 For most of its life, Church’s view—that  CT1  is not subject to mathematical proof, or any sort of mathematical treatment at all—has prevailed. But some such as Shoenfield, (and Church himself), have observed that  CT2  is subject to intuitive proof despite the lack of a mathematically precise definition of “computable” (Shoenfield 1967: 120). Shoenfield’s view of CT2 has been seconded by Kripke. All that is needed, Kripke says, is “just a direct argument” (2013). Moreover, Kripke and others (e.g., Mendelson 1990) have challenged the received view that CT1 is not amenable to mathematical treatment. Accordingly, the discussion of CT1 divides into two parts: The first part is the evidence usually adduced for CT1 on the assumption that it is a definition or hypothesis not amenable to proof; and the second part is the evidence derived from recent attempts to support CT1 by mathematical arguments. The first sort of evidence we may term “external” and the second “internal”. The emphasis in what follows is on Church’s role in these developments.