 Although we have defined addition and multiplication for both ordinals and cardinals, their features make it hard to make sense of subtraction or division. First, the non-commutativity of the ordinal operations, and the triviality of the cardinal operations, makes it hard to define meaningful inverses of these operations. (If it’s possible to find an ordinal or cardinal that can be added or multiplied to a first one to get a second one, then it is usually possible to find infinitely many that can.) But more importantly, the conceptual ideas of counting (whether by order type or by bijection) don’t really allow for negatives or fractions, the way that the conceptual ideas of measuring and geometry do (considered in section  3.2).  Counting involves treating each element as discrete and unique, and there is no way for multiple elements to combine to yield nothing (as subtraction requires) or to yield a unit (as division does). However, measurement (e.g., of distance) involves a sort of structure on the thing being measured so that some measurements can be fractions of others, or can point in the opposite direction, which yields meaningful notions of division and subtraction.