Kleene re his  Introduction to Metamathematics would disagree that metamathematics isnt a topic for mathematics But Minsky  seems to agree with you never mind hes not too convincing with his intuitive argument using appeals to intuitive arguments In his discussion of an effective procedure aka 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 eg 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   they amount in my view to a satisfactory refutation of many such objections p   Minsky goes on to state One cannot expect to prove Turings his name for the ChurchTuring 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  Boolos and Jeffrey   are more constructive Although this thesis Churchs 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 effectiveTo 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 Churchs 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   Thus the sun always rises because the sun has always risen Truth by enumeration animal logic Bertrand Russell would have a fieldday 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 isnt mechanical then it must be intuitive Say what that the world of computation is divided into two mutuallyexclusive sets the intuitive and the mechanical Sure sounds like dualism to me  intuitive minds with souls vs brainsmade 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 th dimension into our D spacetime 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 dont know I think they are the same  wvbailey