TITLE: Applications of intuitionistic logic
QUESTION [9 upvotes]: Can anybody enlighten me about the applications of intuitionistic logic? I am familiar with this system only by G.Takeuti's book, where it is described as one of the examples of axiomatic systems of logic. Is it possible to explain to a non-specialist why this system deserves studying?
I suspect that this is easier to explain on the example of propositional intuitionistic calculus, so if somebody could cast a light on this, this would be especially valuable.

REPLY [10 votes]: The internal logic of a topos is inherently intuitionistic, not classical. This means that if we want to "prove a fact in all topoi (satisfying some conditions)," we should generally look for constructive arguments.
Even if we don't care about the internal structure of a topos, this is still worth thinking about since we may then be able to "de-toposify" the result we get to arrive at a purely classical fact (usually involving sheaves somehow). For example, see this answer of Ingo Blenschmidt where he shows roughly that a particular constructive argument about rings yields the classical fact that for $X$ a reduced scheme and $\mathcal{F}$ an $\mathcal{O}_X$-module of locally finite type, $\mathcal{F}$ is locally free iff its rank is constant.