The SOM and the Symbolic

December 20, 2011 § Leave a comment

Symbols are at the roots of intelligence, language, or culture.

Once, if these roots will have been revealed and described, one could control any of those puzzling phenomena. For instance, we would understand how to create intelligence, how to do science in the best (if not correct) way, how to optimally develop and plan our society.

All of the claims and hopes listed above are victims of a particular misunderstanding: that symbols are entities that could be naturalized. The symbol is not a thing that you could write down, or something which you could carry around.

The Setting

In order to develop the relation between the symbolic and abstract associative structures (for which we take the self-organizing map as a salient example), we first have to identify our hypothesis about symbols, and second we should describe the problem as it is perceived in the fields of “computational intelligence,” or “artificial intelligence.”

Above we said that the symbol is not a thing that you could write down or carry around. This is true for the symbolon, σύμβολον, the name-giving mythological object, it is true for the x in mathematics, or the Ferrari driven by a young Italian male. It is true despite you can read the opposite in the dictionaries.

The etymological roots of the classic languages are much more precise. In Latin, the symbolum meant a sign, but also a mark, token, or, of course, symbol, while in ancient Greek the sumbolon meant “a sign by which one infers something.” It is not possible to carry around a symbol because it is not possible to carry around a sign, as we know since the great works of Charles S. Peirce [1].

Yet, what do we carry around and write down? Simply shapes, or more abstract, forms, I would say. Those forms serve as a kind of anchor that serves several purposes, and it can provide this services only through a significant and signifying double-articulation. The form itself just allows for a doubtless identification through modeling and concept matching. The particular form of the form is not important at all, yet it is not completely arbitrary. The reason is the involvement of modeling, which needs a particular context. Different symbols need different kinds of contexts to a different extent. A Ferrari needs more external context than a Kanji sign, perhaps.

The whole process from a particular identifiable form up to the process of matching concepts we have to call “symbol.” Yet, similar things could be said about signs. Symbols are situations, mediated events, of a particular quality. The difference between symbols and signs is just a difference in the degree of fixation in the chain of interpretation. That fixation is not based on particular absolute necessities, of course, it is dependent on a cultural dynamics, say contingencies, eventually making their way into deontic contexts.

Our hypothesis is that particularly symbols, and in a strong difference to the chain of signification by “signs,” provide a chain of references that ends up in the material world again.

This hypothesis distinguishes our position from many other writings (if not any) about symbols, including Cassirer’s. Only a very short note is indicated about Cassirer’s position(s) here. Albeit our position is sometimes quite similar to Cassirer’s, for example with respect to the role of externalization of mental processes, or the role of doing and acting [2, p.239], we reject the role he assigned to philosophy [3, p.110] as a “philosophy of man” and kind of a “tool” (he doesn’t use this term) to provide insight into human culture as an “organic whole.” We also reject the strong apriori of logics that he assumes.

„Unter ‚symbolischer Prägnanz‘ soll also die Art verstanden werden in der ein Wahrnehmungserlebnis, als ‚sinnliches‘ Erlebnis, zugleich einen bestimmten nicht-anschaulichen ‚Sinn‘ in sich faßt und ihn zur unmittelbaren konkreten Darstellung bringt. [4, p.235]

For us, that assumption of a direct “immediate” representation is deeply inappropriate, if not referring to an impossibility. It neglects the primacy of interpretation and the primacy of modeling. We always remember to the fact that it is this primacy that saves us from being deterministic machines. Despite his reference to perception, Cassirer remains blind against both, not quite surprising and fully consistent with his idealist attitude.

Cassirer misunderstands symbols as schemes, precisely because he is blind against modeling. His (Anti-Kantian) claim that any kind of reference to the world is dependent on symbolization thus just starts from the false end. He can’t conceive of different grades of symbolization, he can’t conceive the relation (and the difference) between concepts, notions, words and symbols, he can’t talk about the relation between immanence and the virtual (it seems that there is only immanence for him), and so on. His position is straightforwardly developed as an idealistic philosophy of signs. Due to his almost hyper-idealistic attitude as induced by the claim of “immediatedness”, Cassirer also can’t understand the relation between the mental and the communal, or shortly, the issue of rule-following in cultural settings. For all these reasons, Cassirer’s philosophy of symbols is not of any relevance for our considerations here, despite he is still serving as a kind of inevitable measure for many approaches in the philosophy of the symbolic.

Intention and Hypothesis

It is pretty clear, so to speak, by intuition, that the role of symbols is vital for any understanding of higher mental processes, whether they are related to a brain or to a machine. Cassirer’s proposal that animals do / can not have symbols at their disposal, is deeply misleading, as well as the link between consciousness and the symbolic, which he thought to be the essence of human spirit.

There are many myths and misconceptions around concerning the relationship between mental processes and symbols. It does not start with the Whorf-hypothesis [5] (syntax determines condition of experience) and does not end with the unfortunate model of “mentalese,” developed by Fodor [6] and Pinker [7], among others. Despite Fodor vigorously refutes (or tries to do so) the position of Whorf, the reason for their failure (Fodor’s as well as Whorf’s) is the same. Both obviously assume that symbols can exist and that they can even exist outside of usage.

Symbols belong to class of immaterial entities that we usually call pointer. As such, the concept of symbol is related to the concept of reference. Again, we meet van Fraassen [8] here, but of course, the question is much older and posed by many philosophers from Aristotle [9] to Kant [10]. Famous also the serious critique that Wittgenstein addressed to Russell’s way of introducing symbols in the Principia Mathematica as early as in the Tractatus [11].

Here, we should briefly stay at the usage of the concept of the notion of “symbols” in everyday contexts of the human Lebensform. We already met the Ferrari that is not only a symbol for luxury, but also for machism. All around our cities we find thousands of pictograms, which could be considered themselves as symbols, but which also are a symbol in their totality. Of course, we find symbols not only in the cities, but also in villages, and, as Cassirer correctly observed, everywhere in social intercourse and in human culture. This includes fields like logics and science, of course. Many people know about the meaning of the logical symbol ¬ standing for negation, or the mathematical symbol ℕ, standing for natural numbers. Mathematics is full of symbols, it is apparently even a science about systems of symbols.

From these observations we may derive that there is one single property that is shared among all sorts and instances of symbols (it is not the only one, of course): Symbols serve as an abbreviation. A symbol indicates that there is a strict reference between two entities, of which one of them, the subject of the symbol, is taken to be comparatively persistent, i.e. unchangeable. Without this persistence and stability subjects can not be symbolized. The persistence of symbols is much stronger than the persistence required by signs. Actually, it is so strong that one could misunderstand symbols as a grammatical entity. Just this was the critique by Wittgenstein of Russell’s Principia.

Yet, we are allowed to take this notion of persistent stability of the subject of symbols as an indicator for a quasi-materiality that indeed could instantiate as matter, but that also could be rendered into something like strong cultural structures. The traditional point-of-view, the way how things appear, may well be taken as a material substance.

It is this play, unfolding between quasi-materiality and reference, that after all allows for mathematics and its proofs. The same link, however, also easily gives rise for the misunderstanding that logics could by applied to reality.
The subjects of symbols need to be stable up to quasi-materiality. Symbols themselves are just representing that stability then, and thus even the reference represented by symbols takes quasi-material form. Nevertheless, symbols should not be conceived as “indices,” or in Peirce’s terms, as firstness. An index is a simple sign, incapable to continue the chain of reference, and so also not back to quasi-materiality. An index is not an abbreviation, it is an equivalence by definition. Sometimes, also a set of individual indices are called “an index,” e.g. in computer science. Such a set then could be again symbolized.

Above we said that the concept of “symbol” describes a particular process of referencing that is rooted two-fold in quasi-materiality. Here we have to be more precise; we  replace quasi-materiality by irreversibility. This allows us to subsume matter and actions, or purposes, respectively. We can see now that a symbol is like a sign with a purpose, only that normal signs never have a dedicated purpose. To express this even more sharpened (becoming thus a bit imprecise) we could say that  symbols imply reference, while signs imply interpretation and modeling. Using signs is more complicated, at least for higher-order signs (see Peirce’s classification) such like rhea-signs, than it is for symbols. Yet, the direct comparison is actually not quite feasible, they simply play different roles, and both are necessary in “thinking.”

If you already read the chapters about modeling (generalized model, model as category) you know about the salient role that we assign to the model. For us it is a practical and transcendental primary. From that perspective, symbols are not pre-cursors of signs, but formally degenerate semi-signs.

This result is quite important for, as well as fully consistent with the view we will develop in the chapter about generalized conditions. It also nicely fits to the observable role of mathematics and logics in our human Lebensform.

Our position here, particularly the aspect of linking symbols to action, is similar to that of Robert Cowles [12], who developed his perspective following and extending the work of Lev Vygotsky [13]. Yet, Cowles does not discern symbols and signs, which for us are clearly distinct. Else, he does not recognize the aspect of irreversibility in the concept of actions, and thus also not their quasi-material characteristics and inherent relationships to logics. Presumably mainly due to these missing distinctions, Cowles replaces the symbol-grounding-obstacle by the question “How do semiotic symbols come to play a role in thinking?” We again meet the old question cited frequently here about the reference of signs and its relationship to meaning. After all, we think that this question is almost as ill-posed as the symbol-grounding-problem, just from the opposite direction. Remarkably enough, though Cowles insists on a central role of interpretation, he does not further investigate the structure of it and henceforth modeling does not play any role in his proposals.

The Pseudo-Problem(s)

Everybody being active in the area of machine-learning and/or computational intelligence, or artificial intelligence knows about the so-called symbol grounding problem. In [11] we can read:

According to a widely held theory of cognition called “computationalism,” cognition (i.e., thinking) is just a form of computation. But computation in turn is just formal symbol manipulation: symbols are manipulated according to rules that are based on the symbols’ shapes, not their meanings. How are those symbols (e.g., the words in our heads) connected to the things they refer to? It cannot be through the mediation of an external interpreter’s head, because that would lead to an infinite regress, just as looking up the meanings of words in a (unilingual) dictionary of a language that one does not understand would lead to an infinite regress. [14]

“Formal manipulation” of symbols just means that symbols are replaced by other symbols according to an axiomatically founded system of possible equations between expressions. Nothing is added or removed through such a “manipulation.”

The problem was first (and more exactly) posed by Stevan Harnad [15], by raising the question:

How can the semantic interpretation of a formal symbol system be made intrinsic to the system, rather than just parasitic on the meanings in our heads? How can the meanings of the meaningless symbol tokens, manipulated solely on the basis of their (arbitrary) shapes, be grounded in anything but other meaningless symbols?

In his article, where he notably sets the first header as “From Behaviorism to Cognitivism,” Harnad also proposes a solution which comes in two parts.

[…] (1) “iconic representations” , which are analogs of the proximal sensory projections of distal objects and events, and (2) “categorical representations”, which are learned and innate feature-detectors that pick out the invariant features of object and event categories from their sensory projections. Elementary symbols are the names of these object and event categories, assigned on the basis of their (non-symbolic) categorical representations.

So far, we barely agree on his perspective, despite at a first look it seems appealing. Even the problem statement is weird. Why should anyone try a “semantic interpretation of a formal system”? We should not forget, that the distinction between syntax, semantics and pragmatics has been introduced by a convinced positivist (and behaviorist): Charles Morris. Only today, a few people start to correct this extremely harmful distinction (Peter Janich, Robert Brandom).

Harnad seemingly believes that his solution is also a sufficient one and thus completely overlooks the necessity of some communality, even if it is a restricted one. A single brain, or a single simulation of an artificial neural network can not symbolize anything. Assuming so instead , Harnad commits the same categorical mistake as it is unfortunately quite abundant in computer sciences and cognitive sciences and which ultimately leads to the private-language-fallacy.

The misunderstanding is even visible in the label itself. Symbols can’t be grounded at all. There is neither “ground” as a plane of stability nor a “location” as kind of an origin, from where symbols could emerge. Only the—mistakenly through naturalization reduced—versions, being more “somethings” than symbols anymore could be thought of to have a point of origin or an apriori fixed reference. As we already said, symbols require communality despite their quasi-materiality. Communality can not provide “origins,” except perhaps in radical bureaucrazies, or dictatorships. Note that both organizational forms are characterized by extensive bodies of written, i.e. positively determined rules.

Symbols and Associative (Quasi-)Matter

Symbols imply a compound consisting from quasi-material and immaterial aspects. In that they are indeed indispensable, though in a way quite different from what Cassirer proposed. They link us to irreversibility (deliberately avoiding here the notions of “world”, “reality”, or even “materiality”). Metaphorically, they provide us a “handle” attached to the world, allowing us to carry around that world (not the symbols!). Their character is “dual” in several directions.

By means of their materiality/immateriality duality they provide us the possibility to relate to the world, which relates them closely to associative structures. Without associative structures there are no (primary) models, hence ; on the other hand, without symbols there is no possibility to create compound models.

The relationship between models and symbols is rich, complicated and co-generative. In the chapter about associativity we saw that models and associativity overlap considerably. So, the pas de deux turns into a men…

First, we have seen that modeling requires the assignment of “properties” in order to instantiate measurement and initial—also often called “raw”—data. This assignment thus refers to symbols that are defined outside and apriori to modeling. Primary models then have to establish well-separated classes by means of an idealization process; these ideal classes are based on empirical classes, or, in other words, intensions. Next, these ideals have be named, which again needs symbols from the outside. Once the ideals have been named they can be used, and if this usage both will be repeated and provides stability due to sufficient anticipative power they eventually may be symbolized. This symbolization is both contingent and communal. Yet, they should not be misunderstood as carriers of common sense.

A sufficiently large and rich collection of symbols, or the capability for simulative instantiation of symbols, then allows for higher-order modeling. Higher-order modeling, however, rests on the steam of symbols, not on physical data. High-order modeling increasingly draws on relations between symbols and also returns increasingly idealized results, until finally there are only immanent relations left. Such constructs may be called concepts.

Conclusion: Filling the Gap

It is very important always to remember the interdependency of models, symbols and concepts. There is no possible actualization of a “pure” concept, or a “pure” model. For example, research in physics heading towards a “Grand unified theory” are bare nonsense just for structural reasons, and if actually carried out, utter and dangerous hubris. The impossibility of an actualization of “pure” concepts, models, or symbols lead us to the further result that those labels denote transcendental entities.

It may seem that the interdependency mentioned above can be “sidestepped” only in case of the primary models that are built solely through the associative power of networks on the level of the mere body, whether as software or as neuroware. Yet, it is not really a sidestepping; in its individuation it is just drawing itself on an ontogenetic apriori. Any individual is embedded into an evolutionary/ historical process that itself represents the dynamics between models, symbols and concepts, just on a larger time scale, or on stacks of those.

As a kind of conclusion we could state that our results may be regarded also as a reconstruction of the linkage between concepts and their references. We have seen that the growing of that linkage requires kind of an abstract compart-mentalization into models (as categories), intensions, names, semi-signs, and signs, including a particular dynamics between those parts. It now becomes quite lucid why mathematics is so important, for science, but also for the society at large. Without the semi-signs and its investigation there would not only be a chasm between primary models and concepts, concepts as we know it would even be absent. “Concepts” are freely floating radicals, giving rise to a manifold of metamorphoses (Homer, and Ovid, then playing with it). We even have a name for such conditions: the mythical age. As a corollary, we also could say that phantasy emerges where symbols become volatile or even retreat. In our investigation about vagueness we will find that words take a particular role in corroborating the fixation of symbols.

Last, but not least, this growing is a social (and sociogenetic) process. Actually, it should not be necessary to remember that the clarification of the relationship between model, symbol and concept is essential for any progress towards a machine-based epistemology.

This whole landscape between the body, its modeling, the symbols and the concepts we now can incorporate into the rather different perspective of generalized conditions. This arrangement describes the ultimate border for any onto-epistemic action, which we will investigate here.

  • [1] Charles S. Peirce
  • [2] Ernst Cassirer, Philosophie der symbolischen Formen. Band II, Darmstadt 1977
  • [3] Ernst Cassirer, Versuch über den Menschen. Meiner Verlag, Hamburg 2007, S. 110.
  • [4] Ernst Cassirer, Philosophie der symbolischen Formen. Band III, Darmstadt 1982
  • [5] John B. Carroll (ed.), Language, Thought, and Reality: Selected Writings of Benjamin Lee Whorf, MIT Press. Boston 1964.
  • [6] Jerry Fodor, The Modularity of Mind, 1985.
  • [7] Steven Pinker, The Language Instinct. 1995.
  • [8] Bas van Fraassen, „Putnam’s Paradox: Metaphysical Realism Revamped and Evaded.“ Philosophical Perspectives, vol.11. Blackwell, Boston 1997. S. 17-42.
  • [9] Aristotle
  • [10] Kant
  • [11] Ludwig Wittgenstein, Tractatus Logico-Philosophicus. [1918]
  • [12] Robert Cowles, Semiotic Symbols and the Missing Theory of Thinking, in: Tony Belpaeme, Stephen J. Cowley, Karl F. Macdorman (eds.), Symbol Grounding, Benjamins Publ., Amsterdam 2009. pp.107-126.
  • [13] Lev S. Vygotsky, Thought and Language. Alex Kozulin (ed.), Revised Edition, MIT Press, Boston 1986.
  • [14] Symbol Grounding, Wikipedia, available online
  • [15] Stevan Harnad (1990) The Symbol Grounding Problem. Physica D 42: 335-346. available online


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