Tim, I think I'm starting to understand what you're saying. However, it
still seems that anything you can do with intuitionistic logic, toposes,
etc., can also be done with classical logic and set theory. (I'm not
confident about this, but see my previous post in reponse to Bruno.) Maybe
it's not as convenient or "natural" in some cases (similar to how modal
logic can be more convenient than explicitly quantifying over possible
worlds even when they are equivalent), but if one is not already familiar
with intuitionistic logic and category theory, is it really worth the
trouble to learn them?

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For example, posets can certainly be studied and understood using
classical logic. How much does intuitionistic logic buy you here?