To get a clearer view of the relation between primary arithmetic and primary algebra consider the following extremely simple algebraic expression:

In this expression the variable name appears as an operand name.  In functional terms, is called an argument name, but it’s best to avoid the potentially confusing connotations of the word argument here, since it also refers in logical discussions to a more or less specific pattern of reasoning.

As we discussed, the algebraic variable name indicates the contemplated absence or presence of any arithmetic expression taking its place in the surrounding template, which expression is proxied well enough by its formal value, and of which values we know but two.  Thus, the given algebraic expression varies between these two choices:

The above selection of arithmetic expressions is what it means to contemplate the absence or presence of the operand in the algebraic expression   But what would it mean to contemplate the absence or presence of the operator in the algebraic expression

That is the question I’ll take up next.

cc: Systems Science • Structural Modeling • Ontolog Forum • Laws of Form • Cybernetics