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The Emergence of Quaternionic, Octonionic and Clifford Algebra Structure
From Laws of Multiboundary Form
Rough Draft, not for distribution
Ben Goertzel1, Onar Aam, Tony Smith, Kent Palmer
1
Computer Science Dept., College of Staten Island and IntelliGenesis Corp.1. Introduction
This paper describes a new kind of abstract algebra called multiboundary algebra, and describes in detail a specific multiboundary algebra called Ons algebra, which has
been developed with two purposes in mind:
The interconnectedness of these two purposes is evident if one believes that quaternion, octonion and Clifford algebras are in fact essential to the structure of the physical, psychic and essential universe.
Multiboundary algebra is a deviation from the standard formalism of abstract algebra, which has multiple operators but only one type of parenthese. In multiboundary algebra we have multiple boundary operators and a sensitive interdependence between boundaries and operator rules. In philosophical terms, we may say that whereas the standard approach embodies an objectivist approach to the universe, the multi-boundary approach embodies a relativist or subjectivist approach – not a nihilist approach in which there are no rules, but rather, an approach in which the rules depend on the context one is in; in which each algebraic entity has its own "universe" of rules, valid within its own space.
Ons algebra is a relatively simple multiboundary algebra generalizing the single-boundary algebra presented by G. Spencer-Brown in Laws of Form. As Laws of Form leads to Boolean algebra, so Ons algebra leads to quaternions, octonions and Clifford algebras. The somewhat greater complexity of Ons algebra is compensated by the increased richness of the structures obtained.
2. Laws of Form
Every investigation has a multitude of starting points; and every presentation needs to choose just one. The simplest place to start the present exposition is perhaps Laws of Form by G. Spencer-Brown, a beautiful and audacious book in which the author attempts to lay out the simplest possible framework within which the emergence of complex form can be modeled and studied. It posits a formal system containing one mark
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called a "distinction" which can interact with other marks of the same kind in two ways:
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--| ---| = ---|
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The introduction of a single mark with two modes of interaction is, it is shown, sufficient to give rise to an infinite variety of possible forms. The introduction of variables x, y, etc. taking values of either distinction or void (non-distinction), allows one to write equations involving distinction. The algebra of distinctions turns out to be equivalent to Boolean algebra.
Laws of Form goes on to develop a logic of paradox in terms of infinite forms; here I will be interested in moving in a different direction – namely, extending Laws of Form to give rise to more interesting algebras than Boolean algebra. In particular, I will show how to extend Laws of Form to deal with algebras such as quaternions, octonions and Clifford algebra. These algebras are arguably essential to the structure of the physical universe and the psyche as well.
Of course, the extensions are not as simple as the Laws of Form formalism, but some additional formal complexity is needed to attain more complex structures. Just as Laws of Form aims to be the minimal framework for the emergence of Boolean form, so the present framework aims to be the minimal framework for the emergence of quaternions, octonions and Clifford algebra.
Before getting started, it is worthwhile to introduce a typographically easier stand-in for the visually elegant Laws of Form notation. As Lou Kauffmann has suggested, the mark of distinction may be denoted
[]
and the two methods of interaction may be written
[[]] =
[][] =
This is not as "clean" as the original notation, in that it takes what is fundamentally a unified entity, a boundary, and breaks it up into two parts. However, typing in the original notation rapidly becomes cumbersome; the modification is a necessary concession to the one-dimensional nature of modern typography and word-processing.
Note that yet a different notation for the same rules would be
[] * [] =
[] + [] = []
I.e., the expression of the two combinatory operations as nesting and adjacency is intuitive and philosophically meaningful but not mathematically necessary. In the following I will refer to these two operations as either "interpenetration" versus "coexistence" or "multiplication" versus "addition".
3. Multi-Boundary Algebra
The development here will remain within the framework of boundaries that interact in two different ways; the Laws of Form algebra will be enriched, however, by the addition of more types of boundaries. This development is eccentric with regard to conventional mathematics, which contains only one type of parenthese for grouping entities, but numerous operations for combining groups and ungrouped entities. However, introducing several types of boundary is a perfectly viable alternative to introducing several types of operator, and is in some ways, we shall see, a more powerful alternative.
Notation for multiple boundaries is fairly difficult to come by, though the typewriter keyboard provides a number of alternatives, e.g. ( ) , [ ], { }, < > . For talking about multi-boundary algebra in the abstract I will use the notation (k k) to refer to a boundary of type k, but this notation is not good for working out concrete examples.
The interior of a boundary will be called a "space," and the operator
(k k) * b
will be understood to mean that the entity b is placed in the space demarcated by the boundary (k k). Multi-boundary algebra envisions a universe in which various simple and composite entities coexist and interpenetrate within various spaces, and in which the results of coexistence and interpenetration depend on the composition of the entities involved, and the space in which the entities exist.
Formally, a general multi-boundary algebra consists of
The local rule sets must be consistent with the global rule set R. The individual rule sets must be consistent with the global rule set and with the local rule sets. The local rule set applying to the interaction of two boundary forms is determined by the the boundary that they most locally belong to; e.g. if we have [ < a + b> ] then the meaning of the operation + is determined by the rule set for < > and not the rule set for [ ].
Multiboundary algebra, in general, is an extremely general framework, much like Universal Algebra (but more general). The specific multiboundary algebras to be discussed here will involve the standard two operators, + and *, plus an additional operator ^ dealing with temporality. Furthermore, in all the specific algebras to be discussed here, the + operator will be commutative, and will be associative in regard to the ordinary parenthese ( ), meaning that + can be commuted across an arbitrary number of arguments, e.g. a + b = b + a, a + b + c + d = d + c + b + a = d + b + a + c, etc. In more formal language, we will be dealing with +-polycommutative (+,*,^) multiboundary algebras (where "*i -polycommutative means, in general, that all rule sets in the multiboundary algebra hold the operator * i to be commutative and associative with regard to ( ) ).
From the definition of multiboundary algebra, we see exactly what the difference is between the standard mathematical technique of having one boundary (the parenthese) and numerous operators, and the present alternative of having few operators and multiple boundaries. The notion of space-dependent rule sets has no parallel in standard mathematics, and breaks down the barrier between algebraic rules and algebraic formulas in a very interesting way, which is only explored here very partially.
3. Ons Algebra
Ons algebra begins with two types of boundary, which are denoted {} and []. The boundary {} is called an On, and is thought of as a particle of being; the boundary [] is called a Chronon, and is thought of as a particle of time. These interpretations are metaphorical and ontological rather than physical, as will be seen in the following.
The Chronon or "hard boundary" [] obeys laws similar to those of Laws of Form, but reversed:
[] + [] = [][] =
[] * [] = [[]] = []
Within the Laws of Form algebra, the + and * operators are completely symmetrical, so that the hard boundary may actually be considered identical to Spencer-Brown’s mark of distinction. Note that the (+,*) notation and the adjacency/nesting notation are, as suggested above, identical in meaning, and will be used interchangeably in the following.
The On boundary {} is also called a "reified boundary" and obeys the rule
{} + {} = {}{} = []
The reified boundary may be thought of as a boundary which has lost some of its possibility for collapse. Note that there is no rule for the collapse of forms like {{}},
{{{}}}, {{}{{}}}, etc. Such forms are considered as distinct from each other, and may be used to model such structures as the natural numbers (cf. the von Neumann construction, 1 = {}, 2 = {{}}, 3= {{{}}}, 4 = {{{{}}}}, etc.).
These three rules are part of the global rule set of Ons algebra.
There is an interesting and useful relationship between {}, [] and mod 4 arithmetic. Consider the correspondence
0
1 {}
2 []
3 {}[]
Denoting the blank space by V for void, we have
Proposition 1: The algebra ({ V, {}, [], {}[] },+) is isomorphic to Z 4
Proof: Consider the above correspondence.
N+0=N follows from a+V=a
1+1=2 follows from {}{} = []
1+2=3 follows from {} + [] = {}[], i.e. by definition
2+2=4 follows from [][]=V
2+3 = 1 follows from []{}[]={}[][]={}
3+3=2 follows from []{}[]{} = [][]{}{} = []
QED
Next, we must specify the interaction of {} and [] boundaries under * . These rules are not part of the global rule set of Ons algebra; rather, they appear in the rule set of []. They apply to entities living inside a hard boundary.
{}*{} + {}*{} = {{}} {{}} = []
{}*[] + {}*[] = { [] }{ [] } = []
[]*{} + []*{} = [{}] [{}] = []
These rules combined with the previous ones give us the following consequences:
Proposition 2: Inside a hard boundary,
a) [a] [a] = a a
b) {a}{a}= [a a]
where the variable a may have values V, {} or []
Proof: Let a = V, then
[a][a] = [][] = V
a a= V V = V
{a}{a}= {}{} = []
[a a] = [V V] = []
Let a = [], then
[a][a] = [[]][[]] = [][] = V
a a= [][] = V
{a}{a}= {[]}{[]} = []
[a a] = [[] []] = []
Let a = {}, then
[a][a] = [{}][{}] = []
a a= {} {} = []
{a}{a}= {{}}{{}} = []
[a a] = [{}{}] = []
QED
These relations are extremely useful and in fact, will be needed more generally. The following rules extend Proposition 2 and are placed in the rule set of the hard boundary []:
[a] [a] = a a
{a}{a} = [a a]
for any form a. Proposition 2 shows that these rules are consistent with the previous rules.
The preceding rules deal with general sums inside hard boundaries; the following rule deals with general products inside hard boundaries:
[a1 + … + an ]* [b1 + … + bn ] =
[a1 + … + an +
b1 + … + bn +
a1 ^ b1 + … + a1 ^ bn +
a2 ^ b1 + … + a2 ^ bn +
… +
an ^ b1 + … + an ^ bn ]
The operator ^ determines temporal precedence, and is dealt with as follows. First, the global rule set of Ons algebra contains the rule:
Second, any particular hard boundary may augment this rule set with its own assignment of [] or V to specific forms that occur within it. A hard boundary that provides individual rules for ^ will be called an "ordering boundary." A linear order for a set G of forms is an ordering boundary that defines the ^ operator on G in such a way that
are obeyed.
Is this general form for * within hard boundaries consistent with the specialized * formulas given above for combining {} and []? Yes: we may check that
[]*[] = [V]*[V] = [V+V + V^V] = [], as above
[[]]*[] = [ [] + V + []^V] = [[]] = [], as above where we have [[]]*[]=[]*[]=[]
[[]]*[[]] = [ [] + [] + []^[] ] = [ [][] ] = [], as above where we have
[[]] * [[]] = [] * [] = []
[{}]*[{}] = [ {} + {} + {}^{} ] = [[]] = [] as above
These definitions lead up to the following:
Proposition 3: Let a be any form in Ons algebra, formed by combination of {} and [] according to the operators + and *. Then, within a hard boundary, the following equation holds for any form a:
either a+a=V or a+a = [].
Proof: For any form a, let S(a) denote the number of symbols in the shortest expression in Ons algebra that is equivalent to a. The proof will proceed by induction on S(a). If S(a)=1 then the result follows from Proposition 2. Otherwise, suppose that the result holds for all propositions b with S(b)=1,2,…,S(a)-1. Then, I will show that it must hold for a.
The form a must be of one of the following forms: [b], {b}, b+c, b*c, where b and c have the property S(b)<S(a), S(c)<S(a), so that the result is known to be true for b and c. Each case will be treated separately.
Suppose a = [b], then a a = [b] [b] = b b = V or []
Suppose a = {b}, then a a = {b} {b} = [b b] = either [V] = [] or [[]] = V
Suppose a = b+ c, then a a = b c b c = b b c c = either V V = V or [] [] = V
Suppose a = b*c, then a = b*c = [s] where s is a certain sum, so the result follows by the first case above
QED
4. The Emergence of Q, O and Cl(n)
How do these multiboundary algebras lead to quaternion, octonion and Clifford algebras? It’s relatively simple. However, three additional boundaries will be required along the way.
The first is the soft boundary, denoted < >, which is to be thought of as a "permeable membrane," supporting free distributivity. The global rule for soft boundaries is:
a * <b + c> = a * b + a * c
<b + c> * a = b * a + c * a
Treatment of nondistributive algebras may require consideration of left-soft and right-soft boundaries, which allow distribution in only one direction; but the present considerations require only soft boundaries that support distribution on both sides.
Furthermore, there is a special rule for combination of hard boundaries within soft boundaries. Namely, within a soft boundary, we have the "negation rule"
[a + []] + [a] = V
Essentially, what this says is that, within a soft boundary, [] acts like a negative. Note that the general annihilating rule "a + a = either V or []" does not hold within soft boundaries, but only within hard boundaries. The negation rule is a kind of substitute.
Next, we will also need the "spatializing" boundary, denoted <_ _>, which is characterized by the following "spatialization rule":
<_ [] + a _> = <_ _>
Recalling the identification of the set {V,{},[],{}+[]} with arithmetic modulo 4, we note that the spatializing boundary has the effect of projecting Z4 into Z2 , by mapping [] into V and {}+[] into [], i.e. mapping 2 into 0 and 3 into 1.
Finally, we will need what I call the "vector boundary", denoted <! !>.
A vector boundary imposes a linear ordering on the elements within it, and the interaction between elements follows the ordering. It is convenient in fact to write a vector boundary using vector notation, and if we have <! a b c d !> where a<b<c<d by the boundary’s individual definition of ^, then we may write simply (a,b,c,d) as a shorthand. On the other hand, if a and b are simultaneous in this expression, and both occur before c, which occurs before d, then we would have (a+b,c,d). The operations + and * are defined on vector boundaries in the expected fashion, i.e. vectorially:
(a,b) + (c,d) = (a+b,c+d)
(a,b)*(c,d) = (a*d,b*d)
This is a very different way of dealing with order than is embodied in the definition of * for hard boundaries.
Where G is any set of forms, let h(G) denote the collection of all entities of the form [s], where s is a finite sum formed from elements of G; and let h(G,k) denote the collection of all elements of h(G) with exactly k elements. Similarly, let v(G) denote the collection of all entities of the form <! s !> where s is a sum of elements of G; and let v(G,k) be defined accordingly
Proposition 4: Let G be a hard-bounded set of forms of cardinality n; and assume that all hard boundaries are ordering boundaries that impose a common linear order on the elements of G. Then, h(G) under the operation
a x b = [ <_ a + b _> + a^b ]
is isomorphic to the Discrete Clifford Group DCLG(n), and s(h(G)) is isomorphic to the Discrete Clifford Algebra DCL(n)..
Proof:
To be filled in J
Proposition 5: Let G be a hard-bounded set containing 3 forms; and assume that all hard boundaries are ordering boundaries that impose a common linear order on the elements of G. Then h(G,1) under the operation
a x b = a + b + G
is isomorphic to the quaternion group, and s(h(G,1)) is isomorphic to the integer quaternion ring.
.
Proof:
To be filled in
Proposition 6: Let G be a hard-bounded set containing 3 forms; and assume that all hard boundaries are ordering boundaries that impose a common linear order on the elements of G. Then h(G,2) under the operation
a x b = [ <_ a + b _> + a^b ]
is isomorphic to the quaternion group, and s(h(G,2)) is isomorphic to the integer quaternion ring.
Proof:
To be filled in
Proposition 7: Let G be a hard-bounded set containing 7 forms; and assume that all hard boundaries are ordering boundaries that impose a common linear order on the elements of G. Then h(G,1), under the operation
a x b = a + b + G
is identical to the octonion group up to sign, i.e., it is a "quasi-octonionic" algebra.
Proposition 8: Let (K,x) be a quasi-octonionic algebra (i.e., K has 7 elements interacting via the operation x), and consider v(K,7), in which each vector boundary demarcating a set of elements of K has its own individual linear ordering of the elements of K. Let M denote the collection of 7 elements of v(K,7) with the following property: no two elements of M contain the same element of K in the same position in their ordering. Then the algebra (M, *) is isomorphic to the octonions, and s(M) is isomorphic to the integer octonion ring.
Proof:
To be filled in