":Kleene (re his (1952) Introduction to Metamathematics) would disagree that metamathematics isn't a topic for mathematics. But Minsky (1967) seems to agree with you, never mind he's not too convincing with his ""intuitive argument"" using appeals to ""intuitive arguments"". In his discussion of ""an effective procedure a.k.a. algorithm"" (his definition, in fact) he says the matter is subjective:
""Different people may not agree on whether a certain procedure should be called effective. Perhaps there are processes, we might suppose, which simply cannot be described in any formal language, but which can nevertheless be carried out, e.g., by minds. One might even argue that those important, but still mysterious, functions which make minds superior to all presently known mechanical processes must, by their intuitive nature, escape any systematic description.

""Turing discusses some of these issues in his brilliant article, 'Computing Machines and Intelligence' [1950] .. they amount, in my view, to a satisfactory refutation of many such objections."" (p. 107)

Minsky goes on to state:
""One cannot expect to prove Turing's [his name for the Church-Turing] thesis, since the term 'naturally' relates rather to human dispositions than to any precisely defined quality of a process. Support must come from intuitive arguments..."" (p.108)

Boolos and Jeffrey (1974, 1999) are more constructive:
""Although this thesis ('Church's thesis') is unprovable, it is refutable, if false. For if it is false, there will be some function which is computable in the intuitive sense, but not in our sense. To show that such a function is computable in the intuitive sense, it would be sufficient to give instructions for computing its values for all (arbitrary) arguments, and to see that these instructions are indeed effective....To show that such a function is not computable in our sense, one would have to survey all possible Turing machines and verify (by giving a suitable proof) that none of them compute that function.... Then if Church's thesis is false, it is refutable by finding a counterexample; and the more experience of computation we have without finding a counterexample, the better confirmed the thesis becomes."" (italics in original, p. 20)

""Thus, the sun always rises because the sun has always risen."" Truth by enumeration: animal logic. Bertrand Russell would have a field-day. At least Boolos and Jeffrey offer us a ""road map"" of how to go about approaching the problem. But they fall into the same trap: they are asserting that if a computational procedure isn't mechanical then it must be intuitive. Say what? that the world of computation is divided into two mutually-exclusive sets: ""the intuitive"" and ""the mechanical""? Sure sounds like dualism to me ... intuitive 'minds' with 'souls' vs 'brains'made of gooey mechanical stuff ... interesting... most (but not all) philosphers of mind have given up on 'souls' ... maybe here is how one disproves or proves dualism? To wit: 

What would be interesting would be if someone could prove that the truth of the Thesis is formally undecidable in any construct whatever, ""intuitive or otherwise"" forever unknowable to man or machine or God. One little [dualism] difficulty: what the hell does 'intuitive' really mean? [maybe that God is punching holes through the 5th dimension into our 4-D space-time, and is seeding minds with ""notions"" (but not mechanically altering them in the process)?] 

I would call this sort of discussion a 'foundations of mathematics' debate, and therefore an open topic in some domain or other. Whether its 'philosophy' or 'foundations of mathematics' I don't know. I think they are the same.  wvbailey 

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