This answer will focus on two references that may be useful to understand the issues dividing realism and anti-realism.
Rather than looking at this from the perspective of logic it may be more useful to see it from the perspective of various metaphysical disputes such as platonism in mathematics versus intuitionism, realism of the physical world versus ...
Husserlian concept of time and temporality is three-tiered:
Internal time consciousness
And contrary to what you might think they arise from each other in 1 to 2 to 3 fashion, where the fundamental base for a worldly/global time and all time is the internal time the consciousness of the individual.
It is described ...