Beginning in the nineteenth century, mathematics' traditional domains of 'number and figure' became vigorously displaced by altered settings in which former verities became discarded as no longer sacrosanct. And these innovative recastings appeared everywhere, not merely within the familiar realm of the non-Euclidean geometries. How can mathematics retain its traditional status as a repository of necessary truth in the light of these revisions? The purpose of this Element is to provide a sketch of this developmental history.