A most interesting book in this context is [1]. It has introductory discussions of logicism, intuitionism, and formalism by Carnap, Heyting, and von Neumann, an

A most interesting book in this context is [1]. It has introductory discussions of logicism, intuitionism, and formalism by Carnap, Heyting, and von Neumann, and interestingly, they were not aware yet of Gödel's results at that point which were published in the same year. After that, Heyting's "Dispute" is a nice exposition of the different view points, and I find myself agreeing with much of the intuitionist point of view, namely that mathematical reasoning cannot be reduced to formalism (but I don't agree that intuitionistic reasoning is the only allowed form of mathematical reasoning). I mean, Gödel's incompleteness result is as hard a proof of this as you will ever get! And again, in a later piece by Paul Bernays, "On platonism in mathematics", Bernays states: 'Is it possible to draw an exact boundary between what is evident, and what is only plausible? I believe that one must answer [this question] negatively'. In other words: Common sense is a necessary part of mathematical reasoning. Fundamentally, there is no way around this. [1] Philosophy of Mathematics, Selected Readings. https://doi.org/10.1017/CBO9781139171519 https://doi.org/10.1017/CBO9781139171519