Intuitionistic Mathematics and Realizability in the Physical World

Intuitionistic mathematics perceives subtle variations in meaning where classical mathematics asserts equivalence, and permits geometrically and computationally motivated axioms that classical mathematics prohibits. It is therefore wellsuited as a logical foundation on which questions about computability in the real world are studied. The realizability interpretation explains the computational content of intuitionistic mathematics, and relates it to classical models of computation, as well as to more speculative ones that push the laws of physics to their limits. Through the realizability interpretation Brouwerian continuity principles and Markovian computability axioms become statements about the computational nature of the physical world. 1 Intuitionistic understanding of truth Constructive mathematics, whose main proponent was Erret Bishop,1 lives at the fringe of mainstream mathematics. It is largely misunderstood by mathematicians, and consequently by physicists as well. Contrary to the popular opinion, constructive mathematics is not poorer but richer in possibilities of mathematical expression than its classical counterpart. It differentiates meaning where classical mathematics asserts equivalence and thrives on geometric and computational intuitions that are banned by the classical doctrine. In this contribution I explore what constructive mathematics and the related realizability interpretation of intuitionistic logic have to offer to those who are interested in real-world computation. If classical and constructive mathematicians just disagreed about what was true, the matter would be resolved easily. Unfortunately they use the same words to mean two different things, which is always an excellent source of confusion. The origin of the schism lies in the criteria for truth, i.e., in what makes a statement true. Speaking vaguely, intuitionistic logic demands positive evidence of truth, while classical logic is 1Bishop’s constructivism is compatible with respect to classical mathematics, and should not be confused with the intuitionism of L.E.J. Brouwer, which assumes principles that are classically false.