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.