[1] That is, logic and mathematics are not considered analytic activities wherein deep properties of objective reality are revealed and applied, but are instead considered the application of internally consistent methods used to realize more complex mental constructs, regardless of their possible independent existence in an objective reality.
For the intuitionist, this is not valid; the refutation of the non-existence does not mean that it is possible to find a construction for the putative object, as is required in order to assert its existence.
The logical calculus preserves justification, rather than truth, across transformations yielding derived propositions.
The term potential infinity refers to a mathematical procedure in which there is an unending series of steps.
The term actual infinity refers to a completed mathematical object which contains an infinite number of elements.
Most modern constructive mathematicians accept the reality of countably infinite sets (however, see Alexander Esenin-Volpin for a counter-example).
Brouwer opened our eyes and made us see how far classical mathematics, nourished by a belief in the 'absolute' that transcends all human possibilities of realization, goes beyond such statements as can claim real meaning and truth founded on evidence.Intuitionism's history can be traced to two controversies in nineteenth century mathematics.
The first of these was the invention of transfinite arithmetic by Georg Cantor and its subsequent rejection by a number of prominent mathematicians including most famously his teacher Leopold Kronecker—a confirmed finitist.
In an appendix to the second volume, Frege acknowledged that one of the axioms of his system did in fact lead to Russell's paradox.
These controversies are strongly linked as the logical methods used by Cantor in proving his results in transfinite arithmetic are essentially the same as those used by Russell in constructing his paradox.
Hence how one chooses to resolve Russell's paradox has direct implications on the status accorded to Cantor's transfinite arithmetic.
In the early twentieth century L. E. J. Brouwer represented the intuitionist position and David Hilbert the formalist position—see van Heijenoort.
Alan Turing considers: "non-constructive systems of logic with which not all the steps in a proof are mechanical, some being intuitive".
[5] Later, Stephen Cole Kleene brought forth a more rational consideration of intuitionism in his Introduction to metamathematics (1952).
[6] Nicolas Gisin is adopting intuitionist mathematics to reinterpret quantum indeterminacy, information theory and the physics of time.