Constructive Mathematics in Theory and Programming Practice

Abstract

The first part of the paper introduces the varieties of modern constructive mathematics, concentrating on Bishop’s constructive mathematics (BISH). It gives a sketch of both Myhill’s axiomatic system for BISH and a constructive axiomatic development of the real line R. The second part of the paper focusses on the relation between constructive mathematics and programming, with emphasis on Martin-Lof’s theory of types as a formal system for BISH. 1 What is Constructive Mathematics? The story of modern constructive mathematics begins with the publication, in 1907, of L.E.J. Brouwer’s doctoral dissertation Over de Grondslagen der Wiskunde [18], in which he gave the first exposition of his philosophy of intuitionism (a general philosophy, not merely one for mathematics). According to Brouwer, mathematics is a creation of the human mind, and precedes logic: the logic we use in mathematics grows from mathematical practice, and is not some a priori given before mathematical activity can be undertaken. It is not difficult to see how, with this view of mathematics as a strictly creative activity, Brouwer came to the view that the phrase “there exists” should be interpreted strictly and uniquely as “there can be constructed” or, in more modern parlance, “we can compute”. In turn, this interpretation of existence led Brouwer to reject the unbridled use of the Law of Excluded Middle (LEM), P ∨¬P, in mathematical arguments. For example, consider the following statement, the Limited Principle of Omniscience (LPO): ∀a ∈ {0, 1} (a = 0 ∨ a 6= 0) . (1) Here, N = {0, 1, 2, . . .} is the set of natural numbers, {0, 1} is the set of all