The answer you're looking for is in category theory. For a long time people ad hoc recognized that "products" and "sums" exist in many different domains, but th

The answer you're looking for is in category theory. For a long time people ad hoc recognized that "products" and "sums" exist in many different domains, but there wasn't a universal method to construct them. Category theory does just that by treating all of these things as special kinds of categories and reading off universal properties which link rings, algebraic data types, logic, sets, topological spaces, etc. etc.