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

Chapter 2
PRIOR WORK
Boundary mathematics was rst explored as a paradigm <strong>of</strong> mathematical representation
in the context <strong>of</strong> propositional logic. It has been applied extensively to logic
[11, 3, 31, 1, 19, 32] as well as to other systems <strong>of</strong> mathematics, including imaginary
logic [25, 34, 35, 33], algebra [13, 27], self-reference and recursion [36, 20, 16, 25, 21],
control and deduction in computer science [4, 7, 8, 6, 14], and numbers [5, 10, 24].
The applications <strong>of</strong> boundary mathematics to numbers are <strong>of</strong>ten called boundary
numbers. G. Spencer-Brown, Louis Kauman, and William Bricken have each created
systems <strong>of</strong> boundary numbers [5, 10, 24, 23, 22]. Their systems share the basic
characteristics associated with boundary mathematics but dier in detail. In all
<strong>of</strong> them, boundaries serve as the objects <strong>of</strong> the system and as operators on those
objects. <strong>Number</strong>s are created out <strong>of</strong> boundaries and boundaries build into arithmetic
operations. This common basis <strong>of</strong> form allows calculation directly on the forms,
erasing and rearranging to transform an expression. Each <strong>of</strong> these systems are briey
described using their creator's original notation.
2.1 Spencer-Brown <strong>Number</strong>s
The boundary paradigm began with Spencer-Brown's Laws <strong>of</strong> Form in which he
reduced logic to a single distinction [11]. He shows how two fundamental choices <strong>of</strong>
reduction, calling and crossing, produce boolean arithmetic. In his work, Spencer-
Brown draws a distinction as a mark, , shown below in these rules.
Calling
Crossing
:
=
:
=
His number system attempts to adapt calling and crossing to the number domain.
He variously treats space as addition and as multiplication but, for simplicity, only
the former system will be considered [10, 5].