Defined by relationships?

I noticed that behind the scenes most math is defined in terms of relationships, operations, mapping, and interaction. Even things like the ordinary counting numbers—you can find a definition of a non-negative integer in terms of the characteristic unique to all sets that have that many elements. (That sounds a bit circular, doesn't it?) One hot topic tries to treat vast swaths of math in terms of objects and morphisms.

This has been extremely fruitful.

The same sort of approach of defining things in terms of their operations seems to be able to produce quite complex machines, whether physical or organizational. It is very convenient to define people-roles in terms of their interactions and operations: "We need N of type AAZ and M of type CBZ." The more interchangeable the components are, the more efficiently you can feed and run the system—at least in theory. If everyone is identical except for minor education and training differences, you can feed as many of each people-role as you need into the hopper, and shift them around like checkers. (It helps if they have no personal attachments, so you can ship the components around the world at will.)

But this kind of abstraction is well known to induce some push-back from the unique individuals who find themselves thus categorized. And, funny thing, there turn out to be irreducible differences that make people non-interchangeable for some of those people-roles.

And the apparent circularity of some definitions strongly suggests to me that there needs to be some room for some fundamental objects in mathematics. I haven't run across a good definition of a "geometric point" that isn't circular in some way.

And if you only think of defining things by their transformations and mappings, you don't have much mental space for thinking about things like Forms or aseity. (Does that mindset account for the rise of process theology?) People and things are contingent and largely defined by relationships (just don’t try to abstract too much away!), but that doesn't prove everything is.