Iconic Arithmetic Volume I begins with marks drawn in sand, with pebbles held in hand, with notches carved in bone, with unity standing alone. [⇒ Parallel Poetry]

The internal evolution of mathematical science has, in spite of appearance, brought about a closer unity among its different parts...and which has led to what is generally known as the “axiomatic method.”8

21. Difference is not a quantity: G. Bateson (1991) A Sacred Unity p.219.

A fusion is different than a sum because the thirteen stripes are no longer distinguished as separate once they are joined together as the flag. The stripes represent a unity, the Union of the original states.

Here is logician Gottlob Frege’s definition. A fusion consists of objects; it is an aggregate, a collective unity, of them; if so it must vanish when these objects vanish...there can be no empty fusion.14

# Significantly fewer concepts

Perhaps the most challenging new idea is that many of the conventional concepts within arithmetic and algebra are unnecessary in order to conduct the conventional operations of arithmetic and algebra. Arithmetic can be far simpler than what we have been taught. The essential concepts of James algebra do not include zero, commutativity, associativity, arity, one-step-at-a-time processes, and symbolic strings. Instead there is a deep visceral unity across counting, addition, multiplication, power, subtraction, division, roots and logarithms.

⇒ Containment relations only

# Semantic use of void

The absence of a container has meaning. We can interpret absence as numeric zero. An empty container is a unit. Emptiness creates unity. Unity is absence of interior parts. The interior of every container, even those with contents, is pervaded by void. Void-equivalent forms are everywhere and in unlimited supply. Bounded space cannot and does not have properties, unlike the sequencing properties of the spaces between these words and the spaces between the characters of each word. Void is not synonymous with emptiness, it is more like the physical space that is underneath all physical objects. This leads to the fundamental Principle of Void-equivalent Forms (and of the void itself ). Void Equivalence Void-equivalent forms are syntactically inert and semantically irrelevant.

A unity (entity, object) is brought forth by an act of distinction. Conversely, each time we refer to a unity in our descriptions, we are implying the operation of distinction that defines it and makes it possible.

What operation does the round boundary apply to Nothing in order to generate Unity? […] We have only a Boundary around Nothing, together with an interpretation that takes ( ) as unity.

⇒ Existence: From the outside we associate empty containers with unity, with that which has no parts.

⇒ Unification: As a type of unit, [ ] must have different additive properties than ( ), otherwise the two would be indistinguishable. Round-brackets provide a tally via Accumulation, which explicitly forbids deleting replicated units. In contrast, [ ] dominates rather than accumulates. We can express this as an iconic equation, one that reduces apparent multiplicity to unity. The Unification axiom defines the Behavior of Square-Brackets.

[] [] ⇒ []

> unification → unify

Although this axiom is an Equality, it is only used in the direction of simplification. […]

21. a numeric operation, multiplying A by the unit 1: The ancient Greeks did not consider 1 to be a number, rather One is unity, a whole. “Multiplying” by One is an absurd action, since multiplication is replication, not unity. This is another example of how the symbolic disembodiment of arithmetic has removed us from common sense.

The implicit outermost container is the ground upon which blocks sit. The gravitational metaphor is not part of this representation system, but it is convenient to think of blocks as sitting on a table. Since blocks touch the table, the table itself provides a ground of Unity.

There are two voids. One we can talk about; that one is indicated by its boundary. That void is coupled to distinction as the foundation of unity. The other void is destroyed upon mention. The unmentionable is the metaphysical motivation of this volume. The concept of number is already so far removed from its origin that it is essential to regress backwards, from Two to One to Nothing to Silence, in order to find number, unity and absence.

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Spencer Brown's tectonic shift in the structure of formal thought suggests how to extend tally arithmetic into rational and real numbers. The innovation is to construct a second kind of boundary unit that is non-accumulating (Unify) and that cancels the round unit (void Inversion).

The Law of Calling […]