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VON FOERSTER’S THEOREM ON CONNECTEDNESS AND ORGANIZATION: SEMANTIC APPLICATIONS

BENNY SHANON” Department of Psychology, Hebrew University,Jerusalem, Israel

and

HENRI ATLAN Department of Medical Biophysics, Hadassah University Hospital, Jerusalem, Israel

Abstract - A theorem originally proposed as a conjecture by H. von Foerster suggests that the more trivially connected are the elements of a system, the less influence they will have on the system as a whole. Following previous applications of the conjecture in the context of economics, game theory and the theory of automata, applications in the semantic domain are noted. A series of examples in poetic language, visual expression and music presents a variety of cases in which the state of affairs depicted by the conjecture holds. Implications to semantic theory and to the study of metaphor in particular are noted.

In 1976 the cyberneticist Heinz von Foerster, in the context of the theory of systems and the modeling of network automata, raised a proposal that has subsequently been referred to by Dupuy (Dupuy & Robert, 1976; Dupuy, 1982) as the von Foerster conjecture. The conjecture regards the relationship between elements in a system and the behavior of the system as a whole. It states that the more the elements of a system are “trivially” connected, the less will be their influence on the behavior of the entire system. In this context, connections are labeled as trivial when they are rigidly and univocally determined. Given this state of affairs, the more trivial the connection, the more will the constituent elements regard their environment as unaffected by their actions, that is, as being external to them, and as a consequence each element of the system will exhibit a greater degree of “alienation.”

At first blush the von Foerster conjecture seems to be counterintuitive. If the relations between the elements are well-defined, one would expect the function that determines the behavior of the system as a whole to be highly determined, and thus it would seem that in this case the elements would be expected to have increased influence or control on the behavior of the system of which they are part. Similarly, if the relations between the elements are ill-defined, fuzzy, or fluid the behavior of the system as a whole would seem to be not well determined and thus the influence of the elements on the entire system would appear to be small. Given this initial appraisal it would be both interesting and significant to show that the conjecture does, in fact, hold in various domains of application as well as in principle.

*Acknowledgement-The research reported here was supported in part by a grant from the Grace and Milton Rabson Endowment Fund awarded to B. Shanon.

81

x2 Benny Shanon mid Henri Atlan

Atlan, Dupuy, and Koppel (in press) have shown how the conjecture applies in the context of economics, game theory, and automata theory. A mathematical proof of the conjecture has also been offered (Koppel, Atlan, & L)upuy, 1987). The proof consists of two theorems, each defining one of two complementary aspects of the conjecture. The first aspect has to do with the relationship between each element and the state of the whole system. The state of the system is determined on the one hand by the elements as they are connected in different degrees of rigidity, and on the other hand by some deterministic factors: ‘I‘he less uncertain is the behavior of a constituent element, the less its influence on the state of the whole-it makes sense that the element is thus more “alienated.” ‘I‘he second aspect of the conjecture has to do with the communicatiorl between the different constituent elements. ‘I‘he sum ot‘ the influence of all the elements on the whole is proportional to the amount of conlmunic-atioll (technically, mutual information) between them. Given this proof the conjecture may be referred to as the von Foerster theorem (henceforth, VFFI‘).

‘I‘he present discussion marks some further applications of VF’I‘ as they are manifested in the expression of meaning both in language and in the arts. The patterns to be examined here are interesting both because they extend the domain of application of the VF’I‘, and because they have a bearing on some basic issues in semantics.

‘I‘he patterns to be noted have to do with the relationships between the meaning or expressive power of an expression and that of its constituent elements. This is perhaps the fundamental question of the theory of meaning. At the basis of standard semantic analyses is the claim-originally attributed to Frege (1952) ( see also, Montague, 1974)-that the meaning of’ a linguistit expression is a well-defined function of the meaning of’ the words of‘ which it is composed. Given this view, a prime task of the student of semantics is to define these functions, a project standardly referred to as the projection problem (cf., Katz, 1972).

As a standard, the Fregean view of semantics does seem to hold. Thus, consider a simple sentence like (1):

(1) .John eats the banana. How is the meaning of the sentence determined in terms of the meaning of the words of which it is comprised ? This determination seems to be very direct. It is algorithmically determined on the basis of two components. On the one hand, there are the meanings of the individual words as they are defined in the lexicon. On the other hand, there are the prqjection rules that determine the meanings of the sentence on the basis of the meaninCgs of the individual words and the way they are syntactically combined in the sentence at hand. Sentence (I) is inter- preted as it is because on the one hand ‘John,” “eat,” and “banana” have the meanings standardly attributed to them, and on the other hand because of thei syntactic position in the sentence, ‘tJohn,” “eat,” and “banana” take the functional roles of agent, action, and recipient, respectively.

Several points should be made with respect to the determination of meaning just noted. First, the meanings of the individual words are given prior to and independent of the particular context of use of these words in the sentence

von Foerster’s theorem: Semantic applications 83

under consideration. Second, the lexical meaning is fully defined independently of the meanings of any other term in the sentence. A corollary of the above is that all the words in the sentence may be independently replaced by others; the meaning of the sentence will change only inasmuch as the particular change is concerned. It follows, then, that except for the difference in the particular values of their variables the meanings of all sentences generated by the noted substitutions would be on a par.

By way of example consider the following sentences: (2) Mary eats the banana. (3) John eats the apple. (4) David drinks the milk.

Sentence (2) differs from (1) only with regards to the identity of the agent performing the eating, and (3) differs only in terms of the choice of the object eaten, whereas (4) differs in both respects as well as with respect to the type of action taken.

The patterns noted are readily expressed in terms of VFT. The relations between the meanings of the words in our sentences are rigid-they are fully determined by the general and well-defined projection rules. At the same time, the contribution of each word to the meaning of the entire sentence is minimal in that it is confined to a particular specific slot and nothing more. Furthermore, the full substitutability that holds between the words is an expression of what has been labeled, in the social and economic context, “alienation” of the elements that constitute the system.

While the foregoing state of affairs is considered standard, it is not always the case. In particular, the semantics of metaphorical expressions will be noted. The examples to be examined are all taken from the work of the poet Wallace Stevens. Even before considering the poems themselves, let us consider (5) which is the title of one of his poem collections (Stevens, 1971):

(5) The palm at the end of the mind. The phrase presented in (5) may be taken as depicting the poet as a painter.

Even before he writes with words, the poet composes with his mind, and in this he creates as the painter does with the brush that is at the end of his hand. Now, look at the phrase again. In the phrase mention is made of neither a poet nor a painter, nor is a brush noted. Even though the syntactic structure of our phrase is straightforward, the semantic relations between its words is far from being so. The two key nouns “palm” and “mind” are related via an entire web of associations. Given that these associations are not given in the phrase itself, they are not fully determined. Surely, the meanings the dictionary gives for the constituent words contribute to the meaning of the phrase under consideration, but the contribution is not fully characterizable in such simple lexical terms. Consequently, several interesting patterns are noted which on the one hand mark the great and non-predictable contribution of the meaning of each word to the meaning of the entire phrase and on the other hand mark the interactive, non-alienated relationship between the constituent words themselves.

First, the interpretation of (5) is based not only on an association of some meaning with the words as they are known to the reader as a speaker of the

84 Benny Shar~on and Henri AtIan

English language, but also as the individual who is holding a collection of “selected poems” that he or she is about to read. Had the phrase been articulated or encountered in a different context, its interpretation would have been totally different. Second, while the meaning of (5) is, of course, a function of the words of which it is composed, the meaning of the words is itself determined by the meaning of the entire phrase. Given the interpretation of’ the phrase as construed on the basis of the words “palm” and “mind,” a specific, non-standard light is shed back on the meaning associated with the word “end,” namely, one that brings to mind a palette and a brush. Third, this interaction marks how nor1- rigid are the interactive relationships between the words in (5). Given that the meaning of the constituent words is not independent of the meaning of the phrase of which they are part, substitution in the manner noted in the sentences ( t)-(4) is not possible. In the present case, the words are independent neither of each other nor of the phrase as a whole.

‘The contrast between this metaphorical case and the simple literal cases noted before attests to the following complex of patterns. First, the meaning of each word might appear to be-in comparison to standard notions of literal, lexical, or dictionary meaning*-vague and ill-defined. Second, the contribution of the words to the phrase of which they are part is more than it is in the simple literal case. Third, the total resulting semantic effect is by far richer: ‘l‘he phrase presents its reader with freshness, with ;I view of things which is new, with openness that affords the introduction of further, perhaps idiosyncratic meaning, with an involved interaction with the text-sentences like (1)-(h) allow none of these. ‘I‘ogether, the two patterns mark the state of affairs suggested by the two theorems of VF’I‘.? Specifically: ‘I‘he more deterministic is the semantic contribution of a word in the sentence the less is its contribution to the meaning of that sentence, while the more inter-deperldence there is between the meanings of the constituent words the greater is their particular effect on the meaning of the sentence as a whole.

By way of another example, let us consider a particular line which appears in the collection noted; the following is the opening phrase of the poem entitled “The dove in the belly”:

(6) ‘I‘he whole of appearance is a toy. What does this phrase mean? Surely, and in line with the foregoing discussion, in order to answer this question a consideration of the entire poem is required. Given that a literary analysis is not the purpose of’ our discussion, and fol limitations of space, let LIS proceed somewhat artificially and examine this single phrase alone; whether the interpretations to be entertained here are supported by the reading of the entire poem that follows is immaterial in the context of this cognitive discussion.

The presentation of the isolated line to students who had never seen it before

von Foer-ster’s theorem: Semantic applications 85

yielded two main interpretations. One interpretation reads the line as noting the non-reality of life. By this interpretation “appearance” is contrasted with the real, and “toy” marks a non-serious activity. The other interpretation reads the line as noting that the world of human experience is malleable, that it affords playing with. By this interpretation “appearance” is taken to be “whatever appears, ” “whatever is present” and “toy” is an instrument of play. The two interpretations show that the meanings of the constituent words are interrelated. The reading of- “appearance” under one interpretation invokes and is invoked by the interpretation under which “toy” is read, and vice versa. On the one hand, then, the meaning of each word may be said to be less determined, but on the other hand the contribution of the word to the meaning of the entire phrase of which it is a constituent is greater.

The semantic picture just drawn is a case of what has been referred to as “interinanimation” by Richards (1950). The term comes to indicate that in a metaphorical expression the meanings of the two terms (referred to as “topic” and “vehicle,” respectively) bear on one another. While Richards’ work is regularly quoted and elaborated upon, it seems to us that typically, in the cognitive literature, it is not taken as strongly as it should be. Usually (see Black, 1979; Searle, 1979), the interpretation taken is that the conjunction of the terms selects one of the interpretations that can be associated with them in the mental lexicon. A stronger interpretation would be to regard the process not as selection but rather as creation. The conjunction of the constituents of a metaphoric expression do not select amongst meanings that are already given but rather they create new meanings. The examples considered here support this creative interpretation.*

In the poetic phrases presented above the patterns in question are accentu- ated. In them the definition of each word is less defined, more context dependent and less characterizable by the standard dictionary. At the same time the interactions between the words are great. The constituent words are not just put together according to the standard rules of syntax and interpreted by the corresponding projection rules of semantics. Rather, the words are like different colors that together paint a picture. The interactions between them are not clearly defined by any given set of rules, but they are strong to the degree that the entire expression is a whole that cannot be reduced to its mere parts. The resulting expressive effect is rich and unique. All these patterns are noted in the following fragment from the poem “The man with the blue guitar”:

(7) Color like a thought that grows Out of a mood, the tragic robe Of the actor, half his gesture, half His speech, the dress of his meaning, silk Sodden with his melancholy words, The weather of his stage, himself.

*For analyses along such lines in natural, non-poetic discourse the reader is referred to the research of’ “contextual expressions” by Clark (1983), and Clark and Gerrig (1983).

86 Benny Shmon and Henri Atlan

This is not the place to attempt a full critical analysis of the poem. Suffice it to note that the fragment is construed on the basis of various semantic elements, ones pertaining to the realms of mentation, dressing, and acting. It is left for the reader him/herself to verify that all three patterns just noted (lack of clear definition of constituents, their interaction, and the unpredicted, rich overall semantic effect) are all exhibited in the fragment lmder consideration. As already noted these are all the patterns indicated by VFI‘.

The contrast between standard and poetic discourse may also be noted globally. Such a contrast is of interest because it presents the application of the patterns defined by VFI‘ on still another, higher level. Given that the phrases of standard prose are well-defined, their composition is relatively unconstrained. While there are some requirements on the consistency and coherence of entire texts, there is significant latitude in the addition of sentences to a given text OI the deletion of sentences from it. The same applies to the sentences produced in verbal discourse. ‘I-he situation in poetry is markedly different. The phrases of a poem comprise a compact whole. They are strongly intertwined in a structure that precludes either addition OI- deletion. A corollary of this contrast between prose and poetry is the different order of magnitudes of the expressions the) can sustain. Given that the relationships between the sentences of a prosaic text are relatively unconstrained, the entire text cm be very long. In contrast, the highly interrelated phrases of a poem comprise a unified whole and therefore cannot be too long.

The patterns defined by VFT are also encountered in the visual modality. Indeed, the following visual case was presented in the philosophical literature as a model for the linguistic case noted above. I refer to Kichards’ (1950) comparison of the forming of a picture in mosaics on the one hand and water colors on the other hand. The former is analogous to the standard composition of words whereas the latter is analogous to metaphorical expression. Specifically, the elementary units of which mosaics are composed are well-defined and the relationships between them are rigid. Even in the best of cases, the ensuing mosaic picture is rather stilted and frozen. This state of affairs markedly contrasts that presented by water colors. Here both the constituent elements and the relationships between them are ill-defined, but the ensuing picture is rich and vivid.

Together, the two cases present two opposites that once again mark the relationships defined by the VFT. In the case of the mosaic, the relationships of the elements are highly defined but the contribution of each such element to the total ensemble is minimal. In fact, each piece contributes exactly what it is: A uniform patch of color in a particular place. The total picture is just the sum of these patches, and its overall impression is characteristically poor and frozen. In the water color case, by contrast, the constituent elements cannot be easily defined in either hue a- form and there is great flexibility and indeterminacy in the relationships between them. These ill-defined, fluid washes of colw interact, and together they result in a picture that is so much richer and livelier than the one presented by the mosaic.

In sum, the contrast of the two visual cases noted presents the following

van Foerster’s theorem: Semantic applications x7

patterns noted by the two theorems of VFT. First, the more rigid the relationships between the constituent elements, the lesser is their contribution to the expressive power of the entire system of which they are part. Second, the more deterministic is the relationship between the elements the more alienated they are; in that case, the total contribution of the elements is solely a function of their being concatenated, not of any significant interaction between them.

Similar patterns may be noted in architecture. A skyscraper of the “interna- tional style” with its repeating concatenation of columns and windows is akin to the mosaic mentioned earlier in our discussion. Each element (e.g., window) contributes only its being in the particular place and nothing more, there is no interaction between the elements, and the total resulting expressive power is determined and relatively poor. This type of building may be contrasted with the neo-baroque, post-modernistic style as it is most expressedly realized in the works of Venturi (1977). In this architectural work there is a much more playful interaction between the architectonic elements and the overall patterns they define are more complex.

Music also presents a trade-off characterizable by the VFT.* The contrast to be noted is between musical systems in the East and in the West. In the tonal music of the West the individual tones as well as the scales and the intervals are well-defined. The tones have exact pitches, the scales comprise a compact abstract system that does not depend on or vary with its particular instantiation in tones, and the intervals are exact multiples of a unit whose definition is not context dependent. In the East (for example, in India and in Iran), the tones, scales, and intervals are not thus well-defined. The tones are not necessarily of exact pitch, the scales are not defined independently of their musical realization, and the intervals need not be fixed in a predefined abstract set, but rather they may be context dependent with one interval being constrained by the one that follows it. Given this difference in the basic musical elements, there ensues a marked difference in the products that they make, namely, the phrases or motives in the two musical traditions. In the Western tonal tradition, the unit of composition is in essence the sequential concatenation of the tones of which it is composed. At the same time, the melodic phrase or the motive is not of central significance. Beautiful as melodic phrases might be they are not the locus of prime significance; this is the case both within each musical piece and in the overall theoretical framework of the musical tradition in question. The situation in non-Western musics is generally different. There, the phrase or the motive is a level of utmost musical significance; again, this is the case both in each given piece and from a general theoretical point of view. Thus, unlike Western music, Eastern music presents highly structured systems on the level of the phrase or the motive. Moreover, in most cases, the structure is also noted on the rhythmic dimension (examples are the Indian tala and the Arabic mizan; for details see Cohen, 1986). Thus, a set of well-formed phrases or motives is defined. These are relatively short and they are primary both in terms of their musical-

*The musical case is based on discussions with Professor Dalia Cohen, whose stimulating and helpful comments are highly appreciated.

88 Betttt! Shanon antI Hettti Atlatt

theoretical status and in terms of their aesthetic effect. In the Western musical

tradition such a state of affairs is totally illcoIiceiv~ible.

‘l‘he musical case is of f‘urther interest I;ecause it presents a second chapter to

our story. Just as phrases may be composeti of‘ tones and iriterwls, large trlusid

compositions are composed of‘ phrases. C;iven that the motives in the East are in

general highly constrained whereas in the West motives are subject to \‘ery little

constraint, the logic defined 1)) the VP’I‘ would lead 01le to expect the

conipositions of motives into larger musical units to present ;I reversal with

respect to the patterns we have observed so fu-. Indeed, this is the case. (;i\:en its

unconstrained repertory of’ tnelodic motives Western music has created

magnificent structured architectures 011 the large scale. Bv cant rat, the East has

offered richness and maximal sensitivity in nuance olt the small local I-illlge but

it has not generated pir~illel cl-eations 011 the large scale. Given the vt‘rb

structured and highly constrained system of motives, the c.onipositious on the

larger scale do not exhibit sophisticated structures. Whereas in the West a whole

repertory of‘ large-sale structures has l~eri developwi (of‘ which the para-

digmatic case is the sonata fiwni), in the East the longer compositions are

either loosely structured forms or at times eveu mere ongoing corlc.aterlatiolis ot

phrases. A corollary is the strong internal struc‘t ure and closure of entire musical

conlpositions encountered in the West but not in the East. A hlomrt sonata is

c-oniplete in that, its end is determined Iy the musical material that precedes it-

it is ;I completion that could not have occurred earlier in the composition;

likewise, given its end, the sonata cannot be continued f’ul-ther. In the East this is

usually not the case. Pieces are fixed in lieit her respect: ‘Ilie), ni;iv be ended

prior to any envisaged set point, or they may be continued beyond ‘\vliat ~oulti

have been their end. Indeed, in Byzantine music “mosaic cornposition” (Welles/.

1% 1) is encountered in which the same elements may be ordered iu tliftu-ent

permutations thus generating different musical pieces; surely, no such patterns

could apply in tonal music of’ the West.

While the cross-cultural comparison presents marked contrasting polarities,

the basic patterns noted may be employed in the differential ~har~tcterization of’

styles within a culture. Specifically, the paradigmatic contrast between the so-

called classical and romantic styles (cf., Woelf‘flin, 19.50) presents l~~r;~llels to the

patterns noted. Both in the visual arts and in music, in the classic style elements

are well-defined and the total experienced ef’f’ect is less complex. whereas in the

romantic style elements are less defined and more interactive, arid the lvhole

experienced stimulus is more complex. Such stylistic contrasts are, of’ course,

stereotyped polarities. In practice, the various patterns--across cultures,

between styles and in the different levels of’ a coml”)sition--will vary in degree

and exhibit different choices and trade-offs. ‘I‘he present ciiscussion sets itself

only to define the overall principles, and only as these patterns are relevant to

the VFT; the examination of’ details in both language and in the arts is left f’ol

f’urther investigation. It is hoped that the distinctions introduced here in line

with the VFT can, indeed, be applied in the character-izatio~i of’ works in various

domains of artistic expressions-in the characterizatior1 of individual style,

historical <genre, and entire cultural traditions.

van Foerster’s theorem: Semantic applications x9

In closing, the VFT and the phenomena we have examined may be viewed as they bear on a more general, and more fundamental, issue, namely, the relationship between noise and emergent organization. Since previous works by von Foerster (1960) and Atlan (1974), this issue has been the subject of discussion in both physics and biology (Atlan, 1979, 1987; Kauffman, 1984; Vichniac, 1986). It has been pointed out that noise, as a source of indeterminacy, increases complexity and thus enables the emergence of organized patterns. Specifically, noise destroys rigid ordered patterns and thereby generates increasingly complex organizations. Relevant patterns in biology are the role of random mutations in phylogenetic evolution and in the maturation of the immune system in individuals (Edelman, 1987). Taken from a functional point of view, such biological patterns of complexity may be regarded as the expression of meaning (Atlan, 1987). S imilarly, in the physics of spin glasses (Hopfield, 1982; Anderson, 1983), noise is produced by temperature and- rather unexpectedly-it has an important role in the stabilization of emerging patterns (Kirkpatrick, Gelatt, 8c Vecchi, 1983; Amit, 1987). ‘I‘hese ideas have been also implemented in models of neural networks which have subsequently received cognitive interpretation (Hinton 8c Sejnowski, 1986).

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