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

15
Table 2.3: Denition <strong>of</strong> Bricken <strong>Number</strong>s.
<strong>Number</strong>s
0 !
e
1 !
2 !
3 !
e
4 !
e
Operators
a + b ! a b
a
a b ! b
Rules
ee
:
e
=
: =
network. Calculation changes this connectivity to a canonical form.
Bricken deviates from Kauman's forms by implementing inverses as gradients
in the connections. He extends the system with algebraic variables and canonical
equations and uses these extensions to derive algebraic solutions.
As with Kauman numbers, Bricken's do not support any exponential form. The
great advantage to Bricken numbers is that they clearly dene calculation as changes
in connectivity.
2.4 Conclusions
Each <strong>of</strong> the systems described uses distinction to build numbers. The representations
compute in-place by the transformation rules <strong>of</strong> each system. Each operates in
parallel, allowing computation to occur locally within a vast conguration.
Each <strong>of</strong> these three systems cover addition and multiplication <strong>of</strong> natural numbers.
Spencer-Brown's system also has exponentiation, Kauman's system also has
the additive inverse, and Bricken's system has additive and multiplicative inverses.
However, none <strong>of</strong> these are clearly extensible beyond their current scope <strong>of</strong> coverage.
The limited scope <strong>of</strong> these systems restricts their usefulness as computational foundations.
Collectively they suggest that it is possible to construct a boundary system
for numbers with broader scope.