Two concepts F,G are equinumerous if there exists a one-to-one correspondence between the objects that fall under F and G.
Equinumerosity is one the most fundamental building blocks of Gottlob Frege's attempt to reduce mathematics to logic. Reading the definition, though, this struck me:
Isn't it problematic that the concept of one-to-one correspondence is used prior to defining the notion of maps-functions? Or is it preassumed that is a concept familiar through empirical means?
If we assume the first, then that would imply we use mathematical entities (not rigorously defined) to explore the nature of mathematics and this seems to me like circular reasoning.
On the other hand, if we assume the last, this would imply that the whole construction of mathematics has some empirical basis, which seem even more counterintuitive.
Could you help me clarify this confusion?