To study these systems, we cast distinctions into representations
that we can transform, like graphs, planar boundary diagrams, or
linear strings of parentheses. The linear form is most useful since
our writing systems are built around text streams.
Because we often draw distinctions as boundaries, as in the planar or
linear forms above, we often refer to this work as boundary
mathematics or, alternatively, to the notation as boundary
notation.
The laws of calling and crossing reduce broad and deep
structures in a manner consistent with logical arithmetic.
Different laws produce other mathematical systems. Eliminating
calling changes the system from logic to number, while eliminating
crossing instead changes it to sets, as illustrated in the table below:


( ) ( ) = ( ) 

( ) ( ) = ( ) ( ) 
( ( ) ) = 

Logic 

Numbers 
( ( ) ) = ( ( ) ) 

Sets 

MultiSets 
Ways to Reduce Structure
Among these four, the logical system has been most thoroughly
studied and is described in some detail below. The number system
requires additional distribution law but still runs into trouble;
ways of dealing with it are also described below. Sets and multisets
have not been fleshed out and are not covered here.
The logical arithmetic has a corresponding algebra. In Laws of
Form, SpencerBrown gives one axiomatization of this algebra but
more interesting versions have been devised by Bricken and Kauffman.
Each of these systems support the following mapping from standard
notation to boundary notation:
Standard Notation
FALSE TRUE NOT a a OR b a AND b
IF a THEN b ELSE c a IFF b  
Boundary Notation
( ) ( a ) a b ( ( a ) ( b ) )
( ( ( a ) b ) ( a c ) ) ( a b ) ( ( a ) ( b ) ) 
The boundary form of logic, called boundary logic, has a one
to many mapping to standard notation. This collapse of many forms
into a single one eliminates many of the transformations necessary in
the traditional notation. For example, "a IFF b" is equivalent to "IF
a OR b THEN a AND b".
SpencerBrown's axiomatization has historical relevance due to its
inclusion with the arithmetic in Laws of Form. He uses two
axioms, position and transposition, as follows:
Position Transposition 

( ( A ) A ) ( ( A B ) ( A C ) ) 
= = 
A ( ( B ) ( C ) ) 
Bricken's version uses three axioms which are computationally
convenient and conceptually clear. The first two axioms parallel the
laws of the arithmetic, while the third represents the law of the
excluded middle. Bricken gives different names to the axiom and to
each direction of application of the axiom but here we will call them
dominion, involution, and pervasion.
Dominion Involution Pervasion 

A ( ) ( ( A ) ) A ( A B ) 
= = = 
( ) A A ( B ) 
A fascinating aspect of this axiom is that its negation (wrapping a
boundary around each side) has not been shown to work, leading to some
conclusions about void substitution.
Unlike the logic, the numerical applications of Laws of Form use
different structures as well as different axioms. The first approach
holds true to the boundary structures of the logic while the other
deviates from the simplicity of the form by introducing boundary types
(i.e. distinctions among distinctions).
SpencerBrown's approach was to restrict the law of calling and to
introduce distribution as an axiom (in the logic, distribution is a
theorem).
Universe Transfer Reflexion Null Power 

( ( ) ) ( ( A B ) ( A C ) ) ( ( A ) ) ( ) : A 
= = = = 
A ( ( B ) ( C ) ) A ( ) 
Kauffman and Engstrom have each tried to solve problems with
SpencerBrown's numbers. Kauffman (1996) modifies SpencerBrown's
version of the natural numbers by restricting null power to odd spaces
and transfer to even spaces, thus eliminating the need for the
illdefined colon. Engstrom (1996) takes a similar tactic,
restricting the application of transfer to prevent invalid
distributions.
James's solution (1993), in contrast, is to introduce new boundary
types. A subset of these types map onto the evenodd distinction
offered by Kauffman, above, but extend the numerical representation
well beyond the natural numbers.
This algebra uses three types of boundaries: two that are each
other's inverse and a third that acts as the spatial inverse. These
three forms are sufficient to build forms that map onto real, complex,
and transcendental numbers.
Standard Notation
0 1 i Pi
a a 1 / a
a + b a  b a * b a / b
a ^ b a ^ (b) log_b(a)  
Boundary Notation
( ) [ < ( ) > ]
a < a > ( < [ a ] > )
a b a < b > ( [ a ] [ b ] ) ( [ a ] < [ b ] > )
( ( [ [ a ] ] [ b ] ) ) ( ( [ [ a ] ] [ < b > ] ) )
( [ [ a ] ] < [ [ b ] ] > ) 
James's first four axioms mirror define logic when the
boundary types are made identical.
The inversion axiom carries the
system beyond natural numbers but in doing so it introduces problems
associated with infinity, division by zero, and phase redundancy.
Dominion Involution1 Involution2 Distribution Inversion 

A [ ] ( [ A ] ) [ ( A ) ] A [ ( B ) ( C ) ] A < A > 
= = = = = 
[ ] A A [ ( A B ) ( A C ) ]

Kauffman has shown a mapping between distinction laws and knot
theory. In knot theory, string crossings draw a distinction:
Knot theory classifies equivalent knots with unravelling rules based
on where these crossings occur. These rules are variations on
calling, crossing, and distribution of the other forms.
Idempotency 




Crossing 




Distribution 




Copyright © 20002004 by