An Introduction To Cross Media Mapping

(Based on a reading given at the '6e Journee Internationale d'etude de Musique Electroacoustique', June 1976, in Bourges -- France)

Trevor E. Batten
Amsterdam

I. Language As A Basic Concept

To begin with, I'm going to talk about the concept "language", this is chosen partly because it is a phenomenon fundamental to our daily life, and partly because of the present tendency to turn to linguistic models in order to comprehend better the phenomenon "music".

i. Exchanging Signs between Friends

But first we need a definition: A language is a system of communication used between members of a linguistic community.

This is of course tautological, and some form of definition for the concept "communication" would be useful.

While refusing to reduce the entire procedure to a simple stimulus -> response situation, it can be fairly safely stated that communication has taken place between two individuals when A transmits a sign (or set of signs) and B produces a response which is satisfactory to A.

Here "sign" is referred to in its more or less traditional interpretation as a physical signal plus a significance. The mention of the word significance immediately conjures up questions such as "significant to whom? And significant of what?"

So without becoming involved in the complex problems of defining a signal or how it acquires a significance, at least we have a more than strong suspicion that a sign is a subjective phenomenon and that the same signal may in fact form different signs for different individuals.

ii. Signs and Compound Signs

It now becomes possible to expand the given definition of a language; it can be said that a language consists of a basic repertoire of signs, plus a set of rules permitting the formation of compound signs from the basic repertoire.

The use of the word sign implies that every output of the language, must per definition -- have a significance. In addition, compound signs may well have a significance which differs radically from the original components.

Because different signs may be formed form the same signal, or conversely, different signals may be given the same significance, a linguistic community can be defined as a group of individuals where each member assigns the same interpretation to each element within the basic repetoir of signals, and uses the same rules to generate compound signs.

iii. The Inescapable Tautology

This remains a fairly tautological definition, but a moments consideration will show that any description or definition, must either rely on undefined terms and thus remain incomplete, or else become a closed definition -- and therefore tautological.

It is a strong personal belief that tautology plays a fundamental role in communication systems, but the precise nature of this role, and the question whether a tautological statement can be of value or not -- will, like so many other questions, temporarily at least, remain open.

iv. Language as Generalised Concept

Such a definition for language as just stated, includes natural languages, artificial languages, and possibly animal communication systems (dependant on how strongly the conditions requiring the production of compound signs is interpreted) but it certainly includes art languages in the form of painting, sculpture music, drama, etc.

These "art languages" are often divided from "verbal languages on the grounds that the latter have specified interpretations for the repertoire of signs while the former have no specific interpretations.

This division is perhaps worthy of further investigation.

v. Language as Varied Praxis

It was stated earlier that although the test that communication had taken place involved a satisfactory or correct response. This should not be taken as implying that a simple stimulus -> response mechanism was operative.

Even with verbal languages, where the interpretation of individual signs is assumed specified, often several exchanges need to be made before communication can truly be said to have taken place. Sometimes new words are introduced, or existing words need clarification. Statements and answers are gradually modified until the communicants are reasonably satisfied that they have approximately the same idea of what was said.

However, a satisfactory answer is not always elicited, and sometimes even highly unsatisfactory responses may result.

In some cases this results from a difference of opinion (the definition of which will also be left open), but often it is the result of a genuine disagreement over the interpretation of a specific sign or group of signs.

Sometimes, the difference between opinion and interpretation may be difficult to distinguish, for example with such concepts as art, democracy, freedom, etc., opinions regarding their modes of operation and usefulness are almost inextricably interwound with the definition of the word.

On the other hand, statements regarding whether a walk in the rain is pleasant or not, are purely a matter of opinion, and independent of interpretation problems in the sense just mentioned. In such cases as these, judgement of a satisfactory response must be more in terms of it appropriateness to the context than to a correct opinion.

Interesting and complicated as the precise relationship between opinion and interpretation may be, there is insufficient time to discuss it further. However the fact remains that within a so-called linguistic community there are differences in interpretation.

For example, in American English "sidewalk" is equivalent to "pavement" in English english; while "pavement" in American English is equivalent to roadway in English english.

This has the result that an American and an Englishman will produce entirely different responses to a sign composed of the words "automobiles must be parked on the pavement"

These differences in meaning are due either to different interpretations being assigned to the same physical signal, or to apparently equivalent interpretations being assigned to different signals, imply that the general linguistic community of "English speaking people" needs to be further divided into subcommunities which may be referred to as "English dialects".

vi. The Smallest Dialect?

But does a dialect deliniate the smallest group within which there are no more variations between signal and interpretation?

Perhaps a personal experience here can answer the question. One day while talking to my mother I was surprised to hear her suddenly say "Hey, look at that silly tit hanging upside-down nibbling his nuts."

Being a little shocked by this remark, I looked in the direction she was pointing and seeing a bird feeding in the garden, realised that she and I had interpreted the sentence in two completely different ways.

It would seem that the sub-category of dialect may need to be further subdivided into "idiolects". In other words, the linguistic community may have only a single member.

Should this appear to be a hasty decision to be made on the basis of a simple example perhaps it can be justified by considering variations in "style" and the use of personal idioms which most people use in their speech.

vii. Individualism in Word and Thought

It may even be possible that the variety of psycho-linguistic theories is not a result of misunderstanding a single objective reality, but the result of differences in the linguistic strategies used by the authors. In other words, because their individual use of language is different, their theories are different.

Or, one could simply ask why do so many misunderstandings occur, and why is it sometimes so difficult to communicate verbally, if both communicants are using identical rules to assign and combine interpretations.

Another basic assumption that is used to divide art-languages from verbal languages is a belief in the existence of objective concepts external to the language, the purpose of a specific verbal language being to communicate these concepts.

It may be debatable whether or not a musical language refers only to musical concepts, but the belief that it is possible to translate, for example from French to English, or vice-versa appears to imply the existence of a something that can be translated. A significance, as it were, that only needed to be assigned to a new signal in order to be translated.

viii. Problems of Translation

Early attempts at machine translation of texts soon showed that translation was not a matter of interchanging words with similar meaning and arranging them in the correct grammatical order.

For example, for my own interest, I recently used a dictionary to translate all the meanings of the Dutch word "opnemen" into English, then each English equivalent back to Dutch, and finally back to English again.

The most general translation for "opnemen" is; to take up, which can also be found in the construction of the word; i.e. op => up, nemen => take.

However, also listed is the translation "take down (stenography)" as in the English sentence, I'll just take down your address!

The apparent contradiction between "take up" and "take down" can be resolved by realising that simultaneously the address is both "taken up" in memory, and "taken down" on paper. In this case the piece of paper is also the memory.

Clearly, the focus of attention is on one of two different aspects of the same action, or in other words the action is seen from two different viewpoints.

Another translation for "opnemen" is "collect (votes)". The translation for collect is "ophalen", which in turn generates; draw up (a bridge), pull up (blinds), raise (a curtain), weigh (anchor), shrug (ones shoulders), turn up (ones nose), collect (money).

At first sight it would appear that English speakers like to make life more difficult by using different words to specify the same activity when performed with different objects.

A closer inspection shows that while in some sense this is true, it does not apply to all cases -- for example, to 'draw up', to pull up,' to 'raise', or even to 'weigh' a bridge are to some extent interchangeable, and can convey the idea of a bridge being lifted, even if some flexibility of interpretation is required for individual words.

On the other hand, the phrases, to 'shrug a bridge', to 'turn up a bridge' or to 'collect a bridge', apart from having a surrealistic effect, produce images that are both radically different from each other, and from the previous set of phrases

If there were external concepts, existing outside the language -- a kind of disembodied meaning -- waiting to be assigned to a signal in order to be communicated. Then it could be expected that a simple relationship would be apparent between words of different languages.

Instead, one finds complex networks of meaning that are impossible to relate exactly to each other. In fact, there are intersections in one place, and not in another, the network may be shifted to produce a better relation in one area, but the difference grows wider somewhere else.

Bearing in mind that Dutch and English are closely related languages, this basic mismatch between the two conceptual networks would appear to be a serious threat to the belief in a set of external translatable concepts.

ix. Communication Impossible?

R.L. Gregory, in a paper entitled "Will seeing machines have illusions" -- in Machine Intelligence Vol. 1 -- Edinburgh University Press, suggests that if an observer, or intelligent machine, is sent to a totally strange environment, initially the messages sent back may be understandable but totally misleading because the sender is interpreting the new environment according to principles found reliable in the old.

These principles may be totally misleading in the new environment and the observer will therefore need to learn more appropriate methods of interpretation before the returned messages will be truly representative of the environment.

The result of such a learning period may then mean that when the new environment was fully understood by the observer, the messages transmitted back may well be unintelligible because the receivers of the information do not have the conceptual framework necessary to understand these messages.

To be honest, Gregory was actually referring to situations involving visual perception, and trying to draw attention to the role of cognitive processes and prior experience in perceptual operations.

However, the problems of translation may not be so different from the problems of the observer transmitting descriptions of an environment understood by the observer, but not by the receiver of the messages.

I remember at the 1975 'Journee d'E9tude de Musique Electroacoustique' when someone was asked if he would translate his paper into English, he replied that if he tried to do so the result would be something completely different from what he had originally said.

Unless I am mistaken, this was more to a difference of thinking modes in the two languages, than to a basic lack of English vocabulary.

II. Cognitive And Perceptual Spaces

If cognitive processes are involved in perception, it is not unreasonable to assume that perceptual information plays an important role in cognition; in fact if divine intervention is ruled out, then the knowledge of the environment essential for survival, can only be obtained through the sense by communication with others, or as a result of some kind of deductive thinking. Should cognition and perception prove to be inter-related, then a tautological system would seem difficult to avoid.

However, this is a somewhat advanced point in my argument. At present I will be satisfied with having drawn attention to the fact that verbal language not so safe and efficient as our daily reliance on it would appear to presuppose, that it is in fact concerned not with the simple assignment of external concepts to a pre-agreed signal set, but with the more fundamental task of searching for intersections between separate and individual logic systems, that are in turn based on varying differences in perceiving the world in which the authors of these systems live.

But what relation does this have with art, or, to be precise, music? Well one hardly needs to be reminded that the concept music also covers a wide range of different sub-categories that can also eventually be reduced to a group with a single member -- i.e. the individual composer or musician

i. Artistic Dialects

Each composer has his own definition of music, his own concepts of how it operates, how it should be made, what the major problems are, and how they should be solved. This is to be expected, we live in a time of exploration and expansion of new concepts, but we shouldn't let this delude us into thinking that music is by definition a set of systems with non-explicit interpretations and therefore automatically opposed to the explicit codings found in verbal communication.

I have tried to draw attention to the fact that verbal systems are not so explicit as at first might be imagined, but conversely, it should not be forgotten that although it is now unfashionable for music systems to have a set of specific semantic assignments it is no more or no less impossible than with verbal languages. Perhaps the two communication systems are not so radically different.

ii. Artistic Dialogue

Until now, focus has been on the difficulties of communication, and the impression may have been created that I am trying to prove that it impossible. Clearly communication does in fact occur. Or at least the condition that a satisfactory response should result from the transmission of signals is sometimes fulfilled, and so it can be assumed that communication has taken place.

Before the objection is made that this response test is not applied in music, and therefore by my own criteria music is not a communications medium. I refer to musical feedback as manifest by discussion, compositions by other composers, or improvisations.

iii. Mapping Conceptual Space

But what is meant by communication? My personal view is that communication is basically a process concerned with the integration of two separate conceptual models into a single conceptual space, which is common to both systems, or an intersection of them both.

This process occurs on many different levels; it can be seen on an interpersonal level in the verbal interactions involved in discussion, or when making a new acquaintance. The reinforcement of this common space can be seen on a social level in the communal rituals that a society uses to define its own identity. It occurs on the micro level when a bundle of concepts become integrated by a common word. In the theatre, the world models of the individual characters and their interactions may be integrated by the audience. In music, apparently dissimilar structures may suddenly be resolved by a common origin.

Communication is not limited to exchanges between two or more individuals. Whenever an individual has two different concepts or models that become suddenly integrated, it is possible to speak of internal communication.

Such an event may occur in a child, when it relates a tactile sensation to a visual stimulus, by for example, placing an object in its mouth.

Or the process may occur in the electronic music studio -- for example, when a sound is mapped into a waveform by integrating the description in terms of audio perception with a description based on the visual perception of an oscilloscope image.

At last the reason for the title of this paper should be clear. My main interest is in the construction of conceptual models and descriptions: The potential effects of using different descriptions for the same phenomenon, and the results of trying to map, or relate, one description to another.

It is a personal belief that scientific, cultural, and artistic concepts are related to, and possibly derived from, the qualities of the description language used: And that mappings from one description language to another form an integral part of communication and research.

III. The Ontological Nature Of Language

i. Descriptions are Prescriptions!

If concepts are influenced by the description language, it cannot be neutral.

On the negative side, this implies that distortion and misunderstanding are not only to be expected, but almost unavoidable when communicating.

On the positive side, these "distortions" may prove beneficial in breaking through tautologies and generating new insights into old problems.

This is of course a rather large and vaguely defined area in which to explore. The rest of the paper will therefore be concerned primarily with the construction of descriptions, with the intention of building a foundation for further investigation of the specific properties of descriptions, and their interaction.

ii. Description Strategies

There are perhaps three basic strategies for constructing a description, which produce the following basic types:

1. An Operational Description

i.e. a description in terms of the operations performed on; or with the subject, or operations performed by the subject.

These descriptions are often in terms of internal motor-sensory mechanisms, but may be verbalised by such statements as "it's used for opening tins".

However the same strategy is also in formal logic systems where a set of axioms may be used to describe an operator in terms of the change its effect on the elements. In other words a truth table in two valid logic is in fact a description of the operator.

While being aware of the apparent confusion between description and definition, I feel that this is a problem that can be solved best at a later date.

2. An Analytic description

i.e. a description in terms of basic features or qualities that the subject is assumed to have. Such a description would be "a large hairy ball with green ears."

For completeness, relationships like above/below or greater than/smaller than are also included as features.

It may also be useful to distinguish between perceptual analysis where the "sensory image" of the subject is described, and a conceptual analysis where the sensory analyses have been co-ordinated and projected onto the subject.

The psychological difference can perhaps be seen in visual art by comparing a painting with apatial perspective -- which is perceptual, an icon, where visual proportions are derived from conceptualised relations such as "the most important or least important figure in the picture".

Both the history of western painting and the development of visual skills in young children suggest that conceptual analysis is developed before explicit perceptual analysis.

3. An Analogical Description

i.e. something is described in terms of something else. Such a description would be "just like an apple".

The difference between metaphor and analogy is so slight that I propose to ignore it, but it is not unreasonable to suggest that all conceptual or physical models are considered to be examples of analogy.

To the above comments, can be added the remark that given a specific signal there is no a-priori reason for a particular strategy to be used. The signal may be analysed, it may be given an analogical description, for example, "it's just like a scream", or it may be described by its effect, for example, "it gives a pain in my ears". All three descriptions may even be combined to form a single concept of the event.

iii. Compound Descriptions

It can also be seen that a "nesting of descriptions" occurs, i.e. to say "it's just like an apple" implies that a description of an apple exists. Or an analytical description may be in terms of components as for example in a wall which is made of bricks, where the bricks would represent another level of description.

This process is also echoed in formal systems, where for convenience definitions are used as abbreviations by substituting a long statement by a shorter statement or label.

The "nesting" of descriptions reintroduces the problem of tautology by reason of the fact that our conceptual image of the world would appear to be either based on an infinite regress of definitions, or else these definitions must in some sense be mutually supportive, and therefore tautological.

iv. Compounding Errors?

Seen from another standpoint, these interrelations between descriptions may be considered as being similar to computer sub-programs, not only because they are necessary means of organising vast quantities of data and frequently recurring operations, but also by virtue of the fact that one small error within one of these sub-structures could have drastic consequences regarding the performance of the whole structure.

v. The Chicken and the Egg!

However, the area of investigation at present is primarily concerned with the construction of descriptions, and here too a problem of infinite regression appears.

The problem is that an analytical description would seem to be a prerequisite for the other description strategies. For example, unless there is an analytical description for two objects, or situations, which shows them to be alike -- how can one form an analogy for the other?

But if an analytical description is required, it becomes difficult not to ask how the concepts that form the basis of the analysis are distinguished from each other without the use of another analytical description language.

vi. The Basic Analogy

On the other hand, if analogy is taken as being a "basic description principle" it could be said that if A can be used as an analogy for B, then A is a description language for both A and B, so the condition that a common description for both is necessary, becomes fulfilled -- without the need for analysis.

Should this seem like an irrelevant linguistic trick, a cursory glance at child development would show that a child can correctly assign the label "tea cup" to an object, long before he can say "hollow thing, with a handle, used for drinking tea!" So it would seem that identification can be operative without explicit description.

It can also be seen that any individual teacup can be used as an analogy for any other.

IV. Finding The Object -- The Perceptual Machine

i. Separating The Flow

The problem, however, is not completely solved. Still remaining is the question of how it is possible to separate the constant flow of sensory information into discrete objects without the use of analysis. The assumption being that without two discrete objects it would be impossible for one to form a description of the other.

ii. Implicit "Being" and Explicit "Description"

Before going further, it is necessary to make a distinction between the "implicit" language actually used by an organism, and an explicit description such as would be necessary for an observer of the organism.

The situation is similar to that of a machine with an observer, the machine reacts to its inputs and exhibits certain states -- an observer would require a description language to refer to the activities of the machine, while the machine is able to function without a description of itself.

iii. Parallel Filters

It the system has a quantitative input, and produces a non-quantitative output (i.e. the system responds in different ways -- instead of an increase or decrease of a single reaction) -- then it could be argued that an explicit analysis of the input is required, and that this presupposes a description language.

While this may be true for a sequential machine, the parallel use of detectors similar to band-pass filters should permit the automatic analysis of as many states as it was desirable to decode.

A similar mechanism is in fact found in the eye, where special cells, by only reacting to certain frequencies of electromagnetic waves are able to generate the impression of colour.

iv. The Colour Generating Hardware

Note, the emphasis is on generation of colour, and not detection of colour. The reason for the generation is easy to understand if one compares the task of trying to remember a specific tone of grey, with the task of trying to remember a specific colour.

While not having done the necessary research, I suspect that the other sense organs work on similar principles.

v. The Decaying Memory

But the question of how to isolate discrete objects remains. This could perhaps be solved by use of a decaying memory trace which needed to be refreshed constantly -- as a filter for "frequently recurring" and "not frequently recurring" sensory patterns.

If a filter like this was used in conjunction with the system of band-pass type filters just described, the relatively constant sensory pattern remaining in the memory would form the object, while noise and transitory effects would be automatically lost.

vi. Pavlov and Freud

Association and conditioning would be fairly simple to explain within such a model, due to the fact that the entire contents of the memory would represent the object, and only later experience would show which implicit features were relevant, and which were irrelevant.

It may be possible that concepts such as cause and effect, and implication are at least partly derived from this basic mechanism.

vii. Internal and External

Assuming that such a system was able to correlate between the various sets of sensory information by reducing all information to a common internal code, then it should be possible for the organism to distinguish between external "tangible" space and its own internal "intangible" space; "objects" and "operation" would then be definable in terms of non-motor sensory co-ordinates and motor-sensory co-ordinates, and therefore only expressible in terms relative to the organism itself (as observed by Piaget in young children).

This is only a rough sketch, and is not intended to deny the role of motor-sensory co-ordinates in perception, but it does indicate that at least in theory the apparent preconditions for analogy can be satisfied.

viii. Sticking on the Labels

Although it is impossible to state when, or why, speech developed -- the association of verbal labels with sensory co-ordinates should be a less difficult problem than that which it provides for later generation, namely the problem of decoding the exact interpretation of a verbal label.

ix. Looking Under the Label

The fact that any member of a class of objects, may be used as an analogy for any other member of that class, when combined with the constancy of a verbal label as opposes to the transitory nature of the sensory co-ordinate to which it refers -- could be a source of complex philosophical problems, such as the existence of paradigm cases, etc.

The Classical Greek preference for the conceptual label and their dislike of the troublesome sensory information, which formed the basis of the split between Western and Eastern modes of thinking may have lead to scientific advances -- but we must remain wary of the danger of becoming trapped in our own descriptions.

x. The Mystery Explained?

Any conceptual systems should be capable of dividing the observed phenomena in the world into two categories, i.e. phenomena that are explainable and those that are not.

Logically it is impossible to explain the unknown, except in terms of the known.

V. Defining The Object -- Refining The Analogy

i. The Problem with Analogy

Primitive analogy has one great limitation as a method of explanation namely it is impossible to specify in which sense the analogy is true. For example, a statement such as "a cabbage is like a rose" could cause considerable misunderstanding which could be removed by stating "a cabbage has a similar structure to a rose."

ii. Back to the Chicken -- Or the Egg!

Suddenly the old problem reappears, how is the concept 'structure' to be derived without an analytical description language, and how are the concepts that form this language derived without an initial starting point.

Without going into complex detail, I would suggest that these concepts are derived initially from a description of man's own non-directed play activities, after the discovery that he had by accident "made something".

In other words, the process of making a model comes first, and only afterwards is it discovered what this is a model of.

iii. A Basic Discovery

For example, a man idly making marks on a surface with a burnt stick, suddenly discovers it "looks like" a bull; by further experiment it becomes possible to draw a lion etc. His own activities would then present the controlled situation necessary to develop a description language, and he would be able to analyse the real subject in terms of the operations involved in producing the model.

iv. The Magical Power of Models

Initially, because his basic conceptual system was operating on principles of analogy the possibility of confusion between the actual object and the image would be great.

This would lead to the development of "sympathetic magic", the power of which would provide the incentive to continue experimenting with the production of models, which would in turn generate more analytical concepts.

v. From External to Internal

However, because I have given a visual example, it must not be implied that these are purely visual processes. It would seem impossible to analyse pitch structures, for example, until pitch had been identified as being a function of the dimension of the sound-producing object, even to the point that the relative lengths of these objects form the basis on which the pitch structures are organised.

What is implied, is that the invention of written music would be almost impossible without the prior invention of instruments-irrespective of whether the notation was specific to the instrument, or mapped into a single hypothetical "objective" space as occurred in the West.

Also implied, is that knowledge of the method of sound production is important for the "syntactic" interpretation of sound structures, for example on the simple level of distinguishing the interplay of elements by correctly assigning timbral qualities to their sources. One can therefore suspect that electronic tape music will continue to be an exotic science-fiction sound effect for the majority until the public has had more experience with the means of production.

It is also possible that in instrumental music, knowledge of the motor-sensory activities involved in sound production may be linked in some cases to "semantic" interpretation.

vi. Developing the Experiment

The availability of analytical description permits the development of experiment -- which basically consists of manipulating a described object and testing to see of there is a change in the description as a result of the operation.

This constancy, or lack of constancy, in the description before and after the operation, is of fundamental importance, and forms a basis for defining the geometry of the system under investigation.

vii. Building the Geometry of Conceptual Space

In fact, in formal systems, a geometry is defined by the changes in the description space that result from the application of an operator. These changes being defined by the construction of the axioms.

Experiment is, therefore, essentially a search for the geometry of the system.

viii. The Crucial Description

The role of the description language is of vital importance under these conditions, for example, an unchanged description may mean that the subject is unchanged, or, that the description fails to register the change. It becomes impossible to prove that the object has not changed, and only possible to search for a new description, or to admit that a specific time no such language exists.

It can be seen in this context that the most difficult task for the public when faced with contemporary music is to decide which description, out of a wide range of possibilities, should be used to interpret the specific composition they are listening to.

Also implied is that a search for a single meta-language with which to analyse every type of music is in fact a misdirected task, and that a phenomenological approach related to the actual description used by the composer would be more useful.

ix. Inconsistency and Completeness

It may happen, with an experiment, that a feature that is detectable but not describable is generated by an operation. Such a feature can only be given an operational description in terms of the specific situation in which it was first discovered.

The next step is to test other descriptions to see if they can be applied to the problem of isolating the new feature.

Assuming that such a language is found, this feature forms an intersection between the old operational description and the new analytical description so that a new item of knowledge is generated -- namely operation X results in a change in parameter Y.

Thus knowledge in this case, resulted from the discovery of "incompleteness" in the original description (or the "inconsistency" inherent in a situation that is both changed and unchanged) -- the application of a new description, and the final integration of the different descriptions in order to resolve the original inconsistency.

x. Truth or Correlation?

In this sense, "objectivity" clearly refers not to a "real world" but to the ability to make correlations between different 'description languages'. So that the distinction between 'scientific' and 'non-scientific' models may be found in the degree of consistency within the model and the possibility of correlation with other descriptions.

Under these conditions, if all known explanations can be mapped into a single description or a single description can explain all known phenomena, then this description is assumed true.

It can clearly be seen, that if all known phenomena can be explained in terms of 'God', the 'ether', the 'orbits of electrons', or little green men from Mars, then these concepts are assumed true, until proved incomplete or inconsistent.

VI. Developing The Immaterial World

Until now, consideration has been given to the development of conceptual geometries based on physical experiment, or derived form physical signal patterns such as might be manifested by a musical structure or a visual composition.

i. From Activity to Rule

However, the verbal labels used for objects and operations, by reason of their abstract nature, need to be manipulated by rule, as opposed to the physical operations of earlier systems.

ii. From Observation to Invention

The rules by which abstract systems operate must presumably be derived, initially, from observations and experiments in the physical world.

However, another characteristic of such formal systems, is, that instead of preceding from observation to hypothesis, the procedure can be reversed:

Thus the basic mode of operation becomes "if a system (or geometry) was based on these basic principles (or axioms) then this would be the result."

iii. The Inevitable Tautology

As demonstrated earlier, the removal of inconsistencies and incomplete descriptions not only leads directly to an increase in knowledge but also to tautology.

For example, if a formal system is used to describe all musical structures, and this description is then used by composers, the prophecy becomes self fulfilling -- as for example with traditional tonality -- which was presumably assumed to be an objective system.

iv. Changing the Rules

Although, to a certain, ambiguity may be useful in breaking out of tautological systems, this is not encouraged in formal systems -- but they do have one advantage.

Because they are not bound by physical laws -- it is possible to change the rules (or basic axioms) on which the system is based.

In this manner it becomes possible to explore new geometries, some of which may be found applicable to specific situations.

v. The Formal Arts

In this sense, formal systems are somewhat similar to the systems found in music and the other arts. The constructor of the system is permitted to build his own world, and to explore the consequences.

If these consequences appear to have counterparts in the so-called "real world", so much the better, if not then it was still an interesting game to have played.

Published in Nieuwsbrief 56, Systeem Group Nederland -- Oct/Nov 1977.