History of constructivism in the 20 th century

In this survey of the history of constructivism, more space has been devoted to early developments (up till ca 1965) than to the work of the last few decades. Not only because most of the concepts and general insights have emerged before 1965, but also for practical reasons: much of the recent work is of a too technical and complicated nature to be described adequately within the limits of this article. Constructivism is a point of view (or an attitude) concerning the methods and objects of mathematics which is normative: not only does it interpret existing mathematics according to certain principles, but it also rejects methods and results not conforming to such principles as unfounded or speculative (the rejection is not always absolute, but sometimes only a matter of degree: a decided preference for constructive concepts and methods). In this sense the various forms of constructivism are all ‘ideological’ in character. Constructivism as a specific viewpoint emerges in the final quarter of the 19th century, and may be regarded as a reaction to the rapidly increasing use of highly abstract concepts and methods of proof in mathematics, a trend exemplified by the works of R. Dedekind and G. Cantor. The mathematics before the last quarter of the 19th century is, from the viewpoint of today, in the main constructive, with the notable exception of geometry, where proof by contradiction was commonly accepted and widely employed. Characteristic for the constructivist trend is the insistence that mathematical objects are to be constructed (mental constructions) or computed; thus theorems asserting the existence of certain objects should by their proofs give us the means of constructing objects whose existence is being asserted. L. Kronecker may be described as the first conscious constructivist. For Kronecker, only the natural numbers were ‘God-given’; all other mathematical objects ought to be explained in terms of natural numbers (at least in algebra). Assertions of existence should be backed up by constructions, and the properties of numbers