Frege’s reflections on the nature of numbers, presented in his Grundlagen der Arithmetik (1884), provide the guiding theme for this dissertation. His characterization of numbers as logical objects was sharply criticized by Paul Benacerraf (1965), who suggested that since mathematicians (as mathematicians) work within a structural framework, philosophical research on mathematics should preserve this structural aspect of mathematics. Benacerraf’s arguments, as well as earlier works by the Bourbaki group, gave raise to several structuralist research projects, one of the most interesting of which is developed by Stewart Shapiro. We present the ontological and epistemological basis of Shapiro’s socalled ante rem structuralism, as well as some of the debates from the late 19th and early 20th centuries which throw some light on the conceptual presuppositions of Shapiro and Benacerraf. In this connection we emphasize the work of the German mathematician Richard Dedekind and of the French group Nicholas Bourbaki.