The Miracle of Comparing

November 11, 2011 § Leave a comment

Miracles denote the incomparable.

Since comparing is so abundant in our thinking, we even can’t think of any activity devoid of comparing, miracles also signify zones that are truly non-cognitive. Curiously, not believing in the existence of such non-cognitive zones is called agnostic.

Actually, divine revelations seem to be the only cognitive acts that are not based on comparisons. In such kinds of events, we directly find some entity in the world or in our mind, without cause, notably. We simply and suddenly can look at it, humble so. In some way, the only outside of the comparison is the miracle, which is not really an outside, since it is outside of everything. Thinking and comparing do not have a proper neighbor, in order to use a Wittgensteinian concept. Thus, we could conclude that it is not really possible to talk about it. Without any reasonable comparable, thinking itself is outside of anything else. We just can look at it, silently. Obviously, there is a certain resemblance to the event of a miracle. Maybe, that’s the reason for the fact that there are so much misunderstandings about thinking.

Of course, we may build models about thinking. Yet, this does not change very much, even if we apply modern means in our research. On the other hand, this also does not reduce the interestingness of the topic. Astonishingly, even renowned researchers in cognitive sciences as Robert Goldstone (Indiana) feel inclined to justify their engagement with the issue of comparison [1].

It might not be immediately clear why the topic of comparison warrants a whole chapter in a book on human thinking. […] In fact, comparison is one of the most integral components of human thought. Along with the related construct of similarity, comparison plays a crucial role in almost everything that we do.

We fully agree to these introductory words, but Goldstone, as so many researcher in cognitive sciences, proceeds with remarkable trivia.

Furthermore, comparison itself is a powerful cognitive tool – in addition to its supporting role in other mental processes, research has demonstrated that the simple act of comparing two things can produce important changes in our knowledge.

In contrast to Goldstone we also will separate the concept of comparison completely from that of similarity. In his article, Goldstone discusses comparison only through the concept of similarity.

Thinking is closely related to consciousness, hence to self-consciousness, as the German philosopher Manfred Frank argued for [2]. Probably for that reason humans avoid to call mental processes in animals “thinking.” Anyway, for us humans thinking is so natural that we usually do not bother with the operation of comparing, I mean with the structure of that operation. Of course, there is rhetoric, which teaches us the different ways of comparing things with one another. Yet, this engagement is outside of the operation, it just concerns its application. The same is true for mathematics, where always a particular way of comparing is presupposed.  In contrast, we are interested in the structure of the operation of comparing.

Well, our thesis here does not, of course, follow the route of the miraculous, at least not without a better specification of the situation. Miracles are a rather unsuitable thing to rely on as a (software) programmer.

What we will try here is to clarify the anatomy of comparisons. Indeed, we distinguish at least three basic types.

The Anatomy of Comparison: Basic Ingredients

Comparisons are operations. They imply some kind of matter, providing a certain storage or memory capacity. Without such storage/memory-matter, no comparison is possible. It is thus not perfectly correct to speak about the anatomy of comparisons, since any type of comparison is a process in time.

Let us first consider the basic case of a pairwise comparison. Without loss of generality, we take an apple (A) and an orange (B) that we are going to compare. What are, in a first account, the basic elements of such a comparison?

As the proverb intends to convey, what can’t be compared that can’t be compared. In order to compare two entities A and B, we have to assign properties that can be applied to both of the entities. Take “COLOR” as an example here. So, first we assign properties…

Fig.1a

In a second step we select just those of the properties that shall (!) be applied to both. There is no necessity that a particular feature is common to both items. Actually, the determination of those sets is a free parameter in modeling. Yet, here we are just interested in the actuality of two such aligned sets (“vectors”).

Fig.1b

Those selected properties represent subsets with respect to their parent sets. Given those two subsets a(j.) and b(m.) we now can align the vectors of properties. In data analysis, such vectors are often called feature vectors.

Fig.1c

Any comparison then refers to those aligned feature vectors. The more features in such a vector the more comparisons are possible.

In diagnostic/predictive modeling a particular class of operations is applied to such feature vectors, mainly in order to determine the similarity, or its inverse, the so-called distance. We will see in the chapter about similarity what “distance” actually means and why it is deeply problematic to apply this concept as an operationalization of similarity in comparisons.

Before we start with the typology of comparison we should note that the “features” can be very abstract, depending on the actual items and their abstractness to be compared. Features could be taken from physical measurement, from information theoretic considerations, or from any kind of transformation of “initial” features. Regardless the actual setting, there will always be some criteria, even if we use an abstract similarity measure based on information theory, as e.g. proposed by Lin [3].

A second note concerns the concept of the feature vector and its generality. The feature vector does not imply by itself a particular similarity measure, e.g. a distance measure, or a cosine similarity. In other words, it does not imply a particular space of comparison. Only the interaction of the similarity functional with the feature vector creates such a space. Related to that we also have to emphasize again that the sets aj and bm need not be completely identical. There might be features that are considered as being indispensable for the description of the item at hand. Such reasons are, however, external to a particular comparison, opening just a further level. In a larger context we have to expect feature vectors that are not completely matching, since similarity includes the notion of commonality. To measure this requires differences, hence the difference in the feature sets. In turn this requires a measure for the (dimensional) difference of potential solution spaces.

We will discuss all these issues in more detail in the chapter about similarity. There we will argue that the distinctions of different kinds of similarity like the one proposed by cognitive psychologist Goldstone in [1] into geometric, feature-based, alignment-based, and transformational similarity are not appropriate.

Yet, for now and first and here, we will focus on the internal structure of the comparison as an event, or an operation.

Three Types of Comparison

Type I: Comparison within a closed Framework

Type I is by far the most simple of all three types. It can be represented by finite algorithms. Hence, type I is the type of comparison that is used (almost exclusively) in computer sciences, e.g. in advanced optimization techniques or in data mining.

In this type the observables are data (“givens”), the entities as well as their features. The salient example is provided by the database. Hence, the space of potential solutions is well-defined, albeit its indeterminacy and its vast size could render it into a pseudo-open space.

We start with basic propertization, exactly as in the general case shown above in Fig.1:

Fig.2a

From these, feature vectors are set up into a table-like structure for column-wise comparison. This comparison is organized as a “similarity function,” which in turn is an operationalization of the concept of similarity. Note that this structure separates the concepts of “comparison” and “similarity,” the latter being set as a part of the former. It is quite important not to conflate comparison and (the operationalization of) similarity

Fig.2b

Based on the result of applying the similarity function to the table of feature vectors, a particular proposal can be made about the relations between A and B.

Fig.2c

The diagram clearly shows that the proposal can’t be conceived as something “objective,” which could be extracted from an “outside” reality. Such a perspective is widely believed to be appropriate in the field of so-called “data mining.” Obviously, proposals are heavily dependent on feature vectors and the similarity functional. Even in predictive model, where the predictive accuracy can be taken as a corrective or defense against proliferate relativism, the proposal is still dependent on the selected features. As many different selections allow for an almost equal predictive accuracy, there isn’t objectivity either.

From the simple case we can take the important lesson that there is no such thing as a “direct” comparison, hence also no “built-in” objectivity, even not on those low levels of cognition.

Fig.2d

We also can see that the structure of comparison comprises three levels of abstraction. This structure further applies to simple translations and topics like transdisciplinary discourses, i.e. the task of relating domain specific vocabularies, where any of them are supposed to consist from well-defined singular terms.

The features that are ultimately used as input into the similarity function are often called the “model parameters”, or also “dimensions.” In philosophical terms we suggest them to relate closely to Spinoza’s Common Terms.

.

Type II: “Inverted” goal-directed Comparison

The second type is quite different from the type I. Here we do not start with data completely defined also on the level of the properties. Instead, the starting point for processes of type II is determined by the proposal and rather coarse entities, or vaguely described contexts. Hence, it is kind of an inverted comparison.

The diagram for the first step doesn’t look quite spectacular, yet, in some sense it is also quite dramatic in its emptiness.

Fig.3a

The second step is based on habits or experience; abstract properties and some similarity function is selected before the comparison.

Fig.3b

These are then applied to the large or vaguely given entities. By means of this top-down projection a relation between A and B appears. Only subsequent to establishing that relation we can start with a forward comparison!

Fig.3c

The final step then associates the initially vague observations, now constructively related, to the initial proposal.

Fig.3d

As we already said, in this process the properties are taken from experience before the are projected onto amorphous observations. In Spinoza’s terms, those properties are “abstract terms”. Essentially, the projection can be also conceived as a construction. In the structure shown above, fiction and optimization are co-dependent and co-extensive. We also could simply call it “invention,” either of a solution or of a problematics, just as you prefer to call it.

The same process of “optimized fiction”, or “fictional optimization,” is often mistaken as “induction from empiric data.” Using the schemes of figure 3 we easily can understand why this claim is a misunderstanding. Actually, such an induction is of course not possible, since their is no necessity in any of the steps.

About the role of experience: The selection of “abstract terms” viz. “suggested properties” is itself based on models, of course. Yet, these models are far outside of the context induced by the observables A, B.

We should note that type-I and type-II comparisons are usually used in a constant interplay, resulting in an open, complex dynamics. This interplay creates a new quality, as Goldstone remarks in [1] as his final conclusion:

When we compare entities, our understanding of the entities changes, and this may turn out to be a far more important consequence of comparison than simply deriving an assessment of similarity.

Goldstone fails, however, to separate similarity and comparison in an appropriate manner. Consequently, he also fails to put categorization into the right place:

Despite the growing body of evidence that similarity comparisons do not always track categorization decisions, there are still some reasons to be sanguine about the continued explanatory relevance of similarity. Categorization itself may not be completely flexible.

Such statements are almost awful in their production of conceptual mess. Our impression is that Goldstone (as many others) does not have the concept at his disposal that we call the transition from probabilistic description to propositional representation.

Type III: Comparison of Populations

The third type, finally, describes the process of comparison on the level of populations. The most striking difference to type I + II concerns the fact that there is no explicitly given proposal, which could be assigned to some kind of input data. Instead, the only visible goal is (long-term) stability. We could say that the comparison is an open comparison.

Let us start with the basic structure we already know as results from type-I and type-II.

Fig.4a

Now we introduce two significant elements, population and, as a consequence, time. Indeed, these two elements mark a very important step, not the least with regard to philosophical concepts like idealism. For instance, Hegel’s whole system suffers from the utter negligence of population and (individual) time.

By introducing populations we also introduce repetition and signal horizons. Yet, even more important, we dissolve the objecthood that allowed us to denote two entities as A and B, respectively, into a probabilistic representation. In other words, we replace (crisp) symbols by (open, borderless) distributions. In natural evolution, the logical sequence is just the other way round. There, we start with populations as a matter of fact.

Comparing A and B means to compare two populations A(n) and B(m). Instead of objects, singulars or concepts we talk about types, or species. Comparing populations also means to repeat (denoted by “::n”) the comparison between approximate instances of A(n), aj and  approximate instances of B(m), bk.

It is quite obvious that in real situations we never compare uniquely identifiable, apriori existing objects that are completely described. We rather always have to deal with populations of them, not the least due to the inevitability of modeling, even with regard to simple perception.

Fig.4b

Comparing two populations does not result in just one proposal, instead we are faced with a lot of different ones. Even more, the set of achieved proposal can not be expected to be constant in its actuality, since not all proposals arrive at the same time. We then could try to reduce this manifoldness by applying a model, that is, by comparing proposals. Yet, doing so we are faced with both a certain kind of empirical underdetermination and a conceptual indeterminacy.

It is this indeterminacy that causes a qualitative shift in the whole game. It is not the proposals any more that are subject of the comparison. The manifold of proposals lift us to the level of the frequency distribution upon all proposals. Since comparing proposals can not refer to other information than that which is present in the population, deciding about proposals turns into an investigation about the influence of the variation above A(n) or B(m), or the influence of the similarity functional.

The comparison of populations obviously introduces an element of self-referentiality. There are two consequences of this. First, it introduces an undecidability, secondly, comparing populations induces an anisotropy, a symmetry break within them. Compare two populations and you’ll get three. Since this changes the input for the comparison itself, the process either develops the perlocutionary aspect of pragmatic stability as a content-free intention, or the whole game disappears at all.

This pragmatics of induced stability can be described by using the concept of fitness landscape.

Fig.4c

The figure above could be conceived as a particular segment of evolutionary processes. For us, it is so natural that things are connected and related to each other that we can hardly image a different picture. Yet, we could revert the saying that evolution is based on competition or competitive selection. Any population that does not engage in the evolutionary comparison game will not develop the pragmatics of stability and hence will disappear more soon than later.

Practical Use

The three types of comparison that we distinguished above are abstract structures. In a practical application, e.g. in modeling, some issues have to be considered.

The most important misunderstanding would be to apply those abstract forms as practical means. Doing so one would commit the mistake not only to equate the local and the global, but also to claim a necessity for this equality.

Above we introduced the principle of aligned feature lists that need to be common for both instances that we are going to compare. Note that we are comparing only two (2) of them! From such a proposal one can not conclude that all items out of a set of available observations are necessarily compared using exactly the same list of features in order to arrive at a particular classification. As Wittgenstein put it, “there is no property all games have in common which distinguishes them from all the activities which are not games.” (cited after Putnam [4, p.11]) This, of course, does not exclude a field of overlapping feature lists suitable for deciding whether something is a game or not. Of course, there is no unique result to be expected. The crispness of the result of such comparison depends on the purpose and its operationalization in modeling, mainly through the choice of the similarity measure.

Yet, such a uniqueness is never to be expected unless we enforce it, e.g. by some sort of formal axiomatics as in mathematics, or mathematical statistics, or, not so different from that, by organizational constraints. If our attempt to create a model for a particular classification task requires a lot of features, such uniqueness is excluded even then if we would use the same list of features for creating all observations. On the other hand this does not mean either that we could not find a suitable result.

The Differential and Ortho-Regulation

Finally, we arrive at the differential as the most sophisticated form of comparison. We consider the differential as the major element of abstract thinking. Here, we neither can discuss the roots of the concepts nor the elaborated philosophy that Gilles Deleuze developed in his book Difference and Repetition [5]. We just would like to emphasize that much of this work has been influenced it.

The differential is not a tool to compare given observables in order to derive a proposal about those observables. Instead, when playing the “Differential Game” we are interested in the potentially achievable proposals. Of course, we also start with a population of observations. Those observations are yet not in any pre-configured, or observable relationship. The diagrams that we will develop in the series below look very different from those we found for the comparisons.

The starting point. Given a set of observations and the respective propertization that we derived according to intellectual habits, our forms of intuition, we may ask which proposals, statements or solutions are possible?

Fig.5a

The first step is to replace immutable properties by a more dynamic entity, a procedure. This procedure could be taken as the tool to create a particular partition in the observations {O}, or as a dynamic representation of possible equivalence classes on {O}. We also could call it a model. Note that models always imply also a usage, or purpose.

The interesting thing now is that procedures can be conceived as consisting from rules and their parameters, or in the language of mathematical category theory, of objects and their transformations. The parameters are much like variables, but from the perspective of any particular partitioning, or say, instance, the parameters are constants. This  scheme has been originally invented, or at least it has been written down, by Lagrange in the late 18th century. Most remarkably, he also observed that this scheme can be cascaded. The parameters on the abstract level can be taken as new, quasi-empirical “observations,” and so on.

Fig.5b

The important part of this schemes are indeed the free parameters, which are, we have to remember that, also constants. If we now play around with these pfree parameters, we can construct different partitions from {O}, but this also means, that by varying the parameters we can create a proposal beyond such partitioning, or solutions regarding some request (again upon {O}).

Fig.5c

Of course, what now becomes possible is the simulation game. Which statement we are actually going to construct is again a matter of habits. Let us call this the forms of construction. Astonishingly, this structure has been overlooked completely so far (with one exception), it went also unnoticed for Immanuel Kant and all of his fellows in philosophy.

Fig.5d

Given this scheme we would like to emphasize that there is no direct path from observations to statements. Hence, habits and conventions become active at two different positions in the process that allows us to speak about an observation. This is already true for the most simple judgments about {O}, indicated by S(0). Again, this has not been recognized by epistemology up to date.

Fig.5e

Finally, we apply the Lagrangian insight to our scheme. Forms of intuition as well as forms of construction are, of course, not just constants. They are regulated, too. This results in the induction of a cascade. Since this regulation of mental forms (of intuition, or construction, respectively) does not refer to {O}, but instead to the abstraction itself (mainly the selection of the parameters), it appears as if this secondary mental forms are in a different dimension, not visible within the underlying activity of comparing. Thus we call it ortho-regulation.

Fig.5f

It is actually quite surprising that there are whole philosophical schools that deny the (cascaded) vertical dimension of these processes. One of the examples is provided by Jacques Derrida’s work. At many places throughout his writings he comes up with rather weird ideas to preserve the mental flatness. One of them is the infamous “uninterpretable trace” (grm.: Spur).

A common misunderstanding is committed by many scientists influenced by positivism (who of them is not?) concerning the alternatives S(k). Determinists claim that the step from abstraction to the solution is unique, or at least determined as a well-defined alternative of a finite set. Doing so, they deny implicitly the necessity for orthoregulation, hence they also deny any form of internal freedom as well as the importance of conventions (see below). This paves the way for the nonsensical conviction that the choice between S(k) can be computed algorithmically (as a Turing-Machine computes). The schemes above clearly show that such a conception must be considered as seriously deficient.

The following scheme may be taken as an abbreviated form for the phenomenon of (abstract) thinking.

Fig.6

Quite important, the differential is isomorphic to the metaphor. Actually, we are convinced that metaphors are not a linguistic phenomenon. Metaphors are the direct consequence of necessary structures in thinking and modeling.

Architectonics of Comparison

Comparisons belong, together with the classifications that are based on them, to the basic elements of cognition and higher-level mental processes. Thus they may be taken as a well-justified starting point for any consideration of the conditions for epistemic processes.

Astonishingly, such considerations are completely absent in science as well as the humanities (and their mixed forms). The only vaguely related references that can be found—and there are really on very few of them—are from the field of (comparative) linguistics or literature science. In linguistics it is not the structure abstract operation of comparison that is focused [6]; here comparison is taken as a primitive and then applied to linguistic structures. One of the research paradigms is given by the case of adjectives like smaller, higher etc. What is studied there is the structure of the application of the operation of comparing, not the operation of comparing itself.  In comparative literature science, however, an interesting note can be found. In his inquiry of the writings of Jean Paul, a German Romanticist, Coker [7, p.397] distinguishes different types of comparisons, at least implicitly, and relates a particular one directly to imagination, a result that we can confirm through our formal analysis:

“The imagination is a structure of comparison through which desire can realize its infinite nature, always transcending finite givens.”

Besides this really rare occurrence, however, comparisons are always taken in its most simple and reduced form, the comparison along a numerical scale. This type is even more primitive than our simplest type shown in Fig.2. It is true that all of our three types ultimately are based on the primitive type, yet, considering normal thinking, the reduction to the primitive case is inappropriate. The language is full of comparisons and comparative moves, where we call it metaphor. For more than 100 years now linguists are reasoning about metaphors in largely inappropriate ways precisely because they impose a reduced concept of comparison.

A reference to a more elaborated and rich concept of comparison is missing completely, in cognitive sciences as well as in computer science. Even the field of metaphorology did not contribute a clear structural view. So we conclude that the problematics of comparison seems to play a role only in improper proverbs about apples and oranges, or apples and pears.

Hence, the two cornerstones of any type of comparison remained undetected, propertization and ortho-regulation. We propose that these are the elements of an  architectonics of comparison. The propertization will be discussed in the chapter about modeling, so we can turn to the phenomenon of ortho-regulation.

Ortho-Regulation

The concept of ortho-regulation becomes visible only if we take two approaches serious: rule-following (Wittgenstein) and the differential (Deleuze). The first step is the discovery of the Forms of Construction. In a second step, symmetry considerations lead us to the cascaded view.

The notion of “forms of construction” may appear as trivial and well-known. Yet, it is usually applied as a concept used while thinking, in the sense of a particular way to construct something, not as a basic concept constitutive for thinking (e.g. Quine in [8], or Sandbothe in [9]); for example, it is spoken about “forms of construction of reality.” In contrast to that we consider “forms of construction” here as a transcendental principle of thinking.

Orthoregulations are rules that organize rules. Wittgenstein dismissed such an attempt, since he feared an infinite regress. Since then this remained the received view. Nevertheless, we think that this dismissal has been devised to hastily. There is no thread through an infinite regress because the rules on the level of orthoregulations are neither based on nor directed towards observations {O} about facts. The subject of ortho-regulative rules are rules. In other words, their empirical basis is not only completely different, but also much smaller and much more difficult to learn. Ortho-regulative rules can not be demonstrated as readily as, say, how to follow an instruction.

The cascade is thus not an infinite one, rather, it stops presumably quite soon. To derive rules Rx about rules from a more basic body of rules Rb, you need a lot of instances or observations about Rb. There are less hawks than mice. Proceeding in the chain of rules about rules soon there are not enough observations available anymore to derive further regularities and rules. We agree to Wittgenstein’s claim that rule-following must come to an end, yet for a different reason. Accordingly, the stopping point where rule-following becomes impossible is not the fear of the philosopher, it is a point deeply buried in our capability to think, precisely because thinking is a bodily, hence empirical activity. Note that “our” here means “human stuffed with a brain.” This point is a very interesting one, which quite unfortunately we can not investigate here. As a last remark on the subject we would like to hint to Leibniz’ idea of the monad and the associated concept of absolute interiority.

Another discussion we can not follow is of course Kant’s notion of the form of intuition. We simply are not fluent enough to develop serious arguments from a Kantian perspective. We find it, however, remarkable, that he missed the counterpart of the rising branch of abstraction. In some way, we guess, this could be the reason for his prevailing difficulties with the (ethical) notion of freedom, which Kant considered to be in an antinomic relation to being determined [e.g. 10]. His categorical imperative is a weak argument, since Kant had to introduce it actually much like an axiom. He was quite desperate about that, as he expressed this in his last writing [11]. His argument that the capability to choose one’s own determination reflects or implies freedom is at least incomplete and does not work in our times any more, where we know crazy things about the brain. Our analysis shows that this antinomic contrast is misplaced.

Instead freedom arises inevitably with thinking itself through the necessity of applying forms of construction. There is no necessity of any kind to choose a particular statement from all potential alternatives S(k) (see Fig.5e). Note that this choice is indeed actualizing a potential therefore. Furthermore, it is not only a creative act, though not without being bound to rules, it is also an act that can not be completely determined any kind of subsequent model. Hence, it is actions that are introducing virtuality into the world by virtue of creating statements in a non-predictable way. Saying non-predictable, one should not think that there could be some kind of measurement that would allow to render this choice or creation predictable. It is non-predictable because it is not even in the space of predictable things.

Freedom is thus not an issue of quantum mechanics, as Kauffman tries to argue so hard [12]. It is also not an issue of human jurisdiction or any other concept of human society. Above all, freedom is nothing which could be created or prepared, as Peter Bieri [13] and other analytic philosophers believe. A reduction to the probabilistic nature of the world would be circular and the wrong level of description. Quite to the contrast to those proposals we think that freedom it is a necessary consequence of abstraction in thought. Since any kind of modeling that is not realized as body (think of adaptive behavior of amoebas or bacterias) implies abstraction, it makes perfectly sense even in philosophical terms to say that even animals have a free will. Everyone who lives with a cat knows about that. We can also see that freedom is directly related to the intensity in the cognitive domain. They do so as long as they are performing abstract modeling. No thoughts, so no freedom, no expression of will, so no cognitive capacity. Being a machine (whether from silicon or from flesh), so no will and no cognitive capacity.

While forms of intuition can be realized quite easily on a computer as a machine learning algorithm, this is not possible for forms of construction. It is the inherently limited cascade of ortho-regulations on the one side, and the import of conventions through those that create a double-articulation for rule-following, that point towards a transcendental singularity. It is not possible to speak formally or clearly about this singularity, of course. Maybe we could say that this singularity is a zone where the being’s exteriority (conventions) directly interferes with its interiority (the associative power of the body). It feels a bit like a wormhole in space, since we find entities connecting that normally are far apart from each other. We also could call it a miracle, no problem with that.

Fortunately enough, there is also a perspective that is closer to the application. More profane we could also simply say (in an expression near the surface of the story) that freedom exists because those brains form “minds” in a community, where those “minds” need to be able to step onto the Lagrangian path of abstraction. We do not need the concept of will for creating freedom, it is just the other way round.

Consequences for Epistemology

Orthoregulation and the underlying forms of construction are probably one of the most important concepts for a proper formulation of epistemology. Without the capability for ortho-regulation we will not find autonomy. A free-ranging machine is not an autonomous being, of course, even if it “develops” “own” “decisions” if it is put into a competitive swarm with similar entities.

The concept of ortho-regulation throws some light onto our path towards machine-based epistemology. Last but not least, it is a strong argument for a growing self-referential system of associative entities that are parts of a human community.

This article was first published 11/11/2011, last revision is from 28/12/2011

  • [1] Robert L. Goldstone, Sam Day, Ji Y. Son, Comparison. in: Britt Glatzeder, Vinod Goel, Albrecht von Müller (eds.), Towards a Theory of Thinking – Building Blocks for a Conceptual Framework. Springer New York. pp.103-122.
  • [2] Manfred Frank, Wege aus dem Deutschen Idealismus.
  • [3] Dekang Lin, An Information-Theoretic Definition of Similarity. In: Proceedings of the 15th International Conference on Machine Learning ICML, 1998, pp. 296-304.
  • [4] Hilary Putnam, Renewing Philosophy. 1992.
  • [5] Gilles Deleuze, Difference and Repetition. Continuum Books, London, New York 1994 [1968].
  • [6] Scott Fults, The Structure of Comparison: An Investigation of Gradable Adjectives. Diss. University of Maryland, 2006.
  • [7] Coker, William N. (2009) “Narratives of Emergence: Jean Paul on the Inner Life,” Eighteenth-Century Fiction: Vol. 21: Iss. 3, Article 5. Available at: http://digitalcommons.mcmaster.ca/ecf/vol21/iss3/5
  • [8] WVO Quine, Two Dogmas of Empiricism,
  • [9] Mike Sandbothe (1998), The Transversal Logic of the World Wide Web,11th Annual Computers and Philosophy Conference in Pittsburgh (PA), August 1998; available online
  • [10] Rudolf Eisler, Freiheit des Willens, Wörterbuch der philosophischen Begriffe. 1904. available online.
  • [11] Immanuel Kant, Zum Ewigen Frieden.
  • [12] Stuart A Kauffman (2009), Five Problems in the Philosophy of Mind. Edge.org,  available online.
  • [13] Peter Bieri, Das Handwerk der Freiheit, Hanser 2001.

۞

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