Figure 2-8 shows a comparison of conventional group theoretic axioms for arithmetic to those of unit-ensembles. There is structure and function hidden within each notation, so the comparison is not intended to demonstrate conceptual simplicity. Rather the comparison shows the understanding needed in order to conduct the operations of arithmetic in each of these abstract theories.

Although we have been associating structural manipulation with numeric multiplication, the conventional axioms of multiplication and addition have not been used in the development of unit-ensembles. Figure 2-8 also shows a hybrid system in which the axioms of group theory are expressed in the language of unit ensembles. With a little boundary thinking, a more intuitive, physically natural axiom structure for integers and the operations of arithmetic is available. Volume II of Iconic Arithmetic compares the boundary mathematics approach to the currently popular formal definitions of numbers by Frege, Peano and other twentieth century philosophers of arithmetic.

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William Bricken, Iconic Arithmetic Volume I: The Design of Mathematics for Human Understanding (Unary press, 2019), p. 58– .

> Existence is not the comparison of a form to the absence of a form. It is rather a **comparison** between the outside of an empty container and the inside. Outside, each and every container represents *not Nothing*. From the outside we associate empty containers with unity, with that which has no parts. (William Bricken, Iconic Arithmetic Volume I: The Design of Mathematics for Human Understanding (Unary press, 2019), p. 169.)