Church’s thesis and its relation to the concept of realizability in biology and physics

Abstract

An attempt to characterize the physical realizability of an abstract mapping process in terms of the Turing computability of an associated numerical function is described. Such an approach rests heavily on the validity of Church’s Thesis for physical systems capable of computing numerical functions. This means in effect that one must investigate in what manner Church’s Thesis can be converted into an assertion concerning the nonexistence of a certain class of physical processes (namely, those processes which are capable of calculating the values of numerical functions which are not Turing-computable). A formulation which may be plausible is suggested, and it is then shown that the truth of Church’s Thesis in this, form is closely connected with the “effectiveness” of theoretical descriptions of physical systems. It is shown that the falsity of this form of Church’s Thesis is related to a fundamental incompleteness in the possibility of describing physical systems, much like the incompleteness which Gödel showed to be inherent in axiomatizations of elementary arithmetic. Various implications of these matters are briefly discussed.