A model is a realization of a mathematical formalism. So ordinary arithmetic is a model of ordinary algebra. That is, the algebra of the integers, the rational numbers, and the real numbers is realized by the arithmetic of the integers, the rational numbers, and the real numbers, respectively. Are there other models of ordinary algebra? Yes. One alternate model in particular is a simple opposite of ordinary arithmetic and deserves the name alternate arithmetic.
The one-to-one correspondence between ordinary arithmetic and alternate arithmetic is as follows:
|Property||Ordinary Arithmetic||Alternate Arithmetic|
What is alternate arithmetic good for? Ordinary arithmetic implicitly assumes beginning with nothing and adding something. So the number N means 0+N. Alternate arithmetic assumes beginning with everything and subtracting something. So the alternate number N means 1/N. That is, ordinary arithmetic is additive and alternate arithmetic is subtractive. The square of opposition in quantification logic presents something similar. None and some form an additive logic. All and not all form a subtractive logic.
Instead of using alternate symbols, we may reinterpret the symbols of ordinary arithmetic. In this way, alternate arithmetic looks exactly like ordinary arithmetic but means something opposite. Or we may simply write the alternate number 1/N as N by abuse of notation.