In 1936 an American (Alonzo Church) and a Briton (Alan M. Turing) published independently (as is often coincidence in science) basics of a new branch in Mathematics (and logic): computability or recursive functions (later to be developed into Automata Theory).
The authors confined themselves to dealing with computations which involved "effective" or "mechanical" methods for finding results (which could also be expressed as solutions (values) to formulae). These methods were so called because they could, in principle, be performed by simple machines (or human-computers or human-calculators, to use Turing's unfortunate phrases). The emphasis was on finiteness: a finite number of instructions, a finite number of symbols in each instruction, a finite number of steps to result. This is why these methods were usable by humans without aid of an apparatus (with exception of pencil and paper as memory aids). Moreover: no insight or ingenuity were allowed to "interfere" or to be part of solution seeking process.
What Church and Turing did was to construct a set of all functions whose values could be obtained by applying effective or mechanical calculation methods. Turing went further down Church's road and designed "Turing Machine" – a machine which can calculate values of all functions whose values can be found using effective or mechanical methods. Thus, program running TM (=Turing Machine in rest of this text) was really an effective or mechanical method. For initiated readers: Church solved decision-problem for propositional calculus and Turing proved that there is no solution to decision problem relating to predicate calculus. Put more simply, it is possible to "prove" truth value (or theorem status) of an expression in propositional calculus – but not in predicate calculus. Later it was shown that many functions (even in number theory itself) were not recursive, meaning that they could not be solved by a Turing Machine.
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).
Quantum mechanics is branch of physics which describes microcosm. It is governed by Schrodinger Equation (SE). This SE is an amalgamation of smaller equations, each with its own space coordinates as variables, each describing a separate physical system. The SE has numerous possible solutions, each pertaining to a possible state of atom in question. These solutions are in form of wavefunctions (which depend, again, on coordinates of systems and on their associated energies). The wavefunction describes probability of a particle (originally, electron) to be inside a small volume of space defined by aforementioned coordinates. This probability is proportional to square of wavefunction. This is a way of saying: "we cannot really predict what will exactly happen to every single particle. However, we can foresee (with a great measure of accuracy) what will happen if to a large population of particles (where will they be found, for instance)."