Constructivism in mathematics

Constructivism is not a matter of principles: there are no specifically constructive mathematical axioms which all constructivists accept. Even so, it is traditional to view constructivists as insisting, in one way or another, that proofs of crucial existential theorems in mathematics respect constructive existence: that a crucial existential claim which is constructively admissible must afford means for constructing an instance of it which is also admissible. Allegiance to this idea often demands changes in conventional views about mathematical objects, operations and logic, and, hence, demands reworkings of ordinary mathematics along nonclassical lines. Constructive existence may be so interpreted as to require the abrogation of the law of the excluded middle and the adoption of nonstandard laws of constructive logic and mathematics in its place.

There has been great variation in the forms of constructivism, each form distinguished in its interpretation of constructive existence, in its approaches to mathematical ontology and constructive logic, and in the methods chosen to prove theorems, particularly theorems of real analysis. In the twentieth century, Russian constructivism, new constructivism, Brouwerian intuitionism, finitism and predicativism have been the most influential forms of constructivism.