So far I have discussed expansions of concepts that do not involve the addition of new objects. Now I will examine whether it is possible to generalize this discussion to external expansions, where new objects are involved. The point I suggest here is basically a formalist one, claiming that words such as “-3” and “√(-1)” can play a crucial role in external expansions, but this approach is given a new sense here: the important external expansions such as negative and complex numbers are viewed as the result of stretching the identity relation. I shall then move to the debate between Frege and the formalists, trying to find a way to retain the intuitions of both sides. Corresponding to the transition from words to objects that occurs in external expansions, there is a subtler transition in which constraints on potential entities are transformed into axioms on a well-defined realm of objects. This is the subject of the third section of this chapter.

This leaves us with the question of where to start. Do we assume, like Kronecker, that the natural numbers are at the basis of all expansions of numbers? It seems to me that this assumption is not necessary. One way that we can begin, which is probably not the only possible way, is to construct the ordinal numbers and the relations between them, hoping that further expansions will take us to richer structures.