  Doing mathematics in a categorical framework is almost always radically different from doing it in a set-theoretical framework (the exception being working with the internal language of a Boolean topos; whenever the topos is not Boolean, then the main difference lies in the fact that the logic is intuitionistic). Hence, as is often the case when a different conceptual framework is adopted, many basic issues regarding the nature of the objects studied, the nature of the knowledge involved, and the nature of the methods used have to be reevaluated. We will take up these three aspects in turn.