A Calculus of Number Based on Spatial Forms
by
Jeffrey M. James
A thesis submitted in partial fulfillment of
the requirements for the degree of
Master of Science in Engineering
University of Washington
1993
Abstract
A calculus for writing and transforming numbers is defined. The calculus
is based on a representational and computational paradigm, called boundary
mathematics, in which representation consists of making distinctions out of
the void. The calc*ulus uses three boundary objects to create numbers and
covers complex numbers and basic transcendentals. These same objects
compose into operations on these numbers. Expressions transform using
three spatial match and substitute rules that work in parallel across
expressions. From the calculus emerge generalized forms of cardinality and
inverse that apply identically to addition and multiplication. An imaginary
form in the calculus expresses numbers in phase space, creating complex
numbers. The calculus attempts to represent computational constraints
explictly, thereby improving our ability to design computational machinery
and mathematical interfaces. Applications of the calculus to computational
and educational domains are discussed.
Dedicated to my very first mathematics teacher, my father.
TABLE OF CONTENTS
List of Figures
List of Tables
Chapter 1 Introduction
1.1 Minimalism ...................................................
1.2 The Calculus .................................................
1.3 Conclusions ..................................................
Chapter 2 Prior Work
2.1 SpencerBrown Numbers ........................................
2.2 Kauffman Numbers .............................................
2.3 Bricken Numbers ..............................................
2.4 Conclusions ..................................................
Chapter 3 Boundary Mathematics
3.1 Introduction ..................................................
3.2 Forms ........................................................
3.2.1 Void ...................................................
3.2.2 Distinction ............................................
3.2.3 Multiple Boundaries ....................................
3.3 Transformation ...............................................
3.3.1 Match and Substitute ...................................
3.3.2 Properties .............................................
3.4 Conclusions ..................................................
Chapter 4 Instance and Abstract
4.1 Introduction .................................................
4.2 The TwoBoundary Calculus ....................................
4.2.1 Elements of the Calculus ...............................
4.2.2 Equivalence Axioms ....................................
4.3 Natural Numbers ..............................................
4.3.1 Addition ...............................................
4.3.2 Multiplication .........................................
4.4 Cardinality .................................................
4.4.1 Dominion ...............................................
4.4.2 Generalized Cardinality ................................
4.5 Conclusions ..................................................
Chapter 5 Inverse
5.1 Introduction .................................................
5.2 The ThreeBoundary Calculus ..................................
5.2.1 Elements of the Calculus ...............................
5.2.2 Equivalence Axioms .....................................
5.3 The Additive Inverse .........................................
5.3.1 Properties of the Additive Inverse .....................
5.3.2 Integers ...............................................
5.3.3 Calculation ............................................
5.4 Multiplicative Inverse .......................................
5.4.1 Properties of the Multiplicative Inverse ...............
5.4.2 Rationals ..............................................
5.4.3 Division by Zero .......................................
5.4.4 Calculation ............................................
5.5 Inverse and Cardinality ......................................
5.5.1 Negative Cardinality ...................................
5.5.2 Fractional Cardinality .................................
5.6 Conclusions ..................................................
Chapter 6 Phase
6.1 Introduction .................................................
6.2 Phase .......................................................
6.2.1 Phase Independence .....................................
6.2.2 Oscillation ............................................
6.2.3 Cardinality of J .......................................
6.3 Multiple Value ...............................................
6.4 Exponentials and Logarithms ..................................
6.5 Transcendentals ..............................................
6.5.1 Euler's Number, e ......................................
6.5.2 Pi .....................................................
6.5.3 Trigonometric Functions ................................
6.6 Conclusions ..................................................
Chapter 7 Future Work
7.1 Introduction .................................................
7.2 Coverage......................................................
7.3 Practical ....................................................
7.4 Extensions ...................................................
7.5 Conclusions ..................................................
Chapter 8 Applications
8.1 Introduction .................................................
8.2 Hardware .....................................................
8.3 Computer Algebra .............................................
8.4 Mathematical Interface .......................................
8.5 Education ....................................................
8.6 Conclusions ..................................................
Glossary
Bibliography
Appendix A Conversion
A.1 Numbers ......................................................
A.2 Functions ...................................................
A.3 Number Formats ...............................................
A.4 Standard Postulates ..........................................
A.5 Formulas .....................................................
Appendix B Examples
B.1 Multiplication ...............................................
B.2 Square Root ..................................................
B.3 Fractions ....................................................
B.4 Algebraic Distribution .......................................
B.5 Quadratic Formula ............................................
LIST OF FIGURES
1.1 Stages of Mathematical Representation ...........................
3.1 The Void ........................................................
3.2 A Distinction ...................................................
3.3 Content and Context of a Distinction ............................
3.4 Two Choices for Further Distinction .............................
4.1 Examples of Involution ..........................................
4.2 Examples of Distribution ........................................
5.1 Examples of Inversion ...........................................
8.1 Technology and Representation ...................................
8.2 Visual Interpretations of Boundary Numbers ......................
LIST OF TABLES
1.1 Simple Calculations in Standard and Boundary Forms ..............
1.2 Mapping of Numbers ..............................................
1.3 Mapping of Functions ............................................
1.4 Axioms of the Calculus ..........................................
1.5 Theorems of the Calculus ........................................
2.1 Definition of SpencerBrown Numbers .............................
2.2 Definition of Kauffman Numbers ..................................
2.3 Definition of Bricken Numbers ...................................
4.1 The TwoBoundary Calculus .......................................
5.1 The ThreeBoundary Calculus .....................................
6.1 Transcendentals in the Calculus .................................
7.1 Rules for Boundary Differentiation ..............................
PREFACE
This work attempts to clarify mathematics, to make complex ideas accessible
to those with less mathematical training. In this thesis, I have reduced
number and operator representation down to three forms, revealing a
simplicity to numbers. I hope this work may aid those who have struggled to
understand numbers and computation, as well as elementary algebra, for I
surely consider these topics in new light.
I am not a mathematician, so writing a thesis about mathematics has been
a challenge for me. For years, I have struggled to understand what research
in mathematics is all about that I might contribute to its continuing
progress. Alas, essential concepts continue to elude me and I remain
uncertain about what mathematicians are actually doing. Nonetheless, I
believe my work to have some relevance to them and have tried to describe
it in a manner that would avail mathematical researchers to make use of it,
though my audience also includes the less mathematically inclined.
I have benefitted from many discussions with friends and colleagues
during the development of this material. Most recognize some insight here,
though I am not certain whether they liked the material itself or just the
idea of trying to make mathematics easier to understand.
I thank William Bricken with my heart and soul. William introduced me
to boundary mathematics and has served as my mentor throughout this
research_his guidance and direction has made it all happen.
I thank the rest of my thesis committee: Judith Ramey, William Winn, and
Steven Tanimoto. Each of them played unique roles in the development of
this material. I was privileged to have had such excellent support for my
research.
I credit my zest for research to Penelope Sanderson, with whom I studied
as an undergraduate at the University of Illinois. Penny challenged me with
interesting research problems and gave me opportunities to present my
findings to receptive audiences. I am forever grateful to her.
I thank the students and staff of the Human Interface Technology Labo
ratory and its director, Dr. Thomas A. Furness, for providing a stimulating
environment for developing these ideas. I thank Kimberley Osberg, Dav Lion,
and Greg Woodward in particular for their personal support. I thank US West
for funding part of this research and I thank all members of the consortium
for supporting the lab.
Though I wrote this document at the HIT Lab in Seattle, I "polished the
turd" at Interval Research in Palo Alto. I thank Dick Shoup, the Natural
Computing group, and everyone at Interval for their support of my research.
I extend my warmest appreciation to Ann Miller, with whom I shared the
joy of discovering many of the ideas herein. Ann taught me humility and
compassion and will always be in my heart.
Finally, I thank Laura Grant, the love of my life. I could never have
finished this thesis without her intellectual and emotional support. Though
it took a while, I finally managed to "swallow the toad!"
Jeffrey Mark James
Human Interface Technology Laboratory
October 14, 1993
Chapter 1
INTRODUCTION
Numbers are simpler than they appear. This thesis presents a calculus of
number that demonstrates this simplicity.
We commonly represent and manipulate numbers with a centuriesold
notation comprised of Arabic digits, decimal point, plus sign, minus sign,
times sign, fraction bar, a few other operators, and some constants.
Typical numbers are 0, 1, 2, 10, 1, 3/7, 2.9, sqrt(2), e^2, i*Pi. The
standard Arabic notation represents ten numbers directly as digits and a
few more as symbolic constants. All others are constructed out of these
using arithmetic operations. Many of the operations used to build numbers
are implicit in the representation, such as the magnitude gain in place
value. Hiding information in this manner achieves conciseness but leaves
the forms abstract and dissociated from their own behavior.
These traditional representations are but one embodiment of numbers.
Formal alternatives exist but generally lack the scope and conciseness of
the standard form. Other representations are useful because they display
interesting properties of numbers. Conway Numbers, for instance, build real
numbers out of a single object, the biset [12]. This thesis presents a
representation of numbers, called boundary numbers, in which numbers and
arithmetic operations are built out of three objects.
1.1 Minimalism
The calculus in this thesis progresses representation of number towards a
minimal basis. It begins with the most fundamental acts of representation.
By starting from scratch, the calculus avoids the conceptual baggage that
has been introduced into standard notation throughout its development.
The calculus is an attempt to minimize the implicit constraints on form
manipulation to just those necessary to define the system of mathematics.
Other constraints should be explicit in the forms. Commutativity, for
example, would be explicit if commutative functions displayed symmetry
between their arguments and noncommutative functions did not. By making
constraints explicit, much "knowledge" of mathematics can be embodied in
the representation.
Standard notation has abstracted away the behavior of its forms so that
knowledge of their behavior has become implicit. For example, the notation
abstracts away magnitudes of the digits, making 1 + 1 = 2 not altogether
obvious but requiring memorization. A model of this disembodiment is shown
in Figure 1.1. The standard notation represents the conceptual objects of
numbers and functions but not the computational objects upon which
constraints are imposed.
Solution Loop
Activity
application
/\ 
  problem representation
 \/
conceptual objects
/\ 
  translation and control
 \/
computational objects
/\ 
\__/ calculation
Figure 1.1: Stages of Mathematical Representation.
When using math, we work through the solution loop shown in Figure 1.1.
We represent elements of a problem as conceptual objects in a mathematical
notation. We manipulate these objects according to calculation rules,
usually through separate computational objects. Guided by solution
techniques, we manipulate the conceptual objects into forms that lend
insights into the problem.
Layers of mathematical objects are necessary because most mathematical
forms cannot be "directly" manipulated with a concise set of rules.
Comprehensively defined manipulation often requires intermediate forms.
Moreover, these intermediate forms can be completely disparate from the
original form, making the translation itself complicated.
For example, with digit magnitude implicit, numbers are often added by
use of separate representations. Many computers use bits as computational
objects because bits can be physically added by digital circuitry. Many
people learn to add by using objects which carry the mathematical
constraints explicitly, such as counting blocks. The counting blocks serve
as computational objects upon which calculation can be directly performed.
Because standard notation does not embody the transformation constraints,
surrogate representations are necessary for machines and children, both of
which require explicit declarations.
Standard notation conceals mathematical behavior in ways besides hiding
digit magnitudes. The equations on the left side of Table 1.1 collectively
reflect mathematical knowledge which, when known, make the equalities
obvious. Visually, they are not obvious: the features of the forms provide
no clues as to their manipulative constraints. The forms carry none of
this knowledge, a failure which places unnecessary demands on users of
mathematics.
Table 1.1: Simple Calculations in Standard and Boundary Forms.
Standard Boundary
1 + 1 = 2 oo = oo
21 + 11 = 32 ([b][oo])o([b][o])o = ([b][ooo])o
2 3 = 6 ([oo][ooo]) = oooooo
32 = 3 3 (([[ooo]][oo])) = ([ooo][ooo])
x + 0 = x x = x
x * 1 = x ([x][o]) = x
x * 0 = 0 ([x][]) =
x^1 = x (([[x]][o])) = x
x^0 = 1 (([[x]][]) ) = o
1/x = x^1 (<[x]>) = (([[x]][]))
x^y * x = x^(y+1) ([(([[x]][y]))][x]) = (([[x]][yo]))
The boundary calculus shifts the focus of representation from the
conceptual objects of numbers and functions to the computational objects
where the dynamics of the forms can be directly captured. It makes a clear
separation between the fundamental constraints on number manipulation and
the concepts that direct this manipulation. The equations on the right side
of Table 1.1 are the same equalities translated to the boundary calculus.
Here, b represents the base of the placevalue system, i.e. b=oooooooooo.
The expressions are longer because they are constructed out of very basic
objects, built into structures that reveal information about their own
transformational constraints. The three axiomatic transformations given in
the next section are all that is required to demonstrate the boundary
equalities.
The mechanics of number manipulation are simple in the boundary
calculus. More complicated concepts are not "builtin" but instead appear
as directed use of it. Techniques for managing numbers, such as separating
magnitude or naming small integers, exist independent of the axioms of this
system. The calculus provides a crisp mathematical substrate within which
to construct and illustrate mathematical concepts that were previously
interwoven with the symbolic rules themselves.
1.2 The Calculus
This thesis defines a few mathematical forms and how they transform, i.e.
it defines a calculus. These forms represent numbers, as opposed to logic
or sets. And these forms are based on a representational paradigm that is
spatial. In short, this thesis defines a calculus of number based on
spatial forms.
This material assumes a minimalist approach to representation called
boundary mathematics [3, 17]. The ideas were initiated by G. SpencerBrown
[11] and have been extended in the domain of numbers by SpencerBrown
himself, as well as Louis Kauffman [22, 24] and William Bricken [5, 9].
This work extends theirs by introducing general forms for cardinality and
inverse, and by defining phase. This work also gives a functional
interpretation of boundaries not found elsewhere, treating them as
logarithmic functions.
In boundary mathematics, representation begins with empty space, a
complete lack of structure, void. Upon this void, distinctions are made
that impose structure and intention. Distinction may be thought of as a
boundary that delineates space. Boundaries may be nested and collected to
form different configurations. A configuration of boundaries represents a
mathematical expression that can be transformed according to match and
substitute rules, as defined for the mathematical system [11, 3].
The boundary numbers described here use more than one distinction, a
practical move which as yet has not been philosophically justified in the
paradigm. Multiple boundaries force a distinction that is not a cleaving of
the space; instead, it is a differentiation of the boundaries. Here, this
distinction is accomplished with geometric characteristics: the boundaries
are drawn round 'o', square '[]', or pointy '<>'. To hold contents, they
are also drawn as delimiters: '(A)', '[A]', ''.
These three boundaries are the fundamental forms of the calculus of
number. Alphabetic characters are also used to denote unknown values and
constants. Equality is expressed as usual, denoting configurations of equal
numerical value.
Natural numbers can be represented by collecting a single boundary, as
in a tally system. Call this boundary instance and draw it round.
Counting proceeds by accumulating empty instance boundaries, as 'o', 'oo',
'ooo',... so on. These numbers add by collecting them in space, no
computation is necessary. A second boundarycall it abstract and draw it
squareenables formation of multiplication and power functions. For
example, the multiplication 2*3 appears in boundary notation as
'([oo][ooo])' and the cube of two, 2^3, appears as '(([[oo]][ooo]))'.
A third boundarycall it inversecan be used to construct additive and
multiplicative inverses, and other inverse structures such as roots.
Inverse extends the calculus to the integers by serving as the additive
inverse, to the rationals by serving as the multiplicative inverse, and to
algebraic irrationals in other combinations. A full sampling of boundary
numbers is shown in Table 1.2 and the boundary operations are shown in
Table 1.3.
Calculation on these forms is completely defined by three axioms:
involution, distribution, and inversion, shown in Table 1.4. Involution
defines instance and abstract to be functionally opposite. Distribution
defines them to additionally have a distributive relationship. Inversion
defines the inverse boundary to behave as an inverse operator. The three
axioms govern all transformations of boundary expressions.
The calculus constructs general forms that are not available in the
standard notation. Using involution and distribution, a repeated form such
as 'AAA' can be transformed into an equivalent expression that requires
only one reference to that form, as '([A][ooo])'. This transformation is
called cardinality and is shown in Table 1.5. This means of counting
applies to repeated addition and to repeated multiplication; it is
therefore considered to be a generalized form of cardinality.
The inverse boundary is also unparalleled in standard notation. It is
used to form the additive inverse as well as the multiplicative inverse
(see Table 1.3). All of the properties of this generalized inverse apply
to both of these contexts. These properties include collection of inverses,
cancellation with itself, and relocation, all shown in Table 1.5.
Table 1.2: Mapping of Numbers
Standard Boundary
0
1 o
2 oo
1
2
1/2 (<[oo]>)
2*3 ([oo][ooo])
2/3 ([oo]<[ooo]>)
4 1/2 oooo(<[oo]>)
3^2 (([[ooo]][oo]))
3^(1/2) (([[ooo]]<[oo]>))
243 ([b][([b][oo])oooo])ooo
10^6 (([[b]][oooooo]))
Table 1.3: Mapping of Functions
Standard Boundary
a a
a
1/a (<[a]>)
a+b ab
ab a
a*b ([a][b])
a/b ([a]<[b]>)
a^b (([[a]][b]))
a^(b) (([[a]][]))
log_b(a) (([[a]]<[b]>))
Table 1.4: Axioms of the Calculus.
Involution ([A]) = A = [(A)]
Distribution (A[BC]) = (A[B])(A[C])
Inversion A =
Table 1.5: Theorems of the Calculus.
Cardinality A...A = ([A][o...o])
Dominion [] A = []
Inverse Collection =
Inverse Cancellation < > = A
Inverse Promotion <(A[B])> = (A[])
Phase Independence [<(A)>] = A[<()>]
J Cancellation [][] =
Its forms can be interpreted so as to extend the calculus to complex and
transcendental numbers. Taking cardinalities of the inverse produces radian
values that can act as complex numbers when their manipulation is further
restricted. The instance and abstract boundaries act as exponential and
logarithmic functions, respectively. By interpreting these to be of base
e, basic transcendental values and trigonometric functions can be formed.
These interpretations are included in Tables 1.2 and 1.3.
1.3 Conclusions
The calculus represents many types of numbers but it covers only part of
number mathematics. It lacks larger structures for doing mathematics and
requires much practical support to make it useful. There are many
additions that can be made to the calculus and areas to which the content
can extend. Future work with the calculus will expand and enhance the
ideas presented here.
The calculus presents a new paradigm of number representation that
challenges how mathematics is currently done. This material may impact the
way mathematics is done physically in hardware, logically in software, and
conceptually in an interface with mathematics. It may also impact how
mathematics is taught, principally because it serves as an alternative to
the standard notation.
Though the boundary forms are not immediately comprehensible,
familiarity makes them legible. And since many of the transformations are
easily visualized, the boundary forms eventually appear quite natural.
Recall Table 1.1 in which calculations in standard notation are paralleled
in boundary notation. Many of the boundary calculations are immediate,
while others require only a few applications of the axioms. Appendix A
contains more comparisons and Appendix B contains examples, including a
derivation of the quadratic formula. Overall the boundary calculations are
much simpler, indicating that standard form makes computation_and therefore
understanding_unnecessarily complex.
Chapter 2
PRIOR WORK
Boundary mathematics was first explored as a paradigm of mathematical
representation in the context of propositional logic. It has been applied
extensively to logic [11, 3, 31, 1, 19, 32] as well as to other systems of
mathematics, including imaginary logic [25, 34, 35, 33], algebra [13, 27],
selfreference 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 of boundary mathematics to numbers are often called
boundary numbers. G. SpencerBrown, Louis Kauffman, and William Bricken
have each created systems of boundary numbers [5, 10, 24, 23, 22]. Their
systems share the basic characteristics associated with boundary
mathematics but differ in detail. In all of them, boundaries serve as the
objects of the system and as operators on those objects. Numbers are
created out of boundaries and boundaries build into arithmetic operations.
This common basis of form allows calculation directly on the forms, erasing
and rearranging to transform an expression. Each of these systems are
briefly described using their creator's original notation.
2.1 SpencerBrown Numbers
The boundary paradigm began with SpencerBrown's Laws of Form in which he
reduced logic to a single distinction [11]. He shows how two fundamental
choices of reduction, calling and crossing, produce_boolean arithmetic. In
his work, SpencerBrown 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].
With space as addition, the natural numbers are easily_formed. The void
acts as zero and counting proceeds by accumulating marks: '()', '()()',
'()()()',... and so on. Numbers are added by spatial collection whereas
multiplication and power operations are composed using marks.
SpencerBrown's numbers are summarized in Table 2.1.
Table 2.1: Definition of SpencerBrown Numbers.
Numbers Operators Rules
0 > a+b > a b (()) =
1 > () a*b > ((a)(b)) ((at)(bt)..) = ((a)(b)...)t
2 > ()() a^b > (b):a ((a)) = a
3 > ()()() ():a = ()
He gives two axioms for calculating on these forms, universe and
transfer. With these axioms he derives many theorems, including reflexion
and null power. Universe and null power are his numerical versions of
calling and crossing.
___
__ 
Universe  =
_______ _______
__ __  __ __ 
Transfer at bt = a bt
___
__ 
Reflexion a = a
__ __
Null Power :a = 
In addition to the mark, he uses a vaguely defined colon to disambiguate
conflicting results which would equate 0^1 > '():()' and 2 > '()()'.
This colon does not completely resolve the problems it was introduced for.
These numbers add by spatial collection. For example, 3 + 2 = 5.
3 + 2 Given
()()() + ()() Number Rewrite
()()()()() Addition Rewrite
5 Number Rewrite
Numbers multiply by the form a*b > '((a)(b)'. The form reduces by
transfer, which distributes the whole of one number throughout the units of
the other. For example, 3*2 = 6.
3 * 2 Given
()()() * ()() Number Rewrite
((()()())(()())) Multiplication Rewrite
((((()()()))((()()())))) Transfer, t = (()()())
()()() ()()() Reflexion (3x)
6 Number Rewrite
__
Powers are expressed as a^b > '(b):a'. This form creates a repeated
multiplication where the arity is determined by the exponent and the base
is introduced as the arguments using transfer. For example, 3^2 = 9.
3 ^ 2 Given
()()() ^ ()() Number Rewrite
(()()) ()()() Power Rewrite
((()()())(()()())) Transfer, t = ()()()
((((()()()))((()()()))((()()())))) Transfer, t = (()()())
()()() ()()() ()()() Reflexion (4x)
9 Number Rewrite
SpencerBrown's system has two major problems. First, transfer is not
fully compatible with spatial match and substitute: it requires that the t
match the entire context. For example, the previous problem produces a
conflicting result if transfer matches only part of the context.
3 ^ 2 Given
()()() ^ ()() Number Rewrite
(()()) ()()() Power Rewrite
((())(())) ()() Transfer, t = ()
()()() Reflexion (2x)
3 Number Rewrite
The other problem with this system_is its treatment of zero. A
multiplication by zero should reduce to zero, as (()(x)) = (()) = .
Similarly, a zero exponent should reduce to one, as () y = ().
SpencerBrown performs both_of 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 of an
expression. The colon was his attempt to disambiguate this paradox but its
use was not thoroughly defined.
SpencerBrown'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 Kauffman Numbers
Kauffman'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 of contents.
The object '*' represents a unit and the boundary '< >' denotes a magnitude
of its contents. Here, it is assumed to be a doubling, <*> = **. Counting
proceeds in Kauffman numbers as *, <*>, *<*>,... and so on, as a base2
system. These numbers add by concatenation. They multiply by replacing the
units of one number with the entirety of the other, denoted by the
insertion AiB . Kauffman numbers are summarized in Table 2.2.
Table 2.2: Definition of Kauffman Numbers.
Numbers Operators Rules _ __ _ _
0 > a+b > ab <*> = ** <*> = ** ** = ** =
1 > * a*b > a/.*>b _ _ _
2 > <*> _ >* = *>* >* = *>* w* = *w
3 > <*>* a > a =
4 > <<*>> < > = >< = * = *
Kauffman uses an overbar to represent the additive inverse, 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 this system uses string rewrite rules, ordering
is significant. The rules on the left of Table 2.2 are for doubling and
inverse, whereas the rules on the right handle sequencing.
In this system, numbers add by concatenation, i.e. spatial collection
in one dimension. For example, 3+2 = 5. The result is transformed to a
canonical form using the string rewrite rules.
3 + 2 Given
<*>* + <*> Number Rewrite
<*>* <*> Addition Rewrite
<*><*>* w* = *w
<**>* >< =
<<*>>* ** = <*>
5 Number Rewrite
Multiplication is performed by replacing one expression for the units in
the other. For example, 3x2 = 6.
3 x 2 Given
<*>* x <*> Number Rewrite
<*>* i <*> Multiplication Rewrite
<<*>><*> Multiplication Substitution
<<*>*> >< =
6 Number Rewrite
Subtraction is performed by multiplying by the negative unit and then adding.
The result is reduced using the rewrite rules with the negative unit. For example,
81 = 7.
8  1 Given
<<<*>>>  * Number Rewrite
<<<*>>> + /* Subtraction Replacement
<<<*>>> /* Addition Rewrite
<<<*>>/*>* >/* = /*>*
<<<*>/*>*>* >/* = /*>*
<<<>*>*>* *>/* = >*
<<<*>*>* <> =
7 Number Rewrite
Though this system is limited to the integers, Kauffman has defined
another boundary as the opposite of the magnitude boundary, such that
()=<(A)>=A. The string rewrite rules for this system are quite
extensive due to the many permutations of the characters and are not
included here.
Alternatively, the extended Kauffman numbers can be defined spatially
with three rules. The definition below uses his delimiter form of the
inverse rather than the overbar form shown above [24].
Double AA =
Inverse A[A] =
Half () = <(A)> = A
Kauffman numbers represent the integers and the system computes basic
arithmetic operations except division and powers. Addition by collection is
immediate, as is multiplication by substitution. Reduction is done locally
by string rewrite rules.
2.3 Bricken Numbers
Kauffman's work was given a different twist by William Bricken, who
interpreted his basic forms as networks [5]. The network form provides
singular references and a clear representation of the structures being
manipulated. Of particular benefit, singular reference allows
multiplication by stacking, providing an algebraic form. Bricken numbers
are summarized in Table 2.3.
Table 2.3: Definition of Bricken Numbers.
Numbers Operators Rules
0 > / / a+b > /b/ /===/ > /o/
/a/
1 > // /o/ > /o/
a*b > /a/b/ /o/ / \/
2 > /===/
___
3 > /===/
Bricken numbers are networks whose value is determined by their
connectivity. The network includes a context and ground, shown as lines
above and below the network. Calculation changes this connectivity to a
canonical form.
Bricken deviates from Kauffman'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 Kauffman numbers, Bricken's do not support any exponential form.
The great advantage to Bricken numbers is that they clearly define
calculation as changes in connectivity.
2.4 Conclusions
Each of the systems described uses distinction to build numbers. The
representations compute inplace by the transformation rules of each
system. Each operates in parallel, allowing computation to occur locally
within a vast configuration.
Each of these three systems cover addition and multiplication of natural
numbers. SpencerBrown's system also has exponentiation, Kauffman's system
also has the additive inverse, and Bricken's system has additive and
multiplicative inverses. However, none of these are clearly extensible
beyond their current scope of coverage. The limited scope of 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.
Chapter 3
BOUNDARY MATHEMATICS
3.1 Introduction
This chapter describes the principles of boundary mathematics, upon which
this thesis builds the calculus of number.
Boundary mathematics is a representational and computational paradigm
for mathematics, based on the concept of distinction. The paradigm was
initially described by SpencerBrown but the term comes from Bricken's
interpretation of his work in which a distinction is drawn as a
topologically closed boundary [3]. (While SpencerBrown equates the
concepts of distinction and boundary, his notation fails to exploit this.
Drawn as a boundary, distinction appears more complete than SpencerBrown's
mark and can be interpreted in linear form as paired delimiters.) The
boundary mathematics paradigm prescribes forms of representation as well as
their mechanism of transformation.
3.2 Forms
Boundary mathematics is based on spatial forms. Representation begins with
complete lack of form and structure: empty space or void. Structure is
added to this void by cleaving the space, by drawing a distinction. Each
distinction adds structure, cleaving new spaces where further distinctions
can be made.
Drawing a distinction is independent of the dimension of the space. For
the purposes of representation, one dimensional distinctions are
typographical delimiters, two dimensional distinctions are closed loops on
a page, and three dimensional distinctions are solid objects.
3.2.1 Void
All forms stand in contrast to lack of form, void. See Figure 3.1. To begin
with anything but the void would assume too much about representation. The
void imposes no structure or intention on forms which are made. In
contrast, linear, tokenbased representations impose a great deal of
structure.
Figure 3.1: The Void.
_____
/ \
 
\_____/
Figure 3.2: A Distinction.
The void plays a vital role in boundary mathematics. It is anywhere and
everywhere; it is where all distinctions are made. The void does not go
away: it permeates all boundary representations [3].
Equivalences with the void can be easily mistaken for typographical
errors when written linearly. The equivalence form A = B puts no explicit
boundary around the expressions being equated so a void equivalence such as
A= may at first appear erroneous. Since void expressions are legitimate
in boundary mathematics, these expressions should be recognized as valid.
3.2.2 Distinction
Distinction is the primary form of boundary mathematics. A distinction
cleaves space, imposing structure and intention upon it. Distinctions serve
both a syntactic and semantic role. The meaning lies in making the
separation.
Figure 3.2 shows a distinction in the twodimensional space of this
page. The shape and scale of this distinction are irrelevant. The
distinction serves only to separate the content of the distinction from the
context in which it was made. The content and context of a distinction are
labeled in Figure 3.3.
Additional distinctions are made with respect to the first distinction.
Distinction is an action; the result of distinction is an object for
further action. There are two ways to make a second distinction: in the
content or context of the first distinction.
These choices are shown in Figure 3.4.
_________
/ \
context  content 
\_________/
Figure 3.3: Content and Context of a Distinction.
_____
/ ___ \ ___ ___
/ / \ \ / \ / \
\ \___/ / \___/ \___/
\_____/
Figure 3.4: Two Choices for Further Distinction.
Distinctions are drawn in the figures as boundaries. Here, they are
written textually as paired delimiters, (). Nested distinctions appear as
(()) and collected distinctions appear as ()():
3.2.3 Multiple Boundaries
Boundary mathematics uses a single distinction to represent logic [11, 3],
but the mathematics in this thesis uses three boundaries. Having different
boundaries requires drawing a separate distinction between the boundaries
aside from the cleaving of space. The separate distinction between the
boundary types is one of convenience, to advance boundary numbers and no
philosophical justification will be attempted. In this thesis, a
distinction and a boundary are not synonymous since (...), [...], and <...>
all represent spatial distinctions but a separate orthogonal distinction is
made between them that is not a cleaving of space.
3.3 Transformation
The dynamic element of a system of boundary mathematics is how to
manipulate forms, how to transform boundary expressions. A boundary
expression is essentially a network. The connectivity of the network
determines its value. Equivalences between connectivities defines the
system of mathematics.
3.3.1 Match and Substitute
Equivalences are described by equivalence rules which transform by
algebraic match and substitute. The rules given to define a system are its
equivalence axioms.
An equivalence rule consists of two or more boundary templates which
include template variables. To apply a rule to a target expression,
replace each template variable with an expression so that one of the filled
templates matches some part of the target expression. Another filled
template from the rule can then substitute for the matched template in the
target expression.
For example, the rule ((A)) = A includes two templates which use the
template variable, A. The target expression (((oo))o) can be matched by the
left template by replacing the A with ooto produce ((oo))=oo. Substituting
the right template for the left template in the target expression
transforms it into (ooo), essentially erasing two boundaries.
3.3.2 Properties
Transformations in boundary mathematics have some special properties. Match
and substitute can operate in parallel and at all levels of granularity in
an expression. For example, the rule ((A)) = A applies to the form
'(((())()))' twice; both applications can occur simultaneously without
conflict to reduce it to ().
In these boundary numbers, the equivalence rules are bidirectional.
The rule itself states no direction of preference, though one may be
imposed for reduction purposes. Therefore, a reduced expression is
mathematically equivalent to many more complicated forms. Because functions
are constructed with the same boundaries as are numbers, the specification
of a calculation is equivalent to a reduced result. The specification is
the resulttransformation just changes the form that it is in.
3.4 Conclusions
Boundary mathematics is a representational and computational paradigm based
on spatial forms. Boundary notation looks and acts differently than
standard notation, using spatial constraints and spatial properties rather
than rote rules.
Spatial forms differ fundamentally from standard forms. Space does not
impose ordering constraints; associativity and commutativity are not
necessary as no order or arity has been imposed in the first place.
Nonrepresentation has meaning and contributes to computation in ways
unparalleled in standard systems. The void acts as a builtin identity that
is always available. Spatial match and substitute operates at all parts of
an expression simultaneously, an inherently parallel computation mechanism.
These properties make boundary mathematics a unique and worthwhile paradigm
of representation and computation.
Many concepts generally attributed to standard mathematics are notably
absent from boundary mathematics and viceversa. Boundary notation makes no
distinction between objects and operations upon them, as both are built out
of the same forms. Standard mathematics does not allow void substitution
but boundary mathematics does. Boundary rules tend to embody symmetry of
function whereas standard mathematics addresses identity elements and
rearrangement properties. The tradeoffs are many.
Boundary mathematics is exotic enough to seem obscure and of unclear
advantage. This thesis will demonstrate the advantage of the boundary
mathematics paradigm by defining in it a calculus that simplifies number
representation and calculation.
Chapter 4
INSTANCE AND ABSTRACT
4.1 Introduction
This chapter introduces boundary numbers by defining a twoboundary
calculus. The twoboundary calculus maps to natural numbers, complete with
addition, multiplication, and power functions. The calculus also builds a
structure without parallel in standard notation: a generalized cardinality
form that applies to both addition and multiplication.
In this calculus, space is treated as addition. A single distinction
represents natural numbers by repeating the distinction for the value.
This distinction shall be called instance and drawn as a circular boundary
when empty 'o' or as parentheses otherwise '(A)'. An empty instance forms
the natural number one; it is the unit for counting. Nestings of instance
are not natural numbers; these forms will be characterized in Chapter 6.
A second distinction builds with the first to make multiplication and
power functions. This distinction shall be called the abstract and drawn as
a square boundary when empty '[]' or as square brackets otherwise '[A]'.
An empty abstract forms a nonreal number called a black hole. Nestings of
abstract produce numerical values beyond the scope of this thesis and will
not be discussed here.
Two axioms define the relationship between instance and abstract. The
involution axiom defines them as functional inverses. When either boundary
is immediately nested within the other, where nothing lies between them,
both boundaries can be erased. The distribution axiom defines a further
relationship between them: when an abstract boundary lies within an
instance boundary, the instance boundary and abstract boundary and the
content between them can be threaded across the contents of the abstract
boundary. The visual forms in Section 4.2.2 make these rules visually
apparent. The two axioms define the calculations necessary to reduce
multiplication and power expressions to the canonical structure of natural
numbers.
The two boundaries also construct a generalized form of cardinality.
Cardinality rewrites a repeated reference using a single reference combined
with a count of the repetition. In standard notation, a repeated sum is
rewritten as a product with the count, e.g. x+...+x = nx, and a repeated
product is rewritten as an exponent of the count, e.g. x*...*x = x^n. In
the boundary calculus, cardinality takes the same form in addition and in
multiplication. Because of this dual use, the boundary form of cardinality
is considered to be a generalized form of cardinality.
This chapter begins by formally defining the twoboundary calculus.
4.2 The TwoBoundary Calculus
The twoboundary calculus is introduced by defining the set of possible
constructions, the elements of the calculus, and by defining axioms to
state their equivalences.
4.2.1 Elements of the Calculus
Elements of the calculus are constructed out of the void using two boundary
distinctions: instance and abstract.
Definition. Let B denote the collection of all wellformed boundary
expressions composed of two boundaries, denoted as (...) and [...]. B is
defined recursively by three rules:
1.The void belongs to B.
2.If a belongs to B, then '(a)' and '[a]' each belong to B.
3.If b1,b2,...,bn (n > 1) belong to B, then the unordered collection
'b1 b2 ... bn' belongs to B.
The first rule establishes the void as the foundational element in the
recursive construction of B. Thus, B contains the void.
The second rule creates elements of B by nesting other elements inside
the two boundaries. The first two elements introduced by this rule are the
boundaries with no content: oand 2: Further distinction introduces the
following elements into B:
(o),([]),[o],[[]],((o)),(([])),([o]),([[]]),[(o)],[([])],[[o]],[[[]]].
The third rule creates new elements of B by collecting previous
elements. These collections are unordered so permutations are irrelevant.
This lack of order means that the form '()[]' is considered identical to
'[]()'. Collection introduces the following elements into B:
oo,o[],[][],o(o),[](o),(o)(o),ooo,oo[],o[][],[][][].
Between the three rules, deep and wide expressions are introduced into
B, including the forms: ([ooo][oo]) and (([[ooo]][oo])).
4.2.2 Equivalence Axioms
Elements in the twoboundary calculus are related by two equivalence
axioms: involution and distribution. The involution axiom defines a
symmetric relationship between the two boundaries.
Axiom 1 (Involution) Instance and abstract are functional inverses:
([A]) = A = [(A)].
Involution allows the removal or introduction of instanceabstract
pairs, including void equivalents when A is void, ([])=[()]= . Involution
does not apply to a pair with something lying between the boundaries; it
can neither introduce or remove a configuration such as (o[ooo]), where 'o'
lies outside of the abstract and more than one instance lies within it.
Acceptable examples of involution are shown in Figure 4.1.
The distribution axiom defines an asymmetric relationship between the
two boundaries.
Axiom 2 (Distribution) An instance around abstract distributes over the
contents of the abstract:
(A[BC]) = (A[B])(A[C]).
Distribution manipulates the modifier form, '(A[...])', composed of
instance, the template variable A, and abstract. It threads the modifier
form over the contents of its inner abstract boundary. Distribution states
that the form can modify these contents collectively or separately. Unlike
involution, distribution matches a pattern lying between the two
boundaries. Examples of distribution are shown in Figure 4.2.
These two axioms define the transformational basis of twoboundary
expressions.
Involution: ([A]) = A = [(A)]
Equivalent Expressions Template Replacement
([oo])
oo A = oo
( ([ooo][oo] ))
(([([ooo][oo])])) A = ([ooo][oo])
[ (o)([(oo) (ooo)])]
[ (o) (oo) (ooo) ] A = (oo)(ooo)
[([(o) (oo)])(ooo) ] A = (o)(oo)
( [o][([oo] [ooo])])
( [o] [oo] [ooo] ) A = [oo][ooo]
([([o] [oo])][ooo] ) A = [o][oo]
([ooo] )
([ooo][o]) A =
(([[(o)]][ooo]))
(([ o ][ooo])) A = o
(( [ooo])) A =
( ooo ) A = ooo
Figure 4.1: Examples of Involution.
Distribution: (A[BC]) = (A[B])(A[C])
Equivalent Expressions Template Replacement
([ooo][o])([ooo][])
([ooo][o ]) A = [ooo], B = o, C =
(([[oooo]][]))
(([[oooo]][])([[oooo]][])) A = [[oooo]], B = , C =
([oo][oooooo])
([oo][ooo])([oo][ooo]) A = [oo], B = ooo, C = ooo
([oo oo][ooo]) A = [ooo], B = oo, C = oo
Figure 4.2: Examples of Distribution.
4.3 Natural Numbers
The elements of the twoboundary calculus map to natural numbers, complete
with arithmetic operations of addition, multiplication, and exponentiation.
Space is treated as addition so that collecting elements adds them.
Natural numbers are constructed simply by accumulating the unit formed by
the empty instance boundary, 'o'. The elements 'o', 'oo', 'ooo',... form
the set of natural numbers, N, within the boundary calculus. The successor
function, S(A) > Ao, inductively builds N from the element o:
4.3.1 Addition
Elements a,b in N add by collection, 'ab'. The set N is closed under
addition because collecting elements can only form an element of equal or
greater cardinality, which still an element of N
Addition is commutative because spatial collection is unordered.
Addition is associative because spatial collection makes no grouping
distinctions for multiple additions. The additive identity is the void: an
element remains uneffected when collected with the void.
When adding these natural numbers, no calculation is necessary. They are
initially phrased in their canonical form. For example, 3 + 2 rewrites as
'ooooo'.
4.3.2 Multiplication
Elements a,b in N multiply by the form ([a][b]). The set N is closed under
the multiplication.
Proof. The product of a in N and initial element, o, reduces to a by
involution, ([a][o])=([a])=a. If the product of a and b, ([a][b]), is in
the N, the product of a and the successor of b is also in the set,
([a][bo]) = ([a][b])([a][o]) = ([a][b])a.
Because addition is closed, this result is in N. Therefore, by induction
all products of a,b in N are in N.
Multiplication is commutative because [a] and [b] are unordered within
the outer instance. Associativity of binary multiplication is shown by two
applications of involution:
([([a][b])][c]) = ([a][b][c])= ([a][([b][c])]):
The multiplicative identity is the unit, o. An element multiplied by
the unit reduces to itself by involution,
([a][o]) = ([a]) = a.
In the boundary calculus, the void represents zero. Accordingly,
multiplication by the void is void, ([a][]) = .
Proof. By distribution, a product with the void is equal to two copies of
the same,
([a][ ]) ([a][ ]) = ([a][ ]).
The equation ee = e is true when e is void or when collection of e is
idempotent. Since collection is assumed additive, ee = e unless e is
void. Therefore, ([a][ ]) = .
Multiplication of natural numbers requires some calculation to return
values to the canonical form of collections of units. This reduction uses
distribution to strip away units one at a time, along with involution to
remove unit multiplications.
3 * 2 Given
ooo * oo Number Rewrite
([ooo][oo]) Multiplication Rewrite
([ooo][o])([ooo][o]) Distribution
([ooo] )([ooo] ) Involution
oooooo Involution
4.4 Cardinality
In the boundary calculus, repeated references to the same element can be
rewritten as a cardinality of that element.
Theorem 1 (Cardinality) Multiple references in the same context may be
reduced to a single reference with multiple units signifying its quantity,
as
A...A = ([A][o...o]).
Cardinality differs from multiplication in that this transformation is
not semantically tied to its context, as it is in multiplication.
Cardinality applies at any depth.
Everything has a count of one, shown directly by involution:
A = ([A]) = ([A][o]).
Here, the lone odenotes the single cardinality of a:Greater
cardinalities can be formed by collecting many units in this space using
distribution of separate single cardinalities. For example, two references
can be distributed into a cardinality of two:
AA = ([A][o])([A][o]) = ([A][oo]).
The induction step to all natural numbers is,
A...AA = ([A][o...o])([A][o]) = ([A][o...oo]).
4.4.1 Dominion
A zero cardinality, '([A][ ])', reduces to the void as a void
multiplication. In this form, the empty abstract, '[ ]', dominates its
context. This element is called a black hole because of this collapsing
property. A general form of this collapsing derives from void
multiplication.
Theorem 2 (Dominion) Elements collected with a black hole are irrelevant,
as
[] A = [].
Dominion is proven from void multiplication with an involution:
[] A = [([] A)] = [([])] = [].
4.4.2 Generalized Cardinality
The cardinality form is independent of its context and can be applied to
repetition in both addition and multiplication. Cardinality applies to all
repetition in the same context, regardless of the form repeated or the
depth of this context. Any collection of identical elements can be counted
this way.
Cardinality applies to addition directly. It applies to multiplication
by counting abstracted elements inside the context of a surrounding
instance boundary. For example, ([a][a])= (([[a]][oo])). Cardinality of
multiplication translates to exponentiation. Cardinality appears in
standard and boundary numbers as:
x + ... + x = n*x versus x...x = ([x][o...o])
x * ... * x = x^n versus ([x]...[x]) = (([[x]][o...o]))
4.5 Conclusions
Though a single boundary represents natural numbers, a second boundary
provides convenient forms for algebraic multiplication and exponentiation.
The involution and distribution axioms completely define computation on the
natural numbers for these functions. The twoboundary calculus is
summarized in Table 4.1.
Table 4.1: The TwoBoundary Calculus.
Numbers Operators Rules
0 > x+y > xy ([A]) = A = [(A)]
1 > o x*y > ([x][y]) (A[BC]) = (A[B])(A[C])
2 > oo x^y > (([[x]][y]))
3 > ooo
The twoboundary calculus constructs a generalized form of cardinality.
This form applies to all situations of repeated reference in space,
including addition and multiplication. It provides a form that is
unavailable in standard notation. Though cardinality is an important
concept, standard notation does not have a form for it that is independent
of context. Cardinality provides a substrate for building a base system and
thus a route to remove the representational inconvenience of the unary
notation for numbers in Table 4.1.
The twoboundary calculus is similar to SpencerBrown's numbers, except
that it uses two boundaries where SpencerBrown uses just one [10]. The
additional boundary disambiguates the situations that his system had
dioculty with. It still includes remnants of his logical arithmetic with
involution and dominion. Ideally, this calculus would be defined as a
numerical arithmetic, from rules such as involution and dominion, and
generalized to algebraic axioms.
Chapter 5
INVERSE
5.1 Introduction
This chapter extends boundary numbers to integers, rationals, and algebraic
irrationals by introducing a third boundary to serve as a generalized
inverse.
The first two boundaries, instance and abstract, remain exactly as
defined in Chapter 4. The first two axioms, involution and distribution,
are also the same.
The third boundary extends this system with a generalized inverse. This
boundary shall be called the inverse boundary and be written as a triangle
when empty, '<>', or as angled brackets otherwise, ''. An inverse
boundary with no content is equivalent to void. The inverse boundary is
its own functional inverse, so nesting inverse inside of inverse cancels
out.
The characteristic property of the inverse boundary is defined by the
inversion axiom. An element and its inverse cancel to the void. Similarly,
an element and its inverse can be introduced from the void. All elements
have an inverse except for the black hole.
Inverse serves directly as the additive inverse (i.e. x) and, in a
construction with instance and abstract, as the multiplicative inverse
(i.e. 1/x). Reduction of both inverses is handled by the inversion axiom,
cancelling a form collected with its inverse. The context of this
cancellation unambiguously determines the semantics as the subtraction
(i.e. xx = 0) or division (i.e. x/x = 1). Because the inverse boundary
serves both addition and multiplication, it is considered to be a
generalized inverse.
This chapter begins by redefining the calculus with three boundaries.
5.2 The ThreeBoundary Calculus
The twoboundary calculus from Chapter 4 is extended with a third boundary
by redefining the set of elements to include it and by adding a third axiom
to define its transformations.
5.2.1 Elements of the Calculus
Elements of the calculus are constructed out of the void using three
boundary distinctions: instance, abstract, and inverse.
Definition. Let B denote the collection of all wellformed boundary
expressions composed of three boundaries, denoted as ();[];and <>:B is
defined recursively by four rules:
1.The void belongs to B.
2.If a belongs to B, then '(a)' and '[a]' each belong to B.
3.If b belongs to B and b != [], then '' belongs to B.
4.If c1,c2,...,cn (n > 1) belong to B, then the unordered collection
'c1 c2 ... cn' belongs to B.
Rules 1, 2 and 4 are identical to those in Chapter 4. The first rule
establishes the void as the foundational element in B. The second rule
creates elements by nesting elements in the instance and abstract
boundaries. The fourth rule creates elements by collecting other elements.
The new, third rule creates elements by nesting other elements in the
inverse boundary. It works identically to the second rule, except that the
black hole is excluded. In other words, <2> is undefined. The first element
introduced by this rule is the inverse boundary with no content: 4: The
distinction rules (2 and 3) introduce the following elements into B from
the void:
(<>),[<>],<<>>,,(),[],<(o)>,<([])>,<()>,<[]>,<<(o)>>.
Between the four rules, deep and wide expressions are introduced into B,
including the forms: <(<[ooo]>)> and (([[]]<[oo]>)).
5.2.2 Equivalence Axioms
Elements in the threeboundary calculus are related by three equivalence
axioms: involution, distribution, and inversion. The involution and
distribution axioms are defined as before.
Axiom 1 (Involution) Instance and abstract are functional inverses:
([A]) = A = [(A)].
Axiom 2 (Distribution) An instance around abstract distributes over the
contents of the abstract:
(A[BC]) = (A[B])(A[C]).
The new axiom of inversion defines transformations with the inverse
boundary.
Axiom 3 (Inversion) An element and its inverse are equivalent to void:
A = .
Inversion matches if the complete contents of an inverse boundary are
found in the context of that boundary. Examples of inversion are given in
Figure 5.1.
Inversion uses void substitution. An element and its inverse can be
introduced anywhere in a boundary expression, so long as both elements are
put into the context. Likewise, an element and its inverse can be removed
from a boundary expression, so long as both reside in the same context.
These three axioms define all of the transformations of the calculus.
5.3 The Additive Inverse
The inverse boundary serves directly as the additive inverse. The additive
inverse extends the natural numbers built in Chapter 4 to the integers.
Adding an element to its additive inverse reduces to the additive
identity, the void. For example, 3 3 = 0 translates to ooo=:The
additive inverse appears generally in standard and boundary numbers as:
x + (x) = 0 versus x = .
Inversion: A =
Equivalent Expressions Template Replacement
oooo
oo A = oo
([ooo])
([ooo][oo]<[oo]>) A = [oo]
<<(ooo)>>
<<(ooo)>><(ooo)>(ooo) A = (ooo)
(ooo) A = <(ooo)>
Figure 5.1: Examples of Inversion.
5.3.1 Properties of the Additive Inverse
The inverse boundary can be manipulated in ways analogous the the additive
inverse. The basic properties of the inverse are outlined by three
theorems: inverse collection, inverse cancellation, and inverse promotion.
The theorem of inverse collection transforms collected inversions into a
single inversion and the theorem of inverse cancellation removes pairs of
nested inverses.
Theorem 1 (Inverse Collection) A collection of inverted elements equals an
inversion of the collection, as
= .
A proof of inverse collection follows from inversion:
= AB = .
Theorem 2 (Inverse Cancellation) The inverse boundary is its own functional
inverse, as
< > = A.
A proof of inverse cancellation also follows from inversion:
<> = <>A = A.
Besides collecting and cancelling, the inverse operation can be
"promoted" over the modifier form, (A[...]). The theorem of inverse
promotion derives this property.
Theorem 3 (Inverse Promotion) An inverse boundary can be promoted over the
composite boundary, (A[...]).
<(A[B])> = (A[]).
Proof. For all A,B,C in B, B != [];
<(A[B])> Given
<(A[B])> (A[ ]) Void Multiplication
<(A[B])> (A[B]) Inversion
<(A[B])> (A[B])(A[]) Distribution
(A[]) Inverse
These three theorems appear in standard and boundary numbers as:
(x) + (y) = (x + y) versus =
(x) = x versus <>= x
(xy) = x(y) versus <(x[y])>= (x[])
5.3.2 Integers
Since the boundary calculus now includes the additive inverse, natural
numbers can be extended to the integers. The instance boundary serves as a
unit for the natural numbers and the inverse boundary inverts that unit.
The negative unit is given by :These numbers add by spatial collection
and multiply by the form using the abstract boundary.
The void and elements o,,oo,,ooo,,... compose the set I of
integers within the boundary calculus. The successor function is S(A) > Ao
and the predecessor function is P(A) > A. From the void, successors to
the void take the form o... and predecessors to the void take the form
. The predecessor function gives , which equals by
inverse collection.
Boundary integers add by collection, as 'ab', and are closed under
addition.
Proof. Elements a,b in I assume one of three forms: the void, a positive
integer, or a negative integer.
1.If either is the void, the sum reduces to the other value, which is in
I.
2.If both are positive integers, the sum is comprised of all units,
which is in I.
3.If both are negative integers, the sum is of the form
which equals by inverse collection. This form is in I.
4.If a and b are of opposite sign and the positive number has fewer
units, the negative number can be split by inverse collection and that
part cancelled out, e.g. o... = o...=.
This result is in I.
5.Otherwise, if a and b must be of opposite sign with the positive
number having more units. In this case, the negative number cancels
directly with part of the positive value, e.g. o...o... = o....
This result is in I.
Addition is commutative because spatial collection is unordered.
Addition is associative because spatial collection makes no grouping
distinctions for multiple additions. The additive identity is still void.
Every element in I has an additive inverse.
Proof. The inverse of the void is the void, <>= ; derived directly by
inversion. All successors to the void, the natural numbers of the form
o..., have as additive inverses the same value wrapped by inverse,
. All predecessors to the void, the natural numbers wrapped by the
inverse boundary as , have as additive inverses the same value
wrapped again by the inverse <>, which reduces to the uninverted
natural number of the form o... by inverse cancellation.
Integers a,b in I multiply with the same form used for natural numbers,
([a][b]). The integers are closed under multiplication.
Proof. Elements a,b in I each assume one of three forms: the void, a
positive integer, or a negative integer. Let c,d in N be the magnitude of
a,b respectively.
1.When either is the void, the product reduces to the void by void
multiplication, which is in I.
2.When both are positive, the product is in I because natural numbers
are closed under multiplication.
3.When one of them is negative, as ([c][]); the product is the
inverse of the product of their magnitudes, <([c][d])>, by inverse
promotion. Since the product of two natural numbers is a natural
number and the inverse of a natural number is an integer, this result
is an integer.
4.When both numbers are negative, as ([][]);the product is the
inverseinverse of the product of their magnitudes, <<([c][d])>>;by
inverse promotion applied twice. This reduces to ([c][d]);the product
of two natural numbers, which is a natural number.
Multiplication is commutative because [a]and [b]are unordered within the
outer instance. Multiplication is associative by two applications of
involution:
([([a][b])][c]) = ([a][b][c]) = ([a][([b][c])]).
5.3.3 Calculation
Calculation with the additive inverse is straightforward, seeking to
eliminate abstract and multiple inverses. Subtraction requires matching of
positive and negative quantities:
3  5 Given
ooo  ooooo Number Rewrite
ooo Subtraction Rewrite
ooo Inverse Collection
Inversion
Multiplication requires management of the inverse.
3 * 2 Given
* Number Rewrite
([][]) Multiplication Rewrite
<([][ oo ])> Inverse Promotion
<<([ ooo ][ oo ])>> Inverse Promotion
([ ooo ][ oo ]) Inverse Cancellation
([ ooo ][o])([ooo][o]) Distribution
ooo ooo Cardinality
5.4 Multiplicative Inverse
The inverse serves as the multiplicative inverse when set between the
instance and abstract boundaries, as '(<[a]>)'.
Multiplying something against its additive inverse reduces to the
multiplicative identity, the unit. For example, 3/3 = 1 translates to
([ooo]<[ooo]>) = (). The multiplicative inverse appears in standard and
boundary numbers as:
x/x = 1 versus ([x]<[x]>) = ()
5.4.1 Properties of the Multiplicative Inverse
The theorems for the additive inverse also apply to the multiplicative
inverse. For simplicity in this discussion, some involution steps are
assumed. For example, when multiplying by a multiplicative inverse,
involution immediately applies:
([a][(<[b]>)]) = ([a]<[b]>).
Inverse collection states that the product of two inverted numbers
equals an inversion of their product. Inverse cancellation states that
multiplicative inverse cancels itself out. These two theorems for
multiplication appear in standard and boundary numbers as:
(1/x)(1/y) = 1/(xy) versus (<[x]><[y]>) = (<[x][y]>)
1/(1/x) = x versus (<[(<[x]>)]>) = x.
Inverse promotion carries the inverse over the modifier form, as
<(a[b])> = (a[]). This theorem translates to standard numbers as the
conversion between the multiplicative inverse of a value and the additive
inverse of its exponent. Inverse promotion for multiplication appears in
standard and boundary numbers as:
(1/x)^y = x^(y) versus (<([[x]][y])>) = (([[x]][])).
5.4.2 Rationals
Because the boundary calculus now includes the multiplicative inverse,
integers can be extended to rationals. The multiplicative inverse of a is
simply (<[a]>): For n,d in I with d != , the element ([n]<[d]>) is in Q,
the set of rationals.
The rationals, a,b in Q add by collection aband are closed under
addition.
Proof. Rational numbers a,b in Q take the form a=([i]<[j]>) and
b=([k]<[l]>), where i,k,j,l in I with j,l != . The sum of these rationals
is ab=([i]<[j]>)([k]<[l]>), which reduces to rational c=([m]<[n]>), where
m=([i][l])([k][j]) and n=([j][l]), by the following derivation.
( [i] <[j]>) ( [k] <[l] >) Given
( [i][l] <[l]><[j]>) ( [k][j] < [j]><[l] >) Inversion
( [i][l] <[l] [j]>) ( [k][j] < [j] [l] >) Inverse Collection
([([i][l])]<[l] [j]>) ([([k][j])]<[([j] [l])]>) Involution
([([i][l]) ([k][j])]<[([j] [l])]>) Distribution
([ m ]<[ n ]>) Rewrite
Since the product of two integers in I is an integer, n is an integer.
Because j and l are nonvoid, their product is nonvoid. Products ([i][l])
and ([k][j]) are also integers. Since the sum of two integers is an
integer, m=([i][l])([k][j])is also an integer. Therefore, c is rational, c
in Q, and addition is closed under the rationals.
Every element in Q has an additive inverse, found simply by wrapping the
inverse boundary around it.
Rationals, a,b in Q multiply by the same form used for natural numbers,
([a][b]). The rationals are closed under multiplication.
Proof. Rational numbers a,b in Q take the form a=([i]<[j]>) and
b=([k]<[l]>), where i,k,j,l in I with j,l != . The product of these
rationals is also in Q because the product of a and b reduces to the
rational ([m]<[n]>), where m=([i][k]) and n=([j][l]).
([ a ][ b ]) Given
([([i]<[j]>)][([k]<[l]>)]) Replacement
( [i]<[j]> [k]<[l]> ) Involution
( [i] [k] <[j] [l]> ) Inverse collection
([([i] [k])]<[([j] [l])]>) Involution
([ m ]<[ n ]>) Replacement
Since the product of two integers is an integer, both m and n are integers.
Because j and l are nonvoid, n is also nonvoid. Therefore, ([m]<[n]>) is
rational and so the product of a and b is rational.
Every element in Q, except for the void, has a multiplicative inverse.
Proof. Rational number a in Q takes the form a=([i]<[j]>), where i,k in I
with j != . Since a is nonvoid, the numerator is also nonvoid, i != . The
multiplicative inverse of a, given by (<[a]>), is rational by the following
reduction back to the rational form.
(<[a]>) = (<[([i]<[j]>)]>) = (<[i]<[j]>>) = (<[i]>[j])
5.4.3 Division by Zero
Dominion, a [] = [], loses information and necessitates excluding <[]> from
the calculus. This restriction prevents formation of division by zero,
(<[]>): This restriction appears in standard and boundary numbers as:
1/0 undefined versus <[]> undefined.
5.4.4 Calculation
Computing a division requires a matching of quantities. When quantities
cannot be matched, the fraction does not reduce. Thus, the calculus
constrains this calculation but does not guide the reduction.
6/2 Given
([oooooo]<[oo]>) Rewrite
([([ooo][oo])]<[oo]>) Cardinality
( [ooo][oo] <[oo]>) Involution
( [ooo] ) Inversion
ooo Involution
5.5 Inverse and Cardinality
The inverse applied to the cardinality form gives negative and fractional
cardinalities.
5.5.1 Negative Cardinality
A negative cardinality is one which cancels out with a positive cardinality
of equal magnitude. A negative cardinality is indicated by the inverse
boundary around the units. For instance, a cardinality of negative two
cancels with a cardinality of two.
([a][oo])([a][]) = ([a][oo]) = ([a][]) =
An inverted multiplicative form is changed to a negative cardinality by
the theorem of inverse promotion.
<([a][o...])> = ([a][])
For addition, this negative cardinality translates to multiplication by
a negative integer. For multiplication, this negative cardinality
translates to exponentiation by a negative integer. These forms appear in
standard and boundary numbers as:
(nx) = (n)x versus <([x][o...])> = ([x][])
(1/x)^n = x^(n) versus (<([[x]][o...])>) = (([[x]][])).
5.5.2 Fractional Cardinality
The multiplicative inverse differs from the additive inverse by placing the
inverse boundary outside of an abstract boundary, instead of inside of it.
Doing the same to the cardinality construction makes fractional
cardinalities.
For example, a halfcardinality is given by ([a]<[oo]>). Collection of
two of these reduces to a:
([a]<[oo]>)([a]<[oo]>) Given
([([a]<[oo]>)][oo]) Cardinality
( [a]<[oo]>[oo]) Involution
( [a] ) Inversion
a Involution
Applied to addition combinations, fractional cardinality builds
fractions. The form ([a]<[oo]>) translates to a/2. In the above proof this
fraction is doubled, as
a/2 + a/2 = 2(a/2) = a(2/2) = a.
Applied to multiplicative combinations, fractional cardinalities build
roots. The form (([[b]]<[oo]>))translates to b^(1/2). In a proof similar to
the above but with a>[b], this root is squared to double the cardinality,
as
b^(1/2) b^(1/2) = (b^(1/2))^2 = b^(2/2) = b.
Since fractional cardinalities form roots, they allow construction of
the algebraic irrationals.
5.6 Conclusions
Though the twoboundary calculus only represents natural numbers, simply
adding an inverse boundary extends the boundary numbers through the
algebraic irrationals. The three axioms completely define computation on
these numbers and functions. The threeboundary calculus is summarized in
Table 5.1.
The inverse boundary serves many inverse functions due to the structures
provided by the instance and abstract boundaries. It is defined in the most
fundamental way that an inverse can be defined, utilizing the void to
cancel inverses to nothing. The spatial formalism makes this inverse
definition possible and powerful. Kauffman defined an inverse boundary of
this sort but did not have the other structures to utilize it in a general
form.
The inverse is powerful but it introduces problems, such as division by
zero. Though the twoboundary calculus was solid, the threeboundary
calculus has a single exception, preventing rules from applying to all
configurations. When performing a computation, constraints on variables
must be propagated through to avoid paradoxes.
Table 5.1: The ThreeBoundary Calculus
Numbers Operators
0 > x > x
1 > o x >
2 > oo 1/x > (<[x]>)
1 > x+y > xy
2 > xy > x
2 > oo x*y > ([x][y])
1/2 > (<[oo]>) x/y > ([x]<[y]>)
2/3 > ([oo]<[ooo]>) x^y > (([[x]][y]))
3^(1/2) > ( ([[ooo]]<[oo]>)) x^(y) > (([[x]][]))
2^(1/3) >( ([[oo]]<[ooo]>)) x^(1/y) > (([[x]]<[y]>))
Rules
([A]) = A = [(A)]
(A[BC]) = (A[B])( A[C])
A =
Chapter 6
PHASE
6.1 Introduction
This chapter extends the boundary calculus to complex numbers and basic
transcendentals by interpreting boundaries functionally and by adding some
manipulative constraints.
The inverse boundary can be put into an arithmetic form to which
cardinality can be applied (the boundary itself cannot be counted). This
form is called the phase element, or J, and it translates to the radian
value iss in standard numbers. Fractional cardinalities of J build into the
complex roots of one, just as fractions of iss determine a radian angle in
the complex plane. This property of J undermines additive space because it
has indeterminate sign and ambiguous magnitude, a complication similar to
using radian numbers to represent complex numbers. This issue is addressed
in terms of cardinality and inverse.
The instance and abstract boundaries act as exponential and logarithmic
functions, respectively. Interpreting these as the natural exponent, e^x
> (x), and the natural logarithm, ln x > [x], introduces these
transcendental functions into the calculus. With these functions, basic
transcendentals such as e and pi can be constructed.
6.2 Phase
When manipulating boundary expressions, the inverse boundary falls into an
arithmetical form that cardinality can apply to.
Recall the concept of negative cardinality, = ([a][]). written as
a=a*(1) in standard notation. In the boundary form, the inverse boundary
has been separated from what it previously contained, into the encapsulated
form []:This form shall be called J:
J = [].
J has a stability unparalleled in the calculus: it uses all three
boundaries exactly once. All other legal configurations of three different
boundaries reduce to void because the abstract and instance boundaries
cancel out. (The sixth case, (<[]>), is undefined.)
[(<>)], ([<>]), <[()]>, <([])>.
6.2.1 Phase Independence
The configuration of three boundaries also serves as phase operator, given
as [<(A)>]: This operator possesses a curious property of independence from
its contents.
Theorem 1 (Phase Independence) In a nesting of abstract, inverse, and
instance boundaries, the contents of the instance can be moved to the
context of the configuration.
Proof. For all A in B,
[<(A)>] Given
[<(A)> ( [ ])] Involution
[<(A)> (A [ ])] Dominion
[<(A)> (A [() <()>])] Inversion
[<(A)> (A [()])(A [<()>])] Distribution
[<(A)> (A )(A [<()>])] Involution
[ (A [<()>])] Inversion
A [<()>] Involution
Phase independence simplifies manipulations with the inverse. In
particular, it simplifies the derivation of negative cardinality, as
= ([<([A])>]) = ([A][]).
Phase independence also gives a different proof of inverse promotion.
Rather than generate, distribute, and cancel (as in the proof in Section
5.3.1), with phase independence, the inverse is wrapped into the phase
operator and moved.
<(A[B])> = ([<(A[B])>]) = (A[<([B])>]) = (A[]).
Using J, two concise and useful forms of inverse promotion become
possible: the inverse jumping inside of the instance boundary and the
inverse jumping outside of the abstract boundary:
<(A)> = ([<(A)>]) = (A[<()>]) = (AJ),
[] = [<([A])>] = [A][<()>] = [A]J.
6.2.2 Oscillation
The phase element has the property of being its own inverse. Only zero
exhibits this property in rational numbers.
Theorem 2 (J Cancellation) J cancels with itself.
[][] = [<([])>] = [<>] = [o] =
That J is its own inverse should be of no surprise, since the inverse
boundary is its own functional inverse and J embodies the inverse.
In light of this, J can be built into an oscillation function,
osc(A)>AJ. Every second application causes the Js to cancel out,
producing the sequence:
> J > > J > > J > ...
Wrapping the elements of this sequence with the instance boundary
produces a more familiar sequence. Instead of starting with void, this
sequence starts with 'o' and uses the oscillation function osc(A)>([A]J),
producing this sequence:
o > (J) > o > (J) > o > (J) > ...
Since (J)= , the function is just a multiplication by negative one.
The sequence translates to:
1 > 1 > 1 > 1 > 1 > 1 > ...
6.2.3 Cardinality of J
The periodicity of this oscillation can be changed by applying cardinality
to J. For example, the halfcardinality of J produces a fourstep
oscillation with the function osc(A)>A([J]< [oo]>):
> ([J]<[oo]>) > J > J([J]<[oo]>) > > ([J]< [oo]>) > ...
Wrapping the elements of this sequence with the instance boundary again
produces a more familiar sequence. The oscillatory function this time is
osc(A)>([A]([J]<[oo]>)):
o > (([J]<[oo]>)) > (J) > (J([J]<[oo]>)) > o > (([J]< [oo]>)) > ...
Interpreting (([J]<[oo]>)) as i reveals the oscillation function to be a
multiplication by i, producing oscillation between 1 and 1, this time in
foursteps:
1 > i > 1 > i > 1 > i > ...
The oscillation function, osc(A)>([A]([J]< [oo]>)), is a halfinverse
operation. Similarly, other cardinalities of J can be taken to produce
other periodicities, forming the roots of unity.
6.3 Multiple Value
A halfinverse operation would be a powerful addition to the boundary
calculus, except that cardinality of J is syntactically unstable. In
particular, J has ambiguous sign and ambiguous (multivalued) cardinality.
Ambiguous sign J =
Ambiguous cardinality = JJ = JJJJ = JJJJJJJJ = ...
The second property follows directly from Jcancellation. The first
property follows from inversion and Jcancellation.
J = J<> = J = J = .
The ambiguous sign makes the cardinalities of J have ambiguous signs
also, disrupting the oscillation points. For example,
([ J ]<[oo]>) Given
([]<[oo]>) Ambiguous Sign
<([ J ]<[oo]>)> Inverse Promotion
The ambiguous cardinality of J occurs at all values.
([ J ] <[oo]>) Given
([JJJ]< [oo]>) J Cancellation
([ J ] <[oo]>)([JJ]< [oo]>) Distribution
([ J ] <[oo]>)([J][oo]<[oo]>) Cardinality
([ J ] <[oo]>)([J]) Inversion
([ J ] <[oo]>) J Involution
J undermines the calculus because of these ambiguities. Cardinalities of
J are useful as they provide complex numbers. Resolving this ambiguity
restores the integrity of the calculus and extends it to complex numbers.
Two steps accomplish this:
1.Convert inverse boundaries to Js. Whenever the inverse boundary is
wrapped by the abstract boundary, as [], pull the inverse out, as
[A]J.
2.Limit J cancellation. Allow Js to cancel only within an instance
boundary and after applying other reductions.
These constraints are not exhaustive but they do approximate the
constraints on collapsing complex numbers to radian values within complex
exponentials.
6.4 Exponentials and Logarithms
The instance and abstract boundaries can be functionally interpreted as an
exponential operation and a logarithm operation, respectively. The base of
these operations is not constrained by the calculus. Though it is not
mandatory to specify, some choices are nevertheless convenient. For
example, using base two or base ten directly provides logarithms and
exponents to that magnitude.
Obviously the "natural" choice is to use base e, creating the
interpretations e^x > (x) and ln x > [x]. With these functions, the
calculus can construct basic transcendental values and the trigonometry
functions.
Multiplication in the calculus can be seen as an adding of logarithms:
xy = e^ln(xy) = e^(ln x + ln y) > ([x][y]).
Exponentiation in the calculus can be seen as a multiplication of a
logarithm:
x^y = e^ln(x^y) = e^(y lnx) = e^(e^(ln(y lnx))) = e^(e^(lny+ln(ln x)))
> (([[x]][y]))
The logarithmic formulas connect multiplicative operations with additive
operations. Because the forms of calculus can be interpreted as logarithms,
these transformations are trivial.
ln(xy) = lnx + lny versus [([x][y])] = [x][y]
ln(x/y) = lnx  lny versus [([x]<[y]>)] = [x]<[y]>
ln(x^y) = y lnx versus [(([[ x]][y]))] = ([[x]][y])
6.5 Transcendentals
Transcendental numbers, such as Euler's number and an algebraic pi, can be
represented in the calculus as can basic trigonometric functions.
6.5.1 Euler's Number, e
The instance boundary acts as an exponential. A double nesting of instance
equals its base, e^e^0 = e > (o):Setting the base of the function to
Euler's number makes this boundary form represent Euler's number. This
nesting does not reduce to any other boundary form, making the
transcendental value incommensurable with previously defined numbers.
6.5.2 Pi, ss
Interpreting the abstract boundary as the natural logarithm makes J =
ln(1) = i*pi. This interpretation fulfills Euler's formula translated to
J wrapped by instance:
1 + e^(i*pi) = 0 versus o([]) =
From J and i, a radian interpretation of pi can be algebraically
constructed.
pi = 1 * i * i*pi
= ([][(([J]<[oo]>))][J])
= (J[J]([J]<[oo]>))
This construction of pi treats it not as a real number but as the radian
unit. Its semantic value comes out of this algebraic construction and has
no relation to its numerical value of pi = 3:14159... .
The numerical values of e and pi in standard mathematics are based on
criteria independent of the manipulations used here (e.g. geometry) and
are not essential to the algebraic behavior of the transcendentals. In the
calculus, they are devoid of numerical value and remain incommensurable
with other quantities. The boundary representations of these basic
transcendental values are given in Table 6.1.
Table 6.1: Transcendentals in the Calculus.
Transcendental Standard Boundary
J i*pi []
i (1)^(1/2) (([J]<[oo]>))
pi 3.1415 (J[J]([J]<[oo]>))
e 2.7183 (o)
6.5.3 Trigonometric Functions
In the calculus, trigonometric functions can be computed symbolically using
the three boundary rules. The function for cosine in standard and boundary
forms appears as:
e^(ix) + e^(ix)
cos x = ________________
2
versus
cos x = ([(([x]([J]< [oo]>)))(([]([J]< [oo]>)))]<[oo]>).
The cosine function behaves as expected, reducing to real results. For
instance, cos pi = 1. Although the derivation can be lengthy and
complicated, it is based entirely on the three axioms and the theorems that
follow from them. Accordingly, trigonometric formulas can be derived from
these axioms.
6.6 Conclusions
The threeboundary calculus supports forms which act algebraically as
complex numbers and basic transcendentals.
Surprisingly, the simple forms of the calculus extend to the more
advanced mathematics of trigonometry and complex exponentials, which
demonstrate the basic concept of manipulating the inverse.
Some simple observations of transcendental numbers arise in this
discussion. The phase behavior of complex exponentials is in no way tied to
the numeric values of e and pi. Assigning them numeric values serves
altogether different purposes (e.g. derivatives).
Undefined and uninterpreted, the base of instance and abstract is
transcendental because (o) is incommensurable with any other boundary
construction. An equality could be introduced that defined this to be
nontranscendental, such as base two with the definition (o) = oo.
Everything in the calculus would still hold as none of it is dependent on
this value.
Chapter 7
FUTURE WORK
7.1 Introduction
The ability to form and manipulate numerical expressions is a small part of
number mathematics and numbers are just one domain of mathematics. The
minimalist techniques of boundary mathematics can be applied to the larger
context of numbers and to additional areas of mathematics. The
representational paradigm may ultimately encompass much of mathematics.
Here, I consider the future of the calculus of number. The advancement
of the calculus may proceed in three directions: towards wider coverage,
that boundary numbers may be welldefined within the structures known to
elementary algebra; towards practicality, that techniques for learning and
working with math can be rediscovered with this new conceptualization; and
towards extended uses of numbers, that integral calculus and other
transformations may be reconceptualized under boundary mathematics. If they
are suociently expanded boundary numbers may find practical use.
7.2 Coverage
The mathematics of number is extensive and detailed. This calculus of
number covers only part of it, showing that algebraic expressions and
arithmetic computation can be done with boundaries. The calculus remains
without many structures that would be required to do number mathematics.
Arithmetic definition. The given definition of the calculus is not an
ideal one because it presents an algebraic definition. A proper definition
would derive the algebraic axioms by generalizing from arithmetic laws on
the basic forms. Choices of axioms need to be assessed and compared for
their utility in describing the dynamics of the forms, working towards
arithmetic laws.
Resolve paradoxes. This definition is not thorough because the
paradoxes of multiplevalue and zero singularity are not completely
resolved. The traditional notation addresses these paradoxes by considering
the domain and range of each function and then limiting rule application
accordingly. This approach does not clearly translate to the boundary forms
because the approach relies heavily on a fairly wide context of forms (i.e.
numerical class) to determine when a rule applies. In the boundary
calculus, the paradoxes must be considered at the level of boundaries, with
as little dependence on number type as possible. A preliminary attempt at
this was made in Chapter 6.
Additional functions. The boundary calculus can express only the basic
arithmetic functions. It leaves out many useful functions found in
elementary algebra. While it includes addition, subtraction,
multiplication, division, exponents, and logarithms, those functions which
utilize set theoretics, such as summations, or those which have
discontinuities, such as absolute value, are not covered.
Structures around expressions. It lacks many structures for doing
ordinary algebra, such as equality and inequality (standard linear
structures were used here). It has no larger system of truth maintenance,
nor a form for abstracting functions. Such basic tenets of mathematics are
sorely necessary for any notation to be remotely useful. Since most of
predicate calculus is well understood, this extension is reasonable.
In this discourse, these issues were avoided by embedding the boundary
forms within traditional linear constructions. Because boundary calculus is
spatial, many of its advantages are lost when it is restricted to linear
constructs. To maintain the spatial advantages, the calculus most cover
this completely, so that the entire notation is spatial.
7.3 Practical
Currently, the boundary calculus is impractical for manipulating numbers,
because traditional support tools and techniques do not apply directly to
its forms. The common knowledge taught in grade school mathematics is
directed towards standard notation and must be reconsidered to use with
boundary numbers (see [29]).
Problem phrasing. Traditional problem representation techniques were
designed around the conceptual objects of standard notation: numbers and
functions. In contrast, the boundary forms use distinction and collection.
It is unlikely that mathematics problems could be phrased directly in terms
of the boundary constructs, though higher compositions of them, such as
numbers and functions, certainly would work. If boundary forms are in any
way fundamental, then thinking in terms of boundaries should provide a
conceptual advantage over traditional forms.
Algorithms. Algorithms for carrying through calculations and finding
solutions must be reinterpreted in this form. Even common arithmetic
routines, such as adding based integers, requires a heuristic to direct the
result to the proper form. In many cases, common heuristics will emerge
that are relatively hidden in the traditional notation. Because the details
of mathematical manipulation differ so much in boundary calculus, the high
level manipulations may fall into remarkably different patterns.
Management of forms. The boundary calculus forces manipulation at the
very lowest level. Constructs that are used regularly should have some sort
of abstraction mechanism for managing repetitive structural operations. For
example, a macro of a base boundary could represent base multiplication:
{A}=([A][oooooooooo]). More complicated constructions will be necessary,
although how to create powerful abstraction mechanisms with spatial forms
is not clear.
Tools. Many people learn and use mathematics with the assistance of
computational aids, such as calculators and computers. These tools allow
powerful interaction directly with the mathematical objects of standard
notation. The boundary calculus suggests that this assistance may hide some
of the fundamental dynamics of numbers from those students. Similar
computational tools can be built for boundary calculus, wrapping in the
preceding techniques for doing mathematics. These tools would need to be
linked with powerful functionality that is independent of notation, such as
graph plotting.
The boundary calculus does not immediately make mathematics easier to
use. Just as numbers are a part of the whole of mathematical
understanding, notation is a part of the whole of mathematical use. The
supporting cast of tools and techniques must be filled in before the
boundary calculus can support full use of mathematics.
7.4 Extensions
Algebra is just the beginning of numerical mathematics: out of algebra
stems differential calculus and integral calculus, out of differential
calculus stems differential equations and partial differentials, and out of
integral calculus stems transformational methods including Fourier
transforms. The fields continue and boundary calculus can potentially
extend to all of them. Boundary numbers may contribute to research and
understanding in these areas, just as mathematical notations have
enlightened problems in the past [26, 2].
Differentials are rather easily expressed in boundary form. Let {A}
represent the differential of A. Let (A) be interpreted as e^A and let [A]
be interpreted as ln(a). The rules defining the inward propagation of the
differential operator, shown in Table 7.1, are quite simple. These can be
expressed more concisely, as the first two rules are equivalent and the
rest are transparencies.
Table 7.1: Rules for Boundary Differentiation.
D1. {(A)} = (A[{A}])
D2. {[A]} = (<[A]>[{A}])
D3. {} = <{A}>
D4. {A B} = {A}{B}
D5. {} =
The chain rule, d(ab) = b da + a db, is easily proved from these rules:
{([A][B])} Given
([A][B][{[A] [B]}]) D1
([A][B][{[A]}{[B]}]) D4
([A][B][(<[A]>[{A}])(<[B]>[{B}])]) D2 (2x)
([A][B][(<[A]>[{A}])])([A][B][(<[B]>[{B}])]) Distribution
([A][B] <[A]>[{A}] )([A][B] <[B]>[{B}] ) Involution
([B][{A}])([A][{B}]) Inversion
Boundary numbers lead into deeper mathematics because they have a
natural interpretation as exponentials. Many extensions of numbers rely on
exponentials, so it is likely that boundary numbers may illuminate some
trends that are not directly noticeable in traditional notation. For
example, boundary numbers provide a framework for considering the
significance of transcendental numbers and their relationships to other
numbers.
These extensions would involve translating current mathematical
knowledge to boundary form, in search of hidden insight. Boundary numbers
provide new perspective on old problems.
7.5 Conclusions
This minimalist effort was motivated by a need to clarify the mathematics
we already know. Behind this goal lies a greater purpose of creating a
solid foundation for mathematics itself_boundary mathematics has this
foundation. Distinction is a fundamental representational act. Networks of
nested and collected distinctions can reduce by only a few basic
possibilities, because the choices are limited. Different choices direct
the initial forms into set theory, natural numbers, or propositional logic.
This representational genesis organizes these domains, mapping them from
the representational starting point of the void. This clear, explicit
organization of mathematical domains is unique to boundary mathematics. It
is the study of the most fundamental dynamics of form.
Chapter 8
APPLICATIONS
8.1 Introduction
The boundary calculus redefines numbers within the paradigm of boundary
mathematics. The calculus may eventually influence the ways we implement
and use mathematics, as well as how we teach it.
When solving problems mathematically, we use three basic
representational stages (introduced in Chapter 1): the problem in its own
terms, the representation of the problem mathematically, and separate
representations for performing calculations. These stages combine into a
basic solution loop for mathematical activity, shown in Figure 8.1
Traditional notation was designed for the cycle between the top two
stages, when representing vaguely understood ideas was more important than
the effectiveness of their representation. As mathematics developed, the
features of the notation evolved to encompass more and more expressive
power. With this progression, calculation with the notation became more
complex, to the point where separate forms were introduced to assist in
calculation [30]. That is, the notation was no longer suitable for direct
calculation, mandating the creation of surrogate forms.
We are taught to compute most arithmetic functions with alternative
forms, such as vertical addition and long division. In symbolic
manipulation software, algebraic expressions are stored in data structures
much different from standard notation, data structures that have been
optimized for the type of computation that needs to be done. Computational
representations are everywhere; they blend in because they are considered
necessary. In fact, they are a discontinuity in representation.
The calculus may simplify the entire solution loop. The critical path
goes through the calculation loop, which presently is quite convoluted. The
fundamental objects of the calculus are boundaries, computation objects
that provide simple and direct computation. The conceptual objects of
numbers and arithmetic functions are structures built out of these
primitives, rather than being the primitives themselves.
Solution Loop
Technology Activity
application
/\ 
interface   problem representation
 \/
conceptual objects
/\ 
software   translation and control
 \/
computational objects
/\ 
hardware \__/ calculation
Figure 8.1: Technology and Representation.
With this new set of primitives, the activities in the solution loop
become more clearly delineated. In particular, control of the calculation
steps, direction, becomes clearly separated from the constraints of the
calculation, terrain. These constraints can be maintained throughout an
algorithm so that all intermediate states are mathematically equivalent.
These primitive dynamics allow more precise control within the
representation than standard forms allow. Any given standard form is not
optimally structured for certain calculations.
Pushing the notation down to computational objects significantly affects
the tools and mechanisms by which we do mathematics. We can streamline the
entire solution loop and the technology that assists us: the hardware that
performs calculations, the software that builds and manipulates conceptual
objects, our interface to these mathematical objects, and the pedagogy of
this entire loop. Each of these areas are profitable applications of this
material and are briefly considered in the following sections.
8.2 Hardware
The way we conceptualize mathematics affects how we implement computational
hardware. Most machine computations are built upon a system of mathematics based
in the higher conceptual objects of the standard notation. The boundary calculus
provides lowerlevel forms from which to base machine computations.
The boundary calculus has computational advantages because it is
inherently parallel. Linear computation models have isolated control
points which sequence computation_forcing these to be parallel violates
their very design, even though mathematics is inherently parallel. The
calculus does not limit control and computation but allows as little or as
much parallelism as a platform will allow.
To implement the calculus in hardware, a network data structure must be
provided in silicon to support the nesting and collecting of boundaries.
It must have means of determining equivalent nodes and creating multiple
reference. It must support the match and substitute mechanism and it must
include a control routine which progresses elements towards a stable,
canonical representation.
A number of issues present challenges to this development. How to
achieve canonical representations must be determined for the broadest
domain of numbers, and constraints must be imposed to remain within that
domain. Algorithmic means of avoiding the paradoxes must be fully worked
out.
A chip that implements the boundary calculus could be based in the five
operations of its definition: set to the void, collect values, wrap with
instance, wrap with abstract, or wrap with inverse. The chip could store
and manipulate algebraic expressions by manipulating unknowns. Two
additional commands provide this function manipulation: create a lambda
variable in a register and replace occurrences of a lambda variable with
another value.
Ironically, these functions are described for a linear machine.
Instruction sets are based on procedural languages within procedural
machines. In contrast, this calculus provides a basis for a spatial
machine. Thorough architectures of spatial machines have recently been
proposed for which the calculus seems compatible [15].
8.3 Computer Algebra
Current computer algebra systems (CAS) adopt data structures which deviate
significantly from the parsing structure of standard algebra. These
structures help optimize manipulations and operations on functions. The
boundary calculus takes this a step further by proposing objects to which
specialized network structures can be designed.
To computer algebra, the calculus represents a significant reducing
technology. Suddenly, less does more. Fewer formulas are required which
ultimately do more. Constraints are easier to follow and simpler to
implement. Additionally, the exponential interpretation makes it
particularly useful for more customized operations, such as integrals and
Fourier transforms where conversion to an exponential form is necessary
anyway.
Because the calculus is minimal, it provides a strong basis for
automatic theorem proving by limiting decision points. As with hardware
application, a key problem is to find a consistent reduction for every
expression, using heuristics to guide the application of rules. The
decision procedure is easier within the calculus simply because there are
fewer constructs to deal with.
The following reductions show that e^(pi/2) = i^i. In boundary form, the two
seemingly different forms actually reduce to an equivalent form.
(([<(J [J]([J]<[oo]>))>][(<[oo]>)])) e^(pi/2)
(([ (JJ[J]([J]<[oo]>)) ][(<[oo]>)])) Inverse promotion
(( JJ[J]([J]<[oo]>) <[oo]>) ) Involution
(( [J]([J]<[oo]>) <[oo]>) ) J cancellation
(( [[(([J]<[oo]>))]][(([J]<[oo]>))])) i^i
(( [J]<[oo]> ([J]<[oo]>) )) Involution
It would be diocult to implement the calculus in software because it
does not cover all the mathematics necessary for a complete mathematics
package. The spatial forms are powerful and immediately useful in isolation
but they conflict with traditional elements of mathematics. Until more
math is similarly expressed in spatial forms, the full power of the
paradigm cannot be exploited.
8.4 Mathematical Interface
The calculus makes an immediate contribution to the visualization of
mathematics because its spatial constructs afford many visual
interpretations. Two interpretations of a*x^2 > ([a]([[x]][oo])) are shown
in Figure 8.2. Because the notation computes inplace, these forms directly
support animation.

_______________________ ( )__
/ ____________ \ / `( )
/ _/ \_ \ / / \
/ / [] \ \ / [ ] [ ]
( [ a ] ( [ [x] ] [()()] ) ) /  / \
\ \_ [] _/ / [ ] [ ] () ()
\ \____________/ /  
\_______________________/ a x
Encircling Boundaries Distinction Networks
Figure 8.2: Visual Interpretations of Boundary Numbers.
The boundary calculus defines constructs spatially from the beginning,
rather than retrofitting prior structures into spatial form. This frees it
from the linear assumptions that corrupt many visual languages.
In breaking from traditional linear methods, the calculus has attained
many features that are essential to a good visual language. First it is
completely visual. There are no hidden semantics except for the axioms
which can be seen in operation and are visually simple. No part of the
calculus involves dissociated textual code; the picture itself completely
and unambiguously describes a mathematical expression.
The calculus also performs computation inplace, a valuable feature of a
visual programming language [18]. Expressions change themselves into new
forms, exactly where they are, in many cases simply by fading in or out
parts of the expression. This computation is naturally parallel and
concurrent, without explicit control points: computation occurs where and
when it can.
The boundary calculus can improve the mathematical interface by making
it visual: expressions are visually specified and computation occurs
visually on the same forms.
8.5 Education
The standard algebraic notation_made of operators and equations_dominates
mathematical experience. As the primary means of communicating algebraic
concepts, it is used in textbooks, on flash cards and posters, on tests and
homework. The notation holds an exclusive role: it is the only common
thread through all mathematical experience and so is commonly confused as
being mathematics itself.
The notation not only serves to implement mathematical concepts, it also
serves as an authority on how these concepts are made to work. The notation
links ideas into common abstractions that serve as the identifying
references for the ideas. All mathematical explanations are ultimately tied
back to these representations; they are relied upon pedagogically as
fundamental to mathematics.
Standard notation conceals knowledge of the relationships between
functions. Formulas specify how they transform: some functions have several
representations while others are unique and some combinations can be
simplified while others cannot. Because the visual form does not convey
these relationships, much external knowledge must be brought to the task of
doing mathematics.
In this way, standard notation fails to support learning mathematics.
Availability and application of rules are not suggested by the forms.
Knowledge of these rules remains distinct and separate from this
mathematical interface. Dissociating knowledge from appearance leaves an
impression that the knowledge is arbitrary and without foundation. Only by
accepting this knowledge on faith can a student bypass the contradictions
of form introduced by this dissociation.
The failure to instantiate mathematical knowledge in its visual forms
further contributes to the misconception that mathematics is a purely
cognitive activity. In doing so, mathematical ability is politely
restricted to only the "gifted" population, another failure of mathematics
education.
The boundary calculus can impact mathematics education. It provides
another reference for the implementation of formal mathematical concepts
and arguably presents a better interface to mathematical behavior.
8.6 Conclusions
The stages of mathematical representation can be readdressed under the
paradigm of boundary mathematics so that the activities of mathematics are
more clearly delineated. The boundary calculus isolates manipulative
constraints, leaving conceptual objects as constructions of the boundary
forms. The delineation gives those particular constructions meaning and
benefit that is usually hidden. Standard choices for organizing numbers are
actually optimized for certain criteria. Opening up representation with
the boundary calculus makes the criteria of representation and the choices
of structure more explicit.
GLOSSARY
abstract A type of boundary in the calculus of number. Abstract forms the
black hole when empty. Written as [] when empty or [A] around boundary
configuration A.
antilogarithm Functional interpretation of the instance boundary, e^x >
(x).
black hole The abstract boundary with no contents. The black hole
dominates its context. Written as [].
boundary Drawing of a distinction. (), [], and <> are types of boundaries.
boundary mathematics A representational and computational paradigm based
on boundary forms that transform by match and substitute.
cardinality A theorem stating that a repeated form can be rewritten using
a single reference. Written as A..A = ([A][o...o]).
collection Spatial justaposition of boundary configurations, as AB.
Collections are unordered, ungrouped, and variary.
configuration Any algebraic boundary expression, including void.
content The space inside of a distinction. e.g. (content).
context The space outside of a distinction. e.g. context().
distinction A cleaving of space. Drawn as a boundary, ().
distribution An axiom that defines the modifier form as distributive over
collection. Written as (A[BC]) = (A[B])(A[C]).
dominion A theorem stating that elements collected with the black hole
are irrelevant. Written as [] A = [].
e A transcendental intepretation of nested instance boundaries, e > (o).
equivalence axiom Two or more templates that define equivalent
expressions. e.g. ([A]) = A
fractional cardinality Expression of a partial cardinality (inverse of
repeated cardinality). Written as ([A]<[o...]>)where the number
of instance marks indicates the demonimator of the reciprocal
multiplying A.
i A complex number formed with a half cardinality of J. Written as i >
(([J]<[oo]>)):
instance A type of boundary in the calculus of number. Instance forms
the unit when empty. Written as owhen empty or (A) around boundary
configuration A.
inverse A type of boundary in the calculus of number. Inverse serves as
a generalized inverse. Written as 4 when empty or around boundary
configuration A.
inverse cancellation A theorem stating that inverse applied twice cancels
out. Written as <>= A.
inverse collection A theorem stating that a collection of two inverted
elements is equivalent to the inversion of the collected elements.
Written as = .
inverse promotion A theorem stating that inverse can be brought inside of
the modifier form. Written as <(A[B])> = (A[]).
inversion An axiom that defines the generalized inverse. Written as
A = .
involution An axiom that defines instance and abstract as functional
inverses. Written as ([A]) = A = [(A)].
J The phase element, J = [].
J cancellation A theorem stating that J cancels with itself. Written as
JJ=
match and substitute Replacement of template variables in an equivalence
axiom so that one of its templates matches part of a given expression.
modifier form A configuration composed of instance around abstract.
Written as (A[B]), with A part of the form, wrapped around contents
B.
natural logarithm Functional interpretation of the abstract boundary,
ln(a) > [a].
negative cardinality Expression of an inverted repetition. Written for
reference A as ([A][]);where the number of units indicates
the magnitude of the cardinality.
phase element A form that captures the inverse in an arithmetic
construction. Defined as J = []. It serves as the inverse when
placed in the modifier form, as (J[A]) = .
phase independence A theorem stating that the contents of the phase
operator can be moved to the context. Written as [<(A)>] = A[<()>].
phase operator A contruction nesting the abstract, inverse, and instance
boundaries around an argument. Written as [<(A)>].
pi An algebraic construction of the radian unit. Written as pi >
(J[J]([J]<[oo]>)).
template A boundary expression that includes zero or more template
variables, e.g. (A[B]).
template variable A form found in templates that can be replaced by any
boundary expression. Template variables are written as uppercase letters.
unit The instance boundary with no contents. It serves as the unit in
cardinality. Written as 'o'.
void Empty space; a lack of structure upon which distinctions are made.
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Appendix A
CONVERSION
A.1 Numbers
The basic numbers types can be expressed in boundary notation. For each
number type below, the general structure of that type is given in boundary
form followed by some examples.
Standard Boundary
Zero 0
Natural n o...o
1 o
2 oo
Integer i n or
1
2
Rational q ([i1]<[i2]>)
2/3 ([oo]<[ooo]>)
1/4 (<[oooo]>)
Algebraic Irrational r (([[q1]][q2]))or r1([r2]([[r3]][r4]))
7^(1/3) (([[ooooooo]]<[ooo]>))
(2^(1/2))/2 (([[oo]]<[oo]>)<[oo]>)
Complex c r1([r2]([J]< [oo]>))
i (([J]< [oo]>))
2 + 4i oo([oooo]([J]< [oo]>))
Transcendental e (o)
pi (J[J]( [J]< [oo]>))
i*pi []
A.2 Functions
The standard algebraic functions of addition, subtraction, multiplication
and division can be formed with boundaries. Powers and transcendental
functions can be formed with the same objects. These are shown below.
Standard Boundary
Identity x x
Inverse x
1/x (<[x]>)
Addition x+y xy
xy x
Multiplication x*y ([x][y])
x/y ([x]<[y]>)
Power x^y (([[x]][y]))
x^(y) (([[x]][]))
x^(1/y) (([[x]]<[y]>))
Transcendental e^x (x)
ln x [x]
A.3 Number Formats
The boundary notation does not restrict the structural format of numbers.
The notation can express numbers in a variety of formats by adapting the
algebraic structure of that format. Below are some examples.
Define b to be the base radix, b = oooooooooo.
Define {A} to be the base boundary, {A} = ([b][A]):
Standard Boundary
Mixed i j/k i([j]<[k]>)
4 2/3 oooo([oo]<[ooo]>)
1 1/2 o(<[oo]>)
Based i*b^1+j*b^2+... i{j...}
17 {o}ooooooo
6243 {{{oooooo}oo}oooo}ooo
Scientific r*b^i ([r]([[ b]][i]))
3*10^8 ([ooo]([[b]][oooooooo]))
6*10^23 ([oooooo]([[b]][foogooo]))
A.4 Standard Postulates
The standard field postulates can be derived directly from the boundary
axioms. Derivations of these postulates are shown below.
Commutativity
x+y = y+x xy = yx
x*y = y*x ([x][y]) = ([y][x])
Commutativity is implicit in the boundary forms because no ordering has
been imposed in the first place. The typographic difference between xy and
yx is not recognized by the boundary notation: containment is meaningful,
ordering is not.
Associativity
x+(y+z) = (x+y)+z xyz = xyz
x*(y*z) = (x*y)*z ([x][([y][z])]) = ([([x][y])][z])
Associativity is similarly implicit in the boundary forms. Addition by
collection is a variary operation, allowing zero or more items to be
collected at a time. Collecting with a collection leaves no evidence of
functional ordering. Multiplication does leave evidence of functional
ordering but this can be erased and replaced by involution.
Distribution
x*(y+z) = (x*y)+(x*z) ([x][yz]) = ([x][y])([x][z])
Distribution of multiplication over addition follows as a special case
of the distribution axiom.
Identity
x+0 = x x = x
1*x = x ([o][x]) = ([x]) = x
The additive identity is implicit in spatial collection as the void. The
multiplicative identity reduces to void within the outer instance boundary
and the resulting unary multiplication reduces to identity.
A.5 Formulas
Common algebraic formulas can be deduced directly from the boundary axioms.
Some are listed below. Many addition formulas have corresponding ones in
multiplication. The pairs are listed together to demonstrate their
similarities in boundary form.
Cardinality
x+...+x = n*x x...x = ([x][o...o])
x*...*x = x^n ([x]...[x]) = (([[x]][o...o]))
Cardinality counts repeated instances in the same space, whether a
collection of x for addition or a collection of [x] for multiplication.
Inversion
x+(x) = 0 x =
x/x = 1 ([x]<[x]>) = ()
Inversion cancels an item and its inverse. It acts on the additive
inverse, cancelling x and , and it acts on the multiplicative,
cancelling [x] and <[x]>.
Inverse Collection
(x)+(y) = (x+y) =
(1/x)*(1/y) = 1/(x*y) ([(<[x]>)][(<[y]>)]) = (<[x][y]>)
Inverse collection accumulates multiple objects within the same inverse
boundary. It carries the additive inverse over an sum and it carries the
multiplicative inverse over a product.
Inverse Cancellation
(x) = x <>= x
1/(1/x) = x (<[(<[x]>)]>)=x
The generalized inverse cancels itself out.
Inverse Promotion
x = (1)*x = ([x][])
1/x = x^(1) (<[x]>) = (([[x]][]))
(x*y) = x*(y) <([x][y])> = ([x][])
1/(x^y) = x^(y) (<[(([[x]][y]))]>) = (([[x]][]))
Inverse promotion converts the additive inverse to a product with
negative one and the multiplicative inverse to a power of negative one. It
provides a straightforward means of moving the inverse around within an
expression.
Powers
x^1 = x (([[x]][o]))=x
x^0 = 1 (([[x]][]))=()
x^m * x^n = x^(m+n) ([(([[x]][m]))][(([[x]][n]))])
= (([[x]][mn]))
(x^m)^n = x^(m*n) (([[(([[x]][m]))]][n]))
= (([[x]][([m][n])]))
x^m * y^m = (x*y)^m ([(([[x]][m]))][(([[y]][m]))])
= (([[([x][y])]][m]))
Powers of one and zero reduce in boundary form by involution and
dominion. The other power formulas are derived by involution and
distribution.
Logarithms
ln(x*y) = ln(x)+ln(y) [([x][y])] = [x][y]
ln(x/y) = ln(x)ln(y) [([x]<[y]>)] = [x]<[y]>
ln(x^y) = y*ln(x) [(([[x]][y]))] = ([[x]][y])
These logarithmic formulas reduce directly by involution.
Radians
1 + e^(i*pi) = 0 o([]) =
Euler's formula works because the form '[]' was defined that way.
Appendix B
EXAMPLES
B.1 Multiplication
Here is an example of multiplication of two integers. To visually simplify
the forms, a few definitions will be made first.
Let b be the base radix, b = oooooooooo. Let {A} be the base function on
A, defined as {A} = ([b][A]). The radix carries into the base function,
b{A} = {oA}. The base function collects, {A}{B}= {AB}, and promotes,
{(A[B])} = (A[{B}]), just like inverse:
{A}{B} = ([b][A])([b][B]) = ([b][AB])= {AB},
{(A[B])} = ([b][(A[B])]) = ([b]A[B]) = (A[([b][B])]) = (A[{B}]).
Using these definitions, the multiplication 23 * 114 can be computed by
making copies at each magnitude, collecting magnitudes, and doing a carry
operation.
23 * 114 Given
{oo}ooo * {{o}o}oooo Number Rewrite
([{oo}ooo][{{o}o}oooo]) Function Rewrite
([{oo}][{{o}o}oooo])([ooo][{{o}o}oooo]) Distribution
{([oo][{{o}o}oooo])}([ooo][{{o}o}oooo]) Promotion
{{{o}o}oooo{{o}o}oooo}{{o}o}oooo{{o}o}oooo{{o}o}oooo Cardinality (2x)
{{{o}o}oooo{{o}o}oooo{o}o{o}o{o}o}oooooooooooo Collection
{{{o}o}oooo{{o}o}oooo{o}o{o}o{o}o}boo Replacement
{{{o}o}oooo{{o}o}oooo{o}o{o}o{o}oo}oo Carry
{{{o}o{o}oooo}oooooooooooo}oo Collection
{{{o}o{o}oooo}boo}oo Replacement
{{{o}o{o}ooooo}oo}oo Carry
{{{oo}oooooo}oo}oo Collection
2622 Rewrite
B.2 Square Root
The following square root calculation is an informed calculation. It does
not represent an algorithm for determining square roots, it merely
demonstrates the equivalence of sqrt(9) and 3.
sqrt(9) Given
(([[ooooooooo]]<[oo]>)) Rewrite
(([[([ooo][ooo])]]<[oo]>)) Cardinality, A = ooo
(([[(([[ooo]][oo]))]]<[oo]>)) Cardinality, A = [ooo]
(( [[ooo]][oo]<[oo]>)) Involution (2x)
(( [[ooo]] )) Inversion
ooo Involution (2x)
3 Rewrite
B.3 Fractions
In standard notation, fractions are added by forming common denominators
and distributing. In the boundary calculus, they are combined by forming
common modifiers and distributing.
1/a + 1/b Given
(<[a]>)(<[b]>) Rewrite
([b]<[b]><[a]>)([a]<[a]><[b]>) Inversion
([ab]<[a]><[b]>) Distribution of <[a]><[b]>
([ab]<[a][b]>) Inverse Collection
([ab]<[([a][b])]>) Involution
(a + b)=(ab) Rewrite
B.4 Algebraic Distribution
This algebraic derivation uses the distribution axiom in ways that appear
different when written in standard form, distributing a+b over subtraction
ab and distributing a and b over addition a + b. These applications all use
the same boundary rule. Expansion of this product can be done in different
ways, this ordering and choice of distributions are but one way.
(a + b)(a  b) Given
([ab][a]) Rewrite
([ab][a])([ab][]) Distribution of [ab]
([a][a])([b][a])([ab][]) Distribution of [a]
([a][a])([b][a])([a][])([b][]) Distribution of []
([a][a])([b][a])<([a][b])><([b][b])> Inverse Promotion (2x)
([a][a]) <([b][b])> Inversion
(([[a]][oo]))<([[b]][oo]))> Cardinality (2x)
a^2  b^2 Rewrite
In the boundary calculus, operations can be done in parallel. Here
simultaneous operations of the same kind are done in parallel and marked
with '(2x)'. The second and third distribution operations can be done in
parallel as can the final inversion and cardinality.
B.5 Quadratic Formula
The quadratic formula gives solutions to equations of the form
ax^2+bx+c = 0. To derive the quadratic formula in the boundary
calculus, this equation will be transformed to this form, which
reveals the solutions of x:
b+sqrt(b^24ac) bsqrt(b^24ac)
(x  _______________)(x  _______________) = 0
2a 2a
The above quadratic appears in boundary form as the following void
equivalent:
([a]([[x]][oo]))([b][x])c =
The objective is to transform it to the following form, revealing
solutions, s1 and s2:
([x][x]) = .
To remove the first coeocient a, introduce its multiplicative inverse from
the void.
([ ]) Involution
([ ]<[a]>) Dominion
Then substitute the quadratic in the black hole of the product.
([ ([a]([[x]][oo]))([b][x])c ]<[a]>) Void substitution
The a coeocient can now be removed. Distribute its inverse and cancel.
([ ([a]([[x]][oo]))]<[a]>)([([b][x])]<[a]>)([c]<[a]>) Distribution
( [a]([[x]][oo]) <[a]>)( [b][x] <[a]>)([c]<[a]>) Involution
( ([[x]][oo]) )( [b][x] <[a]>)([c]<[a]>) Inversion
Now the x^2 has a unary coefficient. Define d and e.
d = ([b]<[a][oo]>)
e = ([c]<[a]>)
Work them into the equation.
(([[x]][oo]))([x][b]<[a]><[oo]>[oo])([c]<[a]>) Inversion
(([[x]][oo]))([x][b]<[a] [oo]>[oo])([c]<[a]>) Collection
(([[x]][oo]))([x][([b]<[a][oo]>)][oo])([c]<[a]>) Involution
(([[x]][oo]))([x][d][oo])e Definition of d and e
Flatten and complete the square.
([x][x])([x][d][oo])e Cardinality
([x][x])([x][d][oo])([d][d])<([d][d])>e Inversion
Now factor it.
([ x][x])([x][d])([x][d])([d][d])<([ d][d])>e Cardinality
([ x][xd])([xd][d])<([d][d])>e Distribution (2x)
([ xd][xd])<([d][d])>e Distribution (2x)
Make a difference of squares.
([xd][xd])<([d][d])><> Inverse Cancellation
([xd][xd])<([d][d]) > Collection
([xd][xd])<(([[([d][d])]])) > Involution
([xd][xd])<(([[([d][d])]]<[ oo]>[oo]))> Inversion
([xd][xd])<(([[(([[([d][d])]]<[ oo]>))]][oo]))> Involution
Define f to clean up the negative square.
f = (([[([d][d])]]<[oo]>))
Put it into the equation and flatten to a product.
([xd][xd])<(([[f]][oo]))> Definition of f
([xd][xd])<([f][f] )> Cardinality
Separate into two factors.
([xd][xd])([xd][f])<([xd][f])><([f][f])> Inversion
([xd][xd])([xd][f])([xd][])([f][]) Promotion
([xd][xdf])([xdf][]) Distribution
([xdf][xd]) Distribution
([x<>][x<>>]) Inverse Cancellation
This gives two solutions.
x = and x = >
Expand these for the recognized solutions. Recall f.
f = (([[([d][d])]]<[oo]>))
First expand the interior of f.
([d][d]) =
= ([([b]<[a][oo]>)][([b]<[a][oo]>)]) Definition of d
= ( [b]<[a][oo]> [b]<[a][oo]> ) Involution
= ([b]<[a][oo]>[b]<[a][oo]>)<([c]<[a]>)> Definition of e
= ([b]<[a][oo]>[b]<[a][oo]>)
<([c]<[a]><[a][oo][oo]>[a][oo][oo])> Inversion
= ([b][b]<[a][oo][a][oo]>)
<([c]<[a][a][oo][oo]>[a][oo][oo])> Collection
= ([([b][b])]<[a][oo][a][oo]>)
<(<[a][a][oo][oo]>[([c][a][oo][oo])])> Involution
= ([([b][b])]<[a][oo][a][oo]>)
<(<[a][a][oo][oo]>[<([c][a][oo][oo])>]) Promotion
= (<[a][oo][a][oo]>[([b][b])<([ c][a][oo][oo])>]) Distribution
= (<([[a][oo]][oo])>[(([[b]][oo]))<([c][a][oo][oo])>]) Cardinality
= (<([[a][oo]][oo])>[(([[b]][oo]))<([c][a][oooo])>]) Cardinality
= (([<[a][oo]>][oo])[(([[b]][oo]))<([c][a][oooo])>]) Promotion
Then determine f.
f = ((<[oo]>[[(([<[a][oo]>][oo])
[(([[b]][oo]))<([c][a][oooo])>])]])) Substitution
= ((<[oo]>[([<[a][oo]>][oo])
[(([[b]][oo]))<([c][a][oooo])>]])) Involution
= ((<[oo]>[([<[a][oo]>][oo])])
(<[oo]>[[(([[b]][oo]))<([c][a][oooo])>]])) Distribution
= ((<[oo]> [<[a][oo]>][oo])
(<[oo]>[[(([[b]][oo]))<([c][a][oooo])>]])) Involution
= (( [<[a][oo]>] )
(<[oo]>[[(([[b]][oo]))<([c][a][oooo])>]])) Inversion
= (<[a][oo]>(<[oo]>[[(([[b]][oo]))<([c][a][oooo])>]])) Involution
Define g as the root expression.
g = ((<[oo]>[[(([[b]][oo]))<([c][a][oooo])>]]))
The solutions are a function of g.
= <([b]<[a][oo]>)(<[a][oo]>[g])> Definition of g
= <([bg]<[a][oo]>)> Distribution
= ([]<[a][oo]>) Promotion
= ([]<[a][oo]>) Collection
> = <> Collection
= f Inverse Cancellation
= <([b]<[a][oo]>)>(<[a][oo]>[g]) Definition of g
= ([]<[a][oo]>)(<[a][oo]>[]) Promotion
= ([g]<[a][oo]>) Distribution
The solutions to the quadratic formula, ([a]([[x]][oo]))([b][x])c, are:
x = ([((<[oo]>[[(([[b]][oo]))<([oooo][c][a])>]]))]<[ffio][a]>),
x = ([<((<[oo]>[[(([[b]][oo]))<([oooo][c][a])>]]))>]<[oo][a]>).
which translate to
b+sqrt(b^24ac) bsqrt(b^24ac)
x = _______________ and x = _______________
2a 2a .