An empty round boundary encloses nothing and can be interpreted as the unit One.
Within the language of boundary forms, we would say that the Whole arises out of framing Nothing. The conceptual foundation of bounded nothing provides a singular perspective from which to understand the formal system of numeric arithmetic. A boundary effectively partitions empty space into two identifiable locations that we call inside and outside. Which is which is purely an artifact of where we place our own viewing perspective. We culturally identify with “outside”.
> nothing is inside unity
Spencer Brown begins Laws of Form with this:
> The theme of this book is that a universe comes into being when a space is severed or taken apart.
It is important to realize that the universe is defined by the severing boundary, not by the apparent pair of spaces. Inside and outside are different perspectives on the same distinction. We cannot, of course, create copies of nothing.
> ( ) is a container, nothing more
Here is Maturana and Varela:
> 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.
Since our fundamental numeric unit can contain other forms, it has both a static and a dynamic reading, as an object and as an operation. It’s convenient to have two alternative representations of round boundaries. We’ll use the token o to emphasize the empty round-bracket as a unit: an object that provides the numeric ground. We’ll use ( ) to emphasize the round boundary as an operation: a process that transforms its contents.
> ( ) = o
What operation does the round boundary apply to Nothing in order to generate Unity? This question can be formulated as a conventional function. What operation (call it ʀ) satisfies the functional form ʀ(0) = 1? Whatever it is, the operator ʀ cannot call upon any number (such as add-two, or divide-by-three), because we do not as yet have numbers. For that matter, we do not yet have logic to draw upon. We are developing the concept of Number from a perspective that does not consider the concepts of Logic or Sets or Multiplicity to be fundamental. We have *only* a Boundary around Nothing, together with an interpretation that takes ( ) as unity.
One interpretation is that ʀ is the successor function, +1. This interpretation uses nesting to mean accumulation, effectively squandering the representational freedom made available by nested containers. Another interpretation of ʀ is the exponential function. ( ) can be interpreted as any base raised to the zero-ith power, which is conventionally defined to be 1.
( ) ☞ 1 #0 = 1
We’ll interpret a round-bracket (A) to be a **power operator** that raises an arbitrary base to the power of the boundary’s contents, #A.
The round boundary could be called an exponential boundary or a power boundary, but these names predispose us to think of James algebra as numeric. *Round* is a convenient label suggesting nothing about the interpretation of the boundary. James algebra makes no distinction between the object *one* and the function *raise-to-the-zeropower* ( ⇒ 0000000010…). Either of these interpretations is available at any time by electing to read the round boundary from the outside as an object or from the inside as a process ( ⇒ Two Concepts Of The Symbolic).