2005, Andrej Bauer, Mathematics and Computation: The Law of Excluded MiddleWhat constructive mathematicians know is that there are mathematical universes in which sets are like topological spaces and properties are like open sets. In fact, these universes are well-known to classical mathematicans (they are called toposes), but they look at them from “the outsideâ€. When we consider what mathematicians who live in such a universe see, we discover many fascinating kinds of mathematics, which tend to be constructive. The universe of classical mathematics is special because in it all sets are like discrete topological spaces. In fact, one way of understanding LEM is “all spaces/sets are discreteâ€. Is this really such a smart thing to assume? If for no other reason, LEM should be abandonded because it is quite customary to consider “continuous†and “discrete†domains in applications in computer science and physics. So what gives mathematicians the idea that all domains are discrete?
