Ben Goertzel
Psychology Department
University of Western Australia
ben@goertzel.org
It is argued that the abstract structure of states of consciousness is given by the octonionic division algebra.
First, the psynet model of mind, presented by the author in previous work, is summarized. This psychological model implies that, under certain reasonable approximations, the mental systems can be modeled as hypercomplex algebras. The question then becomes: Which algebras correspond specifically to conscious systems?
At this juncture the notion of "proprioception of thought" is borrowed from David Bohm's philosophy of mind. It is argued that only conscious mental systems possess proprioception of thought, and that algebraically speaking, proprioception of thought is equivalent to the property of unique division. Hence, conscious systems are represented by division algebras.
This is where advanced mathematics comes into play. The only dimensions in which finite-dimensional division algebras exist are one, two, four and eight; and the only reasonably symmetric division algebras are the reals, the complexes, the quaternions (4-D) and the octonions (8-D). The octonions contain the quaternions, which contain the reals, which contain the complexes. The octonions are taken here as the basic structure of consciousness.
The octonions are generated by an eight-element basis; the seven non-unity elements of this basis are equated with the "magic number" 7 +/- 2 of short-term memory capacity.
The quaternion structure is interpreted as a "perceptual-cognitive-active loop," representing the basic structure of engagement with the world. This structure is seen to lead naturally to a certain type of adaptive learning, analogous to backtracking in artificial intelligence.
The octonion structure is seen to ensue from adding to the quaternions an extra mental process that observes the others. This extra element is called the "inner eye" and is hypothesized to provide for reflexive consciousness and higher-order thought.
1. INTRODUCTION
There are many different "theories of consciousness," and these theories address a variety of different issues. It is not entirely clear what a theory of consciousness is supposed to be. There are physiological theories, which explain the precise nature or general character of the neural processes underlying conscious experience. There are philosophical theories, which explain the nature of consciousess in abstract terms. There are experimental-psychology theories, which explain observed properties of conscious behavior. Finally, there are phenomenological theories, which explore the detailed properties of conscious experience.
The present theory straddles all of these categories, but it certainly does not aspire to be a complete explanation of consciousness. Rather, it aims to derive certain conclusions regarding the abstract structure of consciousness. The abstract structure of consciousness has implications for physiology, philosophy, experimental psychology and phenomenology.
To make this a little more clear, I will distinguish "consciousness" from "awareness." Awareness, as I understand it, is the feeling of "being in the world" or just "being" -- the simple fact of presence, of experience. Awareness, in itself, is just awareness; it has no qualities besides just being awareness. Consciousness is different; it has particular qualities. Consciousness is awareness as modulated by the structure of mind. Thus a "state of consciousness" or "state of mind" may be understood as a particular kind of relationship between awareness and mind. One may study the structure of different states of consciousness without taking a position on the nature of awareness.
The key idea developed here is that the structure of consciousness can be understood in terms of certain abstract mathematical structures: namely, the finite-dimensional division algebras, the real numbers, complex numbers, quaternions and octonions (Cayley algebras). The conceptual connection between division algebras and consciousness was developed collaboratively, in a series of electronic communications involving Onar Aam, Kent Palmer and Tony Smith and the author (1995). The particular way in which the idea is developed here, however, is original with the author. I will derive the division-algebra structure of consciousness from the psynet model of mind, an abstract, system-theoretic framework for studying psychological processes (Goertzel, 1994), together with a single additional axiom.
The discussion throughout is motivated by considerations from transpersonal psychology and Buddhist psychology as well as the "harder" areas of cognitive science and neuropsychology. This may be at first seem an unusual approach, but on careful consideration, it is seen to be necessary. The study of consciousness involves a combination of introspective and scientific evidence. And it is the spiritual traditions, and modern psychologists inspired by them, who have sketched the clearest picture of the actual experience of consciousness, as opposed to its behavioral manifestations.
I will now briefly summarize the paper. Sections 2 and 3 are preparatory: we review the psynet model of mind, and indicate how it suggests a fundamental psychological role for abstract algebras. In Section 4 and 5, then, we turn to the notion of of self-awareness or, in David Bohm's phrase, proprioception of thought (thought's awareness of its own motions). We argue that proprioception of thought extends only through consciousness, and not through the unconscious. By augmenting the psynet model with the assumption that consciousness displays self-proprioception, we arrive at the conclusion that consciousness must involve algebras with the property of unique division. As it turns out, there are only two suitable algebras of this type, the quaternions and the octonions, with three and seven "free" elements respectively. Thus we arrive, quite naturally, at the conclusion that the quaternions and octonions are the structure of consciousness.
The connection between consciousness and division algebras could, in principle, be made in an entirely different way, without reference to the psynet model. Onar Aam, for example, has taken a purely metaphorical and phenomenological view of the same connection (Aam et al, 1995). However, the psynet model provides a very natural entry-point to these issues. It provides general, abstract reasons for considering consciousness as algebraic, and thus leads one to ask questions regarding the specific algebras involved. The emergence of the division-algebra nature of consciousness from the psynet model is an extremely exciting instance of system-theoretic psychology.
Finally, in Section 6, we explore the utility of the quaternion and octonion structures for explaining particular states of mind. First, we relate the seven free elements of the octonionic algebra with the "magic number" 7 +/- 2 of short-term memory capacity. Then we turn to the "loop" structure of the quaternions and relate it to the perceptual-cognitive-active loop which emerges from the neuropsychology of consciousness. In this context, the inverse elements in the quaternions are seen to correspond to the operation of backtracking, as seen in artificial intelligence algorithms. The octonions emerge as the result of adding an extra observing element or "inner eye" to the basic perceptual-cognitive-active loop. Control of this "inner eye" is essential to efficient higher-order learning, and to the maintenance of an effective long-term memory; it is also the goal of most of the world's spiritual traditions.
2. THE PSYNET MODEL
The psynet model is an abstract, system-theoretic model of mind. It can be formulated in mathematical terms, or it can be understood on a more intuitive level. It may be best to begin with an enumeration of the basic principles of the model, bearing in mind that many of the terms involved have not been explained yet:
1. Minds are magician systems residing on graphs
2. The magicians involved are pattern/process magicians
3. Thoughts, feelings and other mental entities are
structural conspiracies, i.e. autopoietic attractors of the mind magician system
4. The structural conspiracies of the mind join together
in a complex network of attractors, meta-attractors, etc.
5. This network of attractors approximates a fractal
structure called the dual network, which is structured according to at least two principles: associativity and hierarchy.
For a detailed exploration of this model and its relevance to various issues in cognitive science, the reader is referred to previous publications. Here we will only review those aspects of the model that are especially pertinent to the algebra of consciousness.
First, what is a "magician system"? This is simply a colorful term for a system of processes which mutually act on and transform each other. The name "magician" arises from the ability of magician processes to transform one another into different forms. One may imagine a community of sorcerers, continually transforming one another into different guises, each guise carrying with it a different collection of magical spells.
A magician system consists of a collection of entities called "magicians" which, by acting on one another, have the power to cooperatively create new magicians. Certain magicians may be paired with "antimagicians," magicians which have the power to annihilate them. At each time step, the dynamics of a magician system consists of two stages. First the magicians in the current population act on one another, producing a provisional new population (called the "raw potentiality"). Then the magician/antimagician pairs in the provisional new population annihilate one another. The survivors are the new population at the next time step.
Magicians may carry out any kind of operations. Of particular interest, however, are those magicians which recognize patterns. Most generally, a pattern in an entity x may be defined as a process which gives rise to x, but is in some sense simpler than x. For instance, if one takes a computational view, and equates simplicity with program length, then a pattern in x becomes any program which computes x and has length shorter than the length of x. Thus algorithmic information theory (Chaitin, 1988) is a special case of general pattern theory.
Using the metaphor of the von Neumann computer architecture, a pattern/process magician may be understood to fall into two parts: a structured collection of data, which varies from magician to magician, and a program, which is the same for all magicians. Magician a acts on magician b in the following way:
1) the program of magician a recognizes patterns emergent between the data of magician a and the data of magician b
2) the program of magician a solves some problem (e.g., an optimization problem) based on the patterns recognized in Step 1
3) magician a creates a new magician based on the result
of Step 2
In the simplest case, the optimization problem in Step 2 is trivial, and Step 3 merely consists of creating a magician whose data manifests all the patterns recognized in Step 1. In this way the patterns that were merely emergent at the previous step are now implemented in individual magicians.
Of course, the distinction between data and programs is somewhat artificial and is not generally applicable to biological and psychological systems. Even so, however, this three-step algorithm provides a very useful and general way of thinking about pattern/process magicians.
Magician systems do have predecessors in system theory: they are related to Varela's (1978) Brownian algebra, Kampis's (1991) component-systems, and, coming from a different point of view, Agha's (1991) and Minsky's (1987) agent systems. There are technical differences between magician systems and these other concepts, which have been discussed elsewhere. The key point, however, is that the psynet model views the mind as a process dynamical system. It is this which distinguishes the psynet model quite sharply from neural network models of the brain, which involve dynamical systems on real vector spaces.
Autopoiesis and Mind
A pattern/process magician system may at first sound like an absolute, formless chaos. Just a bunch of agents acting on each other, recognizing patterns in each other -- where's the structure? Where's the sense in it all?
But this glib analysis ignores something essential -- the phenomenon of mutual intercreation, or autopoiesis (Varela, 1978; Kampis, 1991). Systems of magicians can interproduce. For instance, a can produce a, while b produces a. Or a and b can combine to produce c, while b and c combine to produce a, and a and c combine to produce b. The number of possible systems of this sort is truly incomprehensible. But the point is that, if a system of magicians is mutually interproducing in this way, then it is likely to survive the continual flux of magician interaction dynamics. Even though each magician will quickly perish, it will just as quickly be re-created by its co-conspirators. Autopoiesis creates self-perpetuating order amidst flux.
Some autopoietic systems of magicians might be unstable; they might fall apart as soon as some external magicians start to interfere with them. But others will be robust; they will survive in spite of external perturbations. These robust magician systems are what I call autopoietic attractors, a term whose formal defnition is given in the Appendix. This leads up to the next crucial idea of the psynet model: that thoughts, feelings and beliefs are autopoietic attractors. They are stable systems of interproducing pattern/processes. In CL, autopoietic attractors of the cognitive equation are called structural conspiracies, a term which reflects the mutual, conspiratorial nature of autopoiesis, and also the basis of psychological autopoiesis in pattern (i.e. structure) recognition. A structural conspiracy is an autopoietic attractor whose component processes are pattern/processes.
But structural conspiracy is not the end of the story. The really remarkable thing is that, in psychological systems, there seems to be a global order to these autopoietic attractors. The central claim of the psynet model is that, in order to form a functional mind, these structures must spontaneously self-organize into larger autopoietic superstructures. And perhaps the most important such superstructure is a sort of "monster attractor" called the dual network.
The dual network, as its name suggests, is a network of pattern/processes that is simultaneously structured in two ways. The first kind of structure is hierarchical. Simple structures build up to form more complex structures, which build up to form yet more complex structures, and so forth; and the more complex structures explicitly or implicitly govern the formation of their component structures. The second kind of structure is heterarchical: different structures connect to those other structures which are related to them by a sufficient number of pattern/processes. Psychologically speaking, as will be elaborated in the following section, the hierarchical network may be identified with command-structured perception/control, and the heterarchical network may be identified with associatively structured memory. (While the dual network is, intuitively speaking, a fairly simple thing, to give a rigorous definition requires some complex constructions and arbitrary decisions; a formalization will not be given here due to space considerations.)
A psynet, then, is a magician system which has evolved into a dual network structure. Or, to place the emphasis on structure rather than dynamics, it is a dual network whose component processes are magicians. The central idea of the psynet model is that the psynet is necessary and sufficient for mind. "Psynet" is just a shorthand term for "mind magician system."
The Perceptual-Cognitive Loop
The first effort to treat consciousness in the context of the psynet model was the theory of the Perceptual-Cognitive Loop, given in Chapter 4 of (Goertzel, 1994). The basic idea there was that a great many phenomena may be understood by viewing consciousness as a feedback circuit joining perceptual processes with cognitive processes.
The view of consciousness as a perceptual-cognitive loop has been presented many times, in many different languages, by many different authors. In particular, Edelman's (1988) biological theory of consciousness has had a large influence on the author. For the present purposes, however, it is perhaps better to cite the work of Treisman and Schmidt (1982), who have argued for a two-stage theory of visual perception. Their theory was evolved in the context of the neurophysiology of vision. The two stages are as follows. First, the stage of elementary feature recognition, in which simple visual properties like color and shape are recognized by individual neural assemblies. Then, the stage of feature integration, in which attention focuses on a certain location and unifies the different features present at that location. If attention is not focused on a certain location, the features sensed there may combine on their own, leading to the perception of illusory objects. The ultimate conclusion is a striking one. The role of attention (i.e., of externally-focussed consciousness) is seen as one of combining disparate features into unified wholes.
The viewpoint of Treisman and Schmidt will be very useful here. For, in the language of the psynet model, "unified whole" naturally translates into "autopoietic system." Attention is thus viewed as a process which takes in disparate features from the lower levels of the dual network and, using the network's hierarchical and heterarchical structure as a guide, tries to integrate these features into an autopoietic system. These features may be slightly modified, or grouped with other features that were not "really" there; the important thing is that the result is a coherent system that can survive amidst the natural fluctuations of the evolving mind.
The process by which the brain accomplishes this kind of "coherentization" is not entirely clear. However, it is plain from the logic of the process, as well as from various neurophysiological considerations, that the phenomenon involves some kind of feedback loop between parts of the brain dealing with lower-level, featural information and parts of the brain dealing with higher-level, conceptual information. In CL this idea is lifted from the level of the brain to the level of the mind, and labeled the "perceptual-cognitive loop". The proposed dynamics of this loop are summarized in the following three steps:
1) The perceptual end of the loop sends the cognitive end information consisting of disparate features, with some tentative connections perhaps drawn between them.
2) The cognitive end of the loop modifies the features and links them with other entities stored in memory, seeking to form a coherent autopoietic system integrating the features.
3) The cognitive end sends its conjectural coherentization to the perceptual end, which determines whether the constructed system strays too far from the original features. The perceptual end sends back its report, perhaps containing a few specific suggestions. Unless the report is sufficiently positive, return to Step 2.
In this view, then, consciousness is a process of stepping through the attractor of an autopoietic system, which is itself evolving in order to match its environment (i.e., the information being supplied by the perceptual end of the loop). This is an abstract, schematic model; it is not a detailed model of the neural processes regulating human attention. However, it is concrete enough to guide efforts at neural and psychological modelling.
For reasons that will become clear, in the following we will refer to the Perceptual-Cognitive-Active Loop rather than merely the Perceptual-Cognitive Loop. This is not a theoretical change, but merely a terminological improvement. As noted in (Bisiach and Berti, 1987), there is evidence for a crucial role for premotor neurons in consciousness. The old term "Perceptual-Cognitive Loop" was intended to encompass low-level interactions between perception and motor processes; the new term simply makes this explicit.
3. THE MATHEMATICS OF MAGICIAN SYSTEMS
It is not hard to represent the magician system dynamic mathematically. Let S denote a set, to be called the space of magicians. Then S^{*}, the space of all finite sets composed of elements of S, with repeated elements allowed, is the space of magician systems associated with S. One may write
System_{t+1} = A[System_{t}] (1)
where System_{t} is an element of S^{*} denoting the magician population at time t, and A is the "action operator," a function mapping magician populations into magician populations.
Let us assume, for a first formalization, that all magician interactions are binary, i.e., involving one magician acting on another to create a third. In this instance the machinery of magician operations may be described by a binary algebraic operation *, so that where a, b and c are elements of S, a*b=c is read "a acts on b to create c." The case of unary, ternary, etc. interactions may be treated in a similar way or, somewhat artificially, may be constructed as a corollary of the binary case.
The action operator A may be decomposed as
A[X] = F[ R[X] ] (2)
where R is the "raw potentiality" operator and F is a "filtering" operator. R is formally given by
R[System_{t}[ = R[{a_{1},a_{2},...,a_{n}(t)}]
= { a_{i}*a_{j} | i,j = 1,...,n(t)} (3)
The purpose of R is to construct the "raw potentiality" of the magician system System_{t}, the set of all possible magician combinations which ensue from it. The role of the filtering operator F, on the other hand, is to select from the raw potentiality those combinations which are to be allowed to continue to the next time step. This selection may be all-or-none, or it may be probabilistic.
To define the filtering operator formally, let P^{*} denote the a space of all probability distributions on the space magician systems S^{*}. Then, F is a function which maps S^{*} x S^{*} into P^{*}. The probability of a combination a_{i} * a_{j} of surviving the filtering process F will be called the "weight" of the combination.
Magician systems, thus defined, are ageometric, or, to use the chemical term, "well-mixed." But one may also consider "graphical magician systems," magician systems that are specialized to some given graph G. Each magician is assigned a location on the graph as one of its defining properties, and magicians are only allowed to interact if they reside at the same or adjacent nodes. This does not require any reformulation of the fundamental equations given above, but can be incorporated in the filtering operator, by setting F so that the magician combination a_{i} * a_{j} has zero probability unless a_{i} and a_{j} occupy appropriate relative locations. The dual network, an essential component of the psynet model, is a magician system which resides on a special kind of graph.
Magician Systems as Abstract Algebras
It is somewhat enlightening to express magician systems in terms of abstract algebra. I will describe a magician algebra (S,+,*), where S is the same space mentioned above.
The operation * I will interpret as magician action;
so that a*b means the magician resulting from a acting on b.
It remains to given an appropriate definition for the operation +. But this is trivial. Suppose we construct a mapping V which associates with each element a of S^{*} a vector v of integers, where v_{i} indicates the number of copies of magician a_{i} in a. It is clear that V is one-to-one, so that we may define the sum of two elements of S^{*} as a + b = V^{-1} ( V(a) + V(b)), where the sum V(a) + V(b) is ordinary coordinate-wise addition of vectors. This definition of addition has the side-effect of giving a natural interpretation of the annihilation of magicians: if b annihilates a then we may say b = -a, in which case the simultaneous presence of a and b results in the empty set. It is convenient to call the magician -a the "antimagician" of a. If one makes the natural assumption -a * b = - (a * b), the introduction of antimagicians makes the algebraic system (S,+,*) an algebra of hypercomplex numbers.
For instance, suppose one has a magician system consisting of 3 copies of magician a, 2 copies of magician b, and 4 copies of magician c. This system may be represented by the expression 3a + 2b + 4c. And, in this notation, the "raw potentiality" resulting from the system unrestrainedly acting on itself is given by the expression (3a + 2b + 4c) * (3a + 2b + 4c). For what the distributive law says is that each element of the first multiplicand will be paired exactly once with each element of the second multiplicand. The production of, for instance, 12 a * c's from the pairing of the first 3a term with the last 4c term makes perfect sense, because each of the three a magicians gets to act on each of the four c magicians.
More generally, suppose that the initial state of a magician system is represented by the linear combination System_{0} = z_{0} = c_{1}a_{1} + ... + c_{N}a_{N}, where the c_{i} represent the populations of the various magicians, and suppose that the filtering operator F is the identity, i.e. F R = R. Then, where System_{t} = z_{t}, the trajectory of the magician system may be obtained from the simple iteration
z_{t} = z_{t-1}^{2} (5)
And if one has a magician system with a constant external input, one gets the slightly more complex equation
z_{t} = z_{t-1}^{2} + c (6)
where c is a magician system that does not change over time. Introducing a filtering operation modifies the equation significantly, yielding
z_{t} = f(z_{t-1})^{2} + c (6)
This formulation immediately reveals the close relationship between magician systems and Julia sets. Suppose one has a system of five magicians called 0, 1, i, -1 and -i, with a commutative multiplication obeying i * i = -1. This particular magician system, when used as the algebra for an hypercomplex number space, yields the complex numbers; and the equation for an unfiltered, externally-driven magician system is then just a quadratic iteration in the complex plane, familiar from the theory of Julia and Mandelbrot sets. The Julia set corresponding to a given "environment" value c contains those initial magician system states which do not lead to a situation in which some magician is copied infinitely many times; and it also contains all values c which are limit points of stable environment values of this type. The boundary of the Julia set thus demarcates the viable realm of initial system states from the non-viable realm (note that what we call the Julia set, some authors call the "filled Julia set," reserving the term "Julia set" for the boundary of our Julia set).
There are numerous interesting relationships between magician systems and standard concepts from abstract algebra. However, we will have little need to delve into this topic here. For instance, the algebra S/a, consisting of the right cosets of a under *, represents, intuitively speaking, the whole magician system as seen from the point of view of a. It consists of a set of equivalence classes, where two magicians are equivalent if they are transformed to the same thing by a. If a is a pattern-recognizing magician, then the equivalence classes consist of magicians which have identical structure, as judged by a; i.e., which are identical from a's point of view.
Also, it should be observed that an autopoietic attractor of a system is, clearly, a subalgebra of the algebra defining the whole system. This point follows immediately from the fact that every element in the autopoietic system must be able to be produced by other elements in the system. However, it is not clear whether or not the converse is true: whether every subalgebra can be an autopoietic attractor. There are two problems here. First, with autopoiesis: just because every element a subalgebra can be produced by other elements in the subalgebra, this doesn't imply that this production can be carried out in a way that preserves the system as a whole. Second, with attraction: even if a subalgebra is autopoietic, this is no guarantee that "nearby" algebras will tend to evolve into this algebra under magician dynamics.
4. PROPRIOCEPTION OF THOUGHT
The psynet model states that consciousness creates autopoietic magician systems. The discussion of the previous section shows that magician systems can be expressed as abstract algebras. Thus, according to the psynet model, we may say that consciousness creates abstract algebras. The question then becomes: which algebras?
We are not quite yet ready to answer this question, however. In order to get to the point where this question may be addressed, we will draw inspiration from a somewhat unlikely-sounding direction: the philosophical thought of the quantum physicist David Bohm, as expressed (among other places) in his book Thought as a System (1988).
Bohm views thought as a system of reflexes - - habits, patterns - - acquired from interacting with the world and analyzing the world. He understands the self- reinforcing, self- producing nature of this system of reflexes. And he diagnoses our thought- systems as being infected by a certain malady, which he calls the absence of proprioception of thought.
Proprioceptors are the nerve cells by which the body determines what it is doing - - by which the mind knows what the body is doing. To understand the limits of your proprioceptors, stand up on the ball of one foot, stretch your arms out to your sides, and close your eyes. How long can you retain your balance? Your balance depends on proprioception, on awareness of what you are doing. Eventually the uncertainty builds up and you fall down. People with damage to their proprioceptive system can't stay up as long as as the rest of us.
According to Bohm,
... [T]hought is a movement - - every reflex is a movement really. It moves from one thing to another. It may move the body or the chemistry or just simply the image or something else. So when 'A' happens 'B' follows. It's a movement. All these reflexes are interconnected in one system, and the suggestion is that they are not in fact all that different. The intellectual part of thought is more subtle, but actually all the reflexes are basically similar in structure. Hence, we should think of thought as a part of the bodily movement, at least explore that possibility, because our culture has led us to believe that thought and bodily movement are really two totally different spheres which are no basically connected. But maybe they are not different. The evidence is that thought is intimately connected with the whole system. If we say that thought is a reflex like any other muscular reflex - - just a lot more subtle and more complex and changeable - - then we ought to be able to be proprioceptive with thought. Thought should be able to perceive its own movement. In the process of thought there should be awareness of that movement, of the intention to think and of the result which that thinking produces. By being more attentive, we can be aware of how thought produces a result outside ourselves. And then maybe we could also be attentive to the results it produces within ourselves. Perhaps we could even be immediately aware of how it affects perception. It has to be immediate, or else we will never get it clear. If you took time to be aware of this, you would be bringing in the reflexes again. So is such proprioception possible? I'm raising the question....
The basic idea here is quite simple. If we had proprioception of thought, we could feel what the mind was doing, at all times -- just as we feel what the body is doing. Our body doesn't generally carry out acts on the sly, without our observation, understanding and approval. But our mind (our brain) continually does exactly this. Bohm traces back all the problems of the human psyche and the human world -- warfare, environmental destruction, neurosis, psychosis -- to this one source: the absence of proprioception of thought. For, he argues, if we were really aware of what we were doing, if we could fully feel and experience everything we were doing, we would not do these self-destructive things.
An alternate view of this same idea is given by the Zen master Thich Nhat Hanh (1985), who speaks not of proprioception but of "mindfulness." Mindfulness means being aware of what one is doing, what one is thinking, what one is feeling. Thich Nhat Hanh goes into more detail about what prevents us from being mindful all the time. In this connection he talks about samyojama - - a Sanskrit word that means "internal formations, fetters, or knots." In modern terminology, samyojama are nothing other than self- supporting thought- systems:
When someone says something unkind to us, for example, if we do not understand why he said it and we become irritated, a knot will be tied in us. The lack of understanding is the basis for every internal knot. If we practice mindfulness, we can learn the skill of recognizing a knot the moment it is tied in us and finding ways to untie it. Internal formations need our full attention as soon as they form, while they are still loosely tied, so that the work of untying them will be easy.
Self- supporting thought systems, systems of emotional reflexes, guide our behaviors in all sorts of ways. Thich Nhat Hanh deals with many specific examples, from warfare to marital strife. In all cases, he suggests, simple sustained awareness of one's own actions and thought processes - - simple mindfulness - - will "untie the knots," and free one from the bundled, self- supporting systems of thought/feeling/behavior.
Yet nother formulation of the same basic concept is given by psychologist Stanislaw Grof (1994). Grof speaks, not of knots, but rather of "COEX systems" - - systems of compressed experience. A COEX system is a collection of memories and fantasies, from different times and places, bound together by the self- supporting process dynamics of the mind. Elements of a COEX system are often joined by similar physical elements, or at least similar emotional themes. An activated COEX system determines a specific mode of perceiving and acting in the world. A COEX system is an attractor of mental process dynamics, a self- supporting subnetwork of the mental process network, and, in Buddhist terms, a samyojama or knot. Grof has explored various radical techniques, including LSD therapy and breathwork therapy, to untie these knots, to weaken the grip of these COEX systems. The therapist is there to assist the patient's mental processes, previously involved in the negative COEX system, in reorganizing themselves into a new and more productive configuration.
Reflexes and Magicians
What does this notion of "proprioception of thought," based on a neo-behaviourist, reflex-oriented view of mind, have to do with the psynet model? To clarify the connection, we must first establish the connection between reflexes and magicians. The key idea here is that, in the most general sense, a habit is nothing other than a pattern. When a "reflex arc" is established in the brain, by modification of synaptic strengths or some other method, what is happening is that this part of the brain is recognizing a pattern in its environment (either in the other parts of the brain to which it is connected, or in the sensory inputs to which it is connected).
A reflex, in the psynet model, may be modelled as the interaction of three magicians: one for perception, one for action, and one for the "thought" (i.e. for the internal connection between perception and action). The "thought" magician must learn to recognize patterns among stimuli presented at different times and generate appropriate responses.
This view of reflexes is somewhat reminiscent of the triangular diagrams introduced by Gregory Bateson in his posthumous book Angels Fear (1989). I call these diagrams "learning triads." They are a simple and general tool for thinking about complex, adaptive systems. In essence, they are a system-theoretic model of the reflex arc.
Bateson envisions the fundamental triad of thought, perception and action, arranged in a triangle:
THOUGHTThe logic of this triad is as follows. Given a percept, constructed by perceptual processes from some kind of underlying data, a thought process decides upon an action, which is then turned into a concrete group of activities by an action process. The results of the actions taken are then perceived, along with the action itself, and fed through the loop again. The thought process must judge, on the basis of the perceived results of the actions, and the perceived actions, how to choose its actions the next time a similar percept comes around.
/ \
PERCEPTION -- ACTION
Representing the three processes of THOUGHT, PERCEPTION and ACTION as pattern/process magicians, the learning triad may be understood as a very basic autopoietic mental process system. Furthermore, it is natural to conjecture that learning triads are autopoietic attractors of magician system dynamics.
The autopoiesis of the system is plain: as information passes around the loop, each process is created by the other two that come "before" it. The attraction is also somewhat intuitively obvious, but perhaps requires more comment. It must be understood that no particular learning triad is being proposed as an attractor, in the sense that nearby learning triads will necessarily tend to it. The claim is rather that the class of learning triads constitutes a probabilistic strange attractor of magician dynamics, meaning that a small change in a learning triad will tend to produce something that adjusts itself until it is another learning triad. If this is true, then learning triads should be stable with respect to small perturbations: small perturbations may alter their details but will not destroy their basic structure as a learning mechanism.
Pattern, Learning and Compression
We have cast Bohm's reflex-oriented view of mind in terms of pattern/process magicians. A reflex arc is, in the psynet model, re-cast as a triadic autopoietic attractor of magician dynamics. In this language, proprioception of thought -- awareness of what reflexes have produced and are producing a given action -- becomes awareness of the magician dynamics underlying a given behaviour.
In this context, let us now return to the notion of pattern itself. The key point here is the relation between pattern and compression. Recall that to recognize a pattern in something is to compress it into something simpler - - a representation, a skeleton form. Given the overwhelmingly vast and detailed nature of inner and outer experience, it is inevitable that we compress our experiences into abbreviated, abstracted structures; into what Grof calls COEX's. This is the function of the hierarchical network: to come up with routines, procedures, that will function adequately in a wide variety of circumstances.
The relation between pattern and compression is well-known in computer science, in the fields of image compression and text compression. In this contexts, the goal is to take a computer file and replace it with a shorter file, containing the same or almost the same contents. Text compression is expected to be lossless: one can reconstruct the text exactly from the compressed version. On the other hand, image compression is usually expected to be lossy. The eye doesn't have perfect acuity, and so a bit of error is allowed: the picture that you reconstruct from the compressed file doesn't have to be exactly the same as the original.
Psychologically, the result of experience compression is a certain ignorance. We can never know exactly what we do when we lift up our arm to pick up a glass of water, when we bend over to get a drink, when we produce a complex sentence like this one, when we solve an equation or seduce a woman. We do not need to know what we do: the neural network adaptation going on in our brain figures things out for us. It compresses vast varieties of situations into simple, multipurpose hierarchical brain structures. But having compressed, we no longer have access to what we originally experienced, only to the compressed form. We have lost some information.
For instance, a man doesn't necessarily remember the dozens of situations in which he tried to seduce women (successfully or not). The nuances of the different womens' reactions, the particular situations, the moods he was in on the different occasions -- these are in large part lost. What is taken away is a collection of abstracted patterns that the mind has drawn out of these situations.
Or, to take another example, consider the process of learning a tennis serve. One refines one's serve over a period of many games, by a process of continual adaptation: this angle works better than that, this posture works better so long as one throws the ball high enough, etc. But what one takes out of this is a certain collection of motor processes, a collection of "serving procedures." It may be that one's inferences regarding how to serve have been incorrect: that, if one had watched one's serving attempts on video (as is done in the training of professional tennis players), one would have derived quite different conclusions about how one should or should not serve. But this information is lost, it is not accessible to the mind: all that is available is the compressed version, i.e., the serving procedures one has induced. Thus, if one is asked why one is serving the way one is, one can never give a decent answer. The answer is that the serve has been induced by learning triads, from a collection of data that is now largely forgotten.
The point is that mental pattern recognition is in general highly lossy compression. It takes place in purpose-driven learning triads. One does not need to recognize all the patterns in one's tennis serving behavior -- enough patterns to generate the full collection of data at one's disposal. One only wants those patterns that are useful to one's immediate goal of developing a better serve. In the process of abstracting information for particular goals (in Bohm's terms, for the completion of particular reflex arcs), a great deal of information is lost: thus psychological pattern recognition, like lossy image compression, is a fundamentally irreversible process.
This is, I claim, the ultimate reason for what Bohm calls the absence of proprioception of thought. It is the reason why mindfulness is so difficult. The mind does not know what it is doing because it can do what it does far more easily without the requirement to know what it is doing. Proceeding blindly, without mindfulness, thought can wrap up complex aggregates in simple packages and proceed to treat the simple packages as they were whole, fundamental, real. This is the key to abstract symbolic thought, to language, to music, mathematics, art. Intelligence itself rests on compression: on the substitution of packages for complex aggregates, on the substitution of tokens for diverse communities of experiences. It requires us to forget the roots of our thoughts and feelings, in order that we may use them as raw materials for building new thoughts and feelings.
If, as Bohm argues, the lack of proprioception of thought is the root of human problems, then the only reasonable conclusion would seem to be that human problems are inevitable.
5. THE ALGEBRA OF CONSCIOUSNESS
We are now ready to turn to the central question of the paper: if the purpose of consciousness is to create autopoietic systems, then what sorts of autopoietic systems does consciousness create? The answer to this question might well be: any kind of autopoietic system. I will argue, however, that this is not the case: that in fact consciousness produces very special sorts of systems, namely, systems with the structure of quaternionic or octonionic algebras.
In order to derive this somewhat surprising conclusion from the psynet model, only one additional axiom will be required, namely, that the autopoietic systems constructed by consciousness are "timeless," without an internal sense of irreversibility (an "arrow of time"). I.e.,
In the magician systems contained in consciousness, magician operations are reversible
In view of the discussion in the previous section, an alternate way to phrase this axiom is as follows:
At any given time, proprioception of thought extends through the contents of consciousness
Bohm laments that the mind as a whole does not know what it is doing. I have argued that, on grounds of efficiency, the mind cannot know what it is doing. It is immensely more efficient to compress experience in a lossy, purpose-driven way, than to maintain all experiences along with the patterns derived from them. However, this argument leaves room for special cases in which thought is proprioceptive. I posit that consciousness is precisely such a special case. Everything that is done in consciousness is explicitly felt, in the manner of physical proprioception: it is there, you can sense its being, feel it move and act.
Of course, physical proprioception can be unconscious; and so can mental proprioception. One part of the mind can unconsciously sense what another part is doing. The point, however, is that conscious thought-systems are characterized by self-proprioception. I will take this an an axiom, derived from phenomenology, from individual experience. This axiom is not intended as an original assertion, but rather as a re-phrasing of an obvious aspect of the very definition of consciousness. It should hardly be controversial to say that a conscious thought or thought-system senses itself.
In the context of the psynet model, what does this axiom mean? It means, that, within the scope of consciousness, magician processes are reversible. There is no information loss. What is done, can be undone.
Algebraically, reversibility of magician operations corresponds to division. For, if multiplication represents both action and pattern recognition, then the inverse under multiplication is thus an operation of undoing. If
a * b = c
this means that by acting on b, a has produced this pattern c; and thus, in the context of the cognitive equation, that c is a pattern which a has recognized in b. Now the inverse of a, when applied to c, yields
a^{-1} * c = b
In other words, it restores the substrate from the pattern: it looks at the pattern c and tells you what the entity was that aa recognized the pattern in.
For a non-psychological illustration, let us return, for a moment, to the example of text compression. A text compression algorithm takes a text, a long sequence of symbols, and reduces it to a much shorter text by eliminating various redundancies. If the original text is very long, then the shorter text, combined with the decompression algorithm, will be a pattern in the original text. Formally, if b is a text, and a is a compression algorithm, then a * b = c means that c is the pattern in b consisting of the compressed version of a plus the decompression algorithm. c^{-1} is then the process which transforms c into b; i.e., it is the process which causes the decompression algorithm to be applied to the compressed version of a, thus reconstituting the original a. The magician a compresses, the magician a^{-1} decompresses.
So, proprioception of thought requires division. It requires that one does not rely on patterns as lossy, compressed versions of entities; that one always has access to the original entities, so one can see "what one is doing." Suppose, then, one has a closed system of thoughts, a mental system, in which division is possible; in which mental process can proceed unhindered by irreversibility. Recall that mental systems are subalgebras. The conclusion is that proprioceptive mental systems are necessarily division algebras: they are magician systems in which every magician has an inverse. The kinds of algebras which consciousness constructs are division algebras.
This might at first seem to be a very general philosophical conclusion. However, it turns out to place very strict restrictions on the algebraic structure of consciousness. For, as is well-known in abstract algebra, the finite-dimensional division algebras are very few indeed.
Quaternions and Octonions
The real number line is a division algebra. So are the complex numbers. There is no three-dimensional division algebra: no way to construct an analogue of the complex numbers in three dimensions. However, there are division algebras in four and eight dimensions; these are called the quaternions and the octonions (or Cayley algebra), see e.g. Kurosh, (1963).
The quaternions are a group consisting of the entities {1,i,j,k} and their "negatives" {-1,-i,-j,-k}. The group's multiplication table is defined by the products
i * j = k
j * k = i
k * i = j
i * i = j * j = k * k = -1,
This is a simple algebraic structure which is distinguished by the odd "twist" of the multiplication table according to which any two of the three quantities {i,j,k} are sufficient to produce the other.
The real quaternions are the set of all real linear combinations of {1,i,j,k}, i.e., the set of all expressions a + bi + cj + dk where a, b, c and d are real. They are a four-dimensional, noncommutative extension of the complex numbers, with numerous applications in physics and mathematics.
Next, the octonions are the algebraic structure formed from the collection of entities q + Er, where q and r are quaternions and E is a new element which, however, also satisfies E^{2} = -1. These may be considered as a vector space over the reals, yielding the real octonions. While the quaternions are non-commutative, the octonions are also non-associative. The octonions have a subtle algebraic structure which is rarely if ever highlighted in textbooks, but which has been explored in detail by Onar Aam and Tony Smith (personal communication). The canonical basis for the octonion algebra is given by (i,j,k,E,iE,jE,kE). Following a suggestion of Onar Aam, I will adopt the simple notation I = iE, J = jE, K = kE, so that the canonical basis becomes (i,j,k,E,I,J,K).
Both the real quaternions and the real octonions have the property of allowing division. That is, every element has a unique multiplicative inverse, so that the equation A * B = C can be solved by the formula A = B^{-1} * C. The remarkable fact is that these are not only good examples of division algebras, they are just about the only reasonable examples of division algebras. One may prove that all finite division algebras have order 1, 2, 4 or 8. Furthermore, the only division algebras with the property of alternativity are the real, complexes, real quaternions and real octonions. Alternativity means that subalgebras consisting of two elements are associative. These results are collected under the name of the Generalized Frobenius Theorem. Finally, the only finite algebras which are normable are -- the reals, complexes, real quaternions and real octonions.
What these theorems teach us is that these are not merely arbitrary examples of algebras. They are very special algebras, which play several unique mathematical roles.
Several aspects of the quaternion and octonion multiplication tables are particularly convenient in the magician system framework. Perhaps the best example is the identity of additive and multiplicative inverses. The rule A^{-1} = -A (which applies to all non-identity elements) says that undoing (reversing) is the same as annihilation. Undoing yields the identity magician 1, which reproduces everything it comes into contact with. Annihilation yields zero, which leaves alone everything it comes into contact with. The ultimate action of the insertion of the inverse of A into a system containing A is thus to either to reproduce the system in question (if multiplication is done first), or to reproduce the next natural phase in the evolution of the system (if addition is done first).
Compare this to what happens in a magician system governed by an arbitrary algebra. Given a non-identity element A not obeying the rule A^{2} = -1, one has to distinguish whether its opposite is to act additively or multiplicatively. In the magician system framework, however, there is no easy way to make this decision: the opposites are simply released into the system, free to act both additively and multiplicatively. The conclusion is that only in extended imaginary algebras like the quaternions and octonions can one get a truly natural magician system negation.
Consciousness and Division Algebras
The conclusion to which the Generalized Frobenius Theorem leads us is a simple and striking one: the autopoietic systems which consciousness constructs are quaternionic and octonionic in structure. This is an abstract idea, which has been derived by abstract means, and some work will be needed to see what intuitive sense it makes. However, the reasoning underlying it is hopefully clear.
The notion of reversibility being used here may perhaps benefit from a comparison with the somewhat different notion involved in Edward Fredkin's (Fredkin and Toffoli, 1982) theory of reversible computation. Fredkin has shown that any kind of computation can be done in an entirely reversible way, so that computatinoal systems need not produce entropy. His strategy for doing this is to design reversible logic gates, which can then be strung together to produce any Boolean function, and thus simulate any Turing machine computation. These logic gates, however, operate on the basis of redundancy. That is, when one uses these gates to carry out an operation such as "A and B," enough information is stored to reconstruct both A and B, in addition to their conjunction.
The main difference between these reversible computers and the reversible magician systems being considered here is the property of closure. According to the psynet model, mental entities are autopoietic attractors. In the simplest case of fixed point attractors, this implies that mental entities are algebraically closed magician systems: they do not lead outside themselves. Even in the case of periodic or strange attractors, there is still a kind of closure: there is a range of magicians which cannot be escaped. Finally, in the most realistic case of stochastic attractors, there is a range of magicians which is unlikely to be escaped. In each case, everything in the relevant range is producible by other elements in that same range: the system is self-producing. Fredkin's logic gates do not, and are not intended to, display this kind of property. It is not in any way necessary that each processing unit in a reversible computing system be producible by combinations of other processing units. The contrast with reversible computers emphasizes the nature of the argument that has led us to postulate a quaternionic/ octonionic structure for consciousness. The simple idea that consciousness is reversible does not lead you to any particular algebraic structure. One needs the idea of reversible conscious operations in connection with the psynet model, which states that mental systems are autopoietic attractors of process dynamical systems. Putting these two pieces together leads immediately to the finite division algebras.
Next, a word should be said about the phenomenological validity of the present theory. The phenomenology of consciousness is peculiar and complex. But one thing that may be asserted is that consciousness combines a sense of momentary timelessness, of being "outside the flow of time," with a sense of change and flow. Any adequate theory of consciousness must explain both of these sensations. The current theory fulfills this criterion. Consciousness, it is argued, is concerned with constructing systems that lack a sense of irreversibility, that stand outside the flow of time. But in the course of constructing such systems, consciousness carries out irreversible processes. Thus there is actually an alternation between timelessness and time-boundedness. Both are part of the same story.
Finally, let us move from phenomenology to cognitive psychology. If one adopts the view given here, one is immediately led to the conclusion that the Generalized Frobenius Theorem solves an outstanding problem of theoretical psychology: it explains why consciousness is bounded. No previous theory of consciousness has given an adequate answer for the question: Why can't consciousness extend over the whole mind? But in the algebraic, psynet view, the answer is quite clear. There are no division algebras the size of the whole mind. The biggest one is the octonions. Therefore consciousness is limited to the size of the octonions: seven elements and their associated anti-elements.
The occurence of the number 7 here is striking, for, as is well-known, empirical evidence indicates that the capacity of human short-term memory is about 7. The figure is usually given as 7 +/- 2. Of course, 7 is a small number, which can be obtained in many different ways. But, nevertheless, the natural occurrence of the number 7 in the algebraic theory of consciousness is a valuable piece of evidence. In no sense was the number 7 arbitrarily put into the theory. It emerged as a consequence of deep mathematics, from the simple assumption that the contents of consciousness, at any given time, is a reversible, autopoietically attractive magician system.
6. MODELLING STATES OF MIND
The quaternionic and octonionic algebras are structures which can be concretized in various ways. They describe a pattern of inter-combination, but they do not specify the things combined, nor the precise nature of the combinatory operation.
Thus, the way in which the algebraic structures are realized can be expected to depend upon the particular state of mind involved. For instance, a state of meditative introspection is quite different from a state of active memorization, which is in turn quite different from a state of engagement with sensory or motor activities. Each of these different states may involve different types of mental processes, which act on each other in different ways, and thus make use of the division algebra structure quite differently.
As the quaternions and octonions are very flexible structures, there will be many different ways in which they can be used to model any given state of mind. What will be presented here are some simple initial modelling attempts. The models constructed are extremely conceptually natural, but that is of course no guarantee of their correctness. The advantage of these models, however, is their concreteness. Compared to the general, abstract quaternion-octonion model, they are much more closely connected to daily experience, experimental psychology and neuroscience. They are, to use the Popperian term, "falsifiable," if not using present experimental techniques, then at least plausibly, in the near future. They also reveal interesting connections with ideas in the psychological and philosophical literature.
Perhaps the best way to understand the nature of the ideas in this section is to adopt a Lakatosian, rather than Popperian, philosophy of science. According to Lakatos, the "hard core" of a theory, which is too abstract and flexible to be directly testable, generates numerous, sometimes mutually contradictory "peripheral theories" which are particular enough to be tested. The hard core here is the quaternionic-octonionic theory of consciousness, clustered together with the psynet model and the concepts of thought-proprioception and reversible consciousness. The peripheral theories are applications of the algebras to particular states of consciousness.
An Octonionic Model of Short-Term Memory
Consider now a state of consciousness involving memorization or intellectual thought, which requires holding a number of entities in consciousness for a period of time, and observing their interactions. In this sort of state consciousness can be plausibly identified with what psychologists call "short-term memory" or "working memory."
The most natural model of short-term memory is one in which each of the entities held in consciousness is identified with one of the canonical basis elements {i,j,k,E,I,J,K}. The algebraic operation * is to be interpreted as merely a sequencing operation. While all the entities held in consciousness are in a sense "focuses of attention," in general some of the entities will be focussed on more intensely than others; and usually one entity will be the primary focus. The equation A * B = C means that a primary focus on A, followed by a primary focus on B, will be followed by a primary focus on C.
This model of short-term memory that implies the presence of particular regularities in the flow of the focus of consciousness. The nature of these regularities will depend on the particular way the entities being stored in short-term memory are mapped onto the octonions. For instance, suppose one wants to remember the sequence
PIG, COW, DOG, ELEPHANT, FOX, WOMBAT, BILBY
Then, according to the model, these elements must be assigned in a one-to-one manner to the elements of some basis of the octonion algebra. This need not be the canonical basis introduced above, but for purposes of illustration, let us assume that it is. For convenience, let us furthermore assume that the identification is done in linear order according to the above list, so that we have
pig, cow, dog, elephant, fox, wombat, bilbyThen the theory predicts that, having focussed on PIG and then COW, one will next focus on DOG. On the other hand, having focussed on DOG, and then on the absence or negative of COW, one will next focus on PIG.
i j k E I J K
Of course, the order of focus will be different if the mapping of memory items onto basis elements is different. There are many different bases for the octonions, and for each basis there are many possible ways to map seven items onto the seven basis elements. But nevertheless, under any one of these many mappings, there would be a particular order to the way the focus shifted from one element to the other.
This observation leads to a possible method of testing the octonionic model of short-term memory. The model would be falsified if it were shown that the movement from one focus to another in short-term memory were totally free and unconstrained, with no restrictions whatsoever. This would not falsify the general division algebra model of consciousness, which might be applied to short-term memory in many different ways; but it would necessitate the development of a more intricate connection between octonions and short-term memory.
This octonionic model of short-term memory may be understood in terms of chaos theory -- an interpretation which, one suspects, may be useful for empirical testing. Suppose one has a dynamical system representing short-term memory, e.g. a certain region of the brain, or a family of neural circuits. The dynamics of this system can be expected to have a "multi-lobed" attractor similar to that found by Freeman in his famous studies of olfactory perception in the rabbit. Each lobe of the attractor will correspond to one of the elements stored in memory. The question raised by the present model is then one of the second-order transition probabilities between attractor lobes. If these probabilities are all equal then the simple octonionic model suggested here is falsified. On the other hand, if the probabilities are biased in a way similar to one of the many possible octonion multiplication tables, then one has found a valuable piece of evidence in favor of the present model of short-term memory and, indirectly, in favor of the finite division algebra theory of consciousness. This experiment cannot be done at present, because of the relatively primitive state of EEG, ERP and brain scan technology. However, it is certainly a plausible experiment, and there is little doubt that it will be carried out at some point over the next few decades.
Learning Triads
The previous model dealt only with the holding of items in memory. But what about the active processing of elements in consciousness? What, for example, about states of consciousness which are focussed on learning motor skills, or on exploring the physical or social environment?
To deal with these states we must return to the notion of a "learning triad," introduced above as a bridge between the psynet model and Bohm's reflex-oriented psychology. The
first step is to ask: how might we express the logic of the learning triad algebraically? We will explore this question on an intuitive basis, and then go back and introduce definitions making our insights precise.
First of all, one might write
THOUGHT = PERCEPTION * ACTION
ACTION = THOUGHT * PERCEPTION
PERCEPTION = ACTION * THOUGHT
These equations merely indicate that a perception of an action leads to a revised thought, a thought about a perception leads to an action, and an action based on a thought leads to a perception. They express in equations what the triad diagram itself says.
The learning triad is consistent with the quaternions. It is consistent, for example, with the hypothetical identification
PERCEPTION THOUGHT ACTIONBut the three rules given above do not account for much of the quaternion structure. In order to see how more of the structure comes out in the context of learning triads, we must take a rather unexpected step. We must ask: How does the activity of the learning triad relate to standard problem-solving techniques in learning theory and artificial intelligence?i j k
Obviously, the learning triad is based on a complex-systems view of learning, rather than a traditional, procedural view. But on careful reflection, the two approaches are not so different as they might seem. The learning triad is actually rather similar to a simple top-down tree search. One begins from the top node, the initial thought. One tests the initial thought and then modifies it in a certain way, giving another thought, which may be viewed as a "child" of the initial thought. Then, around the loop again, to a child of the child. Each time one is modifying what came before -- moving down the tree of possibilities.
Viewed in this way, however, the learning triad is revealed to be a relatively weak learning algorithm. There is no provision for "backtracking" -- for going back up a node, retreating from a sequence of modifications that has not borne sufficient fruit. In order to backtrack, one would like to actually erase the previous modification to the thought process, and look for another child node, an alternative to the child node already selected.
An elegant way to view backtracking is as going around the loop the wrong way. In backtracking, one is asking, e.g.: What is the thought that gave rise to this action? Or, what is the action that gave rise to this percept? Or, what is the percept that gave rise to this thought? In algebraic language, one is asking questions that might be framed
THOUGHT * ACTION = ?
ACTION * PERCEPT = ?
PERCEPT * THOUGHT = ?
In going backwards in time, while carrying out the backtracking method, what one intends to do is to wipe out the record of the abandoned search path. One wants to eliminate the thought-process modifications that were chosen based on the percept; one wants to eliminate the actions based on these thought modifications; and one wants to eliminate the new percept that was formed in this way. Thus, speaking schematically, one wants to meet PERCEPT with -PERCEPT, ACTION with -ACTION and THOUGHT with -THOUGHT. Having annihilated all that was caused by the abandoned choice, one has returned to the node one higher in the search tree. The natural algebraic rules for backtracking are thus:
THOUGHT * ACTION = - PERCEPT
ACTION * PERCEPT = - THOUGHT
PERCEPT * THOUGHT = - ACTION
Backtracking is effected by a backwards movement around the learning triad, which eliminates everything that was just laid down.
The view of learning one obtains is then one of repeated forward cycles, interspersed with occasional backward cycles, whenever the overall results of the triad are not satisfactory. The algebraic rules corresponding to this learning method are consistent with the quaternion multiplication table. The division-algebra structure of consciousness is in this way seen to support adaptive learning.
In this view, the reason for the peculiar power of conscious reasoning becomes quite clear. Consciousness is sequential, while unconscious thought is largely parallel. Consciousness deals with a small number of items, while unconscious thought is massive, teeming, statistical. But the value of conscious thought is that it is entirely self-aware, and hence it is reversible. And the cognitive value of reversibility is that it allows backtracking: it allows explicit retraction of past thoughts, actions and perceptions, and setting down new paths.
In the non-reversible systems that dominate the unconscious, once something is provisionally assumed, it is there already and there is no taking it back (not thoroughly at any rate). In the reversible world of consciousness, one may assume something tentatively, set it aside and reason about it, and then retract it if a problem occurs, moving on to another possibility. This is the key to logical reasoning, as opposed to the purely intuitive, habit-based reasoning of the unconscious.
Thought Categories and Algebraic Elements
We have posited an intuitive identification of mental process categories with quaternionic vectors. It is not difficult to make this identification rigorous, by introducing a magician system set algebra based on the magician system algebra given above.
To introduce this new kind of algebra, let us stick with the current example. Suppose one has a set of magicians corresponding to perceptual processes, a set corresponding to thought processes, and a set corresponding to action processess. These sets are to be called PERCEPTION, THOUGHT and ACTION. The schematic equations given above are then to be interpreted as set equations. For instance, the equation
PERCEPTION * THOUGHT = ACTION
means that:
1) for any magicians P and T in the sets PERCEPTION and THOUGHT respectively, the product P*T will be in the set ACTION
2) for any magician A in the set ACTION, there are magicains P and T in the sets PERCEPTION and THOUGHT respectively so that P*T=A
"Anti-sets" such as -PERCEPTION are defined in the obvious way: e.g. -PERCEPTION is the class of all elements R so that P = -R for some element P in PERCEPTION.
In general, suppose one has a collection of subsets S (S_{1},...,S_{k}) of a magician system M. This collection may or may not naturally define a set algebra. In general, the products of elements in S_{i} and S_{j} will fall into a number of different classes S_{m}, or perhaps not into any of these classes at all. One may always define a probabilistically weighted set algebra, however, in which different equations hold with different weights. One way to do this is to say that the tagged equation
S_{i} * S_{j} = S_{k} p_{ijk}
holds, with
p_{ijk} = q_{ijk}^{a} r_{ijk}^{2-a}
where q_{ijk} is the probability that, if one chooses a random element from S_{i}, and combines it with a random element from S_{j}, one will obtain an element from S_{k} is q_{ijk}; and r_{ijk} is the probability that a randomly chosen element from S_{k} can be produced by combining some element of S_{i} with some element of S_{j}.
It is easier to deal with straightforward set algebras than their probabilistic counterparts. In real psychological systems, however, it is unlikely that an equation such as
PERCEPTION * THOUGHT = ACTION
could hold strictly. Rather, it might be expected to hold probabilistically with an high probability (making allowances for stray neural connections, etc.).
Finally, suppose one has a collection of subsets S and a corresponding set algebra. One may then define the
relative unity of this set algebra, as the set 1_{S} of all elements U with the property that U * S_{i} is contained in S_{i} for all i. The relative unity may have an anti-set, which will be denoted -1_{S}. These definitions provide a rigorous formulation of the correspondence between thoughts, perceptions and actions and quaternionic vectors, as proposed in the previous section.
Note that the set algebra formalism applies, without modification, to stochastic magician systems, i.e. to the case where the same magician product A*B may lead to a number of different possible outcomes on different trials.
Octonions and Second-Order Learning
Quaternions correspond to adaptive learning; to learning triads and backtracking. The octonionic algebra represents a step beyond adaptive learning, to what might be called "second-order learning," or learning about learning. The new element E, as it turns out, is most easily interpreted as a kind of second-order monitoring process, or "inner eye." Thus, in the linear combinations q+Er, the elements q are elementary mental processes, and the elements Er are mental processes which result from inner observation of other mental processes.
The three quaternionic elements i, j and k are mutually interchangeable. The additional octonionic element E, however, has a distinguished role in the canonical octonionic multiplication table. It leads to three further elements, I=ie, J=je and K=ke, which are themselve mutually interchangeable. The special role of E means that, in terms of learning triads, there is really only one natural interpretation of the role of E, which is given by the following:
PERCEPTION THOUGHT ACTIONThe meaning of this correspondence is revealed, first of all, by the observation that the systems of elementsi j k
INNER EYE PERCEPTION' THOUGHT' ACTION'
E I J K
(1,i,E,I), (1,J,E,J), (1,k,E,K)
are canonical bases of the subalgebras isomorphic to the quaternions generated by each of them. These quaternionic subalgebras correspond to learning triads of the form
INNER EYE
/ \
PERCEPTION -------- PERCEPTION'
INNER EYE
/ \
THOUGHT --------- THOUGHT'
INNER EYE
/ \
ACTION ----------- ACTION'
The element E, which I have called INNER EYE, can thus act as a kind of second-order thought. It is thought which treats all the elements of the first-order learning triad as percepts: thought which perceives first-order perception, thought and action, and produces modifies processes based on these perceptions. These actions that second-order thought produces may then enter into the pool of consciously active processes and interact freely. In particular, the new perception, thought and action processes created by the inner eye may enter into the following learning triads:
THOUGHT'
/ \
PERCEPTION' -- ACTION
THOUGHT
/ \
PERCEPTION' -- ACTION'
THOUGHT'
/ \
PERCEPTION -- ACTION'
These are genuine, reversible learning triads, because the sets (1,i,K,J), (1,I,j,K), and (1,I,J,k) are canonical bases for the subalgebras isomorphic to the quaternions which they generate. These, together with the basic learning triad and the three triads involving the INNER EYE, given above, represent the only seven quaternionic subalgebras contained within the octonions.
It is well worth noting that there is no loop of the form
THOUGHT'within the octonion algebra. That is, complete substitution of the results of the inner eye's modifications for the elements of the original learning triad is not a reversible operation. One can substitute any two of the modified versions at a time, retaining one of the old versions, and still retain reversibility. Thus a complete round of second-order learning is not quite possible within the octonion algebra. However, one can attain a good approximation.
/ \
PERCEPTION' -- ACTION'
And, as an aside, the actual behavior of this "complete substitution" loop {I,J,K} is worth reflecting on for a moment. Note that IJ = -k, JK = -i, KI = -j. Traversing the complete substitution loop forward, one produces the anti-elements needed for backtracking in the original learning triad. Thus the INNER EYE has the potential to lead to backtracking, while at the same time leading to new, improved learning triads. There is a great deal of subtlety going on here, which will only be uncovered by deep reflection and extensive experimentation.
In order to completely incorporate the results of the inner eye's observations, one needs to transcend the boundaries of reversible processing, and put the new results (I,J,K) in the place of the old elementary learning triad (i,j,k). Having done this, and placed (I,J,K) in the role of the perceptual-cognitive-active loop, the octonionic process can begin again, and construct a new collection of modified processes.
What is permitted by the three triads generated by (1,i,K,J), (1,I,j,K), and (1,I,J,k) is a kind of "uncommitted" second-order learning. One is incorporating the observations of the inner eye, but without giving up the old way of doing things. The results of these new, uncommitted triads cannot be externally observed until a committment has been made; but the new triads can be used and progressively replaced, while the observing-eye process goes on.
The Construction of Meaning
The learning triad represents a state of mind focussed on external reality and inductive learning. Equally important is the process of fixing things in memory. Insight into this process, from a phenomenological point of view can be found in the literature of the Eastern spiritual traditions. In these traditions, the fixing of definite ideas in memory is seen as an obstruction of the path to enlightenment, to clear inner vision. A deep consideration of transpersonal psychology would take us too far afield here, but some brief comments may be in order.
Philip Kapleau (1980), in his classic book Three Pillars of Zen, wrote as follows:
It is important [in Zen Buddhism] to distinguish the role of transitory thoughts from that of fixed concepts. Random ideas are relatively innocuous, but ideologies, beliefs, opinions and points of view, not to mention the factual knowledge accumulated since birth (to which we attach ourselves) are the shadows which obscure the light of truth.
In this view, it is the systematic nature of thoughts and feelings that obstructs clear inner vision. It is the tendency of thoughts and feelings to adhere to each other and form systems, which are then taken for the Self.
The Japanese Zen master Kosho Uchiyama (1993) is yet more explicit on this point. He says,
Even if a thought of something does actually arise, as long as the thought does not grasp that something, nothing will be formed. For example, even if A (a flower) occurs, as long is it is not followed by thought B (is beautiful), no meaning such as A is B is formed. Neither is it something that could be taken in the sense of A which is B (beautiful flower). So, even if thought A does occur, as long as the thought does not continue, A occurs prior to the formation of meaning. It is not measurable in terms of meaning, and in that condition will disappear as consciousness flows on.
This is remarkably astute introspection. Uchiyama is saying that what obstructs right knowledge, what disrupts zazen, is not thoughts or feelings in themselves, but rather the compounding together of thoughts and feelings into coherent, meaningful wholes. Thought A or thought B in isolation are all right, and are not contradictory to the zazen state of consciousness. One can think "flower," or one can have the detached feeling "beauty." These are not meaningful. But when one binds the two together, one has< engaged the consciousness in an act of making-real, an act of establishing something to be the case, and one has thus distracted consciousness from its steady, river-like flow. Binding "flower" and "beautiful" together is creating a false, miniature unity that distracts consciousness from the true unity of the Self-Universe.
In terms of the learning triad model, the binding-together of "beautiful" and "flower" into a definite whole to be registered in memory is an ACTION. Instead of a physically-oriented action, it is a memory-oriented action. And the loop is completed when this concrete entity "beautiful flower," created by the ACTION corner of the triad, is used to influence the very perception of the flower. Next time, it will not merely be perceived as a flower, it will be perceived as a beautiful flower. This kind of distortion of perceptions based on memories is the very basis of Buddhist psychology (Rao, 1988).
The function of the INNER EYE here is somewhat different than in the induction-oriented learning triad. Instead of backtracking for the purpose of improved learning, here it is used to "backtrack" for the point of view of maintaining proprioception of thought, of keeping in mind what things actually are. Backtracking here replaces a beauty-distorted image of a flower with the original image of the flower itself. This leaves one open to follow other search paths -- to classify the flower as ugly, for instance. But, most importantly, it leaves one open to recognize the distortions of one's own perceptions.
As long as one stays within the reversible algebraic structure of consciousness, one can be aware of the distortions that go into one's perceptions. Once one imprints something on memory, however, one is no longer dealing with a reversible structure, and so the memory trace need not be decomposible into its component parts.
This brings us to the interesting question of the psychology of altered states of consciousness -- in particular, spiritually advanced or "enlightened" states of mind. In traditional Buddhist psychology, in that the spiritually enlightened mind is supposed to have complete proprioception of thought, complete self-understanding and freedom from "ignorance." Yet obviously, since the enlightened mind functions in the world, it must still produce memory traces. The complete resolution of this apparent paradox is a difficult matter, and would take us too far afield here. As a start, however, one may observe that is is possible to store the whole triad in memory -- "flower," "beautiful," and "beautiful flower" instead of just "beautiful flower." In this instance, one is recording the sense impression, the reaction, and the meaningful combination of the two. The prerequisite for doing this is a mind that is trained to consciously distinguish these three items: i.e., to take the whole triangle and treat it as a PERCEPTION. This kind of operation is, formally, within the capacity of the INNER EYE, which creates a new PERCEPTION based on interaction with the PERCEPTION, THOUGHT and ACTION. In practice, however, it is very difficult to achieve.
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