From a Brouwerian Point of View

Abstract

There are a number of loosely related notions and schools of constructivism in mathematics; in some cases there is only an attempt to capture certain constructive notions and procedures in the existing body of mathematics, in other, more fundamental, cases the object is to reconstruct mathematics as a whole within the frame work of a constructivistic philosophy. It is almost a platitude to state that constructivism has always been around in mathematics, and indeed, a, say, eighteenth century mathematician would have accepted the constructivist claim of his twentieth century colleague of the mild variety, i.e. the practical non-dogmatic practitioner, as self-evident and rather commonplace. The issue of constructivism in mathematics only became urgent after the discovery of abstract, non-effective techniques and notions. The watershed is David Hilbert’s famous solution of Gordan’s problem of the finite basis of a family of invariants (1888). This result, which stated that a certain finite collection existed, without giving the least clue how to compute the elements or their number, opened up a new era of abstract mathematics, which was viewed by the majority of the mathematical community as the promised land of generality without the curse of messy details (and hard labour). The paradigm of general non-constructive mathematics became the socalled axiom of choice, a principle which asserted the existence of certain objects – choice functions, that performed literally the task of choosing objects where a human being would confess perplexity. In a simple form the axiom of choice says that, given a family of non-empty sets, there is a function which chooses an element from each set of the family. If we think of a function as an instruction how to produce certain outputs, then this axiom postulates the choice function much as a playwright introduce a deus ex machina: it is there, but we cannot see how. So, either the axiom