 The first part is closely tied to Dedekind’s employment of set-theoretic tools and techniques. He uses these to construct new mathematical objects (the natural and real numbers, ideals, modules, etc.) or whole classes of such objects (algebraic number fields, rings, lattices, etc.), usually in terms of infinite sets. Closely related but even more characteristic, both in foundational and other contexts, is the following: These sets are then endowed with general structural features (order relations, arithmetic and other operations, etc.); and the resulting systems are studied in terms of corresponding higher-level concepts (continuity for the real number system, simple infinity for the natural numbers, unique factorization in algebraic number fields and function fields, etc.). Once more, philosophically relevant is not just that this procedure is infinitary (his acceptance of actual infinities) and non-constructive (the added features are not necessarily grounded in algorithms). A whole variety of relational systems, including many new ones, can now be investigated; and this brings with it a significant extension of the subject matter of mathematics. Mathematics is no longer tied to what is given intuitively or even empirically (concrete numbers of things, geometric magnitudes, physical processes that are observable, etc.); it has become the study of relational systems much more generally (based on a new “logic of relations”,  as Russell and Cassirer will later put it).