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 can not 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 can not 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