I prefer the HoTT/univalent foundations approach, which is about equivalences, which seem to me to capture a deep essence of what mathematics is about. Arithmetic starts with the idea that you can make some marks on the paper and manipulate them according to certain rules, and this somehow corresponds to how many sheep are in your field or what have you. Geometry starts with the idea that you can record some angles and then after your fields have flooded you can put the boundary markers back in places that are somehow the same as the places they originally were. Lists, sets and graphs are formal systems with arbitrary rules - but, crucially, they're equivalent to things we care about, and we can lift results about those formal systems (where they're easy to calculate) back along that equivalence to be facts about those real things (where they're useful).