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

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Table 2.2: Denition <strong>of</strong> Kauman <strong>Number</strong>s.
<strong>Number</strong>s
0 !
1 !
2!
3!
4!
Operators
a + b ! a b
a b ! a i b
,a ! a i
Rules
:
= > = >
< > = > = >
= ><
:
=
:
= = w
:
= w
=
reduce to one, as y = . Spencer-Brown performs both <strong>of</strong> these reductions using
transfer into a mark with nothing inside, as t = .However, this rule undermines
addition space because adding one provides an empty for obliterating the rest <strong>of</strong>
an expression. The colon was his attempt to disambiguate this paradox but its use
was not thoroughly dened.
Spencer-Brown's numbers represent the natural numbers and the system computes
basic arithmetic operations on them except for those requiring an inverse. The system
is simple and concise, using only a single distinction, but it ultimately breaks down.
2.2 Kauman <strong>Number</strong>s
Kauman's number system also represents the natural numbers using boundaries. It
contrasts from other boundary systems in that it transforms by string matching rules
rather than by spatial matching rules.
In his system, a boundary represents magnitude, a doubling <strong>of</strong> contents.
object represents a unit and the boundary denotes a magnitude <strong>of</strong> its contents.
Here, it is assumed to be a doubling, = : . Counting proceeds in Kauman
numbers as ; ; ;::: and so on, as a base-2 system. These numbers add by
concatenation. They multiply by replacing the units <strong>of</strong> one number with the entirety
<strong>of</strong> the other, denoted by the insertion AiB. Kauman numbers are summarized in
Table 2.2.
Kauman uses an overbar to represent the additiveinverse, making the negative
: :
unit. A positive unit and a negative unit cancel out, as = = . Many
transformation rules involve both the inverse and the doubling boundary. Because
The