A Calculus of Number Based on Spatial Forms - University of ...

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rewrites a repeated reference using a single reference combined with a count <strong>of</strong>the
repetition. In standard notation, a repeated sum is rewritten as a product with the
count, e.g. x + :::+x = nx, and a repeated product is rewritten as an exponent <strong>of</strong>
the count, e.g. x :::x= x n . In the boundary calculus, cardinality takes the same
form in addition and in multiplication. Because <strong>of</strong> this dual use, the boundary form
<strong>of</strong> cardinality is considered to be a generalized form <strong>of</strong> cardinality.
This chapter begins by formally dening the two-boundary calculus.
4.2 The Two-Boundary <strong>Calculus</strong>
The two-boundary calculus is introduced by dening the set <strong>of</strong> possible constructions,
the elements <strong>of</strong> the calculus, and by dening axioms to state their equivalences.
4.2.1 Elements <strong>of</strong> the <strong>Calculus</strong>
Elements <strong>of</strong> the calculus are constructed out <strong>of</strong> the void using two boundary distinctions:
instance and abstract.
Denition.
Let B denote the collection <strong>of</strong> all well-formed boundary expressions
composed <strong>of</strong> two boundaries, denoted as(:::) and [:::]: B is dened recursively by
three rules:
1. The void belongs to B.
2. If a belongs to B, then (a) and [a] each belong to B.
3. If b 1 ;b 2 ;:::;b n (n 1) belong to B, then the unordered collection b 1 b 2 :::b n
belongs to B.
The rst rule establishes the void as the foundational element in the recursive
construction <strong>of</strong> B. Thus, B contains the void.
The second rule creates elements <strong>of</strong> B by nesting other elements inside the two
boundaries. The rst two elements introduced by this rule are the boundaries with
no content: and 2: Further distinction introduces the following elements into B:
();(2);[];[2];(());((2));([]);([2]);[()];[(2)];[[]];[[2]]: