## Associativity

December 19, 2011 § Leave a comment

Initially, the meaning of ‘associativity’ seems to be pretty clear.

According to common sense, it denotes the capacity or the power to associate entities, to establish a relation or a link between them. Yet, there is a different meaning from mathematics that almost appears as kind of a mocking of the common sense. Due to these very divergent meanings we first have to clarify our usage before discussing the concept.

#### A Strange Case…

In mathematics, associativity is defined as a neutrality of the results of a compound operation with respect to the “bundling,” or association, of the individual parts of the operation. The formal statement is:

A binary operation ∘ (relating two arguments) on a set* S* is called associative if it satisfies the associative law:

x∘(y∘z) = (x∘y)∘z for all x, y, z ∊ S

This, however, is just the opposite of “associative,” as it demands the independence from any particular association. If there would be any capacity to establish an association between any two elements of S, then there should not be any difference.

Maybe, some mathematician in the 19th century hated the associative power of so many natural structures. Subsequently, modernism contributed its own part to establish the corruption of the obvious etymological roots.

In mathematics the notion of associativity—let us call it *I-associativity* in order to indicate the inverted meaning—is an important part of fundamental structures like “classic” (Abelian) groups or categories.

Groups are important since they describe the basic symmetries within the “group” of operations that together form an algebra. Groups cover anything what could be done with sets. Note that the central property of sets is their enumerability. (Hence, a notion of “infinite” sets is nonsense; it simply contradicts itself.) Yet, there are examples of quite successful, say: abundantly used, structures that are not based on I-associativity, the most famous of them being the Lie-group. Lie-groups allow to conceive of continuous symmetry, hence it is much more general than the Abelian group that essentially emerged from the generalization of geometry. Even in the case of Lie-groups or other “non-associative” structures, however, the term refers to the meaning such as to inverting it.

With respect to categories we can say that so far, and quite unfortunately, there is not yet something like a category theory that would not rely on I-associativity, a fact that is quite telling in itself. Of course, category theory is also quite successful, yet…

Well, anyway, we would like to indicate that we are not dealing with I-associativity here in this chapter. In contrast, we are interested in the phenomenon of associativity as it is indicated by the etymological roots of the word: The power to establish relations.

#### A Blind Spot…

In some way the particular *horror creationes* so abundant in mathematics is comprehensible. If a system would start to establish relations it also would establish novelty by means of that relation (sth. that simply did not exist before). So far, it was not possible for mathematics to deal symbolically with the phenomenon of novelty.

Nevertheless it is astonishing that a Google raid on the term “associativity” reveals only slightly more than 500 links (Dec. 2011), from which the vast majority consists simply from the spoofed entry in Wikipedia that considers the mathematical notion of I-associativity. Some other links are related to computer sciences, which basically refer to the same issue, just sailing under a different flag. Remarkably, only one (1) single link from an open source robotics project [1] mentions associativity as we will do here.

Not very surprising one can find an intense linkage between “associative” and “memory,” though not in the absolute number of sources (also around ~600), but in the number of citations. According to Google scholar, Kohonen and his Self-Organizing Map [2] is being cited 9000+ times, followed by Anderson’s account on human memory [3], accumulating 2700 citations.

Of course, there are many entries in the web referring to the word “associative,” which, however, is an adjective. Our impression is that the capability to associate has not made its way into a more formal consideration, or even to regard it as a capability that deserves a dedicated investigation. This deficit may well be considered as a continuation of a much older story of a closely related neglect, namely that of the relation, as Mertz pointed out [4, ch.6], since associativity is just the dynamic counterpart of the relation.

#### Formal and (Quasi-)Material Aspects

In a first attempt, we could conceive of associativity as the capability to impose new relations between some entities. For Hume (in his “Treatise”, see Deleuze’s book about him), association was close to what Kant later dubbed “faculty”: The power to do sth, and in this case to relate ideas. However, such wording is inappropriate as we have seen (or: will see) in the chapters about modeling and categories and models. Speaking about relations and entities implies set theory, yet, models and modeling can’t be covered by set theory, or only very exceptionally so. Since category theory seems to match the requirements and the structure of models much better, we also adapt its structure and its wording.

Associativity then may be taken as the capability to impose arrows between objects A, B, C such that at least A ⊆ B ⊆ C, but usually A ⋐ B ⋐ C, and furthermore A ≃ C, where “≃” means “taken to be identical despite non-identity”. In set theoretic terms we would have used the notion of the equivalence class. Such arrows may be identified with the generalized model, as we are arguing in the chapter about the category of models. The symbolized notion of the generalized abstract model looks like this (for details jump over to the page about modeling):

eq.1

**U**=usage;

**O**=potential observations;

**F**=featuring assignates on O;

**M**=similarity mapping;

**Q**=quasi-logic;

**P**=procedural aspects of implementation.

Those arrows representing the (instances of a generalized) model are *functors* that are mediating between categories. We also may say that the model imposes potentially a manifold of partially ordered sets (posets) onto the initial collection of objects.

Now we can start to address our target, the structural aspects of associativity, more directly. We are interested in the necessary and sufficient conditions for establishing an instance of an object that is able (or develops the capability) to associate objects in the aforementioned sense. In other words, we need an abstract model for it. Yet, here we are not interested in the basic, that is transcendental conditions for the capability to build up associative power.

Let us start more practically, but still immaterial. The best candidates we can think of are Self-Organizing Maps (SOM) and particularly parameterized Reaction-Diffusion Systems (RDS); both of them can be subsumed into the class of associative probabilistic networks, which we describe in another chapter in more technical detail. Of course, not all networks exhibit the emergent property of associativity. We may roughly distinguish between associative networks and logistic networks [5]. Both, SOM as well as RDS, are also able to create manifolds of partial orderings. Another example from this family is the Boltzmann engine, which, however, has some important theoretical and practical drawbacks, even in its generalized form.

Next, we depict the elementary processes of SOM and RDS, respectively. SOM and RDS can be seen as instances located at the distant endpoints of a particular scale, which expresses the topology of the network. The topology expresses the arrangement of quasi-material entities that serve as persistent structure, i.e. as a kind of memory. In the SOM, these entities are called nodes and they are positioned in a more or less fixed grid (albeit there is a variant of the SOM, the SOM gas, where the grid is more fluid). The nodes do not range around. In contrast to the SOM, the entities of an RDS are freely floating around. Yet, RDS are simulated much like the SOM, assuming cells in a grid and stuffing them with a certain memory.

Inspecting those elementary processes, we of course again find transformations. More important, however, is another structural property to both of them. Both networks are characterized by a dynamically changing field of (attractive) forces. Just the locality of those forces is different between SOM and RDS, leading to a greater degree of parallelity in RDS and to multiple areas of the same quality. In SOMs, each node is unique.

The forces in both types of networks are, however, exhibiting the property of locality, i.e. there is one or more center, where the force is strong, and a neighborhood that is established through a stochastic decay of the strength of this force. Usually, in SOM as well as in RDS, the decay is assumed to be radially symmetric, but this is not a necessary condition.

After all, are we now allowed to ask ‘Where does this associativity come from?’ The answer is clearly ‘no.’ Associativity is a holistic property of the arrangement as a total. It is the result of the copresence of some properties like

- – stochastic neighborhoods that are hosting an anisotropic and monotone field of forces;
- – a certain, small memory capacity of the nodes; note that the nodes are not “points”: in order to have a memory they need some corporeality. In turn this opens the way to think of a separation of of the function of that memory and a variable host that provides a container for that memory.
- – strong flows, i.e. a large number of elementary operations acting on that memory, producing excitatory waves (long-range correlations) of finite velocity;

The result of the interaction of those properties can not be described on the level of the elements of the network itself, or any of its parts. What we will observe is a complex dynamics of patterns due to the superposition of antagonist forces, that are modeled either explicitly in the case of RDS, or more implicitly in the case of SOM. Thus both networks are also presenting the property of self-organization, though this aspect is much more dominantly expressed in RDS as compared to the SOM. The important issue is that the whole network, and even more important, the network and its local persistence (“memory”) “causes” the higher-level phenomenon.

We also could say that it is the quasi-material body that is responsible for the associativity of the arrangement.

#### The Power of a Capability

So, what is this associativity thing about? As we have said above, associativity imposes a potential manifold of partial orderings upon an arbitrary open set.

Take a mixed herd of Gnus and Zebras as the open set without any particular ordering, put some predators like hyenas or lions into this herd, and you will get multiple partially ordered sub-populations. In this case, the associativity emerges through particular rules of defense, attack and differential movement. The result of the process is a particular probabilistic order, clearly an immaterial aspect of the herd, despite the fact that we are dealing with fleshy animals.

The interesting thing in both the SOM and the RDS is that a quasi-body provides a capability that transforms an immaterial arrangement. The resulting immaterial arrangement is nothing else than information. In other words, something specific, namely a persistent contrast, has been established from some larger unspecific, i.e. noise. Taking the perspective of the results, i.e. with respect to the resulting information, we always can see that the association creates new information. The body, i.e. the materially encoded filters and rules, has a greater weight in RDS, while in case of the SOM the stabilization aspect is more dominant. In any case, the associative quasi-body introduces breaks of symmetry, establishes them and stabilizes them. If this symmetry breaking is aligned to some influences, feedback or reinforcement acting from the surrounds onto the quasi-body, we may well call the whole process (a simple form of) “learning.”

Yet, this change in the informational setup of the whole “system” is mirrored by a material change in the underlying quasi-body. Associative quasi-bodies are therefore representatives for the transition from the material to the immaterial, or in more popular terms, for the body-mind-dualism. As we have seen, there is no conflict between those categories, as the quasi-body showing associativity provides a double-articulating substrate for differences. Else, we can see that these differences are transformed from a horizontal difference (such as 7-5=2) into vertical, categorical differences (such like the differential). If we would like to compare those vertical differences we need … category theory! …or a philosophy of the differential!

#### Applications

Early in the 20th century, the concept of association has been adopted by behaviorism. Simply recall the dog of Pavlov and the experiments of Skinner and Watson. The key term in behaviorism as a belated echo of 17th century hyper-mechanistics (support of a strictly mechanic world view) is conditioning, which appears in various forms. Yet, conditioning always remains a 2-valued relation, practically achieved as an imprinting, a collision between two inanimate entities, despite the wording of behaviorists who equate their conditioning with “learning by association.” What should learning be otherwise? Nevertheless, behaviorist theory commits the mistake to think that this “learning” should be a passive act. As you can see here, psychologists still strongly believe in this weird concept. They write: “Note that it does not depend on us *doing* anything.” Utter nonsense, nothing else.

In contrast to imprinting, imposing a functor onto an open set of indeterminate objects is not only an exhausting activity, it is also a multi-valued “relation,” or simply, a category. If we would analyze the process of imprinting, we would find that even “imprinting” can’t be covered by a 2-valued relation.

Nevertheless, other people took the media as the message. For instance, Steven Pinker criticized the view that association is sufficient to explain the capability of language. Doing so, he commits the same mistake as the behaviorists, just from the opposite direction. How else should we acquire language, if not by some kind of learning, even if it is a particular type of learning? The blind spot of Pinker seems to be randomization, i.e. he is not able leave the actual representation of a “signal” behind.

Another field of application for the concept of associativity is urban planning or urbanism, albeit associativity is rarely recognized as a conceptual or even as a design tool. [cf. 6] It is obvious that urban environments can be conceived as a multitude of high-dimensional probabilistic networks [7].

#### Machines, Machines, Machines, ….Machines?

Associativity is a property of a persistent (quasi-material) arrangement to act onto a volatile stream (e.g. information, entropy) in such a way as to establish a particular immaterial arrangement (the pattern, or association), which in turn is reflected by material properties of the persistent layer. Equivalently we may say that the process leading to an association is encoded into the material arrangement itself. The establishment of the first pattern is the work of the (quasi-)body. Only for this reason it is possible to build associative formal structures like the SOM or the RDS.

Yet, the notion of “machine” would be misplaced. We observe strict determinism only on the level of the elementary micro-processes. Any of the vast number of individual micro-events are indeed uniquely parameterized, sharing only the same principle or structure. In such cases we can not speak of a single machine any more, since a mechanic machine has a singular and identifiable state at any point in time. The concept of “state” does neither hold for RDS nor for SOM. What we see here is much more like a vast population of similar machines, where any of those is not even stable across time. Instead, we need to adopt the concept of *mechanism*, as it is in use in chemistry, physiology, or biology at large. Since both principles, SOM and RDS, show the phenomenon of self-organization, we even can not say that they represent a probabilistic machine. The notion of the “machine” can’t be applied to SOM or RDS, despite the fact that we can write down the principles for the micro-level in simple and analytic formulas. Yet, we can’t assume any kind of a mechanics for the interaction of those micro-machines.

It is now exciting to see that a probabilistic, self-organizing process used to create a model by means of associating principles looses the property of being a machine, even as it is running on a completely deterministic machine, the simulation of a Universal Turing Machine.

Associativity is a principle that transcends the machine, and even the machinic (Guattari). Assortative arrangements establish persistent differences, hence we can say that they create proto-symbols. Without associativity there is no information. Of course, the inverse is also true: Wherever we find information or an assortment, we also must expect associativity.

۞

- [1] iCub
- [2] Kohonen, Teuvo, Self-Organization and Associative Memory. Springer Series in Information Sciences, vol.8, Springer, New York 1988.
- [3] Anderson J.R., Bower G.H., Human Associative Memory. Erlbaum, Hillsdale (NJ) 1980.
- [4] Mertz, D. W., Moderate Realism and its Logic, New Haven: Yale 1996.
- [5] Wassermann, K. (2010), Associativity and Other Wurban Things – The Web and the Urban as merging Cultural Qualities. 1st international workshop on the urban internet of things, in conjunction with: internet of things conference 2010 in Tokyo, Japan, Nov 29 – Dec 1, 2010. (pdf)
- [6] Dean, P., Rethinking representation. the Berlage Institute report No.11, episode Publ. 2007.
- [7] Wassermann, K. (2010). SOMcity: Networks, Probability, the City, and its Context. eCAADe 2010, Zürich. September 15-18, 2010. (pdf)

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