We know from Gödel's incompleteness theorems that any significantly powerful axiomatic system is either inconsistent or incomplete, and we even have a few examples where the ZFC comes short. As you try to fill in all the missing patches so to speak, you need more and more axioms.
In physics (relativity and quantum mechanics for example), it's been found that a relational approach (seems) necessary to explain observation. Grete Hermann, for example, introduced the relational concept for quantum mechanics where state and quantity are only meaningful under the context of relations between systems,and where the physical nature of a QM universe is understood as the set of all relations.
Has anyone proposed something of this sort as foundation for mathematics?