I consider the raging issue concerning the extent to which is it possible to rigorously prove (or disprove) that a function is computable if and only if it is recursive. It makes for an interesting case study for the relationship between formal and informal proof, and for the relationship between mathematics and non-mathematical reality. I examine the extent to which Friedrich Waismann's notion of open texture applies to mathematics, using Imre Lakatos's thesis of proofs and refutations, and his historical examples, as a backdrop. The ramifications for the philosophical understanding of Church's thesis are then examined.