It seems to me that in the program Mathematica, things that look like sets are actually lists ands what are called lists are actually functions. (is this so?) So that's what prompts the question. If there is such a theory, what is it called and where can one read about it, say at the level of Halmos's Naive Set Theory?
There has been attempts to base mathematics exclusively on the concept of function rather than on the concept of set. They don't seem to have gathered any wide following, though.
One example is the map theory of Klaus Grue.