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

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For example, a half-cardinality is given by ([a]): Collection <strong>of</strong> two <strong>of</strong> these
reduces to a:
([a])([a]) Given
([([a])][]) Cardinality
([a][]) Involution
([a])
Inversion
a
Involution
Applied to addition combinations, fractional cardinality builds fractions. The
form ([a]) translates to a=2. In the above pro<strong>of</strong> this fraction is doubled, as
a=2+a=2 =2(a=2) = a(2=2) = a:
Applied to multiplicative combinations, fractional cardinalities build roots. The
form (([[b]])) translates to b 1=2 . In a pro<strong>of</strong> similar to the above but with
a=[b], this root is squared to double the cardinality, as
b 1=2 b 1=2 =(b 1=2 ) 2 =b 2=2 =b:
Since fractional cardinalities form roots, they allow construction <strong>of</strong> the algebraic
irrationals.
5.6 Conclusions
Though the two-boundary calculus only represents natural numbers, simply adding an
inverse boundary extends the boundary numbers through the algebraic irrationals.
The three axioms completely dene computation on these numbers and functions.
The three-boundary calculus is summarized in Table 5.1.
The inverse boundary serves manyinverse functions due to the structures provided
by the instance and abstract boundaries. It is dened in the most fundamental way
that an inverse can be dened, utilizing the void to cancel inverses to nothing. The
spatial formalism makes this inverse denition possible and powerful. Kauman
dened an inverse boundary <strong>of</strong> this sort but did not have the other structures to
utilize it in a general form.
The inverse is powerful but it introduces problems, such as division by zero.
Though the two-boundary calculus was solid, the three-boundary calculus has a single
exception, preventing rules from applying to all congurations. When performing