Contents:Søren Brier: Foreword Full Text Claus Emmeche: Defining life as a semiotic phenomenon Abstract David J. Depew & Bruce H. Weber: What Does Natural Selection Have to Be like Abstract Jesper Hoffmeyer: Surfaces Inside Surfaces Abstract Robert Vallée: Cognition et Système, Essai d'Épistémo-praxéologie Abstract Robert Vallée: An Introduction to "Epistemo-praxiology" Abstract Columns Ranulph Glanville: A (Cybernetic) Musing: Varieties of Variety? Full Text Louis H. Kauffman: Virtual Logic - The Calculus of Indications Full Text Reviews Maj-Britt Rosenkilde, Anja Abel Sørensen, Christine Nordentoft and Søren Brier: Review of International Encyclopedia of Systems and Cybernetics Full Text Axel Randrup: Whispering Pond Full Text Mariaelena Bartesaghi: "The Therapy of Dialogical Possibility" Full Text
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A (Cybernetic) Musing: Varieties of Variety?By Ranulph Glanville In this issue's column, I wish to explore a question. The question is, how many varieties of variety are there? Pushed a little, this transforms into what on earth is variety that we can use it as a measure? This is a genuine question, and I have no ready-made answer, although I have been aware of it for some time, first attempting to deal with it in a presentation at the EMCSR in 1988 (which I found I could not commit to paper). I find the notion of variety to be extraordinarily useful in explaining many things about the world I find myself in, especially in terms of limitations (in particular, Bremmermann's limit of transcomputability) and of creativity; the loss of control and the generation of the new, especially in conversations. I have written about these under the heading "The Value of Unmanageability" and mean to come back to them in a later column. As I see it, Ashby makes two quite different uses of his term variety (Ashby 1956). The first (which I shall refer to as e-variety) metaphorically uses a concept from physics-entropy. The second (s-variety) is simply to do with the number of states in a system. So the first of these uses is heavily influenced by Shannon and Weaver's Information Theory (Shannon and Weaver 1949) (hardly surprising, considering when Ashby was developing his concept). The measure of variety is closely analogous to the measure of information that Shannon and Weaver proposed (as a complement to kinetic entropy), but whereas the computation of information depends on the basic informational unit, the bit, variety depends on Ashby's operational and organisational unit, the state. In this incarnation, variety is a measure of how many of the possible states a system could (theoretically/logically) attain which it actually uses. This can be useful in determining "expandability" in a system, ie how many further states it might attain, although the fact that a system does not take up all logically possible states may, itself, be of interest and tell us much, just as this does in information theory-where it can be used, for instance, to help code-breaking. This value of absence seems to be at the very heart of computing: for in (digital) computing the 0's are every bit (!) as important as the 1's: what matters is that every address is fillable, and only then whether it is filled and how full it is. An empty address is as significant as a filled one. Of course, musicians have known this for a long time, filling their pieces with what we sometimes call "telling" silences. John Cage took to a final extreme in his piece 4'32", a totally silent piano piece. Architects, too, would claim that the material they manipulate (space) is essentially the lack of material: massive silences between the stones. Returning to variety, it is the second use that seems to me potentially the richer. Here, Ashby just counts and enumerates the states the system can take. In this case, Ashby is only dealing with one aspect of the equation which constructs his e-variety. He is simply looking at the (number of) actual states that may be attained. Perhaps I should rehearse the difference, hoping to make clear my understanding. Consider a set of traffic lights. Since these differ in how they work according to the country they are in, I will use the British system for convenience. There the lights follow the following sequence: red (code for stop) There are just 4 states in this sequence. However, the possible combinations of 3 light colours is 23, or 8 states (omitted in the British system are green and amber; red and green; red, green and amber all together; and no lights at all-although this last logical possibility is often taken to be code for watch out, there's a problem here-the lights must have broken down!). In the second interpretation Ashby gives to his term variety, it is the number of states actually possible in the system as it operates (ie, in this case 4), rather than the relationship between the actual and the logical (ie 8) number of states possible that is of concern. What interests me is what Ashby uses this (state) measure of s-variety for. In particular, I have a great respect and affection for his "Law of Requisite Variety". There are a number of ways of expressing this Law-for instance, that only Variety can destroy Variety-but I like a simple and direct version: Any controlling system must have at least as much variety as the (controlled) system it is to control. Like many a profound truth, this is, I believe, obvious-now that Ashby has said it. If the controlling system cannot "map" all the states the controlled system may attain, it cannot control its behaviour because some of the behaviours of the controlled system are outside the range of the controlling system. All the controlling system can do, in the case that some behaviour is presented by the controlled system which it cannot map, is to ignore this behaviour: and that is not controlling. (In the case of that wonderful control device, the digital computer, the computer will just crash.) Conversely, when the controlling system has greater variety than the system it controls, there is room for the controlled system to change (expand, develop). This is a point I do not recollect Ashby making, although it is central to any discussion, in terms of variety, of increases in options. Ashby uses this Law to determine both whether any proposed control system has a chance of controlling some system-to-be-controlled, and whether systems are actually in principle controllable. This latter he does with reference to Bremmermann's Constant: a numerical limit invented by the physicist Hans J Bremmermann which, originally, shows what the earth could have computed were it a solid computer operating over its known life (Bremmermann 1962). Of course, such a figure is arbitrary and depends on the actual numbers originally chosen from which to make the calculation. But, this is beside the point: what is important is not the absolute size of the limit, but the fact that there is one. As long as it is enumerable, it does not matter so (as Ashby shows) if the figure is out by millions of times, this is only a tiny error when compared to the actual number Bremmermann gives as the theoretical limit: 1047 bits computed by the earth during its entire existence. Ashby (1964) extends this figure to base it around the known universe and ends up with the assertion that numbers beyond 10100 are beyond the computable. Bremmermann had referred to systems that generate these vast numbers of possible states as transcomputable, since there is no realistic way they can be computed by completely secure means. Ashby shows that even very small systems (such as an array of 20 by 20 lights) are essentially transcomputable (such a board generates 220^20 possible states, ie 2400, which is about 10120, a number vastly bigger, as mathematicians remind us, than 10100). I intend to delve further into the magic of Bremmermann's Constant at a later date. Thus, while the Law of Requisite Variety tells us how much variety is required of one system that is to control another, Bremmermann's Constant, as applied by Ashby (and later, with great enthusiasm, by Stafford Beer) tells us of the limits to what is conceivable within the understandings we currently hold about the universe we inhabit. It turns out that it is very easy to attain variety that vastly exceeds the capacity of the physical universe to compute it. Yet we still try to, and often believe we do, control systems that either are, or are close to, being uncontrollable because they are transcomputable. (That there are many other difficulties in making systems controllable pales into insignificance when compared to this in principle limitation.) Of course, we do have ways of "reducing" the variety of very complex systems, thus making them controllable. These usually involve some form of simplification and approximation: we are willing to trade off the discrete distinction of each observation for the rough approximations that allow us to control. Techniques include those of statistics, probability and fuzzification. In fact, I would postulate that we tend to balance variety against controllability: this can be seen clearly when we consider ways of representing (or perhaps embodying) signals. In the case of audio signals, we have long since traded the infinite variety of the analogue, renowned for terrible problems of control, for the standardised control of the digital, with its increasing but never infinite variety. Generally, our ears tell most of us we have made a good trade-off, albeit that we are strongly influenced by techno-hype. But let us return to the (British) traffic lights. What happens to those who are red-green colour blind? According to the account given above, since red = green (which I write as red-green) they will only distinguish 3 states in the sequence, as follows: red-green I have not indicated the code for each, since it is clear that both stop and go may be simultaneously indicated by the red-green colour even where meaning is communicated by a code. If it were only colour that distinguished the coded commands, it would follow that one of the first tests for a potential driver would be a colour blindness test, but I know of nowhere where this is so. Drivers can and do distinguish between red and green lights by means other than colour. Those I have asked who are colour blind tell me they also use position. That is, there are more states than simply those involved in colour: there is position as well (ie, red is on top, green at the bottom and amber-when present-is in the middle). It happens that there is in this case a mapping between position and colour leading to redundancy and hence to the description of the states of the traffic light system by means of the states associated with only one of the variables-colour. But this co-incidence is serendipitous and certainly not at all logically necessary. And sometimes it is in appropriate. Thus, I hope it is clear that variety depends on how the states are defined. What I suspect looked to Ashby like an absolute measure turns out to depend on our interpretations and ways of setting up what we're looking for! In fact, since it depends on how we define the states the system takes, in a world in which the observer is included such as second-order cybernetics, this means that the variety of a system is open, to some extent at least, to our individual determinations. However, under these circumstances we find we must modify the Law of Requisite Variety. In a second order (control) system, each component is both controlling and controlled. That is, each controls the other. Control does not exist in one or the other, but between them in a circular process. If, as is demanded by the Law of Requisite Variety, the variety of the controlling system must equal or exceed that of the controlled system, it is clear that in a second order system, where each controls the other, the variety of each system must exactly match (eg Glanville 1987, 1994). The variety of the one system must exactly equal that of the other, since each is controlled by that other (ie, each is the other's controlling system). For second order systems, the Law of Requisite Variety must be rewritten accordingly. Thus, in second order systems, it is not possible to leave room for an increase in the controlled system's variety (by having a controlling system with greater variety) while maintaining controllability, because the controlling system cannot have more variety than the controlled since there is ultimately no controlling and no controlled system (in the classical sense). And yet… Systems do increase in variety, they do grow and change. And, if second order cybernetics is to reflect this and to benefit from the notion of variety, it is clear that how we consider variety must, also, change. We can no longer leave room for change (increase) by creating a controlling system that has more variety than the controlled system. It is, however, possible that we can design systems in which variety increases on both sides: either by adding states or through redefinition, that is through defining the states that pertain in a different way. As I said, I suspect that Ashby saw variety as being a constant and, indeed, a quality measured in terms of an absolute quantity, for any system. However, since in second order cybernetics we demand the active presence of the observer and accept his/her involvement in determining the qualities of the system, in second order cybernetics we can expect to be, and anticipate being involved in the process of definition. And since we can look in more than one way, use more than one perspective, we can redefine the system so that we can accommodate change through the way we consider the states attainable. Let me give three examples. he first (mine) is perhaps rather picturesque: it was the first notion I developed when I started studying with Gordon Pask, and I have not presented it before publicly. It is a response to the familiar difficulty (Goedellian in nature) that any universe should contain everything that it needs to contain, yet must be seen from outside if this is to be appreciated. Then the view from outside, that the universe does indeed contain everything it should, should be within the universe, for "contains everything it should" is a property of that universe! I proposed that the boundaries of universes should be seen as having metaphorical hands on them, constantly grabbing that which was outside but should be inside and placing it within the confines of the universe. Thus, the universe is in a state of constant expansion: that which had been outside being absorbed-leaving the outside bare, awaiting another external view (then to be absorbed in turn): variety constantly increasing. The second concerns the notion diversity that Heinz von Foerster developed. Von Foerster suggested that, in a probabilistic system (e-variety is measured, as is entropy, as a function of probabilities between 0 and 1) the range might exceed the value 1. As I remember it, the system would then rescale itself to fit the range 0 to 1. This allowed the system to behave in a manner outside the range of predictions: ie, to take states that were never considered when the system was defined. Variety (in the form of s-variety) could be increased (although the logical total for the system then "rescaled" after accommodating the new). For some reason, this notion was, as far as I know, only reported in an internal BCL memo. The third comes from discussions between Mike Robinson (whose outstanding paper on variety I will introduce when I return to creativity and unmanageability) and myself, in which we proposed two types of complexity. To paraphrase rather inaccurately, the first was bottom up, supporting the belief that there was a determinate variety which leads very quickly to transcomputability (as Ashby showed); the second top down, allowing that complexity is something that can be handled by grouping, that is, we see the system and then decide its variety. This means that variety is determined by the observer. In turn, this alleviates the problem of increases in variety, for the variety that is seen within the system can be redefined and, therefore, accommodated. This, however, means that variety is absolutely not absolute. I did some experimental work based on this notion, reported in Glanville 1984. Why does any of this matter? It matters because of what it suggests for variety. To me it suggests that there are several kinds of variety, or ways of considering variety (in a second order world, these are equivalents). There is the difference that Ashby created in his two uses: variety as the number of states a system can attain (s-variety); and the "entropically inspired" variety as the relationship between the logically possible and the actual number of states a system could achieve (e-variety). Although I now see a connection between these, I remain confused as to why he used the same term to communicate these two different measures. To this confusion I have to add the confusion I find resulting from the need to modify the Law of Requisite Variety that second-order understanding requires and the experience that not only does the variety in a system often increase, but it can (and must) be (re-)defined according to how the included observer chooses to see it. Thus (as the examples show), variety can be seen as needing to be both static and dynamic, both defined by the system and observer determined. Are there other qualities that I have not considered here? I suspect so. Then what are they? How many varieties of variety are there? Has anyone an answer? References
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