S.KRAUSE

System of the World

The headline: "Virus Trackers Find Malware With Google," and the text: “Malware hunters have figured out a way to use the freely available Google SOAP Search API, as well as WDSL, to find dangerous .exe files sitting on thousands of Web servers around the world. Queries can be written to examine the internals of web-accessible binaries, thus allowing the hunters to identify malicious code from across the internet.”

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By itself this is uninteresting, but it serves as an illustration. The headline tries to serve a certain rhetorical function by shocking us: did you know that some bad people are mis-using something we generally consider good? Oh. My! The introductory text explains how this works, and that is what interests me—not the text, nor the specific instance, the explanation—: that meaningful and functional structures are are often built upon simple, almost axiomatic systems, and these higher level structures are hard to predict from those lower level structures. That is a type of weak emergence. This does not really qualify, in fact, it is a simple deductive corollary: Google let's us find files on webservers, .exe files are a type of file, Google lets us find them if they're accessible on a webserver.

Any example of a similar sort would have served as an illustration; there was no particular knowledge value in this one—its selection was arbitrary, and I chose it because when I came across it the matter of systems came back to mind.

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I want to start with a simple, let's say naive, definition of a system. When I think of a system, it is a thing, a whole, but I conceive of it as consisting of elements or parts. This is not to say it is reducible to these parts—that is an analytic step I am not trying to make here—I am simply observing that I cannot conceive a system that does not consist of parts. Given a collection of elements or parts, if they do not relate to each other in some way, or some ways, I would say that I have a collection, perhaps, but not a system. By “relate in some way” I am not trying to specify how they relate—similarity of characteristics (this assumes they have characteristics, that they are in some way individual, etc. ... assumptions I do not wish to make here), hierarchy, temporal-spatial or causal connections, etc—only that if I have two elements, I might have rules that allow me to compare them, relate them, perhaps turn them into another element of the collection. (Aside: note that I am not touching upon the Axiom of Choice here, but it would have something to say about the ability to pick and choose elements for the sake of relating them). So far my naive definition of a system says that a system consists of elements and ways to relate those elements.

It was already clear that a system without elements would go against our common sense notion of a system, and a set of elements without relations between them would just be a mass or collection. There are one or two other limitations that spring to mind that make sense to implement. First, if I have rules for relating or combining some elements, then whatever results from this relating or combining should be considered part of the system as well. Furthermore, if the elements, the relations, or the elements plus relations, separate into unrelated (that is, lacking a way to relate one group of elements with another group) sub-groups, we might want to say that we do not have a system, but perhaps a collection of separate systems.

This definition is intentionally general and somewhat vague. By not describing what elements are or are not, I want to leave open the possibility that elements themselves might be systems in their own right. It was not specified whether there is a finite or infinite number of elements (or rules), or, if infinite, what class of infinitude. I did not specify how the relations work: between two elements? between three or more? is the relation of two elements another element, or something else?

One can relate this definition both to a number of actual example systems as well as to a number of metaphors for how such a system definition works. On this latter point, one might think of elements as nouns and relations as verbs, or elements as words and relations as the rules of morphology or syntax that change or combine them. A less metaphorical and more abstract approach leads to actual systems, primarily those of a mathematical (and/or logical) nature.

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A set is a collection of elements, and has little internal structure; our operations on it, such as dividing it into subsets, creating unions or intersections of sets, adding or subtracting subsets from sets, etc., are actions external to the set itself. A group, on the other hand, is often described as a set with certain (internal) structures, and is considered the most basic of such structures. A group is a set of elements along with a binary operation that takes two elements (a, b) and returns a third element (c); furthermore, this binary operation and the set together must have an identity element, such that when it is combined with any other element, that other element is returned; and the relation has an inverse, a way of undoing it. The integers under addition are a group: there is a set of elements (the integers), a relation (addition), an identity element (0, a+0=a for all a), and an inverse (subtraction). Algebraic rings and fields add relations; a ring is a type of group with a second operation that fulfills certain rules, and a field imposes more rules on this second operation. In the case of groups, rings, and fields, one can speak of them as systems, at least according to our simple definition, whereas sets are too simple to be systems.

In groups, for example, the result of the binary operation is merely another element in the group, and there is little to distinguish individual elements. Taking the integers under addition, for example, all the integers are the same, except that they are individually identified and ordered (1, 2, 3, ...), but it is not at all obvious that in a system all elements need to be alike, of the same type, except for the fact that they are subject to the same rules within the system. In the integers under addition, for example, it does not matter that some integers are odd and some even, some positive, and some negative, some prime, and some composite, etc. It is not that a system consists of like elements (or elements of the same type or class), but that within a system and with respect to the relations of a system, all elements are treated as of the same type.

Instead of beginning with the system, one could work from the bottom up—if one has a set of elements and relations that operate on the elements, one could then treat the elements and relations together as a system.

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For a different type of system, but one which is as logical as mathematical groups, rings, and fields, consider instead formal systems. A formal system is treated as a language, and is as composed of (a finite set of) symbols and a grammar, and these symbols can be combined or transformed in various ways according to the rules of inference. The grammar describes, tautologically, what is grammatical. A combination of symbols is called a formula, and the formula that are well-formed—those that are grammatical—are called theorems. Axioms are the basic theorems from which other theorems are built using the rules of inference, so returning to our simple system definition, a formal system has theorems as its elements and rules of inference as its relations, but ideally beginning with only the axioms, one is able to produce all theorems. The purpose in using a formal system, primarily, is to aid in separating true from false statements; that is to say, if given a statement composed in the symbols of the formal system, can we tell whether the statement is true (and a theorem)? The way to do so is to find a decision procedure, a chain of known theorems connected by the rules of inference that lead to the statement in question.

Indeed, formal systems served as a model for my seemingly naive definition of a system, and represent an extreme and rigid expression of it. A formal system is related to axiomatic systems, a collection of axioms and the deductive rules for applying them. Goals of both formal and axiomatic systems are completeness and consistency—the ability to describe all theorems based on the axioms and rules of logic, and a lack of contradictions. (Aside: Gödel's Incompleteness Theorem limits a formal/axiomatic system's ability to be both complete and consistent). Such axiomatic and formal systems cannot escape a type of rigidity that is either inappropriate or impractical in non-mathematical or logical endeavors.

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A number of everyday objects can be treated as systems. A traditional clock, for example, consists of gears, hands, springs, a frame, etc. Their spatial relationships to one another as well as the laws of physics (traditional mechanics) make a clock a clock rather than a collection of clock parts. The same is true for an automobile. In contrast, a toolbox is often just a collection of tools—hammers, pliers, screwdrivers, tape measures, etc.—but even if well-organized by size, according to which tray things belong, etc., we are unlikely to consider a toolbox a system, which brings us, for example to classical mousetraps, which consist of four or five parts (board, spring, metal beam, and a release (or release-trigger). Mousetraps are well-organized, but it is not organization, but structure that turns them into an effective system, and we might redefine our system as a structured (but not merely organized) collection of objects.

Such simple definition of a system on the one hand implies great breadth, but at the same time it does not lack in analytic and heuristic power. A simpler definition of a system (such as a collection with order, or even one that lacks the idea of a whole with parts) is broad to the point of meaninglessness (both because it does not fit our normal feeling of what a system is or how it behaves, and because it is so encompassing as to include practically all objects recognized as such), but a more narrow one limits us to specific classes of systems: mathematical, logical, formal, social, political, geological, etc. Systems at the level we have now defined enable numerous possibilities. It is a general enough definition to compare systems—somewhat abstractly—across fields or disciplines, or even different systems within a discipline or context. Inter-context it is about meta-systems and meta-systematicity, about not only how systems function or work between fields, but also about what thinking in terms of s systems provides, about how thinking in terms of systems affects such thinking, and how similarities and differences between systems inspire us to rethink a given framework, find commonalities, etc. Intra-context analysis of systems is the more standard analysis, asking which (if either) system is better or a better model, of finding underlying structures or causes, and so on. Working with systems, however, is also productive, especially because practical and theoretical limits to systems provide us with some surprising behavior.

One of the more surprising but practically less interesting (although no less important) theoretical results is Gödel's Incompleteness Theorem, as applicable to formal systems. Betrand Russell's position was that logical paradoxes were a matter of the mixing of syntax and semantics, of structure and meaning; a formal system would limit itself to matters of form, of structure, of syntax—thus the “grammar” of such a system—and eliminate logical paradoxes. Gödel's Theorem shows that in any interesting (sufficiently complex—for example, arithmetic) formal system it is impossible to completely separate form and content, syntax and meaning; his proof relies on demonstrating that it is possible to imbed the system within the system, making it possible for the theorems of the system to—recursively—say something about the system itself. The result is incompleteness; not all true statements can be proved by the axioms and rules of the formal system—there are always some statements unreachable by any decision process. The formal structure studied by Gödel is equivalent to the functioning of cellular automata as well as that of computer programs (tied here to the concept of a Universal Turing Machine), and there are thus analogs to the Incompleteness Theorem in computer science and elsewhere.

A consequence of this is that any purely rational system of knowledge or truth (insofar as it models itself as a formal system), not just mathematics (though that is the ideal for many such attempts), is destined for certain types of failure (guaranteed incompleteness or inconsistency/paradox), though this has nothing to say about systems that do not follow such a model.

A greater question for systems of most stripes is one of emergent structures, which in one of its simplest formulations can be stated as the whole is greater than sum of its parts, though this formulation is quite misleading, since it relies on a certain understanding of sum (as simple arithmetic addition)—on the one hand it is questionable what the use of sum in a non-mathematical (for non-mathematical systems) sense might mean or entail, and on the other it assumes that the operation sum is not part of the whole created by combining the parts. Returning to the clock model above, a clock is not more than the sum of its parts unless one understands parts only as the physical elements—understood as a system, the rules for combining the elements (parts) are indispensable from the clock-system-whole. This does, admittedly, border on a certain phenomenological understanding of matters.

Another look at emergence might ask, are there properties of the whole that cannot be derived logically from the properties of the parts? Much of this approach might be seen as epistemological rather than ontological, and then perhaps more linguistic and conceptual than epistemological in a more abstract sense of what can be known. If we apply the term staircase rather than stairs to the same physical object (ignoring all semiotic complications in the process), is it not less a matter of the object being referenced and more a matter of how we view collective nouns versus simple plurals? When we ask, how many grains of sand must there be before we have a pile (or in reverse, how many can we take away before a pile of sand is merely some grains of sand?), is this a matter of a pile emerging from the collected grains, or it just a linguistic and semiotic problem, one that is partially solved semiotically through a theory of arbitrary reference and furthermore separating signifiers from the (conceptual) signified and any physical instance (the concept of a pile from any particular pile)?

Ontological emergence is a stronger claim as well as a more controversial one, though even here we border on the epistemological. For example, materialist and reductionist claims, either with regard to science in general or the brain-mind divide in particular, are incomplete. There are claimed to be qualities of a higher level system (e.g. the mind) that sits upon a lower level system (e.g. the brain) that cannot be reduced to (that is to say, deduced from) the lower level system, and those properties are then called emergent properties. The question is whether such properties are truly emergent (not actually implied by the rules of the lower level system), or whether the problem is in fact merely that our understanding of the lower (and higher) systems are currently limited, either practically in terms of our ability to understand or analyze the system or due to lack of data or completeness to the system's rules, and that at a later date an improved model will eliminate the supposed emergence. That is to say, claimed ontological emergence is reduced to epistemological incompleteness.

Many forms of emergence, at least those formulated in this fashion, relate at least in a distant fashion to Gödel's Incompleteness Theorem, and keeping that theorem in mind, as well as realizing that it has little if anything to say about informal systems, it should not surprise us if there are structures in a system that cannot be reduced, apparently or easily, to the existing structures we do understand. Of course, this is assuming that we accept such a loose definition of emergent. One might protest that the emergent property must be (in some system dependent way, perhaps) more complex than the normal elements of the system, not just underived from those elements, and this leads us to the classic example of emergence: ants.

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The argument goes as follows. Ant colonies are a system, consisting of individual ants, which happen to fall into a hierarchy and taxonomy, such as workers, drones, and queens. They have differentiated functions by caste as well as social structures, and communicate primarily by pheromones. On the one hand it is clear that an ant colony is more than just a collection or organization of ants; there is significant structure. Furthermore it is clear that individual ants have little intelligence, and are likewise capable of few actions. Yet at the same time the complex behavior of the ant colony seems to suggest a level of intelligence and problem solving skills that is nowhere situated in any given ant, nor does it seem to follow from how one or two ants can communicate or interact. And so we often speak of this type of intelligence as an emergent property.

At a more abstract or mathematical level, certain types of structure seem to emerge from random sets. A simple example is the party problem from Ramsey Theory. Let us define knowing someone such that if A knows B, B also knows A—of course, in the real world, it is possible to know of someone without them returning the favor, but for the sake of this example we will work a symmetric definition. At any gathering of six or more people we are guaranteed to find either three people who are complete strangers or three people who all know each other. The proof is simple, and rests on the so-called pigeon-hole principe, but the result is more general. First, the proof. Let us take six people and select one (A). A either knows or does not know at least three of the other five (if A knows fewer than three, then A does not know three or more, and vice versa), so without loss of generalization, A knows at least three, and let's call three of them B, C, and D. If any of B, C, or D know each other, then they form, with A, three who all know each other, and thus none can know another of the three, and they form three mutual strangers. If we consider the people as vertices in a graph and their knowing/not-knowing as edges between vertices, colored red for knowing, blue for not-knowing, then what we are saying is that for any 2-colored (red-blue) complete graph of six of more vertices, there exists a red triangle or a blue triangle. Most importantly, this generalizes not just to triangles, but to larger structures, the implication being, that even in random or arbitrary collections, orderly structures arise. For complex enough systems, it should not surprise us if structures unrelated to those predicated by the defined relationships between elements arise, though predicting them or being certain that they will arise is a different matter.

political systems
in action

Systems are not merely abstract or academic topics, though it is often within those contexts that they are discussed, but one can also speak of political systems, systems of control, and so on. On the one hand this lends discussion of them relevance, though on the other, especially with regard to one's relationship to a system, as well in discussing systems of power or control, establishing a distanced or neutral perspective can be difficult if not impossible. While being under the yoke of certain systems can be suffocating, an analysis of such systems can find the (logical) limits of a system, its inconsistencies (especially when a more axiomatic model is used), etc. Practically emergent properties (which I'll label such to indicate my uncertainty as to whether the represent true ontological emergence), perhaps those that require metaphorical critical mass, are likely the most fragile of structures, the type that slight disruptions can destroy—not graceful dissolutions, but more of the one minute it's hear, the next it's gone variety.

breaking a system

It turns out that my supposedly naive definition of a system is, more or less, the common definition of a system. I am leaving out nuances such as open system and closed system. The etymology likewise supports this approach, though we have to note the tendency toward our weaker definition (of organized elements) in the etymology and the stronger definition in the usage.

Arrangement is an early stage of order—division and distinction come earlier—but structure is more complex and comes later. A four-part division is common, the first three being necessary for a system, the fourth being problematic though usually included, and this organization tends to go from concrete to abstract, from material to metaphysical. The three examples that interest me most are nearly isomorphic, depending on presentation: Aristotles' four causes, Foucault's four similarities, and common linguistic/language systems (tying in to formal systems again). All consist of four levels, though this begs the systematicity question, for labeling them systems rather than just taxonomies or such presupposes levels or layers, not just types. That is to say, I am taking David Bohm's approach regarding the four causes, that instead of just describing four types of causes, these causes are arranged in a type of hierarchy. If not logically at least in terms of value this is clearly the case in the history of the causes, but Bohm sees logical organization as well. The efficient builds upon the material, the formal takes the external manifestations of the efficient and embodies them in inner structure, and the final cause provides direction or purpose to this structure; while each later cause builds upon the former, the later makes the former meaningful. The same arrangement is modeled by the four types of similarity described by Foucault in his distillation of early modern epistemology: convention, emulation, analogy and sympathy: the first is mere temporal-spatial overlap, the second mirroring, a simple relation, the third structured similarity, and the last metaphysical similarity between the macrocosmos and microcosmos, for example, and it loops back around, for metaphysical similarity is what makes it possible to conclude that convention (as expressed in/as synchronicity, for example) is meaningful in the first place. In language systems we tend to think of there being lexical items, which are combined and composed by way of morpho-syntactic rules, followed by larger patterns and structures (phrases, clauses, sentences, connected sentences, etc.), and finally a level of semantics, of meaning. If we leave out the fourth level, that relating to meaning or purpose, we are left with the elements of a system: parts, relations (or related parts), and structures that describe the whole, though this does bear a certain resemblance to the the goal of formal systems—separating the semantic and relying only on the systacto-grammatical so as to avoid paradox. The meaning level is still the most problematic, the one that is so far beyond the concrete as to be non-empirical, merely assumed—it is a can't there from here situation. But the structural level, the formal, the analogic, the syntactic, is the level of beauty, of aesthetics, and of judgment ... here's to systems.

—July 19 2006