March 20, 2012 § Leave a comment
In this second part of the essay about a fresh perspective on
analogical thinking—more precise: on models about it—we will try to bring two concepts together that at first sight represent quite different approaches: Copycat and SOM.
Why engaging in such an endeavor? Firstly, we are quite convinced that FARG’s Copycat demonstrates an important and outstanding architecture. It provides a well-founded proposal about the way we humans apply ideas and abstract concepts to real situations. Secondly, however, it is also clear that Copycat suffers from a few serious flaws in its architecture, particularly the built-in idealism. This renders any adaptation to more realistic domains, or even to completely domain-independent conditions very, very difficult, if not impossible, since this drawback also prohibits structural learning. So far, Copycat is just able to adapt some predefined internal parameters. In other words, the Copycat mechanism just adapts a predefined structure, though a quite abstract one, to a given empiric situation.
Well, basically there seem to be two different, “opposite” strategies to merge these approaches. Either we integrate the SOM into Copycat, or we try to transfer the relevant yet to be identified parts from Copycat to a SOM-based environment. Yet, at the end of day we will see that and how the two alternatives converge.
In order to accomplish our goal of establishing a fruitful combination between SOM and Copycat we have to take mainly three steps. First, we briefly recapitulate the basic elements of Copycat and the proper instance of a SOM-based system. We also will describe the extended SOM system in some detail, albeit there will be a dedicated chapter on it. Finally, we have to transfer and presumably adapt those elements of the Copycat approach that are missing in the SOM paradigm.
The particular power of (natural) evolutionary processes derives from the fact that it is based on symbols. “Adaptation” or “optimization” are not processes that change just the numerical values of parameters of formulas. Quite to the opposite, adaptational processes that span across generations parts of the DNA-based story is being rewritten, with potential consequences for the whole of the story. This effect of recombination in the symbolic space is particularly present in the so-called “crossing over” during the production of gamete cells in the context of sexual reproduction in eukaryotes. Crossing over is a “technique” to dramatically speed up the exploration of the space of potential changes. (In some way, this space is also greatly enlarged by symbolic recombination.)
What we will try here in our attempt to merge the two concepts of Copycat and SOM is exactly this: a symbolic recombination. The difference to its natural template is that in our case we do not transfer DNA-snippets between homologous locations in chromosomes, we transfer whole “genes,” which are represented by elements.
Elementarizations I: C.o.p.y.c.a.t.
In part 1 we identified two top-level (non-atomic) elements of Copycat
- (1) restricted generalized evolution, and
- (2) concrete instances of domain specific idealizations.
Since the first element, covering evolutionary aspects such as randomness, population and a particular memory dynamics, is pretty clear and a whole range of possible ways to implement it are available, any attempt for improving the Copycat approach has to target the static, strongly idealistic characteristics of the the structure that is called “Slipnet” by the FARG’s. The Slipnet has to be enabled for structural changes and autonomous adaptation of its parameters. This could be accomplished in many ways, e.g. by representing the items in the Slipnet as primitive artificial genes. Yet, we will take a different road here, since the SOM paradigm already provides the means to achieve idealizations.
At that point we have to elementarize Copycat’s Slipnet in a way that renders it compatible with the SOM principles. Hofstadter emphasizes the following properties of the Slipnet and the items contained therein (pp.212).
- (1) Conceptual depth allows for a dynamic and continuous scaling of “abstractness” and resistance against “slipping” to another concept;
- (2) Nodes and links between nodes both represent active abstract properties;
- (3) Nodes acquire, spread and lose activation, which knows an switch-on threshold < 1;
- (4) The length of links represents conceptual proximity or degree of association between the nodes.
As a whole, and viewed from the network perspective, the Slipnet behaves much like a spring system, or a network built from rubber bands, where the springs or the rubber bands are regulated in their strength. Note that our concept of SomFluid also exhibits the feature of local regulation of the bonds between nodes, a property that is not present in the idealized standard SOM paradigm.
Yet, the most interesting properties in the list above are (1) and (2), while (3) and (4) are known in the classic SOM paradigm as well. The first item is great because it represents an elegant instance of creating the possibility for measurability that goes far beyond the nominal scale. As a consequence, “abstractness” ceases to be nominal none-or-all property, as it is present in hierarchies of abstraction. Such hierarchies now can be recognized as mere projections or selections, both introducing a severe limitation of expressibility. The conceptual depth opens a new space.
The second item is also very interesting since it blurs the distinction between items and their relations to some extent. That distinction is also a consequence of relying too readily on the nominal scale of description. It introduces a certain moment of self-reference, though this is not fully developed in the Slipnet. Nevertheless, a result of this move is that concepts can’t be thought without their embedding into other a neighborhood of other concepts. Hofstadter clearly introduces a non-positivistic and non-idealistic notion here, as it establishes a non-totalizing meta-concept of wholeness.
Yet, the blurring between “concepts” and “relations” could be and must be driven far beyond the level Hofstadter achieved, if the Slipnet should become extensible. Namely, all the parts and processes of the Slipnet need to follow the paradigm of probabilization, since this offers the only way to evade the demons of cybernetic idealism and control apriori. Hofstadter himself relies much on probabilization concerning the other two architectural parts of Copycat. Its beyond me why he didn’t apply it to the Slipnet too.
Taken together, we may derive (or: impose) the following important elements for an abstract description of the Slipnet.
- (1) Smooth scaling of abstractness (“conceptual depth”);
- (2) Items and links of a network of sub-conceptual abstract properties are instances of the same category of “abstract property”;
- (3) Activation of abstract properties represents a non-linear flow of energy;
- (4) The distance between abstract properties represents their conceptual proximity.
A note should be added regarding the last (fourth) point. In Copycat, this proximity is a static number. In Hofstadter’s framework, it does not express something like similarity, since the abstract properties are not conceived as compounds. That is, the abstract properties are themselves on the nominal level. And indeed, it might appear as rather difficult to conceive of concepts as “right of”, “left of”, or “group” as compounds. Yet, I think that it is well possible by referring to mathematical group theory, the theory of algebra and the framework of mathematical categories. All of those may be subsumed into the same operationalization: symmetry operations. Of course, there are different ways to conceive of symmetries and to implement the respective operationalizations. We will discuss this issue in a forthcoming essay that is part of the series “The Formal and the Creative“.
The next step is now to distill the elements of the SOM paradigm in a way that enables a common differential for the SOM and for Copycat..
Elementarizations II: S.O.M.
The self-organizing map is a structure that associates comparable items—usually records of values that represent observations—according to their similarity. Hence, it makes two strong and important assumptions.
- (1) The basic assumption of the SOM paradigm is that items can be rendered comparable;
- (2) The items are conceived as tokens that are created by repeated measurement;
The first assumption means, that the structure of the items can be described (i) apriori to their comparison and (ii) independent from the final result of the SOM process. Of course, this assumption is not unique to SOMs, any algorithmic approach to the treatment of data is committed to it. The particular status of SOM is given by the fact—and in stark contrast to almost any other method for the treatment of data—that this is the only strong assumption. All other parameters can be handled in a dynamic manner. In other words, there is no particular zone of the internal parametrization of a SOM that would be inaccessible apriori. Compare this with ANN or statistical methods, and you feel the difference… Usually, methods are rather opaque with respect to their internal parameters. For instance, the similarity functional is usually not accessible, which renders all these nice looking, so-called analytic methods into some kind of subjective gambling. In PCA and its relatives, for instance, the similarity is buried in the covariance matrix, which in turn is only defined within the assumption of normality of correlations. If not a rank correlation is used, this assumption is extended even to the data itself. In both cases it is impossible to introduce a different notion of similarity. Else, and also as a consequence of that, it is impossible to investigate the particular dependency of the results proposed by the method from the structural properties and (opaque) assumptions. In contrast to such unfavorable epistemo-mythical practices, the particular transparency of the SOM paradigm allows for critical structural learning of the SOM instances. “Critical” here means that the influence of internal parameters of the method onto the results or conclusions can be investigated, changed, and accordingly adapted.
The second assumption is implied by its purpose to be a learning mechanism. It simply needs some observations as results of the same type of measurement. The number of observations (the number of repeats) has to exceed a certain lower threshold, which, dependent on the data and the purpose, is at least 8, typically however (much) more than 100 observations of the same kind are needed. Any result will be within the space delimited by the assignates (properties), and thus any result is a possibility (if we take just the SOM itself).
The particular accomplishment of a SOM process is the transition from the extensional to the intensional description, i.e. the SOM may be used as a tool to perform the step from tokens to types.
From this we may derive the following elements of the SOM:1
- (1) a multitude of items that can be described within a common structure, though not necessarily an identical one;
- (2) a dense network where the links between nodes are probabilistic relations;
- (3) a bottom-up mechanism which results in the transition from an extensional to an intensional level of description;
As a consequence of this structure the SOM process avoids the necessity to compare all items (N) to all other items (N-1). This property, together with the probabilistic neighborhoods establishes the main difference to other clustering procedures.
It is quite important to understand that the SOM mechanism as such is not a modeling procedure. Several extensions have to be added and properly integrated, such as
- – operationalization of the target into a target variable;
- – validation by separate samples;
- – feature selection, preferably by an instance of a generalized evolutionary process (though not by a genetic algorithm);
- – detecting strong functional and/or non-linear coupling between variables;
- – description of the dependency of the results from internal parameters by means of data experiments.
Yet, as we explained in part 1 of this essay, analogy making is conceptually incompatible to any kind of modeling, as long as the target of the model points to some external entity. Thus, we have to choose a non-modeling instance of a SOM as the starting point. However, clustering is also an instance of those processes that provide the transition from extensions to intensions, whether this clustering is embedded into full modeling or not. In other words, both the classic SOM as well as the modeling SOM are not suitable as candidates for a merger with Copycat.
Fortunately, there is already a proposal, and even a well-known one, that indeed may be taken as such a candidate: the two-layer SOM (TL-SOM) as it has been demonstrated as essential part of the so-called WebSom [1,2].
Actually, the description as being “two layered” is a very minimalistic, if not inappropriate description what is going on in the WebSom. We already discussed many aspects of its architecture here and here.
Concerning our interests here, the multi-layered arrangement itself is not a significant feature. Any system doing complicated things needs a functional compartmentalization; we have met a multi-part, multi-compartment and multi-layered structure in the case of Copycat too. Else, the SOM mechanism itself remains perfectly identical across the layers.
The real interesting features of the approach realized in the TL-SOM are
- – the preparation of the observations into probabilistic contexts;
- – the utilization of the primary SOM as a measurement device (the actual trick).
The domain of application of the TL-SOM is the comparison and classification of texts. Texts belong to unstructured data and the comparison of texts is exposed to the same problematics as the making of analogies: there is no apriori structure that could serve as a basis for modeling. Also, as the analogies investigated by the FARG the text is a locational phenomenon, i.e. it takes place in a space.
Let us briefly recapitulate the dynamics in a TL-SOM. In order to create a TL-SOM the text is first dissolved into overlapping, probabilistic contexts. Note that the locational arrangement is captured by these random contexts. No explicit apriori rules are necessary to separate patterns. The resulting collection of contexts then gets “somified”. Each node then contains similar random contexts that have been derived from various positions in different texts. Now the decisive step will be taken, which consists in turning the perspective by “90 degrees”: We can use the SOM as the basis for creating a histogram for each of the texts. The nodes are interpreted as properties of the texts, i.e. each node represents a bin of the histogram. The values of the individual bins measure the frequency of the text as it is represented by the respective random context. The secondary SOM then creates a clustering across these histograms, which represent the texts in an abstract manner.
This way the primary lattice of the TL-SOM is used to impose a structure on the unstructured entity “text.”
Figure 1: A schematic representation of a two-layered SOM with built-in self-referential abstraction. The input for the secondary SOM (foreground) is derived as a collection of histograms that are defined as a density across the nodes of the primary SOM (background). The input for the primary SOM are random contexts.
To put it clearly: the secondary SOM builds an intensional description of entities that results from the interaction of a SOM with a probabilistic description of the empirical observations. Quite obviously, intensions built this way about intensions are not only quite abstract, the mechanism could even be stacked. It could be described as “high-level perception” as justified as Hofstadter uses the term for Copycat. The TL-SOM turns representational intensions into abstract, structural ones.
The two aspects from above thus interact, they are elements of the TL-SOM. Despite the fact that there are still transitions from extensions to intensions, we also can see that the targeted units of the analysis, the texts get probabilistically distributed across an area, the lattice of the primary SOM. Since the SOM maps the high-dimensional input data onto its map in a way that preserves their topological properties, it is easy to recognize that the TL-SOM creates conceptual halos as an intermediate.
So let us summarize the possibilities provided by the SOM.
- (1) SOMs are able to create non-empiric, or better: de-empirified idealizations of intensions that are based on “quasi-empiric” input data;
- (2) TL-SOMs can be used to create conceptual halos.
In the next section we will focus on this spatial, better: primarily spatial effect.
The Extended SOM
Kohonen and co-workers [1,2] proposed to build histograms that reflect the probability density of a text across the SOM. Those histograms represent the original units (e.g. texts) in a quite static manner, using a kind of summary statistics.
Yet, texts are definitely not a static phenomenon. At first sight there is at least a series, while more appropriately texts are even described as dynamic networks of own associative power . Returning to the SOM we see that additionally to the densities scattered across the nodes of the SOM we also can observe a sequence of invoked nodes, according to the sequence of random contexts in the text (or the serial observations)
The not so difficult question then is: How to deal with that sequence? Obviously, it is again and best conceived as a random process (though with a strong structure), and random processes are best described using Markov models, either as hidden (HMM) or as transitional models. Note that the Markov model is not a model about the raw observational data, it describes the sequence of activation events of SOM nodes.
The Markov model can be used as a further means to produce conceptual halos in the sequence domain. The differential properties of a particular sequence as compared to the Markov model then could be used as further properties to describe the observational sequence.
(The full version of the extended SOM comprises targeted modeling as a further level. Yet, this targeted modeling does not refer to raw data. Instead, its input is provided completely by the primary SOM, which is based on probabilistic contexts, while the target of such modeling is just internal consistency of a context-dependent degree.)
Just to avoid misunderstanding: it does not make sense to try representing Copycat completely by a SOM-based system. The particular dynamics and phenomenologically behavior depends a lot on Copycat’s tripartite morphology as represented by the Coderack (agents), the Workspace and the Slipnet. We are “just” in search for a possibility to remove the deep idealism from the Slipnet in order to enable it for structural learning.
Basically, there are two possible routes. Either we re-interpret the extended SOM in a way that allows us to represent the elements of the Slipnet as properties of the SOM, or we try to replace the all items in the Slipnet by SOM lattices.
So, let us take a look which structures we have (Copycat) or what we could have (SOM) on both sides.
Table 1: Comparing elements from Copycat’s Slipnet to the (possible) mechanisms in a SOM-based system.
|1.||smoothly scaled abstraction||Conceptual depth (dynamic parameter)||distance of abstract intensions in an integrated lattice of a n-layered SOM|
|2.||Links as concepts||structure by implementation||reflecting conceptual proximity as an assignate property for a higher-level|
|3.||Activation featuring non-linear switching behavior||structure by implementation||x|
|4.||Conceptual proximity||link length (dynamic parameter)||distance in map (dynamic parameter)|
|5.||Kind of concepts||locational, positional symmetries,||any|
From this comparison it is clear that the single most challenging part of this route is the possibility for the emergence of abstract intensions in the SOM based on empirical data. From the perspective of the SOM, relations between observational items such as “left-most,” “group” or “right of”, and even such as “sameness group” or “predecessor group”, are just probabilities of a pattern. Such patterns are identified by functions or dynamic combinations thereof. Combinations ot topological primitives remain mappable by analytic functions. Such concepts we could call “primitive concepts” and we can map these to the process of data transformation and the set of assignates as potential properties.2 It is then the job of the SOM to assign a relevancy to the assignates.
Yet, Copycat’s Slipnet comprises also rather abstract concepts such as “opposite”. Further more, the most abstract concepts often act as links between more primitive concepts, or, in Hofstadter terms, conceptual items of lower “conceptual depth”.
My feeling here is that it is a fundamental mistake to implement concepts like “opposite” directly. What is opposite of something else is a deeply semantic concept in itself, thus strongly dependent on the domain. I think that most of the interesting concepts, i.e. the most abstract ones are domain-specific. Concepts like “opposite” could be considered as something “simple” only in case of geometric or spatial domains.
Yet, that’s not a weakness. We should use this as a design feature. Take the following rather simple case as shown in the next figure as an example. Here we mapped simply triplets of uniformly distributed random values onto a SOM. The three values can be readily interpreted as parts of a RGB value, which renders the interpretation more intuitive. The special thing here is that the map has been a really large one: We defined approximately 700’000 nodes and fed approx. 6 million observations into it.
Figure 2: A SOM-based color map showing emergence of abstract features. Note that the topology of the map is a borderless toroid: Left and right borders touch each other (distance=0), and the same applies to the upper and lower borders.
We can observe several interesting things. The SOM didn’t come up with just any arbitrary sorting of the colors. Instead, a very particular one emerged.
First, the map is not perfectly homogeneous anymore. Very large maps tend to develop “anisotropies”, symmetry breaks if you like, simply due to the fact the the signal horizon becomes an important issue. This should not be regarded as a deficiency though. Symmetry breaks are essential for the possibility of the emergence of symbols. Second, we can see that two “color models” emerged, the RGB model around the dark spot in the lower left, and the YMC model around the bright spot in the upper right. Third, the distance between the bright, almost white spot and the dark, almost black one is maximized.
In other words, and not quite surprising, the conceptual distance is reflected as a geometrical distance in the SOM. As it is the case in the TL-SOM, we now could use the SOM as a measurement device that transforms an unknown structure into an internal property, simply by using the locational property in the SOM as an assignate for a secondary SOM. In this way we not only can represent “opposite”, but we even have a model procedure for “generalized oppositeness” at out disposal.
It is crucial to understand this step of “observing the SOM”, thereby conceiving the SOM as a filter, or more precisely as a measurement device. Of course, at this point it becomes clear that a large variety of such transposing and internal-virtual measurement devices may be thought of. Methodologically, this opens an orthogonal dimension to the representation of data, resembling strongly to the concept of orthoregulation.
The map shown above even allows to create completely different color models, for instance one around yellow and another one around magenta. Our color psychology is strongly determined by the sun’s radiated spectrum and hence it reflects a particular Lebenswelt; yet, there is no necessity about it. Some insects like bees are able to perceive ultraviolet radiation, i.e. their colors may have 4 components, yielding a completely different color psychology, while the capability to distinguish colors remains perfectly.3
“Oppositeness” is just a “simple” example for an abstract concept and its operationalization using a SOM. We already mentioned the “serial” coherence of texts (and thus of general arguments) that can be operationalized as sort of virtual movement across a SOM of a particular level of integration.
It is crucial to understand that there is no other model besides the SOM that combines the ability to learn from empirical data and the possibility for emergent abstraction.
There is yet another lesson that we can take home from the simple example above. Well, the example doesn’t not remain that simple. High-level abstraction, items of considerable conceptual depth, so to speak, requires rather short assignate vectors. In the process of learning qua abstraction it appears to be essential that the masses of possible assignates derived from or imposed by measurement of raw data will be reduced. On the one hand, empiric contexts from very different domains should be abstracted, i.e. quite literally “reduced”, into the same perspective. On the other hand, any given empiric context should be abstracted into (much) more than just one abstract perspective. The consequence of that is that we need a lot of SOMs, all separated “sufficiently” from each other. In other words, we need a dynamic population of Self-organizing maps in order to represent the capability of abstraction in real-life. “Dynamic population” here means that there are developmental mechanisms that result in a proliferation, almost a breeding of new SOM instances in a seamless manner. Of course, the SOM instances themselves have to be able to grow and to differentiate, as we have described it here and here.
In a population of SOM the conceptual depth of a concept may be represented by the efforts to arrive at a particular abstract “intension.” This not only comprises the ordinary SOM lattices, but also processes like Markov models, simulations, idealizations qua SOMs, targeted modeling, transition into symbolic space, synchronous or potential activations of other SOM compartments etc. This effort may be represented finally as a “number.”
The structure of multi-layered system of Self-organizing Maps as it has been proposed by Kohonen and co-workers is a powerful model to represent emerging abstraction in response to empiric impressions. The Copycat model demonstrates how abstraction could be brought back to the level of application in order to become able to make analogies and to deal with “first-time-exposures”.
Here we tried to outline a potential path to bring these models together. We regard this combination in the way we proposed it (or a quite similar one) as crucial for any advance in the field of machine-based episteme at large, but also for the rather confined area of machine learning. Attempts like that of Blank  appear to suffer seriously from categorical mis-attributions. Analogical thinking does not take place on the level of single neurons.
We didn’t discuss alternative models here (so far, a small extension is planned). The main reasons are that first it would be an almost endless job, and second that Hofstadter already did it and as a result of his investigation he dismissed all the alternative approaches (from authors like Gentner, Holyoak, Thagard). For an overview Runco  about recent models on creativity, analogical thinking, or problem solving provides a good starting point. Of course, many authors point to roughly the same direction as we did here, but mostly, the proposals are circular, not helpful because the problematic is just replaced by another one (e.g. the infamous and completely unusable “divergent thinking”), or can’t be implemented for other reasons. Thagard  for instance, claim that a “parallel satisfaction of the constraints of similarity, structure and purpose” is key in analogical thinking. Given our analysis, such statements are nothing but a great mess, mixing modeling, theory, vagueness and fluidity.
For instance, in cognitive psychology and in the field of artificial intelligence as well, the hypothesis of Structural Mapping (STM) finds a lot of supporters . Hofstadter discusses similar approaches in his book. The STM hypothesis is highly implausible and obviously a left-over of the symbolic approach to Artificial Intelligence, just transposed into more structural regions. The STM hypothesis has not only to be implemented as a whole, it also has to be implemented for each domain specifically. There is no emergence of that capability.
The combination of the extended SOM—interpreted as a dynamic population of growing SOM instances—with the Copycat mechanism indeed appears as a self-sustaining approach into proliferating abstraction and—quite significant—back from it into application. It will be able to make analogies on any field already in its first encounter with it, even regarding itself, since both the extended SOM as well as the Copycat comprise several mechanisms that may count as precursors of high-level reflexivity.
After this proposal little remains to be said on the technical level. One of those issues which remain to be discussed is the conditions for the possibility of binding internal processes to external references. Here our favorite candidate principle is multi-modality, that is the joint and inextricable “processing” (in the sense of “getting affected”) of words, images and physical signals alike. In other words, I feel that we have come close to the fulfillment of the ariadnic question this blog:”Where is the Limit?” …even in its multi-faceted aspects.
A lot of implementation work has now to be performed, eventually commented by some philosophical musings about “cognition”, or more appropriate the “epistemic condition.” I just would like to invite you to stay tuned for the software publications to come (hopefully in the near future).
2. It is somehow interesting that in the brain of many animals we can find very small groups of neurons, if not even single neurons, that respond to primitive features such as verticality of lines, or the direction of the movement of objects in the visual field.
3. Ludwig Wittgenstein insisted all the time that we can’t know anything about the “inner” representation of “concepts.” It is thus free of any sense and meaning to claim knowledge about the inner state of oneself as well as of that of others. Wilhelm Vossenkuhl introduces and explains the Wittgensteinian “grammatical” solipsism carefully and in a very nice way. The only thing we can know about inner states is that we use certain labels for it, and the only meaning of emotions is that we do report them in certain ways. In other terms, the only thing that is important is the ability to distinguish ones feelings. This, however, is easy to accomplish for SOM-based systems, as we have been demonstrating here and elsewhere in this collection of essays.
4. Don’t miss Timo Honkela’s webpage where one can find a lot of gems related to SOMs! The only puzzling issue about all the work done in Helsinki is that the people there constantly and pervasively misunderstand the SOM per se as a modeling tool. Despite their ingenuity they completely neglect the issues of data transformation, feature selection, validation and data experimentation, which all have to be integrated to achieve a model (see our discussion here), for a recent example see here, or the cited papers about the Websom project.
-  Timo Honkela, Samuel Kaski, Krista Lagus, Teuvo Kohonen (1997). WEBSOM – Self-Organizing Maps of Document Collections. Neurocomputing, 21: 101-117.4
-  Krista Lagus, Samuel Kaski, Teuvo Kohonen in Information Sciences (2004)
Mining massive document collections by the WEBSOM method. Information Sciences, 163(1-3): 135-156. DOI: 10.1016/j.ins.2003.03.017
-  Klaus Wassermann (2010). Nodes, Streams and Symbionts: Working with the Associativity of Virtual Textures. The 6th European Meeting of the Society for Literature, Science, and the Arts, Riga, 15-19 June, 2010. available online.
- [4 ]Douglas S. Blank, Implicit Analogy-Making: A Connectionist Exploration.Indiana University Computer Science Department. available online.
-  Mark A. Runco, Creativity-Research, Development, and Practice Elsevier 2007.
-  Keith J. Holyoak and Paul Thagard, Mental Leaps: Analogy in Creative Thought.
MIT Press, Cambridge 1995.
-  John F. Sowa, Arun K. Majumdar (2003), Analogical Reasoning. in: A. Aldo, W. Lex, & B. Ganter (eds.), “Conceptual Structures for Knowledge Creation and Communication,” Proc.Intl.Conf.Conceptual Structures, Dresden, Germany, July 2003. LNAI 2746, Springer New York 2003. pp. 16-36. available online.
-  Wilhelm Vossenkuhl. Solipsismus und Sprachkritik. Beiträge zu Wittgenstein. Parerga, Berlin 2009.
March 12, 2012 § Leave a comment
Since the beginnings of the intellectual adventure
that we know as philosophy, elements take a particular and prominent role. For us, as we live as “post-particularists,” the concept of element seems to be not only a familiar one, but also a simple, almost a primitive one. One may take this as the aftermath of the ontological dogma of the four (or five) elements and its early dismissal by Aristotle.
In fact, I think that the concept element is seriously undervalued and hence it is left disregarded much too often, especially as far as one concerns it as a structural tool in the task to organize thinking. The purpose of this chapter is thus to reconstruct the concept of “element” in an adequate manner (at least, to provide some first steps of such a reconstruction). To achieve that we have to take tree steps.
First, we will try to shed some light on its relevance as a more complete concept. In order to achieve this we will briefly visit the “origins” of the concept in (pre-)classic Greek philosophy. After browsing quickly through some prominent examples, the second part then will deal with the concept of element as a thinking technique. For that purpose we strip the ontological part of it (what else?), and turn it into an activity, a technique, and ultimately into a “game of languagability,” called straightforwardly “elementarization.”
This will forward us then to the third part, which will deal with problematics of expression and expressibility, or more precisely, to the problematics of how to talk about expression and expressibility. Undeniably, creativity is breaking (into) new grounds, and this aspect of breaking pre-existing borders also implies new ways of expressing things. To get clear about creativity thus requires to get clear about expressibility in advance.
The remainder of this essay revolves is arranged by the following sections (active links):
As many other concepts too, the concept of “element” first appeared in classic Greek culture. As a concept, the element, Greek “stoicheion”, in greek letters ΣΤΟΙΧΕΙΟΝ, is quite unique because it is a synthetic concept, without predecessors in common language. The context of its appearance is the popularization of the sundial by Anaximander around 590 B.C. Sundials have been known before, but it was quite laborious to create them since they required a so-called skaphe, a hollow sphere as the projection site of the gnomon’s shadow.
Figure 1a,b. Left (a): A sundial in its ancient (primary) form based on a skaphe, which allowed for equidistant segmentation , Right (b): the planar projection involves hyperbolas and complicated segmentation.
The planar projection promised a much more easier implementation, yet, it involves the handling of hyperbolas, which even change relative to the earth’s seasonal inclination. Else, the hours can’t be indicated by an equidistant segments any more. Such, the mathematical complexity has been beyond the capabilities of that time. The idea (presumably of Anaximander) then was to determine the points for the hours empirically, using “local” time (measured by water clocks) as a reference.
Anaximander also got aware of the particular status of a single point in such a non-trivial “series”. It can’t be thought without reference to the whole series, and additionally, there was no simple rule which would have been allowing for its easy reconstruction. This particular status he called an “element”, a stoicheia (pronunciation). Anaximander’s element is best understood as a constitutive component, a building block for the purpose to build a series; note the instrumental twist in his conceptualization.
From this starting point, the concept has been generalized in its further career, soon denoting something like “basics,” or “basic principles”. While Empedokles conceived the four elements, earth, wind, water and fire almost as divine entities, it was Platon (Timaios 201, Theaitet 48B) who developed the more abstract perspective into “elements as basic principles.”
Yet, the road of abstraction does not know a well-defined destiny. Platon himself introduced the notion of “element of recognition and proofing” for stoicheia. Isokrates, then, a famous rhetorician and coeval of Platon extended the reach of stoicheia from “basic component / principle” into “basic condition.” This turn is quite significant since as a consequence it inverts the structure of argumentation from idealistic, positive definite claims to the constraints of such claims; it even opens the perspective to the “condition of possibility”, a concept that is one of the cornerstones of Kantian philosophy, more than 2000 years later. No wonder, Isokrates is said to have opposed Platon’s arguments.
Nevertheless, all these philosophical uses of stoicheia, the elements, have been used as ontological principles in the context of the enigma of the absolute origin of all things and the search for it. This is all the more particularly remarkable as the concept itself has been constructed some 150 years before in a purely instrumental manner.
Aristotle dramatically changed the ontological perspective. He dismissed the “analysis based on elements” completely and established what is now known as “analysis of moments”, to which the concepts of “form” and “substance” are central. Since Aristotle, elemental analysis regarded as a perspective heading towards “particularization”, while the analysis of moments is believed to be directed to generalization. Elemental analysis and ontology is considered as being somewhat “primitive,” probably due to its (historic) neighborhood to the dogma of the four elements.
True, the dualism made from form and substance is more abstract and more general. Yet, as concept it looses contact not only to the empiric world as it completely devoid of processual aspects. It is also quite difficult, if not impossible, to think “substance” in a non-ontological manner. It seems as if that dualism abolishes even the possibility to think in a different manner than as ontology, hence implying a whole range of severe blind spots: the primacy of interpretation, the deeply processual, event-like character of the “world” (the primacy of “process” against “being”), the communal aspects of human lifeforms and its creational power, the issue of localized transcendence are just the most salient issues that are rendered invisible in the perspective of ontology.
Much more could be said of course about the history of those concepts. Of course, Aristotle’s introduction of the concept of substance is definitely not without its own problems, paving the way for the (overly) pronounced materialism of our days. And there is, of course, the “Elements of Geometry” by Euclid, the most abundant mathematical textbook ever. Yet, I am neither a historian nor a philologus, thus let us now proceed with some examples. I just would like to emphasize that the “element” can be conceived as a structural topos of thinking starting from the earliest witnesses of historical time.
Think about the chemical elements as they have been invented in the 19th century. Chemical compounds, so the parlance of chemists goes, are made from chemical elements, which have been typicized by Mendeleev according to the valence electrons and then arranged into the famous “periodic table.” Mendeleev not only constructed a quality according to which various elements could be distinguished. His “basic principle” allowed him to make qualitative and quantitative predictions of an astonishing accuracy. He predicted the existence of chemical elements, “nature’s substance”, actually unknown so far, along with their physico-chemical qualities. Since it was in the context of natural science, he also could validate that. Without the concept of those (chemical) elements the (chemical) compounds can’t be properly understood. Today a similar development can be observed within the standard theory of particle physics, where basic types of particles are conceived as elements analogous to chemical elements, just that in particle physics the descriptive level is a different one.
Here we have to draw a quite important distinction. The element in Mendeleev’s thinking is not equal to the element as the chemical elements. Mendeleev’s elements are (i) the discrete number (an integer between 1..7, and 0/8 for the noble gases like Argon etc.) that describes the free electron as a representative of electrostatic forces, and (ii) the concept of “completeness” of the set of electrons in the so-called outer shell (or “orbitals”): the number of the valence electrons of two different chemical elements tend to sum up to eight. Actually, chemical elements can be sorted into groups (gases, different kinds of metals, carbon and silicon) according to the mechanism how they achieve this magic number (or how they don’t). As a result, there is a certain kind of combinatorianism, the chemical universe is almost a Lullian-Leibnizian one. Anyway, the important point here is that the chemical elements are only a consequence of a completely different figure of thought.
Still within in chemistry, there is another famous, albeit less well-known example for abstract “basic principles”: Kekulé’s de-localized valence electrons in carbon compounds (in today’s notion: delocalized 6-π-electrons). Actually, Kekulé added the “element” of the indeterminateness to the element of the valence electron. He dropped the idea of a stable state that could be expressed by a numerical value, or even by an integer. His 6-π-orbital is a cloud that could not be measured directly as such. Today, it is easy to see that the whole area of organic chemistry is based on, or even defined by, these conceptual elements.
Another example is provided by “The Elements of Geometry” by Euclid. He called it “elements” probably for mainly two reasons. First, it was supposed that it was complete, secondly, because you could not remove any of the axioms, procedures, proofs or lines of arguments, i.e. any of its elements, without corroborating the compound concept “geometry.”
A further example from the classic is the conceptual (re-)construction of causality by Aristotle. He obviously understood that it is not appropriate to take causality as an impartible entity. Aristotle designed his idea of causality as an irreducible combination of four distinct elements, causa materialis, causa formalis, causa efficiens and causa finalis. To render this a bit more palpable, think about inflaming a wooden stick and then being asked: What is the cause for the stick being burning?
Even if I would put (causa efficiens) a wooden (causa materialis) stick (causa formalis) above an open flame (part of causa efficiens), it will not necessarily be inflamed until I decide that it should (causa finalis). This is a quite interesting structure, since it could be conceived as a precursor of the Wittgensteinian perspective of a language game.
For Aristotle it made no sense to assume that any of the elements of his causality as he conceived it would be independent from any of the others. For him it would have been nonsense to conceive of causality as any subset of his four elements. Nevertheless, exactly this was what physics did since Newton. In our culture, causality is almost always debated as if it would be identical to causa efficiens. In Newton’s words: Actioni contrariam semper et aequalem esse reactionem.  Later, this postulate of actio = reactio has been backed by further foundational work through larger physical theories postulating the homogeneity of physical space. Despite the success of physics, the reduction of causality to physical forces remains just that: a reduction. Applying this principle then again to any event in the world generates specific deficits, which are well visible in large parts of contemporary philosophy of science when it comes to the debate about the relation of natural science and causality (see cf. ).
Aristotle himself did not call the components of causality as “elements.” Yet, the technique he applied is just that: an elementarization. This technique was quite popular and well known from another discourse, involving earth, water, air, and fire. Finally, this model had to be abolished, but it is quite likely that the idea of the “element” has been inherited down to Mendeleev.
Characterizing the Concept of “Element”
As we have announced it before, we would like to strip any ontological flavor from the concept of the element. This marks the difference between conceiving them as part of the world or, alternatively, as a part of a tool-set used in the process of constructing a world. This means to take it purely instrumental, or in other words, as a language game. Such, it is also one out of the row of many examples for the necessity to remove any content from philosophy (Ontology is always claiming some kind of such content, which is highly problematical).
A major structural component of the language game “element” is that the entities denoted by it are used as anchors for a particular non-primitive compound quality, i.e. a quality that can’t be perceived by just the natural five (or six, or so) senses.
One the other hand, they are also strictly different from axioms. An axiom is a primitive proposition that serves as a starting point in a formal framework, such as mathematics. The intention behind the construction of axioms is to utilize common sense as a basis for more complicated reasoning. Axioms are considered as facts that could not seriously disputed as such. Thus, they indeed the main element in the attempt to secure mathematics as a unbroken chain of logic-based reasoning. Of course, the selection of a particular axiom for a particular purpose could always be discussed. But itself, it is a “primitive”, either a simple more or less empiric fact, or a simple mathematical definition.
The difference to elements is profound. One always can remove a single axiom from an axiomatic system without corroborating the sense of the latter. Take for instance the axiom of associativity in group theory, which leads to Lie-groups and Lie-algebras. Klein groups are just a special case of Lie Groups. Or, removing the “axiom” of parallel lines from the Euclidean axioms brings us to more general notions of geometry.
In contrast to that pattern, removing an element from an elemental system destroys the sense of the system. Elemental systems are primarily thought as a whole, as a non-decomposable thing, and any of the used elements is synthetically effective. Their actual meaning is only given by being a part of a composition with other elements. Axioms, in contrast, are parts of decomposable systems, where they act as constraints. Removing them leads usually to improved generality. The axioms that build an “axiomatic system” are not tied to each other, they are independent as such. Of course, their interaction always will create a particular conditionability, but that is a secondary effect.
The synthetic activity of elements simply mirrors the assumption that there is (i) a particular irreducible whole, and (ii) that the parts of that whole have a particular relationship to the embedding whole. In contrast to the prejudice that elemental analysis results in an unsuitable particularization of the subject matter, I think that elements are highly integrated, yet itself non-decomposable idealizations of compound structures. This is true for the quaternium of earth, wind, water and fire, but also for the valence electrons in chemistry or the elements of complexity, as we have introduced them here. Elements are made from concepts, while axioms are made from definitions.
In some way, elements can be conceived as the operationalization of beliefs. Take a belief, symbolize it and you get an element. From this perspective it again becomes obvious (on a second route) that elements could not be as something natural or even ontological; they can not be discovered as such in a pure or stable form. They can’t be used to proof propositions in a formal system, but they are indispensable to explain or establish the possibility of thinking a whole.
Mechanism and organism are just different terms that can be used to talk about the same issue, albeit in a less abstract manner. Yet, it is clear that integrated phenomena like “complexity,” or “culture,” or even “text” can’t be appropriately handled without the structural topos of the element, regardless which specific elements are actually chosen. In any of these cases it is a particular relation between the parts and the whole that is essential for the respective phenomenon as such.
If we accept the perspective that conceives of elements as stabilized beliefs we may recognize that they may be used as building blocks for the construction of a consistent world. Indeed, we well may say that it is due to their properties as described above, their positioning between belief and axiom, that we can use them as an initial scaffold (Gestell), which in turn provides the possibility for targeted observation, and thus for consistency, understood both as substance and as logical quality.
Finally, we should shed some words on the relation between elements and ideas. Elsewhere, we distinguished ideas from concepts. Ideas can’t be equated with elements either. Just the other way round, elements may contain ideas, but also concepts, relations and systems thereof, empirical hypotheses or formal definitions. Elements are, however, always immaterial, even in the case of chemistry. For us, elements are immaterial synthetic compounds used as interdependent building blocks of other immaterial things like concepts, rules, or hypotheses.
Many, if not all concepts, are built from elements in a similar way. The important issue is that elements are synthetic compounds which are used to establish further compounds in a particular manner. In the beginning there need not to be any kind of apriori justification for a particular choice or design. The only requirement is that the compound built from them allows for some kind of beneficial usage in creating higher integrated compounds which would not be achievable without them.
Elements may well be conceived as epistemological stepping stones, capsules of belief that we use to build up beliefs. Such, the status of elements is somewhere between models and concepts, not as formal and restricted as models and not as transcendental as concepts, yet still with much stronger ties towards empiric conditions than ideas.
It is quite obvious that such a status reflects a prominent role for perception as well as for understanding. The element may well be conceived as an active zone of differentiation, a zone from which different kind of branches emerge: ideas, models, concepts, words, beliefs. We also could say that elements are close to the effects and the emergence of immanence. The ΣΤΟΙΧΕΙΟΝ itself, its origins and transformations, may count as an epitome of this zone, where thinking creates its objects. It is “here” that expressibility finds its conditions.
At that point we should recall – and keep in mind – that elements should not be conceived as an ontological category. Elements unfold as (rather than “are”) a figure of thought, an idiom of thinking, as a figure for thought. Of course, we can deliberately visit this area, we may develop certain styles to navigate in this (sometimes) misty areas. In other words, we may develop a culture of elementarization. Sadly enough, positivism, which emerged from the materialism of the 19th century on the line from Auguste Comte down to Frege, Husserl, Schlick, Carnap and van Fraassen (among others), that positivism indeed destroyed much of that style. In my opinion, much of the inventiveness of the 19th century could be attributed a certain, yet largely unconscious, attitude towards the topos of the “element.”
No question, elevating the topos of the element into consciousness, as a deliberate means of thinking, is quite promising. Hence, it is also of some importance to our question of machine-based episteme. We may just add a further twist to this overarching topic by asking about the mechanisms and conditions that are needed for the possibility of “elementarization”. Still in other words we could say that elements are the main element of creativity. And we may add that the issue of expression and expressibility is not about words and texts, albeit texts and words potentiated the dynamics and the density of expressibility.
Before we can step on to harvest the power of elementarization we have to spend some efforts on the issue of the structure of expression. The first question is: What exactly happens if we invent and impose an element in and to our thoughts? The second salient question is about the process forming the element itself. Is the “element” just a phenomenological descriptional parlance, or is it possible to give some mechanisms for it?
Spaces and Dimensions
As it is already demonstrated by Anaximander’s ΣΤΟΙΧΕΙΟΝ, elements put marks into the void. The “element game” introduces discernability, and it is central to the topos of the element that it implies a whole, an irreducible set, of which it is a constitutive part. This way, elements don’t act just sign posts that would indicate a direction in an already existing landscape. It is more appropriate to conceive of them as a generators of landscape. Even before words, whether spoken or written, elements are the basic instance of externalization, abstract writing, so to speak.
It is the abstract topos of elements that introduce the complexities around territorialization and deterritorialization into thought, a dynamics that never can come to an end. Yet, let us focus here on the generative capacities of elements.
Elements transform existing spaces or create completely new ones, they represent the condition for the possibility of expressing anything. The implications are rather strong. Looking back from that conditioning to the topos itself we may recognize that wherever there is some kind of expression, there is also a germination zone of ideas, concepts and models, and above all, belief.
The space implied by elements is particular one yet, due to the fact that it inherits the aprioris of the wholeness and non-decomposability. Non-decomposability means that the elemental space looses essential qualities if one of the constituting elements would be removed.
This may be contrasted to the Cartesian space, the generalized Euclidean space, which is the prevailing concept of space today. A Cartesian space is spanned by dimensions that are set orthogonal to each other. This orthogonality of the dimensional setup allows to change the position in just one dimension, but to keep the position in all the other dimensions unchanged, constant. The dimensions are independent from each other. Additionally, the quality of the space itself does not change if we remove one of the dimensions of a n-dimensional Cartesian space (n>1). Thus, the Cartesian space is decomposable.
Spaces are inevitably implied as soon as entities are conceived as carriers of properties, in fact, even if at least one (“1”!) property will be assigned to them. These assigned properties, or short: assignates, then could be mapped to different dimensions. A particular entity thus becomes visible as a particular arrangement in the implied space. In case of Cartesian spaces, this arrangement consists of a sheaf of vectors, which is as specific for the mapped entity as it could be desired.
Dimensions may refer to sensory modalities, to philosophical qualias, or to constructed properties of development in time, that is, concepts like frequency, density, or any kind of pattern. Dimensions may be even purely abstract, as in case of random vectors or random graphs, which we discussed here, where the assignate refers to some arbitrary probability or structural, method specific parameter.
Many phenomena remain completely mysterious if we do not succeed to setup the (approximately) right number of dimensions or aspects. This has been famously demonstrated by Abbott and his flatland , or by Ian Stewart and his flatter land . Other examples are the so-called embedding dimension in the complex systems analysis, or the analysis of (mathematical) cusp catastrophes by Ian Stewart . Dimensionality also plays an important role in the philosophy of science, where Ronald Giere uses it to develop a “scientific perspectivism.” 
Suppose the example of a cloud of points in the 3‑dimensional space, which forms a spiral-like shape, with the main axis of the shape parallel to the z-axis. For points in the upper half of the cloudy spiral there shall be a high probability that they are blue; those in the lower half shall be mostly red. In other words, there is a clear pattern. If we now project the points to the x-y-plane, i.e. if we reduce dimensionality we loose the possibility to recognize the pattern. Yet, the conclusion that there “is” no pattern is utterly wrong. The selection of a particular number of dimensions is a rather critical operation. Hence, taking action without reflecting on the dimensionality of the space of expressibility quite likely leads to severe misinterpretations. The cover of Douglas Hofstadter’s first book “Gödel, Escher, Bach” featured a demonstration of the effect of projection from higher to lower dimensionality (see the image below), another presentation can be found here on YouTube, featuring Carl Sagan on the topic of dimensionality.
In mathematics, the relation between two spaces of different dimensionality, the so-called manifold, may itself form an abstract space. This exercise of checking out the consequences of removing or adding a dimension/aspect from the space of expressibility is a rewarding game even in everyday life. In the case of fractals in time series developments, Mandelbrot conceptualizes even a changing dimensionality of the space which is used to embed the observations over time.
Undeniably, this decomposability contributed much to the rise and the success of what we call modern science. Any of the spaces of mathematics or statistics is a Cartesian space. Riemann space, Hilbert space, Banach space, topological spaces etc. are all Cartesian insofar as the dimensions are arranged orthogonal to each other, thus introducing independence of elements before any other definition. Though, the real revolutionary contribution of Descartes has not been the setup of independent dimensions, it is the “Copernican” move to move the “origin” around, and with that, to mobilize the reference system of a particular measurement.
But again: By performing this mapping, the wholeness of the entity will be lost. Any interpretation of the entities requires a point outside of the Cartesian dimensional system. And precisely this externalized position is not possible for an entity that itself “performs cognitive processes.”2 It would be quite interesting to investigate the epistemic role of externalization of mental affairs through cultural techniques like words, symbols, or computers, yet that task would be huge.
Despite the success of the Cartesian space as a methodological approach it obviously also remains true that there is no free lunch in the realm of methods and mappings. In case of the Cartesian space this cost is as huge as its benefit, as both are linked to its decomposability. In Cartesian space it is not possible to speak about a whole, whole entities are simply nonexistent. This is indeed as dramatic as it sounds.Yet, it is a direct consequence of the independence of the dimensions. There is nothing in the structure of the Cartesian space that could be utilized as a kind of media to establish coherence. We already emphasized that the structure of the Cartesian space implies the necessity of an external observer. This, however, is not quite surprising for a construction devised by Descartes in the age of absolutistic monarchies symbiontically tied to catholicism, where the idea of the machine had been applied pervasively to anything and everything.
There are still further assumptions underlying the Cartesian conception of space. Probably the two most salient ones are concerning density and homogeneity. At first it might sound somewhat crazy to conceive of a space of inhomogeneous dimensionality. Such a space would have “holes” about which one could neither talk from within that space not would they be recognizable. Yet, from theoretical physics we know about the concept of wormholes, which precisely represent such inhomogeneity. Nevertheless, the “accessible” parts of such a space would remain Cartesian, so we could call the whole entity “weakly Cartesian”. A famous example is provided by Benoît Mandelbrot’s warping of dimensionality in the time domain of observations [8,9]
From an epistemological perspective, the Cartesian space is just a particular instance for the standardization or even institutionalization of the inevitable implication of spaces. Yet, the epistemic spaces are not just 3-dimensional as Kant assumed in his investigation, epistemic spaces may comprise a large and even variable number of dimensions. Nevertheless, Kant was right about the transcendental character of space, though the space we refer to here is not just the 3d- or (n)d-physical space.
Despite the success of Cartesian space, which builds on the elements of separability, decomposability and externalizable position of the interpreter, it is perfectly clear that it is nothing else than just a particular way of dealing with spaces. There are many empirical, cognitive or mental contexts for which the assumptions underlying the Cartesian space are severely violated. Such contexts usually involve the wholeness of the investigated entity as a necessary apriori. Think of complexity, language, the concept of life forms with its representatives like urban cultures, for any of these domains the status of any part of it can’t be qualified in any reasonable manner without referring always to the embedding wholeness.
The Aspectional Space
What we need is a more general concept of space, which does not start with any assumption about decomposability (or its refutation). Since it is always possible to proof and to drop the assumption of dependence (non-decomposability), but never for the assumption of independence (decomposability) we should start with a concept of space which keeps the wholeness intact.
Actually, it is not too difficult to start with a construction of such a space. The starting point is provided by a method to visualize data, the so-called ternary diagram. Particularly in metallurgy and geology ternary diagrams are abundantly in use for the purpose of expressing mixing proportions. The following figure 2a shows a general diagram for three components A,B,C, and Figure 2b shows a concrete diagram for a three component steel alloy at 900°C.
Figure 2a,b: Ternary diagrams in metallurgy and geology are pre-cursors of aspectional spaces.
Such ternary diagrams are used to express the relation between different phases where the influential components all influence each other. Note that the area of the triangle in such a ternary diagram comprises the whole universe as it is implied by the components. However, in principle it is still possible (though not overly elegant) to map the ternary diagram as it is used in geology into Cartesian space, because there is a strongly standardized way about how to map values. Any triple of values (a,b,c) is mapped to the axes A,B,C such that these axes are served counter-clockwise beginning with A. Without that rule a unique mapping of single points from the ternary space to the Cartesian space would not be possible any more. Thus we can see that the ternary diagram does not introduce a fundamental difference as compared to the Cartesian space defined by orthogonal axes.
Now let us drop this standard of the arrangement of axes. None of the axes should be primary against any other. Obviously, the resulting space is completely different from the spaces shown in Fig.2. We can keep only one of n dimensions constant while changing position in this space (by moving along an arc around one of the corners). Compare this to the Cartesian space, where it is possible to change just one and keep the other constant. For this reason we should call the boundaries of such a space not “axes” or “dimensions” and more. By convention, we may call the scaling entities “aspection“, derived from “aspect,” a concept that, similarly to the concept of element, indicates the non-decomposability of the embedding context.
As said, our space that we are going to construct for a mapping of elements can’t be transformed into a Cartesian space any more. It is an “aspectional space”, not a dimensional space. Of course, the aspectional space, together with the introduction of “aspections” as a companion concept for “dimension” is not just a Glass Bead Game. We urgently need it if we want to talk transparently and probably even quantitatively about the relation between parts and wholes in a way that keeps the dependency relations alive.
The requirement of keeping the dependency relations exerts an interesting consequence. It renders the corner points into singular points, or more precisely, into poles, as the underlying apriori assumption is just the irreducibility of the space. In contrast to the ternary diagram (which is thus still Cartesian) the aspectional space is neither defined at the corner points nor along the borders (“edges”). In other words, the aspectional space has no border, despite the fact that its volume appears to be limited. Since it would be somehow artificial to exclude the edges and corners by dedicated rules we prefer to achieve the same effect (of exclusion) by choosing a particular structure of the space itself. For that purpose, it is quite straightforward to provide the aspectional space with a hyperbolic structure.
The artist M.C. Escher produced a small variety of confined hyperbolic disks that perfectly represent the structure of our aspectional space. Note that there are no “aspects,” it is a zero-aspectional space. Remember that the 0-dimensional mathematical point represents a number in Cartesian space. This way we even invented a new class of numbers!3 A value in this class of number would (probably) represent the structure of the space, in other words the curvature of the hyperbola underlying the scaling of the space. Yet, the whole mathematics around this space and these numbers is undiscovered!
Figure 3: M.C. Eschers hyperbolic disk, capturing infinity on the table.
Above we said that this space appears to be limited. This impression of a limitation would hold only for external observers. Yet, our interest in aspectional spaces is precisely given by the apriori assumption of non-decomposability and the impossibility of such an external position for cognitive activities. Aspectional spaces are suitable just for those cases where such an external position is not available. From within such a hyperbolic space, the limitation would not be experiencable, a at least not by simple means: the propagation of waves would be different as compared to the Cartesian space.
So, what is the status of the aspectional space, especially as compared to the dimensional Cartesian space? A first step of such a characterization would investigate the possibility of transforming those spaces into each other. A second part would not address the space itself, but its capability to do some things uniquely.
So, let us start with the first issue, the possibility for a transition between the two types of species. Think of a three-aspectional space. The space is given by the triangularized relation, where the corners represent the intensity or relevance of a certain aspect. Moving around on this plane changes the distance to at least two (n-1) of the corners, but most moves change the distance to all three of the corners. Now, if we reduce the conceptual difference and/or the possible difference of intensity between all of the three corners we experience a sudden change of the quality of the aspectional space when we perform the limes transition into a state where all differential relevance has been expelled; the aspects would behave perfectly collinear.
Of course, we then would drop the possibility for dependence, claiming independence as a universal property, resulting in a jump into Cartesian space. Notably, there is no way back from the dimensional Cartesian space into aspectional spaces. Interestingly, there is a transformation of the aspectional space which produces a Cartesian space, while the opposite is not possible.
This formal exercise sheds an interesting light to the life form of the 17th century Descartes. Indeed, even in assuming the possibility of dependence one would grant parts of the world autonomy, something that has been categorically ruled out at those times. The idea of God as it was abundant then implied the mechanical character of the world.
Anyway, we can conclude that aspectional spaces are more general than Cartesian spaces as there is a transition only in one direction. Aspectional spaces are indeed formal spaces as Cartesian spaces are. It is possible to define negative numbers, and it is possible to provide them with different metrices or topologies.
Figure 4: From aspectional space to dimensional space in 5 steps. Descartes’ “origin” turns out to be nothing else than the abolishment or conflation of elements, which again could be interpreted as a strongly metaphysically influenced choice.
Now to the second aspect about the kinship between aspections and dimensions. One may wonder, whether the kind of dependency that could be mapped to aspectional spaces could not be modeled in dimensional spaces as well, for instance, by some functional rule acting on the relation between two dimensions. A simple example would be the regression, but also any analytic function y=f(x).
At first sight it seems that this could result in similar effects. We could, for instance, replace two independent dimensions by a new dimension, which has been synthesized in a rule-based manner, e.g. by applying a classic analytical closed-form function. The dependency would disappear and all dimensions again being orthogonal, i.e. independent to each other. Such an operation, however, would require that the dimensions are already abstract enough such that they can be combined by closed analytical functions. This then reveals that we put the claim of independence already into the considerations before anything else. Claiming the perfect equality of functional mapping of dependency into independence thus is a petitio principii. No wonder we find it possible to do so in a later phase of the analysis. It is thus obvious, that the epistemological state of a dependence secondary to the independence of dimensions is a completely different from the primary dependence.
A brief Example
A telling example4 for such an aspectional space is provided by the city theory of David Grahame Shane . The space created by Shane in order to fit in his interests in a non-reductionist coverage of the complexity of cities represents a powerful city theory, from which various models can be derived. The space is established through the three elements of armature, enclave and (Foucaultian) heterotopia. Armature is, of course a rather general concept–designed to cover more or less straight zones of transmission or the guidance for such–, which however expresses nicely the double role of “things” in a city. It points to things as part of the equipment of a city as well as their role as anchor (points). Armatures, in Shane’s terminology, are things like gates, arcades, malls, boulevards, railways, highway, skyscraper or particular forms of public media, that is, particular forms of passages. Heterotopias, on the other hand, are rather complicated “things,” at least it invokes the whole philosophical stance of the late Foucault, to whom Shane explicitly refers. For any of these elements, Shane then provides extensions and phenomenological instances, as values if you like, from which he builds a metric for each of the three basic aspects. Throughout his book he demonstrates the usefulness of his approach, which is based on these three elements. This usefulness becomes tangible because Shane’s city theory is an aspectional space of expressibility which allows to compare and to relate an extreme variety of phenomena regarding the city and the urban organization. Of course, we must expect other such spaces in principle; this would not only be interesting, but also a large amount of work to complete. Quite likely, however, it will be a just an extension of Shane’s concept.
Freeing the concept of “element” from its ontological burden turns it into a structural topos of thinking. The “element game” is a mandatory condition for the creation of spaces that we need in order to express anything. Hence, the “element game,” or briefly, the operation of “elementarization,” may be regarded as the prime instance of externalization and as such also as the hot spot of the germination of ideas, concepts and words, both abstract and factual. For our concerns here about machine-based episteme it is important that the notion of the element provides an additional (new?) possibility to ask about the mechanism in the formation of thinking.
Elementarization also represents the conditions for “developing” ideas and to “settle” them. Yet, our strictly non-ontological approach helps to avoid premature and final territorialization in thought. Quite to the contrary, if understood as a technique, elementarization helps to open new perspectives.
Elementarization appears as a technique to create spaces of expressibility, even before words and texts. It is thus worthwhile to consider words as representatives of a certain dynamics around processes of elementarization, both as an active as well as a passive structure.
We have been arguing that the notion of space does not automatically determine the space to be a Cartesian space. Elements to not create Cartesian spaces. Their particular reference to the apriori acceptance of an embedding wholeness renders both the elements as well as the space implied by them incompatible with Cartesian space. We introduced the notion of “aspects” in order to reflect to the particular quality of elements. Aspects are the result of a more or less volitional selection and construction.
Aspectional spaces are spaces of mutual dependency between aspects, while Cartesian spaces claim that dimensions are independent from each other. Concerning the handling and usage of spaces, parameters have to be sharply distinguished both from aspects as well as from dimensions. In Mathematics or in natural sciences, parameters are distinguished from variables. Variables are to be understood as containers for all allowed instances of values of a certain dimension. Parameters are modifying just the operation of placing such a value into the coordinate system. In other words, they do not change the general structure of the space used for or established by performing a mapping, and they even do not change the dimensionality of the space itself. For designers as well as scientists, and more general for any person acting with or upon things in the world, it is thus more than naive to play around with parameters without explicating or challenging the underlying space of expressibility, whether this is a Cartesian or an aspectional space. From that it also follows that the estimation of parameters can’t be regarded as an instance of learning.
Here we didn’t mention the mechanisms that could lead to the formation of elements.Yet, it is quite important to understand that we didn’t just shift the problematics of creativity to another descriptional layer, without getting a better grip to it. The topos of the element allows us to develop and to apply a completely different perspective to the “creative act.”
The mechanisms that could be put into charge for generating elements will be the issue of the next chapter. There we will deal with relations and its precursors. We also will briefly return to the topos of comparison.
Part 5: Relations and Symmetries (forthcoming)
-  Wilhelm Schwabe. ‘Mischung’ und ‘Element’ im Griechischen bis Platon. Wort- u. begriffsgeschichtliche Untersuchungen, insbes. zur Bedeutungsentwicklung von ΣΤΟΙΧΕΙΟΝ. Bouvier, Bonn 1980.
-  Isaac Newton: Philosophiae naturalis principia mathematica. Bd. 1 Tomus Primus. London 1726, S. 14 (http://gdz.sub.uni-goettingen.de/no_cache/dms/load/img/?IDDOC=294021)
-  Wesley C. Salmon. Explanation and Causality. 2003.
-  Abbott. Flatland.
-  Ian Stewart Flatter Land.
-  Ian Stewart & nn, Catastrophe Theory
-  Ronald N. Giere, Scientific Perspectivism.
-  Benoit B. Mandelbrot, Fractals: Form, Chance and Dimension.Freeman, New York 1977.
-  Benoit B. Mandelbrot, Fractals and Scaling in Finance. Springer, New York 1997.
-  David Grahame Shane, Recombinant Urbanism, Wiley, New York 2005.
-  Klaus Wassermann (2011). Sema Città-Deriving Elements for an applicable City Theory. in: T. Zupančič-Strojan, M. Juvančič, S. Verovšek, A. Jutraž (eds.), Respecting fragile places, 29th Conference on Education in Computer Aided Architectural Design in Europe
eCAADe. available online.
February 17, 2012 § Leave a comment
The status of self-referential things is a very particular one.
They can be described only by referring to the concept of the “self.”
Of course, self-referential things are not without conditions, just as any other thing, too. It is, however, not possible to describe self-referential things completely just by means of those conditions, or dependencies. Logically, there is an explanatory gap regarding their inward-directed dependencies. The second peculiarity with self-referential things is that there are some families of configurations for which they become generative.
For strongly singular terms no possible justification exists. Nevertheless, they are there, we even use them, which means that the strong singularity does not imply isolation at all. The question then is about how we can/do achieve such an embedding, and which are the consequences of that.
Despite the fact that there is no entry point which could by apriori be taken as a justified or even salient one we still have to make a choice which one actually to take. We suppose that there is indeed such a choice. It is a particular one though. We do not assume that the first choice is actually directed to an already identified entity as this would mean that there already would have been a lot of other choices in advance. We would have to select methods and atoms to fix, i.e. select and choose the subject of a concrete choice, and so on.
The choice we propose to take is neither directed to an actual entity, nor is it itself a actual entity. We are talking about a virtual choice. Practically, we start with the assumption of choosability.
Actually, Zermelo performed the same move when trying to provide a sound basis for set theory  after the idealistic foundation developed by Frege and others had failed so dramatically, leading into the foundational crisis of formal sciences . Zermelo’s move was to introduce choosability as an axiom, called the axiom of choice.
For Zermelo’s set theory the starting point, or if you prefer, the anchor point, lies completely outside the realm of the concept that is headed for. The same holds for our conceptualization of formalization. This outside is the structure of pragmatic act of choice itself. This choice is a choice qua factum, it is not important that we choose from a set from identified entities.
The choice itself proposes by its mere performance that it is possible to think of relations and transformations; it is the unitary element of any further formalization. In Wittgenstein’s terms, it is part of the abstract life form. In accordance to Wittgenstein’s critique of Moore’s problems1, we can also say that it is not reasonable, or more precise: it is without any sense, to doubt on the act of choosing something, even if we did not think about anything particular. The mere executive aspect of any type of activity is sufficient for any a posteriori reasoning that a choice has been performed.
Notably, the axiom of choice implies the underlying assumption of intensive relatedness between yet undetermined entities. In doing so, this position represents a fundamental opposite to the attitude of Frege, Russell and any modernist in general, who always start with the assumption of the isolated particle. For these reasons we regard the axiom of choice as one of the most interesting items in mathematics!
The choice thus is a Deleuzean double-articulation , closely related to his concept of the transcendental status of difference; we also could say that the choice has a transcendental dualistic characteristics. On the one hand there is nothing to justify. It is mere movement, or more abstract, a pure mapping or transformation, just as a matter of fact. On the other hand, it provides us with the possibility of just being enabled to conceive mere movement as such a mapping transformation; it enables us to think the unit before any identification. Transformation comes first; Deleuze’s philosophy similarly puts the difference into the salient transcendental position. To put it still different, it is the choice, or the selection, that is inevitably linked to actualization. Actualization and choice/selection are co-extensive.
Just another Game
So, let us summarize briefly the achievements. First, we may hold that similarly to language, there is no justification for formalization. Second, as soon as we use language, we also use symbols. Symbols on the other hand take, as we have seen, a double-articulated position between language and form. We characterized formalization as a way to give a complicated thing a symbolic form that lives within a system of other forms. We can’t conceive of forms without symbols. Language hence always implies, to some degree, formalization. It is only a matter of intensity, or likewise, a matter of formalizing the formalization, to proceed from language to mathematics. Third, both language and formalization belong to particular class of terms, that we characterized as strongly singular terms. These terms may be well put together with an abstract version of Kant’s organon.
From those three points follows that concepts that are denoted by strongly singular terms, such as formalization, creativity, or “I”, have to be conceived, as we do with language, as particular types of games.
In short, all these games are being embedded in the life form of or as a particular (sub-)culture. As such, they are not themselves language games in the original sense as proposed by Wittgenstein.
These games are different from the language game, of course, mainly because the underlying mechanisms as well as embedding landscape of purposes is different. These differences become clearly visible if we try to map those games into the choreostemic space. There, they will appear as different choreostemic styles. Despite the differences, we guess that the main properties of the language game apply also to the formalization game. This concerns the setup, the performance of such games, their role, their evaluation etc.etc., despite the effective mechanisms might be slightly different; for instance, Brandom’s principle of the “making it explicit” that serves well in the case of language is almost for sure differently parameterized for the formalizatin or the creativity game. Of course, this guess has to be subject of more detailed investigations.
As there are different natural languages that all share the same basement of enabling or hosting the possibility of language games, we could infer—based on the shared membership to the family of strongly singular terms— that there are different forms of formalization. Any of course, everybody knows at least two of such different forms of formalization: music and mathematics. Yet, once found the glasses that allow us to see the multitude of games, we easily find others. Take for instance the notations in contemporary choreography, that have been developed throughout the 20ieth century. Or the various formalizations that human cultures impose onto themselves as traditions.
Taken together it is quite obvious that language games are not a singularity. There are other contexts like formalization, modeling or the “I-reflexivity” that exist for the same reason and are similarly structured, although their dynamics may be strikingly different. In order to characterize any possible such game we could abstract from the individual species by proceeding to the -ability. Cultures then could be described precisely as the languagability of their members.
Based on the concept of strongly singular terms we first proof that we have to conceive of formalization (and symbol based creativity) in a similar way as we do for language. Both are embedded into a life form (in the Wittgensteinian sense). Thus it makes sense to propose to transfer the structure of the “game” from the domain of natural language to other areas that are arranged around strongly singular terms, such as formalization or creativity in the symbolic domain. As a nice side effect this brought us to the proper generalization of the Wittgensteinian language games.
Yet, there is still more about creativity that we have to clarify before we can relate it to other “games” like formalization and to proof the “beauty” of this particular combination. For instance, we have to become clear about the differences of systemic creativity, which can be observed in quasi-material arrangements (m-creativity), e.g. as self-organization, and the creativity that is at home in the realm of the symbolic (s-creativity).
The next step is thus to investigate the issue of expressibility.
1. In an objection to Wittgenstein, Moore raised the skeptic question about the status of certain doubts: Can I doubt that this hand belongs to me? Wittgenstein denied the reasonability of such kind of questions.
-  Zermelo, Set theory
-  Hahn, Grundlagenkrise
-  Deleuze & Guattari, Milles Plateaus
February 16, 2012 § Leave a comment
Formalization is based on the the use of symbols.
In the last chapter we characterized formalization as a way to give a complicated thing a symbolic form that lives within a system of other forms.
Here, we will first discuss a special property of the concepts of formalization and creativity, one that they share for instance with language. We call this property strong singularity. Then, we will sketch some consequences of this state.
What does “Strongly Singular” mean?
Before I am going to discuss (briefly) the adjacent concept of “singular terms” I would like to shed a note on the newly introduced term of “strong singularity”.
The ordinary Case
Let us take ordinary language, even as this may be a difficult thing to theorize about. At least, everybody is able to use it. We can do a lot of things with language, the common thing about these things is, however, that we use it in social situations, mostly in order to elicit two “effects”: First, we trigger some interpretation or even inference in our social companion, secondly, we indicate that we did just that. As a result, a common understanding emerges, formally taken, a homeomorphism, which in turn then may serve as the basis for the assignment of so-called “propositional content”. Only then we can “talk about” something, that is, only then we are able to assign a reference to something that is external to the exchanged speech.
As said, this is the usual working of language. For instance, by saying “Right now I am hearing my neighbor exercising piano.” I can refer to common experience, or at least to a construction you would call an imagination (it is anyway always a construction). This way I refer to an external subject and its relations, a fact. We can build sentences about it, about which we even could say whether they correspond to reality or not. But, of course, this already would be a further interpretation. There is no direct access to the “external world”.
In this way we can gain (fallaciously) the impression that we can refer to external objects by means of language. Yet, this is a fallacy, based on an illegitimate shortcut, as we have seen. Nevertheless, for most parts of our language(s) it is possible to refer to external or externalized objects by exchanging the mutual inferential / interpretational assignments as described above. I can say “music” and it is pretty clear what I mean by that, even if the status of the mere utterance of a single word is somewhat deficient: it is not determined whether I intended to refer to music in general, e.g. as the totality of all pieces or the cultural phenomenon, or to a particular piece, to a possibility of its instantiation or the factual instance right now. Notwithstanding this divergent variety, it is possible to trigger interpretations and to start a talk between people about music, while we neither have to play or to listen to music at that moment.
The same holds for structural terms that regulate interpretation predominantly by their “structural” value. It is not that important for us here, whether the externalization is directed to objects or to the speech itself. There is an external, even a physical justification for the starting to engage in the language game about such entities.
Now, this externalization is not possible for some terms. The most obvious is “language”. We neither can talk about language without language, nor can we even think “language” or have the “feeling” of language without practicing it. We also can’t investigate language without using or practicing it. Any “measurement” about language inevitably uses language itself as the means to measure, and this includes any interpretation of speech in language as well. This self-referentiality further leads to interesting phenomena, such as “n-isms” like the dualism in quantum physics, where we also find a conflation of scales. If we would fail to take this self-referentiality into consideration we inevitably will create faults or pseudo-paradoxa.
The important issue about that is that there is no justification of language which could be expressed outside of language, hence there is no (foundational) justification for it at all. We find a quite unique setting, which corrodes any attempt for a “closed” , i.e. formal analysis of language.
The extension of the concept “language” is at the same time an instance of it.
It is absolutely not surprising that the attempt for a fully mechanic, i.e. apriori determined or algorithmic analysis of language must fail. Wittgenstein thus arrived at the conclusion that language is ultimately embedded as a practice in the life form  (we would prefer the term “performance” instead). He demanded, that justifications (of language games as rule-following) have to come to an end1; for him it was fallacious to think that a complete justification—or ultimate foundation—would be possible.
Just to emphasize it again: The particular uniqueness of terms like language is that they can not be justified outside of themselves. Analytically, they start with a structural singularity. Thus the term “strong singularity” that differs significantly from the concept of the so-called “singular term” as it is widely known. We will discuss it below.
The term “strong singularity” indicates the absence of any possibility for an external justification.
In §329 of the Philosophical Investigations, Wittgenstein notes:
When I think in language, there aren’t ”meanings” going through my mind in addition to the verbal expressions: the language is itself the vehicle of thought.
It is quite interesting to see that symbols do not own this particular property of strong singularity. Despite that they are a structural part of language they do not share this property. Hence we may conceive it as a remarkable instance of a Deleuzean double articulation  in midst thinking itself. There would be lot to say about it, but it also would not fit here.
Language now shares the property of strong singularity with formalization . We can neither have the idea nor the feeling of formalization without formalization, and we even can not perform formalization without prior higher-order formalization. There is no justification of formalization which could be expressed outside of formalization, hence there is no (foundational) justification for it at all. The parallel is obvious: Would it then be necessary, for instance, to conclude that formalization is embedded in the life form much in the same way as it is the case for language? That mere performance precedes logics? Precisely this could be concluded from the whole of Wittgenstein’s philosophical theory, as Colin Johnston suggested .
Performative activity precedes any possibility of applying logics in the social world; formulated the other way round, we can say that transcendental logics is getting instantiated into an applicable quasi-logics. Before this background, the idea of truth functions determining a “pure” or ideal truth value is rendered into an importunate misunderstanding. Yet, formalization and language are not only similar with regard to this self-referentiality, they are also strictly different. Nevertheless, so the hypothesis we try to strengthen here, formalization resembles language in that we can not have the slightest thought or even any mental operation without formalization. It is even the other way round, in that any mental operation invokes a formalizing step.
Formalization and language are not the only entities, which exhibit self-referentiality and which can not defined by any kind of outside stance. Theory, model and metaphor belong to the family, too, not to forget finally about thinking, hence creativity, at large. A peculiar representative of these terms is the “I”. Close relatives, though not as critical as the former ones, are concepts like causality or information. All these terms are not only self-referential, they are also cross-referential. Discussing any of them automatically involves the others. Many instances of deep confusion derive from the attempt to treat them separately, across many domains from neurosciences, sociology, computer sciences and mathematics up to philosophy. Since digital technologies are based seriously on formalization and have been developing yet further into a significant deep structure of our contemporary life form, any area where software technology is pervasively used is endangered by the same misunderstandings. One of these areas is architecture and city-planning, or more general, any discipline where language or the social in involved as a target of the investigation.
There is last point to note about self-referentiality. Self-referentiality may likely lead to a situation that we have described as “complexity”. From this perspective, self-referentiality is a basic condition for the potential of novelty. It is thus interesting to see that this potential is directly and natively implanted into some concepts.
Now we will briefly discuss the concept of “singular term” as it is usually referred to. Yet, there is not a full agreement about this issue of singular terms, in my opinion mainly due to methodological issues. Many proponents of analytical philosophy simply “forget that there are speaking”, in the sense mentioned above.
The analytical perspective
Anyway, according to the received view, names are singular terms. It is said that the reference of singular terms are singular things or objects, even if they are immaterial, like the unicorn. Yet, the complete distinctive list of singular terms would look like this:
- – proper names (“Barack Obama”);
- – labeling designation (“U.S. President”);
- – indexical expressions (“here”, “this dog”).
Such singular terms are distinguished from so-called general terms. Following Tugendhat , who refers in turn to Strawson , the significance of a general term F consists from the conditions to be fulfilled, such that F matches one or several objects. In other words, the significance of a singular term is given by a rule for identification, while the significance of a general term is given by a rule for classification. As a consequence, singular terms require knowledge about general terms.
Such statements are typical for analytical philosophy.
There are serious problems with it. However, even the labeling is misleading. It is definitely NOT the term that is singular. Singular is at most a particular contextual event, which we decided to address by a name. Labelings and indexical expressions are not necessarily “singular,” and quite frequently the same holds for names. Think about “John Smith” first as a name, then as a person… This mistake is quite frequent in analytic philosophy. We can trace it even to the philosophy of mathematics , when it comes to certain claims of set theory about infinity.
The relevance for the possibility of machine-based episteme
There can be little doubt, as we already have been expressing it elsewhere, that human cognition can’t be separated from language. Even the use of most primitive tools, let alone be the production and distribution of them, requires the capability for at least a precursor of language, some first steps into languagability.
We know by experience that, in our mother tongue, we can understand sentences that we never heard before. Hence, understanding of language (quite likely as any understanding) is bottom-up, not top-down, at least in the beginning of the respective processes. Thus we have to ask about the sub-sentential components of a sentence.
Such components are singular terms. Imagine some perfectly working structure that comprises the capability for arbitrary classification as well as the capability for non-empirical analogical thinking, that is based on a dynamic symmetries. The machine wold not only be able to perform the transition from extensions to intensions, it would even be able to abstract the intension into a system of meta-algebraic symmetry relations. Such a system, or better, the programmer of it then would be faced with the problem of naming and labeling. Somehow the intensions have to be made addressable. A private index does not help, since such an index would be without any value for communication purposes.
The question is how to make the machine referring to the proper names? We will see elsewhere (forthcoming: “Waves, Words, and Images“), that this question will lead us to the necessity of multi-modality in processing linguistic input, e.g. language and images together into the same structure (which is just another reason why to rely on self-organizing maps and our methodology of modeling).
Refutation of the analytical view
The analytical position about singular term does not provide any help or insight into the the particular differential quality of terms as words that denote a concept.2 Analytical statements as cited above are inconsistent, if not self-contradictory. The reason is simple. Words as placeholders for concepts can not have a particular meaning attached to them by principle. The meaning, even that of subsentential components, is an issue of interpretation, and the meaning of a sentence is given not only by its own totality, it is also dependent on the embedding of the sentence itself into the story or the social context, where it is performed.
Since “analytic” singular terms require knowledge about general terms, and the general terms are only determined if the sentence is understood, it is impossible to identify or classify single terms, whether singular or general, before the propositional content of the sentence is clear to the participants. That propositional content of the sentence, however, is, as Robert Brandom in chapter 6 of his  convincingly argues, only accessible through their role in the inferential relations between the participants of the talk as well as the relations between sentences. Such we can easily see that the analytical concept of singular terms is empty, if not self-nullifying.
The required understanding of the sentence is missing in the analytical perspective, the object is dominant against the sentence, which is against any real-life experience. Hence, we’d also say that the primacy of interpretation is not fully respected. What we’d need instead is a kind of bootstrapping procedure that works within a social situation of exchanged speech.
Robert Brandom moves this bootstrapping into the social situation itself, which starts with a medial symmetry between language and socialness. There is, coarsely spoken, a rather fixed choreography to accomplish that. First, the participants have to be able to maintain what Brandom calls a de-ontic account. The sequence start with a claim, which includes the assignment of a particular role. This role must be accepted and returned, which is established by signalling that the inference / interpretation will be done. Both the role and the acceptance are dependent on the claim, on the de-ontic status of the participants and on the intended meaning. (now I have summarized about 500 pages of Brandom’s book…, but, as said, it is a very coarse summary!)
Brandom (chp.6) investigates the issue of singular terms. For him, the analytical perspective is not acceptable, since for him, as it the case for us, there is the primacy of interpretation.
Brandom refutes the claim of analytical philosophy that singular names designate single objects. Instead he strives to determine the necessity and the characteristics of singular terms by a scheme that distinguishes particular structural (“syntactical”) and semantic conditions. These conditions are further divergent between the two classes of possible subsentential structures, the singular terms (ST) and predicates (P). While syntactically, ST take the role of substitution-of/substitution-by and P take the structural role of providing a frame for such substitutions, in the semantic perspective ST are characterised exclusively by so called symmetric substitution-inferential commitments (SIC), where P also take asymmetric SIC. Those inferential commitments link the de-ontic, i.e. ultimately socialness of linguistic exchange, to the linguistic domain of the social exchange. We hence may also characterize the whole situation as it is described by Brandom as a cross-medial setting, where the socialness and linguistic domain provide each other mutually a medial embedding.
Interestingly, this simultaneous cross-mediality represents also a “region”, or a context, where materiality (of the participants) and immateriality (of information qua interpretation) overlaps. We find, so to speak, an event-like situation just before the symmetry-break that we ay identify as meaning. To some respect, Brandom’s scheme provides us the pragmatic details of of a Peircean sign situation.
The Peirce-Brandom Test
This has been a very coarse sketch of one aspect of Brandom’s approach. Yet, we have seen that language understanding can not be understood if we neglect the described cross-mediality. We therefore propose to replace the so-called Turing-test by a procedure that we propose to call the Peirce-Brandom Test. That test would proof the capability to take part in semiosis, and the choreography of the interaction scheme would guarantee that references and inferences are indeed performed. In contrast to the Turing-test, the Peirce-Brandom test can’t be “faked”, e.g. by a “Chinese Room.” (Searle ) Else, to find out whether the interaction partner is a “machine” or a human we should not ask them anything, since the question as a grammatical form of social interaction corroborates the complexity of the situation. We just should talk to it/her/him.The Searlean homunculus inside the Chinese room would not be able to look up anything anymore. He would have to be able to think in Chinese and as Chinese, q.e.d.
Strongly Singular Terms and the Issue of Virtuality
The result of Brandom’s analysis is that the label of singular terms is somewhat dispensable. These terms may be taken as if they point to a singular object, but there is no necessity for that, since their meaning is not attached to the reference to the object, but to their role in in performing the discourse.
Strongly singular terms are strikingly different from those (“weakly) singular terms. Since they are founding themselves while being practiced through their self-referential structure, it is not possible to find any “incoming” dependencies. They are seemingly isolated on their passive side, there are only outgoing dependencies towards other terms, i.e. other terms are dependent on them. Hence we could call them also “(purely) active terms”.
What we can experience here in a quite immediate manner is pure potentiality, or virtuality (in the Deleuzean sense). Language imports potentiality into material arrangements, which is something that programming languages or any other finite state automaton can’t accomplish. That’s the reason why we all the time heftily deny the reasonability to talk about states when it comes to the brain or the mind.
Now, at this point it is perfectly clear why language can be conceived as ongoing creativity. Without ongoing creativity, the continuous actualization of the virtual, there wouldn’t be anything that would take place, there would not “be” language. For this reason, the term creativity belongs to the small group of strongly singular terms.
In this series of essays about the relation between formalization and creativity we have achieved an important methodological milestone. We have found a consistent structural basis for the terms language, formalization and creativity. The common denominator for all of those is self-referentiality. On the one hand this becomes manifest in the phenomenon of strong singularity, on the other hand this implies an immanent virtuality for certain terms. These terms (language, formalization, model, theory) may well be taken as the “hot spots” not only of the creative power of language, but also of thinking at large.
The aspect of immanent virtuality implicates a highly significant methodological move concerning the starting point for any reasoning about strongly singular terms. Yet, this we will check out in the next chapter.
1. Wittgenstein repeatedly has been expressing this from different perspectives. In the Philosophical Investigations , PI §219, he states: “When I obey the rule, I do not choose. I obey the rule blindly.” In other words, there is usually no reason to give, although one always can think of some reasons. Yet, it is also true that (PI §10) “Rules cannot be made for every possible contingency, but then that isn’t their point anyway.” This leads us to §217: “If I have exhausted the justifications I have reached bedrock, and my spade is turned. Then I am inclined to say: ‘This is simply what I do’.” Rules are are never intended to remove all possible doubt, thus PI §485: “Justification by experience comes to an end. If it did not it would not be justification.” Later Quine proofed accordingly from a different perspective what today is known as the indeterminacy of empirical reason (“Word and Object”).
2. There are, of course, other interesting positions, e.g. that elaborated by Wilfrid Sellars , who distinguished different kinds of singular terms: abstract singular terms (“triangularity”), and distributive singular terms (“the red”), in addition to standard singular terms. Yet, the problem of which the analytical position is suffering also hits the position of Sellars.
-  Ludwig Wittgenstein, Philosophical Investigations.
-  Gilles Deleuze, Felix Guattari, Milles Plateaus.
-  Colin Johnston (2009). Tractarian objects and logical categories. Synthese 167: 145-161.
-  Ernst Tugendhat, Traditional and Analytical Philosophy. 1976
-  Strawson 1974
-  Rodych, Victor, “Wittgenstein’s Philosophy of Mathematics”, The Stanford Encyclopedia of Philosophy (Summer 2011 Edition), Edward N. Zalta (ed.), http://plato.stanford.edu.
-  Robert Brandom, Making it Explicit. 1994
-  John Searle (1980). Minds, Brains and Programs. Behav Brain Sci 3 (3), 417–424.
-  Wilfrid Sellars, Science and Metaphysics. Variations on Kantian Themes, Ridgview Publishing Company, Atascadero, California  1992.
February 15, 2012 § Leave a comment
If there is such a category as the antipodic at all,
it certainly applies to the pair of the formal and the creative, at least as long as we consult the field of propositions1 that is labeled as the “Western Culture.” As a consequence, in many cultures, and even among mathematicians, these qualities tend to be conceived as completely separated.
We think that this rating is based on a serious myopia, one that is quite common throughout rationalism, especially if that comes as a flavor of idealism. In a small series of essays—it is too much material for a single one—we will investigate the relation between these qualities, or concepts, of the formal and the creative. Today, we just will briefly introduce some markers.
The Basic Context
The relevance of this endeavor is pretty obvious. On the one hand we have the part of creativity. If machine-based episteme implies the necessity to create new models, new hypothesis and new theories we should not only get clear about the necessary mechanisms and the sufficient conditions for its “appearance.” In other chapters we already mentioned complexity and evolutionary processes as the primary, if not the only candidates for such mechanisms. These domains are related to the transition from the material to the immaterial, and surely, as such they are indispensable for any complete theory about creativity. Yet, we also have to take into consideration the space of the symbolic, i.e. of the immaterial, of information and knowledge, which we can’t find in the domains of complexity and evolution, at least not without distorting them too much. There is a significant aspect of creativity that is situated completely in the realm of the symbolic (to which we propose to include diagrams as well). In other words, there is an aspect of creativity that is related to language, to story-telling, understood as weaving (combining) a web of symbols and concepts, that often is associative in its own respect, whether in literature, mathematics, reading and writing, or regarding the DNA.
On the other hand, we have the quality of the formal, or when labelled as a domain of activity, formalization. The domain of the formal is fully within the realm of the symbolic. And of course, the formal is frequently conceived as the cornerstone, if not essence, of mathematics. Before the beginning of the 20ieth century, or around its onset, the formal was almost a synonym to mathematics. At that time, the general movement to more and more abstract structures in mathematics, i.e. things like group theory, or number theory, lead to the enterprise to search for the foundations of mathematics, often epitomized as the Hilbert Program. As a consequence, kind of a “war” broke out, bearing two parties, the intuitionists and the formalists, and the famous foundational crisis started, which is lasting till today. Gödel then proofed that even in mathematics we can not know perfectly. Nevertheless, for most people mathematics is seen as the domain where reason and rationalism is most developed. Yet, despite mathematicians are indeed ingenious (as many other people), mathematics itself is conceived as safe, that is static and non-creative. Mathematics is about symbols under analytic closure. Ideally, there are no “white territories” in mathematics, at least for the members of the formalist party.
The mostly digital machines finally pose a particular problem. The question is whether a deterministic machine, i.e. a machine for which a complete analytic description can exist, is able to develop creativity.
This question has been devised many times in the history of philosophy and thinking, albeit in different forms. Leibniz imagined a mathesis universalis and characteristica universalis as well. In the 20ieth century, Carnap tried to proof the possibility of a formal language that could serve as the ideal language for science . Both failed, Carnap much more disastrously than Leibniz. Leibniz also thought about the transition from the realm of the mechanic to the realm of human thought, by means of his ars combinatoria, which he had imagined to create any possible thought. We definitely will return to Leibniz and his ideas later.
A (summarizing) Glimpse
How will we proceed, and what will we find?
First we will introduce and discuss some the methodological pillars for our reasoning about the (almost “dialectic”) relation between creativity and formalization; among those the most important ones are the following:
- – the status of “elements” for theorizing;
- – the concept of dimensions and space;
- – relations
- – the domain of comparison;
- – symmetries as a tool;
- – virtuality.
Secondly, we will ask about the structure of the terms “formal” and “creative” while they are in use, especially however, we are interested in their foundational status. We will find, that both, formalization and creativity belong to a very particular class of language games. Notably, these terms turn out to be singular terms, that are at the same time not names. They are singular because their foundation as well as the possibility to experience them are self-referential. (ceterum censeo: a result that is not possible if we’d stick to the ontological style by asking “What is creativity…”)
The experience of the concepts associated to them can’t be externalized. We can not talk about language without language, nor can we think “language” without practicing it. Thus, they also can’t be justified by external references, which is a quite remarkable property.
In the end we hopefully will have made clear that creativity in the symbolic space is not achievable without formalization. They are even co-generative.
Let us start with creativity. Creativity has always been considered as something fateful. Until the beginnings of psychology as a science by William James, smart people have been smart by the working of fate, or some gods. Famous, and for centuries unchallenged, the passage in Plato’s Theaitetos , where Sokrates explains his role in maieutics by mentioning that the creation of novel things is task of the gods. The genius as well as concept of intuition could be regarded just a rather close relatives of that. Only since the 1950ies, probably not by pure chance, people started to recognize creativity as a subject in its own right . Yet, somehow it is not really satisfying to explain creativity by calling it “divergent” or “lateral” thinking . Nothing is going to be explained by replacing one term by another. Nowadays, and mostly in the domain of design research, conditions for creativity are often understood in terms of collaborations. People even resort to infamous swarm intelligence, which is near a declaration of bankruptcy.
Any of these approaches are just replacing some terms with some other terms, trying to conjure some improvement in understanding. Most of the “explanations” indeed look rather like rain dancing than a valuable analysis. Recently a large philosophical congress in Berlin, with more than 1200 inscribed participants, and two books comprising around 2000 pages focused on the subject largely in the same vein and without much results . We are definitely neither interested in any kind of metaphysical base-jumping by referring directly or indirectly to intuition, and the accompanying angels in the background, nor in phenomenological, sociological or superficial psychological approaches, tying to get support by some funny anecdotes.
The question really is, what are we talking about, and how, when referring to the concept of creativity. Only because this question is neither posed nor answered, we are finding so much esoterics around this topic. Creativity surely exceeds problem solving, although sometimes it occurs righteous when solving a problem. It may be observed in calm story telling, in cataclysmic performances of artists, or in language.
Actually, our impression is that creativity is nothing that sometimes “is there”, and sometimes not. In language it is present all the time, much like it is the case for analogical thinking. The question is which of those phenomena we call “creative,” coarsely spoken, which degree of intensity regarding novelty and usefulness of that novelty we allow to get assigned a particular saliency. Somehow, constraints seem to play an important role, as well as the capability to release it, or apply it, at will. Then, however, creativity must be a technique, or at least based on tools which we could learn to use. It is, however, pretty clear that we have to distinguish between the assignment of the saliency (“this or that person has been creative”) and the phenomenon and its underlying mechanisms. The assignment of the term is introducing a discreteness that is not present on the level of the mechanism, hence we never will understand about what we are talking about if we take just the parlance as the source and the measure.
The phenomenon of language provides a nice bridge to the realm of the formal. Today, probably mainly due to the influence of computer sciences, natural languages are distinguished from artificial languages, which often are also called formal languages. It is widely accepted, that formalization either is based on formal languages or that the former creates an instance of the latter. The concept of formal language is important in mathematics, computer science and science at large. Instantiated as programming languages, formal languages are of an enormous relevance for human society; one could even say that these languages themselves establish some kind of a media.
Yet, the labeling of the discerned phenomena as “natural” and “formal” always strikes me. It is remarkable that human languages are so often also called “natural” languages. Somehow, human language appears so outstanding to humans that they call their language in a funny paradoxical move a “natural” thing, as if this language-thing would have been originated outside human culture. Today, as we know about many instances of cultural phenomena in animals, the strong dichotomy between culture and nature blurred considerably. A particular driver of this is provided by the spreading insight that we as humans are also animals: our bodies contain a brain. Thus, we and our culture also build upon this amazing morphological structure, continuously so. We as humans are just the embodiment of the dichotomy between nature and culture, and nothing could express the confusion about this issue more than the notion of “natural language.” A bit shamefaced we call the expressions of whales and dolphins “singing”, despite we know that they communicate rather complicated matters. We are just unable to understand anything of it. The main reason for that presumable being that we do not share anything regarding their Lebensform, and other references than the Lebensform are not relevant for languages.
Language, whether natural or formal, is supposed to be used to express things. Already here we now have been arriving in deep troubles as the previous sentence is anything than innocent. First, speaking about things is not a trivial affair. A thing is a difficult thing. Taking etymology into consideration, we see that things are the results of negotiations. As a “result,” in turn, “things” are reductions, albeit in the realm of the abstract. The next difficulty is invoked by the idea that we can “express” things in a formal language. There has been a large debate on the expressive capabilities of formal languages, mainly induced by Carnap , and carried further by Quine , Sneed , Stegmüller , Spohn , and Moulines , among others, up to today.
In our opinion, the claim of the expressibility of formal language, and hence the proposed usage of formal languages as a way to express scientific models and theories, is based on probably more than just one deep and drastic misunderstanding. We will elucidate this throughout this series; other arguments has been devised for instance by Putnam in his small treatise about the “meaning of meaning” , where he famously argued that “analyticity is an inexplicable noise” without any possibility for a meaningful usage. That’s also a first hint that analyticity is not about the the same thing(s) as formalization.
Robert Brandom puts the act of expressing within social contexts into the center of his philosophy , constructing a well-differentiated perspective upon the relation between principles in the usage of language and its structure. Following Brandom, we could say that formal language can not be expressive almost by its own definition: the mutual social act of requesting an interpretation is missing there, as well as any propositional content. If there is no propositional content, nothing could be expressed. Yet, propositional content comes into existence only by a series of events where the interactees in a social situation ascribe it mutually to each other and are also willing to accept that assignment.
Formal languages consist of exactly defined labeled sets, where each set and its label represents a rewriting rule. In other words, formal languages are finite state machines; they are always expressible as a compiler for a programming language. Programming languages organize the arrangement of rewriting rules, they are however not an entity capable for semantics. We could easily conclude that formal languages are not languages at all.
A last remark about formalization as a technique. Formalization is undeniably based on the use, or better, the assignment of symbols to particular, deliberately chosen contexts, actions, recipes or processes. Think of proofs of certain results in mathematics where the symbolized idea later refers to the idea and its proof. Such, they may act as kind of abbreviations, or they will denote abstractions. They also may support the visibility of the core of otherwise lengthy reasonings. Sometimes, as for instance in mathematics, formalization requires several components, e.g. the item or subject, the accompanying operators or transformations (take that as “usage”), and the reference to some axiomatics or a explicit description of the conditions and the affected items. The same style is applied in physics. Yet, this complete structure is not necessary for an action to count as a formalization. We propose to conceive of formalization as the selection of elements (will be introduced soon) that consecutively are symbolized. Actually, it is not necessary to write down a “formula” about something in order to formalize that something. It is also not necessary, so we are convinced, to apply a particular logic when establishing the formalization through abstraction. It is just the symbolic compression that allows to achieve further results which would remain inaccessible otherwise. Or briefly put, to give a complicated thing a symbolic form that lives within a system of other forms.
Finally, there is just one thing we always should keep in mind. Using, introducing or referring to a formalization irrevocably implies an instantiation when we are going to apply it, to bring it back to more “concrete” contexts. Thus, formalization is deeply linked to the Deleuzean figure of thought of the “Differential.” 
-  Rudolf Carnap, Logische Syntax der Sprache, Wien 1934 [2. Aufl. 1968].
-  Platon, Theaitetos.
-  Guilford, Creativity , 1950
-  DeBono about lateral thinking
-  Günther Abel (ed.), Kreativität. Kolloquiumsband XX. Kongress der Deutschen Philosophie. Meiner Verlag, Hamburg 2007.
-  Quine, Two Dogmas of Empiricism.
-  Sneed
-  Stegmüller
-  Spohn
-  Moulines
-  Hilary Putnam, The Meaning of Meaning
-  Robert Brandom, Making it Explicit.
-  Gilles Deleuze, Difference and Repetition.