In George Spencer Brown’s Laws of Form the relation between the primary arithmetic and the primary algebra is founded on the idea that a variable name appearing as an operand in an algebraic expression indicates the contemplated absence or presence of any expression in the arithmetic, with the understanding that each appearance of the same variable name indicates the same state of contemplation with respect to the same expression of the arithmetic.
For example, consider the following expression:
We may regard this algebraic expression as a general expression for an infinite set of arithmetic expressions, starting like so:
Now consider what this says about the following algebraic law:
It permits us to understand the algebraic law as saying, in effect, that every one of the arithmetic expressions of the contemplated pattern evaluates to the very same canonical expression as the upshot of that evaluation. This is, as far as I know, just about as close as we can come to a conceptually and ontologically minimal way of understanding the relation between an algebra and its corresponding arithmetic.