Meaning and Information in Constructive Mathematics
Fred Richman · American Mathematical Monthly · 1982
The publication of Errett Bishop's Foundations of Constructive Analysis presented the working mathematician with a viable alternative to the prevailing interpretation of mathematical statements. The interpretation suggested by Bishop is not wholly new; many of the ideas go back to L. E. J. Brouwer in the early part of this century, and have been refined and embellished by the Dutch intuitionists. But these ideas were never presented in a manner that was convincing to the average mathematician. Their exposition was often accompanied by an implication that classical mathematics was at best built on insecure foundations, if not outright false. The call was for salvage operations, and the recommended action included forbidding the use of widely accepted arguments. People who felt perfectly comfortable with the mathematics they were doing were unimpressed by these appeals. They saw no reason to sacrifice their powerful mathematical tools and theories on the altar of some crackpot notion of legitimacy. Bishop showed that one could adopt a thoroughgoing constructive point of view and still do mathematics as it is usually understood. He did this by appropriating standard mathematical symbolism to carry constructive meaning, rather than introducing a specialized notation, and by developing large areas of rather sophisticated mathematics in a constructive manner. His book can be appreciated by mathematicians unfamiliar with logic or recursive function theory, and avoids the more bizarre intuitionistic notions of choice sequence and bar induction. But Bishop-style constructivists also tend to be polemical, and they are met with some of the same hostility and indifference that Brouwer was. Possibly this situation will improve as the constructivists master their trade and as the revolution in mathematical consciousness, brought about by the advent of the high-speed digital computer, spreads. In the meantime it seems appropriate to try to convey to mathematicians, and to those interested in the philosophy of mathematics, some understanding of what constructive mathematics means to its practitioners. In what follows I shall attempt to do this via a case study. Our starting point is the following seemingly trivial theorem: