Carnap introduced his notion of explication to arrive at concepts that are precise enough to progress science. This means that one has to start from less precise concepts---an understanding of explication inapplicable to mathematics. We show that within mathematics explications are widespread. We discuss examples such as topological spaces, prime numbers, and Imre Lakatos's case-study of the polyhedron. We argue that such explications are needed for the modern mathematical endeavour which takes foundational work serious. E.g., automated theorem proving necessitates the expansion of functions and introduction of conventions which are kept. There also are foundational aspects that might make some type system desirable, also enforcing the distinction between addition on different numbers. Such cases appear to be basic, but they either render advanced topics false or enforce being explicit about several of the implicit assumptions.

We argue that these developments correlate with arguments of productive ambiguity within mathematics. Carnap sees potential explications as \emph{proposals} which are not themselves truth-apt. However, as Carnapian explication is not applicable to such cases, we argue that these mathematical case-studies also motivate a different notion of explication which covers the mathematical cases by allowing precise mathematical concepts to be explicated.