Errett Bishop 1928--1983

"It is no exaggeration to say that a straightforward realistic approach to mathematics has yet to be tried. It is time to make the attempt." With these words Errett Bishop embarked on the task of transforming constructive mathematics from a curiosity, of interest only to logicians, philosophers, and mystics, to the preferred method of thinking for mathematicians in general.

As Hao Wang put it, classical mathematics is a "mathematics of being" while constructive mathematics is a "mathematics of doing." Bishop's central insight was that "to show that an object exists is to give a finite routine for finding it"; the most conspicuous consequence of this is the rejection of what Bishop called "principles of omniscience" such as the ability to decide whether a sequence of integers consists entirely of zeros. Bishop refused to elaborate on what he meant by a finite routine. He pointed to the circularity of defining existence in terms of finite routines, and defining finite routines in terms of the existence of a step in a computational process at which that process halts. Uspenskii and Semenov summarized this position when they wrote: "The concept of algorithm like that of set and of natural number is such a fundamental concept that it cannot be explained through other concepts and should be regarded as [an] undefinable one."

Amid the variety of constructive schools of mathematics, Bishop's is unique in systematically infusing standard mathematical notation and terminology with more meaning, rather than devising special notation to accomodate the finer distinctions required by a constructive approach. This technical device, which makes constructive mathematics more accessible to the practicing mathematician, reflects two fundamental tenets:

Constructive mathematics is not a sideshow, but rather a rival philosophy of how the whole of mathematics should be viewed.
Classical mathematics is our heritage, and while many of its claims are rejected, none are denied.

The first point distinguishes Bishop's approach from classical recursive function theory in which computable functions are regarded as a subclass of a more general class of functions. The second point distinguishes Bishop's approach from intuitionism and Russian constructivism. Both of the latter schools, for example, prove that all functions are continuous. Bishop resisted the temptation to prove classically false theorems, the dramatic feature of other constructive approaches that assures their acceptance as something genuinely different. His contention was that theorems in constructive mathematics are more meaningful than their classical counterparts, not that they simply have a different meaning. Thus, passing from classical to constructive mathematics is a process of generalization akin to passing from the study of abelian groups or Hilbert spaces to the study of nilpotent groups or Banach spaces respectively.

Although it is too early to tell what the ultimate effect of Bishop's ambitious undertaking will be, he has inspired the formation of a community of researchers in constructive mathematics, consisting of mathematicians who, like himself, had no particular background or interest in foundational questions as such. Through the efforts of Bishop, and of the constructive community that he brought into being, his constructive point of view has been applied to such major areas of mathematics as algebraic number theory, algebraic topology, functional analysis, probability and measure theory, differential equations, and commutative algebra.

Last modified May 22, 1997