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

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phrased in their canonical form. For example, 3 + 2 rewrites as :
4.3.2 Multiplication
Elements a; b 2 N multiply by the form ([a][b]): The set N is closed under the
multiplication.
Pro<strong>of</strong>.
The product <strong>of</strong> a 2 N and initial element, , reduces to a by involution,
([a][])=([a])=a: If the product <strong>of</strong> a and b, ([a][b]), isintheN, the product
<strong>of</strong> a and the successor <strong>of</strong> b is also in the set,
([a][b])=([a][b])([a][])=([a][b])a:
Because addition is closed, this result is in N. Therefore, by induction all products
<strong>of</strong> a; b 2 N are in N.
Multiplication is commutative because [a] and [b] are unordered within the
outer instance. Associativity <strong>of</strong> binary multiplication is shown by two applications <strong>of</strong>
involution:
([([a][b])][c])=([a][b][c])=([a][([b][c])]):
The multiplicative identity is the unit, : An element multiplied by the unit reduces
to itself by involution,
([a][])=([a])=a:
In the boundary calculus, the void represents zero. Accordingly, multiplication
by the void is void, ([a][])= :
Pro<strong>of</strong>.
By distribution, a product with the void is equal to two copies <strong>of</strong> the same,
([a][ ])([a][ ])=([a][ ]):
The equation ee = e is true when e is void or when collection <strong>of</strong> e is idempotent.
Since collection is assumed additive, ee 6= e unless e is void. Therefore, ([a][ ])= :