Typology (old: now reposted at two articles)

"Very helpful"

Types appear in every kind of system model, in the roles of a human activity system: in the variables related in a causal loop diagram, in the stocks and flows of System Dynamics, in the entities and events of a data processing system, and the classes of a software system.

The types in these abstract (description time) models represent instances or occurrences in real (run-time) systems. If there is a general system theory, it must include the generalities of how we define types and label them using tokens.

Generally, a type describes one member of a category or set. You can find particular type theories in several domains of knowledge.

• In mathematics, types are assigned to objects in a branch of logic, a formal system, that is an alternative to set theory as a foundation for mathematics.
• In software engineering, types are used to classify and manipulate kinds of data and functions.
• In linguistics, words and phrases are categorized into types based on their function, such as "noun," "verb," or "proposition," which enables formal analysis of linguistic expressions and how they combine in a sentence.

Mathematicians, software engineers and linguists may purposefully create an object to "instantiate" a given type or class, in a precise and complete way.

In everyday communication, most types more are organic, soft-edged or fuzzy. Meaning that the conformance of a thing to a type (especially a complex or compound type with several elements) is a matter of a degree or judgment.

Ideas in my articles about semiotics and systems thinking include that verbal languages help people to forrm hard, precise mathematical types, but they evolved from soft, fuzzy, organic types, which in turn evolved from likening one thing to another.

An organic “type” may correspond to a “predicate” in predicate logic. And connections between types may correspond to “implications”. But nothing that follows depends on you knowing anything of maths, linguistics or predicate logic.

Contents: A glossary. Four distinctions. 14 assertions to be explained. Organic types. Tokenless types and typeless tokens. Description as typification. Corresponding type definitions. The nominalist or conceptualist view of types. Types and sets. The symphony example. Yet more about types. The many-to-manyness of systems and entities.

A glossary

Much of what philosophers write about types, tokens and instances is a tough read. Consider these quotes from the Stanford Encyclopedia of Philosophy (SEP) entry on "types and tokens".

• "The distinction between a type and its tokens is between a general sort of thing and its particular concrete instances." (Surely token not = instance?)
• "Types are generally said to be abstract and unique; tokens are concrete particulars, composed of ink, sound waves, etc.". (Surely those are instances of tokens?)
• "A related distinction is that between a thing, or type of thing, and an occurrence of it—where an occurrence is not necessarily a token" (Does this imply an occurrence is not an instance?)

You might struggle to decode philosophers' essays for various reasons: the obscure language used by academics, examples in which things are words rather than rather than material things, obscure references to other philosophers who use the same words differently, and perhaps above all, the lack of a glossary to disambiguate their text.

So, before embarking on a lengthy discussion, let me define the terms in this triadic graphic.

Thinker: an organism (or AI?) that creates descriptions of things in the course of observing and envisaging them.

Thing: an object or process, entity or event observed or envisaged by a thinker, and describable by typification, such as a plant or a lightening strike.

Token: a name or label for a type. For example: “circle”, “rose bush”, and “Newton's first law”.

Type: an intensional definition of a thing. A quality, category, or a compound of them, that is creates or used to describe things observed or envisaged. It includes features a thing needs to qualify as an instance of a the type.

Note: any token may be associated with a type, but mutual understanding depends on sharing token-type associations. For example:

• Circle: a line drawn around and equidistant from a point.
• Rose bush: a thickly growing, flowering, plant with thorny stems.
• Newton's first law: Force = Mass * Acceleration.

Instance: an example, embodiment or manifestation of a type by a thing.

Note: there are recursive possibilities. A thinker or a type may be seen as a thing to be described. Things and types may be seen as instances of the types "thing" and "type".

Set: a collection of (0, 1 or more) instances that are either typified or enumerated.

Note: a type describes a thing that is an instance of the type; it does not describe the whole collection or set of things, which has attributes of its own.

The concept graph below may help to illuminate the text here.

Related terms

The term "instantiation" may refer to either the existence or the creation of a thing that is correlatable with a given type. You may say a thing is an instance or instantiation of a given type. Or else say, a thing or a process instantiates (creates) an instance of a given type.

Instantiation: the existence or creation of a thing that embodies or manifests a type, or makes or performs an instance of a type.

Occurrence: an instance of a process rather than object type.

Archetype: usually, an iconic model or example that serves as a type. Sometimes, (as in Senge's system archetypes) a generic symbolic type.

Four distinctions

In discussion, we sometimes speak of different concepts (such type and set, or instance and thing) as though they are the same, and sometimes it makes no difference to the discussion. But let me draw four distinctions.

Distinguishing token from type

Typically, we define a type in association with a token. For example, "circle" is a token for a type. We use that token as symbol or proxy for the definition below.

• Token: type
• Circle: a line drawn around and equidistant from a point.

In a circular and curious way, the word "type" is a token for the type "type".

• Type: an intensional definition of a thing. A quality, category, or a compound of them, that we create and use to describe things we observe or envisage.

Distinguishing token from instance

Given this Gertrude Stein line: "Rose is a rose is a rose is a rose", the SEP says Charles Peirce would say there are 3 word types and 10 word tokens.

Using terms as defined here, the line contains 4 instances of the token "rose", 10 instances of the type "word", and no instance (no rose) of the "rose" type.

Distinguishing type from set

People often say that a type defines the set that contains instances of that type. More strictly speaking, a set defined by:

• extension is defined by listing its members (we need no type).
• intension is associated with a type that describes one set member.

For example, the type “planet” includes the features that qualify a thing as a set member. The whole set (all planets) has its own features, such as total members.

And note that a type can represent feature(s) of an imaginary or envisaged thing. As soon as we describe one unicorn, we can envisage there being 0, 1 or more instances of that type, even though the current is empty, and likely to remain so.

Distinguishing instance from thing

People refer to a thing as an instance of a type. But outside of pure mathematics, a type does not describe all the qualities of an individual thing that instantiates the type.

For example, a territory is much more than an instance of the features in a map. And Mars is much more than an instance the type "planet".

More accurately, there is a many-to-many relationship between types and things. One type can be used to describe many things; one thing can be described by many types.

Type >--< Thing

The entity-relationship diagram below illustrates how this can be resolved by positioning an "Instance" as the entity that relate one "Type" in quotes to one [Thing] in square brackets.

In other words, the instantiation or manifestation of a "type" by a [thing] is a phenomenon that is distinct from both of them. For example:

• An instance of the "planet" type is found in the planet-like features of [Mars].
• An instance of the "circle" type is found in the circular shape of [a circus ring].
• An instance of the "symphony" type is found in a performance by [an orchestra].
• An instance of the "tennis match" type is found in a match between [tennis players on a court].

And remember, [one thing] can be an instance of (exemplify, embody or manifest) many different "types".

• [A clock face] manifests the types "circle" and "time measurement display".
• [A plant in my garden] manifests the types "rose bush", "structure", "source of joy".
• [A word in speech or writing] manifests the types "word", "symbol" and "token".
• [You] conform to the types "wonderful", "intelligent" and "human being".

Moreover, you are far more than an instance of those three types. It is misleading to equate you (as thing) with your instantiation of a type. (Sadly devotees fo "identity politics" like to do that).

14 assertions to be explained

The following assertions and principles are discussed in the following sections.

1. Types are primarily an informal biological phenomenon, rather than a formal system.
2. Types evolved out of the ability of organisms to liken perceptions to memories (cf. the use of similes in verbal languages).
3. Types work because they work, not because there is a universal ontology.
4. As Godel’s theorem suggests, some type definitions must be axiomatic or circular.
5. We humans normally pair a token with a type (Type token: type definition).
6. A token alone is not a type, it is meaningless until associated with a type..
7. Actors use types to symbolize features of an observed or envisaged thing.
8. Every coherent (non-paradoxical) definition serves as a type.
9. To pair different types with one token creates an ambiguous homonym.
10. Actors may pair many equivalent or corresponding types with one token.
11. A type exists from when it is conceived, and then in one or more copies, until the last is erased.
12. A type can define a member of an empty set. E.g. “Unicorn”.
13. A type can define a member of a set than can never exist. E.g. “Ice sun”,
14. A type cannot be a paradoxical definition. E.g. Cretan: a honest man who lies.

Organic types

This work starts from the idea that soft, fuzzy, organic types are not degraded versions of hard, precise mathematical types; rather the latter evolved from the former, which evolved from likening one thing to another.

Moreover, being sufficiently alike, like enough, is good enough for biological evolution, and most animal knowledge of the world. And for most social communication, messages have to be only accurate enough, often enough.

Assertion 1: Types are primarily an informal biological phenomenon, rather than a formal system of logic.

Assertion 2: Types evolved out of the ability of organisms to liken perceptions to memories (cf. the use of similes in verbal languages).

Assertion 3: Types work because they work, not because there is a universal ontology.

Assertion 4: As Godel’s theorem suggests, some type definitions must be axiomatic or circular.

Monothetic and precise types

Most mathematics and computing feature hard, strict, monothetic types, where the conformance of thing to a type is binary - yes or no.

A monothetic type describes a thing that has every feature of the type.

Precise type usage excludes things that don't exactly match the feature(s) of the type from being called instances.

More organic types

More organic types tend to be polythetic rather than monothetic, and used fuzzily rather than precisely. Given such a type, the conformance of a thing to the type is a a degree that is less than 100%, and yes or no is a matter of a degree or judgment.

A polythetic type embraces things that share only one of several features of a type. (So, if you want to embrace more things under a polythetic type definition, all you need is to find one new feature in an exsiting set member and extend the type to include it!)

Fuzzy type usage llows things that only approximately match the feature(s) of the type to be called instances. This is normal. We say a circus ring is circular when it is only approximately circular. Outside of mathematics, the judgment of what amounts to a “just noticeable difference” (as psychologists call it) is fuzzy.

Predicate logic deals with including approximate feature matching. If you have an approximation relation (i.e. a binary predicate) you can say A is a B if it is within a certain level of approximation. This is distinct from what fuzzy logic is trying to achieve.

Tokenless types and typeless tokens

Assertion 5: We humans normally pair a token with a type (Type token: Type definition).

Asertion 6: A token alone is not a type, it is meaningless until associated with a type..

Tokenless types

On its own, is a definition a type? Yes, but it is probably not a type of interest.

For example, the final score of a tennis match describes a member of the set of matches that result in the same game set and match score. But who cares? If nobody wants to name it (associate a token with the type), then it is not a "type of interest".

And if it was of interest, somebody would soon invent a token for it.

Typeless tokens

Generally a token, such as "Circle", is meaningless until it is associated with a type. And it can be a homonym, such as "Circle: a line drawn around and equidistant from a point" OR ELSE "Circle: the common social contacts of an individual."

Wittgenstein discussed the use of an apparently typeless token. He said some games share only share some "family resemblances". For example we use the token "game" to describe games as diverse as solitaire, poker, archery and baseball.

Did Wittgenstein mean a type can be large and polythetic, meaning that two instances may share only some features of the type, or even none? Or a type can be used fuzzily, meaning that what conforms to the type well enough is an observer's judgment call?

Or did Wittgenstein not try hard enough to generalize what all games have in common? How about this definition? Game: an activity, performed by one or more actors, that has an outcome those actors regard as win/success or loss/fail.

Description as typification

• “To describe a thing is to classify it”. A J Ayer.

Assertion 7: Actors use types to symbolize features of an observed or envisaged thing.

Assertiion 8: Any coherent (non-paradoxical) definition serves as a type.

Every coherent description of a thing serves as a type, of which any number of instances may be observed or envisaged.

• When you describe a unicorn, you define one member of the unicorn set. Today, that set is empty; one day, an instance of the type may evolve and breed.
• When Senge defined "The tragedy of the commons", he defined one member of the set of system archetypes.
• When cartographers draw a maps of a territory, they define some features of it. But any territory with the same features may be regarded as an instance of the type in the map.
• When a composer writes a symphony score, they define one member of the set of symphony performances. A copy of the score, in any form, is a type. A performance of the score, by any means, is an instance.

A description is firmly in the world of things. It only plays the role of a description when it is created or used as such by an actor. Similarly, a type is firmly in the world of things. It only plays the role of a type when it is created or used by an actor.

We usually think of a description or a type as a passive structure, but a computer program is a large and complex type that is executable.

Equivalent or corresponding type definitions

Assertion 9: To pair different types with one token creates an ambiguous homonym.

In natural language, the word "ring" is homonym that can be used to describe a drawn circle, a finger ring, and a sound - things that share no quality we can think of. Here, our interest is in more controlled languages, with no homonyms. But note:

Assertion 10: Actors may pair many equivalent or corresponding types with one token.

Even in a controlled language, 1 token may relate to N definitions that are equivalent or corresponding definitions. For example, the type "circle" can be encoded in many physical forms, in brains, in sounds, in writing, and in many different languages.

And in one language, the meaning of the type is definable in different, yet equivalent, ways. For example

• "Circle": a round plane figure whose boundary consists of points equidistant from a fixed point.
• "Circle": a continuous curved line which is always the same distance away from a fixed central point.
• "Circle": a closed plane curve consisting of all points at a given distance from a point.

We can derive one definition from another, since they are equivalent or “inter-derivable” types. And we can use any one of them to the same effect.

(Aside: the meta type for a member of the set of three defininitions is "Circle type": a definition of the concept known by the token "circle".)

The three definitions are real, physical. You can locate them. You may remember one by encoding it in your neurochemistry. Later, you may reproduce it at a different time and in a different place, in speech or in writing, and use it to tell somebody how to draw a circle.

Every token that is paired with a sensation, action, or verbal definition serves as a type. To reach agreement about what a type means usually requires verbal communication. And to maintain that agreement over time usually requires records to be persisted.

The nominalist or conceptualist view of types

Assertion 11: A type exists from when it is conceived, and then in one or more copies, until the last is erased.

Suppose you observe he sun and moon have a similar shape, name the shape you visualise as "circle", and define the type thus.

• Circle: a line drawn around and equidistant from a point.

Now you may use the type to describe something else you observe (such as a flower head) as also being circular. Or specify the shape of a thing you envisage (such as a clock face) as being circular.

Q) Did instances of the type exist before life? Yes and no. Circular things existed but there was no token or definition of the type.

Q) Did the type exist before life? No. Not until somebody recognized what Wittgenstein called the “family resemblance” between circular things.

Q) Is the type located in one place? No. It is duplicated in many memories and messages.

Q) What characterizes a type of interest to us? There is at least one token-definition pair, be it verbal (Circle: a line drawn around and equidistant from a point) or neural (Circle: a neural sensation of a circular shape).

The life cycle of a type

Types are transient things; they can be created by actors in messages and memories, and can be lost when memories and messages are destroyed.

Creating a type: The types we use in description were invented by us, and created in a location, whether encoded in a nervous system, in speech, in writing, or in another symbolic form.

Copying a type: Once a type has been created, we can readily make corresponding types by copying it in different places. And by translating it into different symbolic forms - from one code to another code, from a biochemical form, to movements of the mouth, to sound waves, to vibrations in an ear drum, to characters drawn on paper, to dots and dashes in morse code, to radio waves, and so on.

Deleting a type: If it were possible to erase all copies of a type in memories and messages of every kind, then the type would disappear from the universe. However, somebody may reinvent it.

The length and complexity of a type has a big impact on our ability to reinvent it.

Complexity and reinvention

How to measure “complexity” is not universally agreed. But it is widely agreed that organic chemicals are more complex than inorganic, living things are complex than inert things, and (pound for pound) the human brain is the most complex thing we know of.

The more complex things are, the more ways they can differ. Generally speaking, the more complex and/or rare the type, the less likely that it will be reinvented if it lost.

The concept we call "circle" is recorded in countless brains and records. Destroy all memories and records of it, the concept will disappear from the universe, but the chances of it being reinvented are high.

The concept we call "Beethoven's 9th symphony" is complex and rare. Destroy all scores, recordings and memories of them, and the symphony will disappear from the universe. The chances of it being reinvented are vanishingly remote.

Types and sets

A type is a description to which zero, one or more things might conform. It describes one member of a set, even if that set is empty (like the set of men on Mars), is a fantasy (like the set of unicorns), or has only one member (like the set of universes).

Assertion 12: A type can define a member of an empty set. E.g. “Unicorn”.

Assertion 13: A type can define a member of a set than can never exist. E.g. “Ice sun”,

Assertion 14 A type cannot be a paradoxical definition. E.g. Cretan: a honest man who lies.

A type is the idea or concept of a set member. It defines features that members of a set share. For example: "Wolf" is a structural type, the corresponding set is the collection of all wolves. "Wedding" is a behavioral type, the corresponding set is the collection of all wedding ceremonies.

"Circle" is a more abstract type, the corresponding set is the collection of all shapes and things that are circular - near enough. Note the fuzziness of "near enough".

As soon as you have described one thing, you can envisage other things of the same type. For example, having described one unicorn, you can envisage others of the same type.

It is common for people to say a type defines a set of possible instances. But a set is an aggregate of set members, with its own attributes. For example, the “stock item” type does not define the following attributes of the stock item set: most recent sale date, items sold so far, and average shelf life.

Yet more about types

When you read "type", you may think of a primitive data type such as integer or character, a business data type such as account number or balance, an entity type such a customer or product. There is much to know about categories or types.

Complex types

Simple data types can be combined in more complex data type like address and domain name. And to describe a thing by saying it instantiates several types, such as "grumpy", "old", "man", is to create a more complex or compound type.

To say Force = Mass * Acceleration, is to create a complex type that relates simpler types in more complex ways than simple addition.

Quantifying qualities

In science, to be precise, we quantify the degree to which a thing manifests a type (such as "Age in years") by giving a quantitative value to a variable (such as 60).

How about something more difficult pin down, such as "beauty"? Scientists have defined facial beauty in terms of symmetry, proportions etc. Different people may prefer somewhat different measures, but the range is limited. (And I am told that predicate logic allows that a type may accommodate approximately similar things.)

Types of types

It is possible to relate types in a recursive hierarchy. The term "archetype" is used for a generic type (such as "the tragedy of he commons") that may be specialized in more particular types (such a "overfishing of some fishing grounds").

Stable and volatile types

There are countless, if not infinite, ways to group things into families, both overlapping and nested, and there are countess conceivable types. The classes or types we name vary on a scale from stable generalizations to volatile and/or domain-specific types.

• "Number" is token for a stable generalization (duplicated in every language), which enables us to enumerate the things of a type we observe or envisage.
• "Circle" is another universal type conceived by humans. There is no perfect circle in material reality; but there are countless more or less circular things.
• "Planet" is relatively volatile type, whose definition has changed, and is now agreed by a committee.

All words, including for example: "order", "operation" and "policy", can have different meanings in different contexts, domains and vocabularies. The flexibility of natural language its weakness and its strength. The Knowledge structures article discusses controlled vocabularies, taxonomies and ontologies.

Remarks

Here, all examples are verbal. But any structure (verbal, neural, or other) might be used as a token, or used in a type description. ?And things we typify using verbal tokens include:

• sensations we form when observing and envisaging things in reality (like hot and cold),
• verbal descriptions of those sensations
• verbal constructs that do not correspond to sensations (like “algebra”).

Alfred North Whitehead spoke on the topic of abstracting general types from particular instantiations of them. He also said

• "Speech is human nature itself, with none of the artificiality of written language."

However, we can't build or share large and complex knowledge structures or systems using speech alone. That is why historians find the beginnings of human civilization associated with the beginnings of written records, as on clay tablets in Sumeria. And surely also why the development of science and engineering in Europe followed the development of the printing press.

Whitehead also spoke to the automation of systems.

• "Civilization advances by extending the number of important operations which we can perform without thinking of them." (Whitehead again)

Among the most impressive achievements of humankind, has been the systematization and digitization of human activity systems in software.

Related articles

This article sets the scene for articles that explore ways to describe things in large and complex knowledge structures, and ways to describe dynamic systems. If you want to read them in the context of a book, watch this space.

Postscript on many-to-many relations

Thing may be related in structures (or graphs in graph theory) of various kinds.

• Simple 1-1 chains, and circles
• Reasonably simple 1-N hierarchies
• Complex N-N networks

Much of the complexity that we must wrestle with when trying to understand and manage things arises from the N-N relationships we find in how things relate to each other in the world, how types relate to each other in ontologies, and now things and types are related.

Remember: one type can be used to describe many things; one thing can be described by many types.

Below, an N-N relation is represented thus: Type >--< Thing.

In sociology

In the 1970s, the social systems thinker Russell Ackoff spoke of abstract systems and concrete systems - which I call real systems. One abstract system can represent many real systems; one real system can be represented by many abstract systems.

We do sometimes speak of one social entity instantiating one system, as though they are one and the same. But generally, one social system can be realized by many social entities, and one entity may realize many social systems.

Card game type >--< Card school

Many card schools may play one type of card game; and one card school may play many types of card game.

The type, the instance and the social entity that realize it are three different things.

The type - the model - the abstract system - is the collection of structures and behaviors described in a card game rule book.

The instance - the phenomenon - the real system - is the structures and behaviors of a particular card game in progress.

The thing - the social entity - which instantiates a card game type - is a card school - a group actors who may interact not only in many types of card game, but also in eating a pizza.

Symphony >--< Orchestra

One abstract symphony score can be realized by many orchestras in symphony performances. One orchestra can instantiate many more or less equivalent symphony scores in performances.

The score of a symphony can be represented in many more or less corresponding forms. Music symbolized and persisted in the brain is no different in principle from music symbolized and persisted on paper, on a vinyl record, or in a digital memory, except that in neural memory is more fragile and forgettable.

(When I composed music, the primary reason for writing it out on paper was to persist it in a stable form that I could repeat and extend over months, and pick up again years later. The remote possibility that anybody else would play the music was not the motivation.)

A composer, composing music for an orchestra, must persist their private memory in the public and copyable form of a musical score. A symphony score serves as a type. It defines a member of the corresponding set of symphony performances. Each performance, near enough, corresponds to the score.

The information in a symphony score can be decoded and encoded, step by step into

• musicians playing their instruments
• sound waves
• vibrations in the ear drums of listeners
• signals in the aural nerves of listeners
• and perhaps wrinkles in grooves on a vinyl record.

In the steps from instance creator to instance receiver, information may be lost on the way, yet still be enough to be regarded as a successful communication.

Where is the universal, definitive type?

We might say there is none, there are only one or more copies of the symphony score. Or else, we may arbitrarily promote one copy over all the others, by declaring one physical and score as the definitive type against which all others may be compared.

Beethoven produced several - slightly different - musical scores for what is called his 9th symphony. Is the last one the definitive one? Observers may assign the label "definitive" to any version of the score they choose. And every version now appears in many copies, some of them imperfect.

Today's social philosophers might say the lived experience of a listener is definitive. But perhaps most would say the version of score used in a performance is the definitive type for that performance. Thus, every variation of the score must be judged as either a different type or an equivalent type.

I am a pragmatist. It is enough for me that people who share a language can agree, well enough, what a description of a thing means, and agree, well enough, which descriptions are definitive, and create generic type definitions against which other things may be usefully be compared. Because this somewhat haphazard process served mankind well for perhaps a million years.

To a degree, we can escape the fuzziness of natural language by defining dictionaries and quantifying variables. To a degree, we can escape the circularity by defining controlled languages for particular domains of knowledge, using taxonomies and ontologies. But I don't assume or hope we can entirely escape fuzziness and circularity.

In software architecture

In the artificial or designed world of programming, the structures and behaviors of software application are documented in thousands or millions of lines of code. The code itself is an abstract system. To understand and manage such a complex system we need a more abstract model of it.

Interface >--< Application

Generally, one interface can be realized by many applications, and following the principle of interface segregation, one application may have several interfaces (say, one for each type of user).

Abstract class >--< Concrete class

Programmers often separate an abstract class (aka interface or type) from the description of a class that performs the operations declared in that interface. So generally, one abstract class can be realized by many concrete class, and one concrete class can realize many abstract classes

Class >--< Object

In OO programming, one class can be realized by many objects, and one object can realize many classes. An object only exists (is deployed to a computer) after it has been created as an instance of a predefined class, and all classes above it in a class hierarchy.

Some deprecate multiple inheritance

Some programming languages allow an object to instantiate classes in several class hierarchies. Some programmers object to this complexity so much that they deny it is feature in how we describe things in the natural world we live in. And some programming languages don't allow it.

Some deprecate deep class hierarchies are deprecated (source lost)

• "Deep inheritance hierarchies create a lot of coupling between abstractions, creating a tension between reuse and information hiding.? An abstraction is basically exposing its representation by announcing that it inherits from another abstraction, and we should all know the kind of maintenance headaches you have when you don't practice information hiding.
• Thankfully, the tide seems to be turning, and people are beginning to realize that type extension is not so great after all, and that "mere" aggregation is often preferable.?If you are an Ada95 or C++ programmer who programs "the pure object-oriented way" by creating deep inheritance hierarchies, then YOU ARE MAKING A HUGE MISTAKE.? You're going to spend all your time just compiling."

Related articles

This article sets the scene for articles that explore ways to describe things in large and complex knowledge structures, and ways to describe dynamic systems. If you want to read them in the context of a book, watch this space.