Constructivism, computability, and physical theories

This dissertation is an investigation into the degree to which the mathematics used in physical theories can be constructivized. The techniques of recursive function theory and classical logic are used to separate out the algorithmic content of mathematical theories rather than attempting to reformulate them in terms of "intuitionistic" logic. The guiding question is: are there experimentally testable predictions in physics which are not computable from the data?;The nature of Church's thesis, that the class of effectively calculable functions on the natural numbers is identical to the class of general recursive functions, is discussed. It is argued that this thesis is an example of an explication of the very notion of an effectively calculable function. This is contrary to a view of the thesis as a hypothesis about the limitations of the human mind.;The extension to functions of a real variable of the notion of effective calculability is discussed, and it is argued that a function of a real variable must be continuous in order to be considered effectively calculable (herein: the Borel-Brouwer thesis). The relation between continuity and computability is significant for the problem at hand. The results of a well-designed experiment do not depend critically upon the precise values of the relevant parameters. Accordingly, if the solution to a problem in mathematical physics depends discontinuously upon the data, it cannot be regarded as an experimentally testable prediction of the theory. The principle that the testable predictions of a physical theory cannot be singular is known as the principle of regularity. This principle is significant, because (by the Borel-Brouwer thesis) discontinuities generate non-computability, but (by the principle of regularity) they also disqualify a prediction from being experimentally testable.;A mathematical framework is set up for discussing computability in physical theories. This framework is then applied to the case of quantum mechanics. It is found that, due to the use of unbounded (hence discontinuous) operators in the theory, noncomputable objects appear, but predictions which satisfy the principle of regularity are nevertheless computable functions of the data.