The definition of bijection is much more interesting than comparing cardinals. Many everyday use cases where (structure-preserving) bijections make it clear that two apriori different objects can be treated similarly.
More generally, mathematics is experimental not just in the sense that it can be used to make physical predictions, but also (probably more importantly) in that definitions are "experiments" whose outcome is judged by their usefulness.
More generally, mathematics is experimental not just in the sense that it can be used to make physical predictions, but also (probably more importantly) in that definitions are "experiments" whose outcome is judged by their usefulness.