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Fred Richman -

is distinguished by the fact that those engaged in it maintain an awareness of their relationship to the objects in the mathematical universe, and are concerned with manipulating those objects rather than merely describing them. Following Hao Wang we may say that is a "mathematics of doing" while classical mathematics is a "mathematics of being." Constructivists feel that the mathematics they do is more meaningful, their theorems more incisive and informative, because of this perspective.

The current resurgence of interest in has two principal sources. One is Errett Bishop's "Foundations of constructive analysis". This book, in which Bishop expounds a constructive philosophy of mathematics and develops several areas of mathematics, using standard notation, from that point of view, can be appreciated by mathematicians who are unschooled in logic and recursive function theory. The other major influence has been the digital computer. The notion of computation in principle has become familiar through computer programming, making the idea that pure mathematics should be based upon computation, should have "numerical meaning", more attractive.

There are several identifiable schools of , all tracing their ancestry, in part, to L. E. J. Brouwer. Each takes its own view of the relationship of the mathematician to the mathematical universe, often based upon a particular ontological commitment. The school of Markov admits only mathematical objects that are, ultimately, finite strings of symbols from a finite alphabet. This results in a list of all partial functions of positive integers, from which follows such nonclassical theorems as that every real valued function on the unit interval is (pointwise) continuous. This list may also be used to produce a counterexample to the Heine-Borel theorem.

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