>>9826216ask euler and gauss or almost any mathematician before the 20th century
sure axioms are good and necessary but the art of mathematics didn't arise because galois sat down and said, "let me arbitrarily call something a field that which has properties suspiciously resembling our ordinary numbers and voila", those categorizations came after the discovery of fields
To continue the example of algebraic structures, people were studying symmetries of "groups" and the properties of certain kinds of "fields" long before someone sat down and said "this is what a field is". More fundamentally, mathematicians do a lot of mathematics with little to no explicit involvement of ZFC explicitly, most mathematicians probably don't even care
