I'm thinking specifically of Brouwer's theorem that all functions R->R are continuous. I know this isn't a theorem of intuitionist logic, because Boolean logic is a special case, and it's clearly not true in that framework. My question is if there is a nice example of an intuitionist logic in which the theorem is true. I know that intuitionist logics can have interesting interpretations. For example, they might tell me "where" something is true, or maybe "how long" until something is true. In Sheaves in Geometry and Logic, Saunders Mac Lane constructs a topos in which Brouwer's theorem is true. Unfortunately, I don't see a nice interpretation of it -- instead of being built upon a topological space, it's built upon the entire category of topological spaces! Unfortunately, I don't see a nice interpretation of the corresponding intuitionist logic. Saunders Mac Lane mentions that there are lots of topoi in which Brouwer's theorem holds -- maybe the logic associated with one of them have a logic with some sort of natural interpretation? Saunders Mac Lane prefaces the section on this by mentioning that Brower treated real numbers as "free choice sequences" -- a mathematician only has an incomplete information about a real number (which is really a Cauchy sequence). No concrete interpretation of this leaps out at me -- does someone know of an actual concrete intuitionist logic that corresponds to this?