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No one succeeded to prove that a function must be recursive in order to be effectively calculable. This is (as Post noted) a "working hypothesis" supported by overwhelming evidence. We don't know of any effectively calculable function which is not recursive, by designing new TMs from existing ones we can obtain new effectively calculable functions from existing ones and TM computability stars in every attempt to understand effective calculability (or these attempts are reducible or equivalent to TM computable functions).
The Turing Machine itself, though abstract, has many "real world" features. It is a blueprint for a computing device with one "ideal" exception: its unbounded memory (the tape is infinite). Despite its hardware appearance (a read/write head which scans a two-dimensional tape inscribed with ones and zeroes, etc.) – it is really a software application, in today's terminology. It carries out instructions, reads and writes, counts and so on. It is an automaton designed to implement an effective or mechanical method of solving functions (determining truth value of propositions). If transition from input to output is deterministic we have a classical automaton – if it is determined by a table of probabilities – we have a probabilistic automaton.
With time and hype, limitations of TMs were forgotten. No one can say that Mind is a TM because no one can prove that it is engaged in solving only recursive functions. We can say that TMs can do whatever digital computers are doing – but not that digital computers are TMs by definition. Maybe they are – maybe they are not. We do not know enough about them and about their future.
Moreover, demand that recursive functions be computable by an UNAIDED human seems to restrict possible equivalents. Inasmuch as computers emulate human computation (Turing did believe so when he helped construct ACE, at time fastest computer in world) – they are TMs. Functions whose values are calculated by AIDED humans with contribution of a computer are still recursive. It is when humans are aided by other kinds of instruments that we have a problem. If we use measuring devices to determine values of a function it does not seem to conform to definition of a recursive function. So, we can generalize and say that functions whose values are calculated by an AIDED human could be recursive, depending on apparatus used and on lack of ingenuity or insight (the latter being, anyhow, a weak, non-rigorous requirement which cannot be formalized).
Sam Vaknin ( http://samvak.tripod.com ) is the author of Malignant Self Love - Narcissism Revisited and After the Rain - How the West Lost the East. He served as a columnist for Central Europe Review, PopMatters, and eBookWeb , and Bellaonline, and as a United Press International (UPI) Senior Business Correspondent. He is the the editor of mental health and Central East Europe categories in The Open Directory and Suite101.