Interpretations of Laws of Form

Jeffrey James


The transformations in Laws of Form can be interpreted as different systems of mathematics. Basic mathematical systems are derived from the fundamental dynamics of distinction. This page describes these derivations for logic, numbers, imaginary logic, and knots. It also presents a short meta-mathematics of logic, sets, and numbers.

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.

( ( ) ) ( )
Graph Planar Linear

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.

Meta-Mathematics

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 Multi-Sets

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 multi-sets have not been fleshed out and are not covered here.

Other mathematical interpretations do not fall into the above framework, including imaginary logic and knot theory.

Logic

The logical arithmetic has a corresponding algebra. In Laws of Form, Spencer-Brown 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".

Spencer-Brown'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 )

Kauffman derives the logical algebra from a single axiom, based on the Robbins problem (see also McCune's solution) and Huntington's axiom:

Huntington ( ( A ) B ) ( ( A ) ( B ) ) = A

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.

As you would expect, each axiom system can be shown to prove the others. These proofs can be found here: Cross-Derivation of Boundary Logic Axiomatizations.

Numbers

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).

Spencer-Brown'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 Spencer-Brown's numbers. Kauffman (1996) modifies Spencer-Brown's version of the natural numbers by restricting null power to odd spaces and transfer to even spaces, thus eliminating the need for the ill-defined 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 even-odd 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 ) ]

The first four axioms define a logical system when the two boundaries are made identical. Otherwise, they form natural numbers as with Spencer-Brown's system above. The Inversion axiom extends this system from the natural numbers to integers when used at zero depth, to rationals when used one level deep, and to irrationals when used otherwise. 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.

Imaginary Logic

The laws of the logical arithmetic do not preclude additional elements in the system. In particular, there can be an element @ such that the following equality holds:

@ = ( @ )

Such an element leads to a contradiction with the real axiom set. We can relax Bricken's axiom set by replacing pervasion, which enforces the law of the excluded middle, with a distribution axiom or an occultation rule as in the following set:

Dominion
Involution
Occultation
A ( )
( ( A ) )
( ( A ) B ) A
=
=
=
( )
A
A

Other formulations of imaginary logic have four or more values, rather than three shown above.

Knots

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


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Richard Shoup, all rights reserved.