Implicitly Defining Mathematical Terms

This chapter expounds the view that mathematical implicit definitions, i.e., systems of axioms as well as abstraction principles, underpin some aspects of pluralism in mathematics. Firstly, it takes in account Horwich’s “uniqueness problem” concerning implicit definitions and examines it particularly in the case of mathematical systems of axioms as well as abstraction principles. Meanings accrued to the definienda are not uniquely determined, and multiple systems of objects may satisfy the system of axioms or the abstraction principle in question. Multiplicity in ontology is one of the aspects of pluralism which characterize the ways in which we implicitly define mathematical terms. Systems of axioms as well as abstraction principles are taken to play a foundational role in building up mathematical theories. However, the chapter turns the focus of interest to a structural view of mathematical implicit definitions. A structural account of mathematical implicit definitions supports a view of modest pluralism. A structuralist is not interested in foundations. In this view, systems of axioms as well as abstraction principles implicitly define structures. Shapiro’s ante rem structuralism involves implicit definitions in procedures of disclosure of certain mathematical structures. Structures have an ontological status independent of their exemplary cases. In the penultimate section, the chapter takes into account the case of satisfiable abstraction principles. Such principles are taken to implicitly define certain structures independently of the “virtues” or “defects” that characterize the principles themselves. In particular, satisfiable abstraction principles that preclude each other, e.g., Hume’s Principle and Nuisance Principle, describe alternative structures with different exemplary systems of objects.