper). The thesis carries no implication concerning the extent of what machines are capable of achieving (even digital machines acting in accordance with “explicitly stated rules” (Gregory 1987)). Because, among a machine’s repertoire of basic operations, there may be operations that a human being working by rote with paper and pencil cannot perform.