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Rough Draft, for comments only
Ben Goertzel
November 1996
On. Say on. Be said on. Somehow on. Till nohow on.
Said nohow on.
-- Samuel Beckett, Worstward Ho
The question I am going to address here is a very simple
and basic one: How does structure emerge from
nothingness?
This is a philosophical question, but it is also a scientific question. It is important in fundamental physics, where one has the question of the origin of the universe, and the question of the emergence of spacetime from pregeometric structure. And it is important in psychology, where one has the problem of consciousness. The conscious moment arises "out of nothing" and then disappears again; this process of fluctuation is the essence of our daily experience, but we understand very little about it.
In order to address the emergence of structure from void, I will propose a particle of being called an on. ons are not particles in spacetime, they exist prior to space and time; they are not really mental forms either, as they exist prior to the structures that define mind.
These ideas are inspired, in concept, by an ongoing e-mail dialogue with Tony Smith, Kent Palmer and Onar Aam. In detail, they are particularly inspired by Tony's "quantum set theory" and Onar's ruminations on the relation between XOR and octonions; as well as by William Hoffman's ideas about symmetric difference and dialectical psychology.
In fact, dear reader, if you are not either Tony, Kent or Onar, you run a serious risk of not understanding what on earth I am talking about in this rough-draft essay. Eventually I will write up a more didactic presentation with all the missing background filled in!
I will begin with a description of the basic properties of ons, which are relatively few in number.
The first is that any two ons are either identical, or nonidentical. Two ons that are nonidentical do not interact; two ons that are identical may interact.
The second is that ons may extend themselves around groups of ons: pulling groups of ons within themselves, one might say. Once an on does this, it is no longer a simple on, it is a metaon. Unless otherwise qualified, the word "on" will generally be taken to refer to simple ons, and the word "metaon" for metaons; it should not be forgotten, however, that a metaon is merely a simple on that has extended itself.
There are three different types of metaon; and the interactions of ons are dependent upon what kind of metaon the ons reside in.
The most basic metaon is the open metaon. Unless otherwise specified, it should be assumed that a group of metaons lies within an open metaon. In an open metaon, two identical metaons annihilate each other when they come into contact with each other.
The other two types of metaon are called set metaons and bag metaons. When ons are within a set or bag metaon, they lose their power to annihilate each other, and they interact with each other in different ways instead. Within a set metaon, two identical ons collapse into a single on. Within a bag metaon, two identical ons remain two; they do not collapse nor do they annihilate, just as if they were not identical.
Finally, there are antions, which can join ons within metaons. Antions are just like ons: they can be identical or different from each other, and they can extend themselves to form metaons. When two identical antions meet inside an open metaon, they annihilate each other; and when two identical antions meet inside a set or bag metaon, they behave as would two identical ons.
Given an on and an antion, the two either match or do not. If they match, they annihilate each other. The annihilation of on/antion pairs works in any kind of metaon.
Notationally, if x is an on -x will be used to denote a metaon that matches x; and where S is a metaon, -S will be used to denote a metaon containing antions corresponding to each on in S.
Metaons can interact, just like simple ons. The definition of the interaction of a group of metaons is as follows: it is the metaon formed by allowing each on in any of the metaons in the group to interact with every other on in any of the metaons in the group. So for instance, if metaon with six elements interacts with a metaon with four elements, the resulting metaon will have twenty-four elements, prior to any cancellation that might occur.
When two metaons of the same type interact the result is a metaon of that type. To determine the result when metaons of different type interact, one must introduce the notion of order and ask which one is the "actor" of the transaction: the type of the result is then the type of the actor.
In terms of set theory, the product S*T of the interaction of two set metaons S and T is in fact nothing but the symmetric difference of the sets S and T. It is everything that is in either S, or T, but not both. For if something is in both, it will annihilate itself when the two sets are brought together. The appearance of symmetric difference here is interesting and important; for, as William Hoffmann has noted, the symmetric difference is the natural mathematical implementation of the philosophical idea of dialectics. The interaction of two metaons is actually a kind of dialectical synthesis, a la Hegel.
Finally, metaons can be formed out of metaons, resulting in a nested hierarchy of metaons within metaons within metaons..., ultimately bottoming out in the lowest level of simple ons and antions. Interaction of these higher-level metaons is defined in the same way as for the simplest metaons.
Finally, I will sometimes need to refer to the PowerSet of a metaon M, meaning the set metaon that contains all set metaons formed from elements of M; or the PowerBag of a metaon M, meaning the bag metaon that contains all bag metaons formed from elements of M; or the PowerOpen of M, which is an open metaon containing all open metaons formed from elements of M. Note that the PowerBag of M is not inclusive of the PowerSet of M, because there is a difference between a set metaon and a bag metaon that happens to contain only one copy of each element inside it.
Ons are related to two different, previously proposed system-theoretic models; my own magician systems and the form dynamics of G. Spencer-Brown's Laws of Form. They are a specialization of magician systems and an extension of form dynamics.
A magician system is a collection of magicians, where a magician is defined as something that transforms magicians into other magicians. This is a very general construction which is related to hypercomplex algebras and has been used to model the dynamics of the mind. In computer science terms, a magician system is a special kind of "agent system," in which there is no communication except via acts of intertransformation. And ons are, in turn, a special kind of magician system, in which interactions are defined, not by an arbitrary "magician multiplication table," but by the three kinds of on dynamics (annihilation, collapse and noninteraction) induced by the three kinds of metaons.
Spencer-Brown's form dynamics, varied and fleshed out by Francisco Varela, Louis Kauffmann and others, is based on the single mark
__ |
which is called a "form," "mark", or "distinction." This mark is essentially the same thing as an on. However, the Spencer-Brown notation is typographically cumbersome, and my subsequent development is quite different from that of Spencer-Brown and his followers, and so it seems justified to introduce a new terminology.
In Spencer-Brown's original work, forms are taken to interact in two ways: one way which is hierarchical and annihilatory
____ __ | = | |
and one way which is heterarchical and involves collapse
__ __ __ | | = |
These correspond to the behavior of identical ons in open and set metaons respectively.
However, Kauffmann, in his work on form dynamics, sometimes relaxes the rules of collapse and annihilation altogether, allowing hierarchical and heterarchical coexistence without collapse. For instance, he allows a construction such as
_________ __ __ | | | |
to exist without collapsing to nothing as it would under Spencer-Brown's rules. In the present language, this particular construction would be called a bag metaon, containing two identical ons.
So, we see that Spencer-Brownian form dynamics contains the three kinds of on interaction: annihilation, collapse and coexistence. It also embodies the notion that metaons are formed by ons extending themselves: the same mark is used for set boundaries and for set elements. The ideas presented here could have been presented in Spencer-Brownian angle; that they are not is really a matter of convenience rather than philosophy.
I will now use ons to tell a story of the emergence of structure from nothingness. I call this story a "cosmogony" -- a story of the emergence of a universe from the void. Whether this universe is physical, mental or something else is not important at this level of abstraction.
This cosmogony could be told in purely philosophical form, or in purely mathematical form. Or, for that matter, it could be cast in the form of a parable! I have opted to begin with philosophy, introduce mathematics along the way, and wind up with a mathematical formulation at the end.
The story depends on two principles, which I will state first in philosophical form:
Absence is inevitably reified
Universal interpenetration: Any group of entities will interpenetrate and form a composite entity
The first principle, the reification of absence, means that nothingness quickly turns into something: in fact as soon as it is called "nothingness" it is no longer nothingness, it is a thing called "nothingness."
The second principle, interpenetration, means that contradictions are not fundamental. While two entities may appear to be irreconcilable opposites, in fact they may be considered to exist together. The universe is large enough to contain both X and ~X.
This cosmogonic story is a sequence of events, but note that it is not a sequence of events in physical time. It is a sequence of events in metaphysical time; it exists on a level prior to the existence of physical time.
In this story, the universe begins with Void. Nothingness.
And then there comes into the nothingness a single on. I call this first on 1.
Before the emergence of the first on, the Nothingness was simply Nothingness. But now it has a character: it is something different than 1. It is not an on. Now that it has a character it merits a name; we may call it 0. This is the inevitable reification of absence.
But now we have two entities, 0 and 1. These are opposites, but by the principle of universal interpenetration, they can join together, forming a new entity "0 and 1 together." Also, 0 is now different than Nothingness; it is "Nothingness." We need a new entity to denote the outside of our collection of entities, true Nothingness. We now have four entities in our universe: 0, 1, "0 and 1 together," and Nothingness. "Nothingness," Being, Being-and-Nothingness, and Nothingness.
At this stage in our cosmogonic evolution, we now have a set of ons, or in other words, a set metaon.
What next? What we have now are two types of ons. We have Being ons and "Nothingness" ons. The two different types of on are defined, not by any particular intrinsic qualities, but by the mere fact that they are different from each other. With this in mind, let us introduce a new notational convention. Instead of using different symbols for different kinds of ons, like B and N or 0, 1 and 2, let us use different positions. Let us consider that we now have two coordinates: Being and "Nothingness." Being is the first coordinate, "Nothingness" is the second. Thus, in this system, Being (which we formerly called 1) is now represented as 10 -- a 1 in the Being place and a 0 in the "Nothingness" place; a single on on and no "Nothingness" on. "Nothingness" (which we formerly called 0) is now 01: a single Nothingness on, no Being on. Being-and-Nothingness is 11; two ons, one of each type. And true Nothingness (or, one might say, truer "Nothingness"; "Nothingness" which is separated from true Nothingness by one less level of abstraction) is 00.
In terms of set theory, what we have done in our cosmogonic evolution, so far, is to move from the set {0,1} to its PowerSet, the set of all subsets of {0,1}, including the whole set (represented by 11) and the empty set (represented by 00).
In terms of Indian logic, on the other hand, what we have done is to go from 1=true and 0=false to the two additional truth-values 11 = true and false, and 00=neither true nor false.
Also, it should be noted at this stage that, in terms of our binary strings 11, 01, 10 and 00, the on set interaction rule is equivalent to the rule of exclusive or or XOR. In other words, let s(S) denote the binary string representation of a set of ons S. Then in general we have
Now let us continue our story of cosmogonic evolution. Let us apply the rule of interpenetration again. We find that every subset of the set {11,01,10,00} is to be considered as a valid entity in its own right, including the empty set which again represents Nothingness (the old Nothingness 00 having become a concretized "Nothingness"). In a coordinate representation, we may relabel as before:
This process can be continued indefinitely. Mathematically, one is looking at an iteration of the form
We have completed the first stage of our cosmogony. Metaons, by interpenetration and absence-reification, naturally give rise to their PowerSet metaons. In this way, more and more complex forms naturally emerge from the initial on, which itself emerged as a reification of the Void.
Ons are essentially set-theoretic in nature, but they also lead to a variety of structures that are plainly algebraic. This can be seen by looking at any stage of the S[n] construction given above, but for a variety of reasons, the stage S[3] represented by binary strings of length 4 is a particularly interesting case.
In this case, one may proceed by dividing the set of strings into two parts: those containing a 1 in the position corresponding to 00, and those containing a 0 in the position corresponding to 00. Let us call the first part Part[3], not only because it is a part, but also because in the physics application to be described below it will be used to model particles. Let us call the second part, which will have to do with antiparticles, Anti[3].
Note that this same construction applies beyond S[3]. In general, one can divide S[n] into two categories Part[n] and Anti[n], by choosing any coordinate, and distinguishing strings that are 0 in that coordinate from those that are 1 in that coordinate.
Part[3] consists of the strings
i = 011 j = 101 k = 110 I = Ei = 100 J = Ej = 010 K = Ek = 001 E = 111 1 = 000
Note that what this XOR multiplication table does not give are the signs of the octonion multiplication table. The symmetric difference gives a symmetric multiplication table, whereas the octonions are antisymmetric. Formally, let L(x) denote the linear subspace containing the octonion x. Let str(x) denote the string assigned to an octonion basis element x by the above correspondence; and if s is a string denoting a basis element, let oct(s) denote the octonion corresponding to s. Then what we have is that, for any two octonion basis elements x and y,
Finally, we have been working with Part[3], but the same correspondence can be done for any n. Part[2] generates the quaternions, Part[1] generates the complex numbers, and in general Part[n] will generate an hypercomplex algebra of order 2^n.
The generation of the octonionic multiplication table from ons is very natural and intuitive, but the final step -- the introduction of linear combinations -- is somewhat worrying and inelegant. One wonders whether it is really necessary to step out of the on framework and introduce linear combinations.
As it turns out, it is not necessary to do so: one can generate discrete linear combinations without stepping outside of the on framework, simply by using bag metaons and antions. In this way, the introduction of arbitrary mathematical operations is avoided, and everything is accomplished in the same basic framework. Linear combination is seen as a particular manifestation of on dynamics.
A linear combination involves the two basic vectors space operations: the summation + of two vectors, and the multiplication of a (real or complex) scalar by a vector. I will show that both of these operations can be taken care of using ons.
First, the operation + signifies the simultaneous presence of two things: x+y means that x and y are both there. This is the same thing accomplished by grouping x and y into a metaon. And under our definition of metaon interaction, there is no difference between taking the linear combination product (x+y)*(w+z) and taking the "set metaon interaction" {x,y}*{y,z}.
On the other hand the scalar multiple of a vector x indicates the number of copies of x that are present: 2x means 2 copies of x, 3x means 3 copies of x,a nd so forth. If the scalar coefficient of x is an integer, then, one can unproblematically deal with linear combinations like 2i+j by using bag metaons. 2i+j corresponds simply to the bag metaon {i,i,j}, and interactions are to be worked out accordingly. Summation works perfectly well with bag metaons instead of set metaons, and so we find that linear combinations with positive linear coefficients are represented by bag metaons with the ordinary definition of metaon interaction.
Next, what about the case where the coefficient of an element is a negative integer? Can this also be interpreted in a natural way in terms of on dynamics? I believe that it can; all that is required is the introduction of antions. In other words, the vector "-x" is to be considered an antion that matches x. Thus when -x comes into contact with x, interaction occurs according to the ordinary laws of on dynamics, and the two annihilate. On/antion annihilation, as indicated above, takes place even within set or bag metaons.
We have thus reconstructed general integer-coefficient linear combinations, using bag metaons and antions. The integer linear combinations generated by S are given by the bag metaon {PowerBag[S], PowerBag[-S]}. In this way, for example, we can generate the integral octonions from Part[3].
The integral octonions are, I have argued elsewhere, a very important structure for the mind. The quaternions are the structure of ordinary states of consciousness, and the octonions are the structure of deeply insightful, intuitive states of consciousness. The reason for the psychological importance of these algebras is their unique mathematical properties. Of all the algebras that are going to emerge from on dynamics according to the processes described here, it is only the algebras of Part[3], Part
It so happens that the octonions are the most symmetric such structure, and furthermore the only one with the property of unique division. So if one wishes to explain the emergence of the octonions from Part[3], one may view Part[3] as generating a host of different algebraic structures, including the octonions; and one may then posit some additional selective process that causes the octonions to emerge from this pool of possibilities.
One might think to incorporate negatives into the picture as well, by using Anti[3] as well as Part[3]. In this direction, one might observe that, moving back to the four-bit representation, what we have above is
i = 1011 j = 1101 k = 1110 I = Ei = 1100 J = Ej = 1010 K = Ek = 1001 E = 1111 1 = 1000
which is naturally complemented by the encoding
-i = 0011 -j = 0101 -k = 0110 -I = Ei = 0100 -J = Ej = 0010 -K = Ek = 0001 -E = 0111 -1 = 0000
This representation of negatives uses the first bit to connote sign, as in the standard representation of sign in the machine language of digital computers. Using this representation we have for instance
In fact, the structure of Anti[3] is exactly the same as the structure of Part[3], and so it is perfectly valid to use Anti[3] to encode the negatives of the octonion basis elements. The trouble is that, once one moves to this encoding, the symmetric difference no longer matches the octonion multiplication table.
I do not yet know how to account for the antisymmetry of the octonion multiplication table in a natural way in terms of on dynamics. I suspect, however, that in order to do so one must introduce the notion of time into on interactions. What follows in this section are some unfinished thoughts in this regard.
The operation x XOR y is symmetric, and does not distinguish between the role of x and the role of y. It represents an interaction in which both x and y exist at the same time. But perhaps it is also useful to consider the case where x and y are separated in time. Then one may introduce the rule that, if x precedes y, the product x*y is the same as it would be if x and y existed at the same time; but if x follows y, then the result is instead the antion matching the ordinary interaction product x*y, i.e. -x*y. In symbols, x[t-1] * y[t] = - (x[t] * y[t]) = -(x[t] * y[t+1]).
The idea here is that an antion corresponds to an on moving backwards in time -- an idea that resonates with the idea, in modern physics, that antiparticles are particles moving backwards in time. One then finds that, when the metaon Part[3] is interpreted in a temporal context, antisymmetry is a natural consequence.
This recourse to time still does not yield the exact octonion multiplication table -- there are other antisymmetric multiplication tables that are generated by Part[3] under XOR. But it brings one a good way closer.
Octonions are important in physics, but in order to model particle interactions in this universe we also need something more: we need Clifford algebras. One way to get these out of the current framework is to look at the emergence of functions.
A function, in mathematics, is represented as a set of ordered pairs. Each pair contains one element from the domain set and one element from the range set. What distinguishes a function from a mere relation is the fact that each element from the domain set is included only once. In on terms, function and relatiosn mapping ons to ons are themselves special kinds of set metaons.
Next, where T is any metaon, etc., define F[T] to be the collection of functions that intersects T either in range or in domain. That is, a function f is in F[T] if it maps some element of T somewhere, or if it maps something into an element of T. This is the extended function space associated with T.
The law of self-annihilation is assumed to be obeyed by functions, just as by simple ons; i.e., the rule
It turns out that one can use the set metaons F[T] to construct the standard algebras of particle interaction. This observation is a reformulation of the "quantum set theory" of Tony Smith (which indeed has inspired much of the development here).
The quantity of interest for physics is the open metaon
In other words, what we have in this symmetric difference is the collection of functions that map Part[3] onto itself, plus the collection of functions that map Anti[3] onto itself. This set generates the even Clifford algebra of the 256-element Clifford algebra Cl[0,8], which, as Tony Smith shows, can be used to generate the algebras of particle interaction in our universe, using linear combination (which has already been taken into account above). Similar calculations for S[1] and S[2] yield algebras for particle interaction in other universes, lower in dimensionality and less interesting than our own.
The interesting point here, from a theoretical physics perspective, is that we have derived all particle interactions from a single underlying "particle", the on, with very minimal assumptions on the nature of ons. On dynamics as developed here mixes up physics and mathematics: even such operations as linear combination are reduced to particle dynamics.
The reduction of linear combinations to particle dynamics has interesting physical implications beyond the derivation of the algebras of particle interaction. For linear superposition plays an important, and largely unexplained, role in quantum theory.
In quantum physics, the state of a system is understood as the "linear superposition" of a number of different possible states. In the Many-Universes interpretation, each of these distinct states corresponds to a different possible universe. In the present language, this linear superposition is obtained by allowing the states of the system in the different universes (represented as bag metaons) to interact with one another. Interaction on the on level leads to the results that we identify with summation (at least when dealing with the integer lattice subspace of a vector space).
And so it turns out that, in the on perspective, the many universes are not separate at all -- this is a fallacy of interpretation. In fact the many universes all interact with one another, according to the dynamics of a bag metaon. A system has a state in each universe, and then these different states are all allowed to interact with each other, resulting in the ultimate state of the system. The "probability" of each state is represented by the number of copies of that state in the bag -- the number of times that state has been created by the dynamics at the previous time.
(This section needs work!)