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On the one hand, there is Voevodsky's subtle and beautiful univalence axiom. Fuccillo charters cruise ship for customers' 5-day Bahamas trip and other sales gimmicks see story Posted by.
On the other hand, we have higher inductive types, which provide direct, logical descriptions of some of the basic spaces and constructions of homotopy theory: The univalence axiom implies, in particular, that isomorphic structures can be identified, a principle that mathematicians have been happily using on workdays, despite its incompatibility with the "official" doctrines of conventional foundations. We believe that univalent foundations will eventually become a viable alternative to set theory as the "implicit foundation" for the unformalized mathematics done by most mathematicians.
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