Intuitionism

Constructivism is often identified with intuitionism, although intuitionism is only one constructivist program.

Intuitionism and infinity Among the different formulations of intuitionism, there are several different positions on the meaning and reality of infinity.

A peculiarity of intuitionism is that it rejects some mathematical ideas considered valid according to other definitions.

Indeed, Hilbert would lose his "gifted pupil" Weyl to intuitionism — "Hilbert was disturbed by his former student's fascination with the ideas of Brouwer, which aroused in Hilbert the memory of Kronecker".

In intuitionism, the term "explicit construction" is not cleanly defined, and that has led to criticisms.

In particular, while other philosophies of mathematics allow objects that can be proved to exist even though they cannot be constructed, intuitionism allows only mathematical objects that one can actually construct.

Intuitionism was created, in part, as a reaction to Cantor's set theory.

Intuitionistic logic was developed by Heyting to study Brouwer's program of intuitionism, in which Brouwer himself avoided formalization.

The fixed-point theorem is, as he originally stated it, false in intuitionism, and Brouwer disavowed it, proposing instead alternative versions to be constructively proven.

The paradoxes, Kleene discusses Intuitionism and Formalism in depth.

There is thus an asymmetry between a positive and negative statement in intuitionism.