Constructive truth in practice
Douglas Bridges · 1998
Abstract In this chapter, which has evolved over the last ten years to what I hope will be its perfect Platonic form, I shall first discuss those features of constructive mathematics that distinguish it from its traditional, or classical, counterpart, and then illustrate the practice of that distinction in aspects of complex analysis whose classical treatment ought to be familiar to a beginning graduate student of pure mathematics. My experience shows that a typical mathematician believes that constructive mathematics is characterized by either a rejection of the law of excluded middle from logic or else a rejection of the full axiom of choice from set theory. In fact, although some authors (notably Richman (1996)) seem to endorse the former characterization, the pioneers of constructivism—Brouwer, Markov, Bishop-all arrived at their rejection of the law of excluded middle (and hence, implicitly, at a rejection of the axiom of choice—see later in this chapter) as a consequence of their insistence that the phrase there exists be interpreted strictly as we can construct.