Intuitionism

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

Brouwer rejected the concept of actual infinity, but admitted the idea of potential infinity.

"According to Weyl 1946, 'Brouwer made it clear, as I think beyond any doubt, that there is no evidence supporting the belief in the existential character of the totality of all natural numbers ... the sequence of numbers which grows beyond any stage already reached by passing to the next number, is a manifold of possibilities open towards infinity; it remains forever in the status of creation, but is not a closed realm of things existing in themselves. That we blindly converted one into the other is the true source of our difficulties, including the antinomies – a source of more fundamental nature than Russell's vicious circle principle indicated. 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." (Kleene (1952): Introduction to Metamathematics, p. 48-49)

Intuitionism's history can be traced to two controversies in nineteenth century mathematics.

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.

Less readable than Goldstein but, in Chapter III Excursis, Dawson gives an excellent "A Capsule History of the Development of Logic to 1928".
Although not directly germane, in his (1923) Brouwer uses certain words defined in this paper.
From van Heijenoort's commentary it is unclear whether or not Herbrand was a true "intuitionist"; Gödel (1963) asserted that indeed "...Herbrand was an intuitionist". But van Heijenoort says Herbrand's conception was "on the whole much closer to that of Hilbert's word 'finitary' ('finit') that to "intuitionistic" as applied to Brouwer's doctrine".
Definitive biography of Hilbert places his "Program" in historical context together with the subsequent fighting, sometimes rancorous, between the Intuitionists and the Formalists.
In a style more of Principia Mathematica – many symbols, some antique, some from German script. Very good discussions of intuitionism in the following locations: pages 51–58 in Section 4 Many Valued Logics, Modal Logics, Intuitionism; pages 69–73 Chapter III The Logic of Propostional Functions Section 1 Informal Introduction; and p. 146-151 Section 7 the Axiom of Choice.
A secondary reference for specialists: Markov opined that "The entire significance for mathematics of rendering more precise the concept of algorithm emerges, however, in connection with the problem of a constructive foundation for mathematics....[p. 3, italics added.] Markov believed that further applications of his work "merit a special book, which the author hopes to write in the future" (p. 3). Sadly, said work apparently never appeared.