Constructive Trend

in mathematics, a mathematical view that regards the study of constructive processes and constructive objects as the fundamental problem of mathematics.

By the end of the 19th century a nonconstructive, set-theoretic approach appeared in mathematics, which was considerably developed by K. Weierstrass, R. Dedekind, and especially G. Cantor. Its aim was the construction of a theory of sets that was to be the foundation of all mathematics. In this theory, in accordance with Cantor’s statement that “the essence of mathematics consists in its freedom,” great arbitrariness was allowed in introducing sets that were then considered as completed “objects.” At the beginning of the 20th century, however, antinomies were discovered—that is, contradictions that showed that one cannot combine objects into sets in an unrestricted manner. Attempts to overcome the difficulties that had arisen proceeded along the lines of an axiomatization of set theory—that is, set theory was to be converted into an axiomatic discipline similar to geometry. This amounted to a choice of axioms that would provide a foundation for mathematics while avoiding known antinomies.

The first attempt along these lines was undertaken by E. Zermelo, who published his system of axioms for set theory in 1908. The known antinomies of set theory did not occur in Zermelo’s system, although there was no guarantee against the subsequent appearance of contradictions. The problem arose of ensuring that axiomatically constructed set theory be free of contradictions. D. Hilbert posed this problem and attempted to solve it. His basic idea was to completely formalize axiomatic set theory and to treat it as a formal system. The task of establishing the noncontradictory nature of the theory would then be reduced to proving the formal unprovability of formulas of a specified kind. Such a proof would have been a convincing argument for constructive objects, that is, for formal proofs. Thus it would fit within the framework of constructive mathematics.

In 1931, K. Gödel proved that the goal set by Hilbert was unattainable. Nevertheless, the means proposed by Hilbert is of great interest. This means is metamathematics, the constructive science of formal proofs, a branch of constructive mathematics. Hilbert’s program can be characterized as an unsuccessful at-tempt to establish set-theoretic mathematics on the basis of constructive mathematics, whose reliability he did not doubt. Hilbert himself must be considered one of the founders of constructive mathematics.

The constructive trend can be considered as a branch of intuitionism. Intuitionism was founded by L. E. J. Brouwer, whose program consists in investigating mental (mathematical) constructions. The closeness of the constructive trend to intuitionism is apparent in the way in which both disciplines view disjunctions and existence theorems, as well as in their treatment of the law of the excluded middle. The principal difference between constructivists and intuitionists is that constructivists, un-like intuitionists, do not think of their constructions as a purely mental pursuit. Moreover, intuitionists argue about certain “freely formed sequences” and consider the continuum as a “medium of free formation” and thus involve nonconstructive objects. The constructive trend in mathematics has led to the formation of a separate discipline, constructive mathematics.