Analogical Thinking, revisited. (II)

March 20, 2012 § Leave a comment

In this second part of the essay about a fresh perspective on

(II/II)

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.

Crossing over

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

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.

We already described the generalized architecture of modeling as well as the elements of the generalized model in previous chapters.

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.

SOM-based Abstraction

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 [3]. 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.)

The Transfer

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.

Copycat extended SOM
 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.”

Conclusions

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 [4] 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 [5] 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 [6] 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 [7]. 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).

Notes

1. see also the other chapters about the SOM, SOM-based modeling, and generalized modeling.

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.[8]  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.

  • [1] Timo Honkela, Samuel Kaski, Krista Lagus, Teuvo Kohonen (1997). WEBSOM – Self-Organizing Maps of Document Collections. Neurocomputing, 21: 101-117.4
  • [2] 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
  • [3] 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.
  • [5] Mark A. Runco, Creativity-Research, Development, and Practice Elsevier 2007.
  • [6] Keith J. Holyoak and Paul Thagard, Mental Leaps: Analogy in Creative Thought.
    MIT Press, Cambridge 1995.
  • [7] 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.
  • [8] Wilhelm Vossenkuhl. Solipsismus und Sprachkritik. Beiträge zu Wittgenstein. Parerga, Berlin 2009.

.

Tagged: , , , , , , , , , , , , , ,

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

What’s this?

You are currently reading Analogical Thinking, revisited. (II) at The "Putnam Program".

meta

%d bloggers like this: