Description as typification

Is there really no such thing as a fish?

A chapter in “the book” at Last updated 06/10/2021 14:13


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The chapter below presents a type theory that underpins how we describe things. It is relevant to the challenge of defining a “domain-specific language” for doing business,


It declares principles related to creating and using the descriptive types we use to characterize things. To describe something(s) observed or envisaged is to typify it/them. To typify something(s) observed or envisaged is to describe it/them.


It discusses fuzziness in how well real-world things and phenomena instantiate types. People speak of the relationship between types and individuals. Arguably, the relationship is better described as between types and the (near enough) instantiations of those types by individuals.


It further explores the many-to-many relationship between types and individuals. A type (say, “unicorn”) can exist without any related individuals, or else be instantiated/realized by one or more individuals. An individual (think, a proton at the beginning of the universe) can exist with any type to describe it; today, an individual may instantiate/realize zero, one or more defined types


For those with a mathematical or philosophic bent, this chapter answers questions like: When or where was the set of numbers created?  the set of even numbers? the set of human beings? the set of fish?


How we know what we know.. 1

Where does mathematics come from?. 1

The discreteness of things in descriptions of the world. 1

Abstracting descriptions from reality. 1

Abstracting types from individuals. 1

Relating types to instances of them.. 1

Relating types to sets. 1

Relating descriptive types in a network structure. 1

Relating descriptive types in a hierarchy. 1

Is there really no such thing as a fish?. 1

More about types. 1

Family resemblances. 1

Mathematics as a tool based on typification. 1

The evolution of formal types from fuzzy typification. 1

Our type theory. 1

The constructivist view of types. 1

Conclusions and remarks. 1



How we know what we know (repeat)

For sure, there is a reality out there. Systems thinking is much about what we know of reality and how we describe it. That subject is more generally called epistemology. Several authors have proposed triangular views of epistemology, in which they relate three concepts to each other.



Epistemological triangle

Ogden and


Semiotic Triangle

References <are symbolised by> Symbols <stand for> Referents

Charles Peirce

Triadic sign relation

Interpretants <understand objects from> Signs <represent> Objects

Karl Popper

Three worlds

Mental worlds <produce> Products of the mind <describe/predict> Physical realities

Pierre Bordieu

Three relations of knowledge

Knower <social> Knowledge <epistemic> Known


This work is rooted in biology rather than linguistics. It uses a triangle that seems both easier to understand and more useful than the triangles above. First, here is an example.


·       Architects <observe and envisage> Buildings

·       Architects <create and use> Architectural drawings

·       Architectural drawings <represent> Buildings


Our triangular relation can be seen in a wide range of biological examples. Consider what animals know of reality and how they describe it. Flies smell food. Once mice memorize a route through your house, they will never forget it. Birds sound alarm calls. Honey bees communicate about pollen locations.


From biochemistry to symbolic language


Cells <sense> Food items

Cells <create and use> Biochemical signals

Biochemical signals <represent> Food items

Fruit flies <smell> Fruit odours

Fruit flies <create and use> Biochemical signals

Biochemical signals <represent> Fruit odours

Animals <observe> Things

Animals <form and recall> Memories

Memories <represent> Things

Mice <observe> Routes

Mice <form and recall> Route Memories

Route Memories <represent> Routes

Social animals <observe> Things

Social animals <create and use> Messages

Messages <represent> Things

Birds <observe and envisage> Predators

Birds <create and use> Alarm calls

Alarm calls <represent> Predators

Observers <observe and envisage> Aspects of reality

Observers <create and use> Descriptions

Descriptions <represent> Aspects of reality

Honey bees <observe and envisage> Pollen locations

Honey bees <act in and read> Dances

Dances <represent> Pollen locations


The epistemological triangle below generalizes from the examples above.


·       Thinkers <observe and envisage> Phenomena

·       Thinkers <create and use> Models.

·       Models <represent> Phenomena.



“Real-time learning, deep understanding, reasoning requires true cognitive abilities that none of the Second Wave AI programs have.” Data economy


To speak of AI overtaking general human intelligence is to underestimate us – not just our sensor and motor apparatus, but our abilities to typify phenomena we observe and envisage - to create, record, communicate and understand descriptions of reality - to create theories about the world – to test theories by creating and running experiments - to study and deploy knowledge of maths, physics, mechanics, electronics, chemistry, biology and psychology.



The meaning(s) of a descriptive model, recorded in a memory or a message, are found not in the memory or message itself, but in the encoding and decoding processes of the encoding and decoding thinkers. A descriptive model is always subjective in the sense it is formed by one or more thinkers. But it may also be called objective in so far as its accuracy can be confirmed by empirical, logical and social verification.

Where does mathematics come from?

In many ways, this book is about common sense, but of the kind that sometimes has to be spelled out, as below.

·       There was no description before life; the universe existed long before thinkers created any description of it.

·       We cannot understand reality directly; we can understand only descriptions we make of it.

·       Descriptions, being abstractions from what they represent, are not all equally true, valid or useful.

·       There is no absolute truth, only degrees of truth; to say I am six foot tall is near enough true for most practical purposes.

·       No description can be complete, because then it would be the aspect of reality it describes.

·       A description typifies what it represents (Ayer).

·       Having described one thing, that description applies to all other things we observe as being near-enough similar.


A description is a generic type that defines one member of a set of near-enough similar things in terms of properties they share. Even if there is one member in that set (one universe?), we can envisage more.


The meanings of words are not fixed, other than in definitions we give them and share in specific “bounded contexts”. Consider the type “human”. There between 10 and 20 or more recognised human species. Some lists don't include Denisovans while some don't have Homo naledi, a hobbit-sized human species discovered in Indonesian caves, perhaps because they look more like chimpanzees than us.


Consider the less ambiguous type “number”. Where numbers they come from? They emerged out of Darwinian evolution.

·       Animals evolved to recognise similarities between things. Even honey bees can do that, perceive a quantity of similar things and communicate that quantity to other bees.

·       Humans evolved to label similar things as being of a named type (e.g. "child").

·       Human gave a name to the quantity of things they judge to instantiate a type (e.g. the number of children in a family); and thus invented numbers.


Our ability to typify things in words led to the development of set theory that mathematicians now see as fundamental.

·       Humans defined a named type by listing properties shared by most instances; and thus invented polythetic types.

·       Human so far formalized some type definitions that to be counted as an instance, an individual must have all its properties; and thus invented monothetic types.

·       Humans capitalised on monothetic types by inventing set theory.


I am not convinced linguistic philosophy or semiotics have much add to the story told here. Again, the types we give names to are not fixed, other than in definitions we give them and share in specific “bounded contexts”.

The discreteness of things in descriptions of the world

The universe may be viewed as an ever-unfolding process, in which space and time are continuous. However, when perceiving, remembering and describing phenomena, we divide the universe into discrete chunks.


Differentiating things

Most obviously, we see bounded solids (fish in a sea, planets in orbit) as discrete entities. We divide a forest into discrete trees, though some may share roots. Thousands of years ago, musicians divided the continuous spectrum of sound into the discrete notes of a musical scale. Hundreds of years ago, Isaac Newton divided the continuous spectrum of sunlight into first five, and later seven, discrete colors.


2/1. “The most fundamental concept in cybernetics is that of ‘difference’, either that two things are recognisably different, or that one thing has changed with time.” “We assume change occurs by a measurable jump.” Ashby, 1956


The concept is more generally fundamental, since for us to distinguish one sensation from another there must be what Ernst Weber (a 19th century experimental psychologist) called a “difference threshold”, that is, the minimum amount by which the intensity of a stimulus must be changed in order to produce a noticeable variation in sensory experience. Today, psychologists call it a “just noticeable difference”.


By observation and classification, we differentiate between:

·       things bounded physically and separated in space (fish in a sea, planets in orbit around a sun)

·       logical types of things that share one or more qualities (family, category, class, species, chemical element)

·       things of the same type (individuals of a species, atoms of a chemical element, members of a football team)

·       quantities of a quality (measures of height, width, depth, weight or volume)

·       changes in the state of a thing (from hot to cold, or asleep to awake)

·       generations of things (from parent to child, or version 1 to version 2).


Klaus Krippendorff, a student of Ashby, wrote that "Differences do not exist in nature. They result from someone drawing distinctions and noticing their effects.” and “Bateson's ‘recognizable change’ [is] something that can be recognised and observed."


How to describe things?

To help us remember discrete things, and describe a thing to others (its position in space, a family or generation it belongs to, a quality it has or a state it is in) we differentiate discrete:

a)     types of things, or qualities (different species, or colors),

b)     instantiations of a type (the different individuals of a species, or appearances of a color in different rainbows),

c)     states of a thing that changes over time (the progressive baby, child and adult states of a person, the cyclical on/off states of a light).


Which of these descriptive tools we use depends on the context. Are “caterpillar” and “butterfly” different types or states of an insect? Does the word “yellow” signify a type of color, or an instantiation of the color type (a member of the set of rainbow colors)? To put it another way, those three descriptive tools are not mutually exclusive.

To describe something to others, we may encode our knowledge in a physical structure of matter or energy, a symbol, that others can read. Say, you leave your office door open to symbolize, to convey the information, that you are open to visitors. You assume visitors can a) differentiate between the open and closed states of the door and b) know the meaning of the code you used when you opened or closed the door.

Abstracting descriptions from reality

Abstracting descriptions from things and phenomena that we observe and envisage is so important to human existence that we have many words for doing it.


Things in reality

are represented in

descriptive structures we construct

in memories and messages

Phenomena we

observe and envisage




property types



give values to

variables or attributes



are characterized by





We can understand and discuss things in reality only in terms of descriptions we construct of them. Descriptions are structures an animal or machine can correlate with what is described. When a bird sounds an alarm call; other birds correlate that sound with the presence of some danger. [GB1] When we use words, many are proxies for physical entities in the real world.


Description in a

typifying assertion

Things in reality

instantiate the type


“Roses are colored”

A display of rose varieties

More particularly

“Some roses are red”

A bunch of red roses

And more particularly

“This rose is red”

One red rose


We could replace the words in the right-hand column of the table by photographs of roses (another kind of description) or by live broadcast pictures (still a description). However close we get to discussing reality itself, we never quite get there.

Abstracting types from individuals

Here, types are individual phenomena that emerged from biological evolution, that exist fuzzily as "resemblances" in our minds, are formalized using language, are shared via social communication, and are confirmed or denied through trial and error.


In mathematics, a type is usually defined by some properties that are necessary and sufficient to identify an individual as exemplifying the type. E.g a triangle is three-sided polygon whose internal angles add up to 180 degrees.


In natural language, a type is often more loosely defined. E.g. The type “mother” may be defined as “birth giver”, “child nurturer” “partner of father” and “female genetic ancestor”. No one of the four property types is sufficient to define the type, and not all are necessary to refer to an individual as a “mother”.


The looseness of type definition surfaces in Wittgenstein’s notion of “family resemblances”, in George Lakoff’s "women, fire and dangerous things" (1987), as an "idealized cognitive model", and in type theory as a “polythetic type”.


Since you and I are interacting by means of words in this document, we will discuss descriptive types using words. A verbal type can be presented as composed of two parts: a type name, and an explanatory definition.


Type name

Type elaboration

“Even number”

a number divisible by two”.

“Triangle corner”

“an angle between the two lines in a corner of a triangle”


“an animal with feathers and a beak”

“Bird of prey”

“a bird which feeds on other animals”


“a bird of prey with a wide wing span”


The type name is a short-hand for the elaboration. To use the type name is to assume your listener already knows the elaboration, or one that is similar enough for practical use.


When we say “this bird is an eagle”, we hope the message receiver understands what we mean. The advantage? Using type names saves a lot of communication time and effort. The disadvantage? The risk that a message receiver associates a different type elaboration with the same type name.


Even if we say: “This bird is an animal with feathers and a beak which feeds on other animals and has a wide wing span” we still depend on our listeners sharing our understanding of the descriptive types we use. In natural language, this shared understanding, acquired largely by trial error, is imperfect, just good enough, often enough.


Together, the type name and elaboration make an intensional definition or predicate statement of this form.


Intensional definition pattern

Predicate statement type

Intensional definition example

Predicate statement instance

A thing

of the named type

is a thing of a more general type

with these particular features.

A thing

of the even number type

is a number

which is divisible by two.


To describe a thing is to typify it in terms of types already understood

“No statement which refers to a ‘reality’ transcending the limits of all sense-experience can possibly have any literal significance” Chapter 1 of “Language truth and logic” A J Ayer.


How do we develop an understanding of words? We start by associating words with sensations of things in the world. We associate “Mum” and “Dad” with individual people. Later we realize there are many Mums and Dads out there, so the words are type names. Later still, we realize others define the Mum and Dad types differently, biologically, sociologically, or both.


Gradually, we build a network that loosely relates words we use to typify things. The word network in our minds is surely analogous to way a dictionary is written. One word is defined by relating it to other words, more by “family resemblances” than by strict equality.


At base or center of the word network in our brain are words we associate with our sense-experience of the world around us. When we are taught that “a planet is a heavy body of matter in orbit around a star”, the description would meaningless if we could not relate it to some experience of matter and motion.


What we don’t do is learn words by starting from a handful of axiomatic abstract super types like “occurrent” (or event in time) and “continuant” (or entity in space), then build  up a gigantic taxonomy or ontology in which every words we learn is defined as a more particular subtype of the generic supertypes (or base types). An ontological hierarchy of this kind is an artificial device we impose on words when defining a domain-specific language

Relating types to instances of them

We have many words not only for types (properties, features, concepts, qualities, characteristics and attributes) but also for instantiating a type. We speak of embodying, exhibiting, exemplifying, manifesting or realizing a type.


Thing in reality


type name

Tells us the thing

A circus ring


manifestly has a diameter the same in all directions

A play


realizes the script written by Shakespeare

A rose bush


exhibits the features “thorny, flowering, and bushy”

A bird


embodies a bird of prey with a wide wing span


A bird that instantiates the “eagle” type should embody the property type “wing span in metres” in a particular pair of wings, which are (say) one metre in width.


Aide: sloppily and confusingly, we tend to say “property” when referring to a general type and when referring to a particular value of a type.

Relating types to sets

People create and use types with no knowledge of, or reference to, set theory. Every description constructed in the mind is a type, even if there is only one thing, or nothing, that conforms to it. Every verbal type is simply a predicate statement we use to describe a thing or a state of a thing.


Types do feature in set theory - a branch of logic most often applied to mathematical concepts. There several set and type theory variations. The aim of here is not to explain one particular theory. It is to make some general observations about how types relate to sets.


A set theory typically begins with this assertion or relation “a thing can be a member of a set”. The set, the collection of members, can be described in one or both of two ways, by extension or intension.


Extensional definition

In basic set theory, a set is identified with its members, and defined by listing its members.



Set name


Extension {members}

Rainbow colors

{red, orange, yellow, green, blue, indigo, violet}

My friends

{Jo, Mary, Tom}

Polygon types

{triangle, quadrilateral, pentangle…}

This triangle’s corner angles

{60, 70, 50}


To distinguish type from set is to distinguish description from reality. However, when naming the members of a set (e.g. My friends) we use words in place of real-world things. And the named set members may be descriptive types (e.g. polygon names) or descriptive variable values (e.g. angles). In other words, our descriptions are themselves realities we can describe.


Intensional definition

With reference to set theory, every type is an intensional definition of a set member. Some say “a type defines a set”. More accurately, it describes one member of a set. It says nothing about the total number of set members; that property of the whole set is invisible in the type.


Note that though every type defines a set member, there are fantastic and impossible types like “flying elephant” that will forever be associated with an empty set.


Moreover, there are sets definable by extension for which no satisfactory type can be defined. Consider the set of lamp posts you have touched (or even the set of fish discussed later). The only way to define a type for these sets is to have, or imply, a list of all set members.


Which came first, type or set?

The question is not whether we should take an intensional or extensional view of sets. The question is whether we place set theory ahead of type theory, or type theory ahead of set theory. And since “knowledge is a biological phenomenon” there is no doubt about the answer.


There is more than one set theory and type theory. A pure mathematician thinks of a set as having a fixed number of members, which exist forever and everywhere. Because their interest is in purely abstract types like “number” and “triangle”. An applied mathematician, stepping out into the world where we typify physical entities and events (as in data analysis), thinks of a set as a dynamic, having a variable number of members, since members can leave and join the set.


The first kind of set theory makes no sense when it comes to types like “planet” and “customer”. It forces you into a metaphysical view of the universe, which is simply unnecessary. And once you give type theory precedence over set theory, the nonsense evaporates. Describers, descriptive types, and describable things can all be placed in time and space. We’ll return to this later in the chapter.

The many-to-many association between types and sets

Obviously, members of several different sets (or subsets) can conform to one generic type.


One generic type name

Set name

Set in reality

Flag color

US flag colors

{Red, White, Blue}

Chinese flag colors

{Red, Yellow}

Indian flag colors

{Orange, White, Green, Blue}


Conversely, one member of a set can conform to different types. In this example, the two different types imply different operations for recognizing a member of the same set.


Type name

Type elaboration

Set in reality

Doubled number

A number exactly divisible by two

{The set of even numbers}

Even number

A number greater than an odd number by one


Fixed and variable sets

Some mathematicians speak only of sets with a fixed number of members, such as the set of angles in the three corners of a triangle, or the infinite set of prime numbers. To them, a set is fixed forever, it is the collection of all members that ever exist across all space and time.


Others speak of real-world sets with a variable number of members. And we can, in the intensional definition of a set member, define limits in space and time, such as the set of roses in my garden today, or the set of customers identified in a database table.


Type name

Type elaboration



A homo sapiens who has lived, is living, or will live.


Person alive now

Extends “Person” with the constraint that “now” is after birth and before death.


Person of interest

Extends “Person” with the constraint that the Person is recorded in our database.



Suppose your friends and my friends are the same people, then, when defined by extension, there is only one set.


Type name

Type elaboration

One set in reality

Your friend

A person whose number is in your phone right now.

{You, me, and seven other people}

My friend

A person whose number is in my phone right now.


Now suppose you add or remove a friend from your phone list. A purist mathematician might say there was one set, and your new set of friends is a different set (so one set has been divided into two). Another may say there were always two sets, and now, your set’s membership has changed.


In describing the natural and business world, people usually think of sets as variable or dynamic, meaning that a set (of friends or anything else) can gain and lose members.


Bear in mind

People commonly think of a set as containing members that conform to an intensional definition, and usually assume a set has several or many members. Note however:


·       there are fantastical types with no set member (like "flying elephant")

·       there are sets with no satisfactory type (like the fish set below)

·       there are types that describe one and only one thing.


Above all remember every description we construct of a thing is a type, even if there is only one thing, or nothing, that conforms to it.

Relating descriptive types in a network structure

Things in the real world are relatable to each other in an extraordinary variety of ways. This chapter does not explore how types in a given domain of knowledge are related to each other by one-to-many and many-to-many associations in a network structure. E.g.


·       One customer <places> one or more orders

·       One order <is placed by> one customer <for> one or more product types.


If you want to learn about entity-relationship modelling of this kind, you probably can’t do better than study relational data analysis and common data model patterns. If only there was space in this book to explore those!

Relating descriptive types in a hierarchy

When people are faced with a complex network of things in the world, they often impose a hierarchy on them. They do this to help them understand and/or manage the network.


A descriptive hierarchy is a tree structure that divides one node (the base or root node) into two or more nodes, and then may subdivide those nodes several times, until a last division into elementary or atomic nodes. Things in the world can be classified by assigning them to the atomic nodes.




Base node

Node 1

Atomic node 1.1



Atomic node 1.2




Node 2

Atomic node 2.1



Atomic node 2.2




Three varieties of descriptive hierarchy are distinguished below.

Composition hierarchies

A composition hierarchy is constructed by relating a larger/longer composite type or thing to smaller/shorter component types or things. E.g.


·       Body < organ < cell < organelle < molecule

·       Document < section < paragraph < sentence < word

·       Business < division < department < team


Dividing a system into subsystems, and composing a system from subsystems, are fundamental ideas in systems analysis and design.

Generalization hierarchies

A generalization (aka class) hierarchy may be constructed from the bottom up by grouping things that share one or more common features into a type, then abstracting supertypes from subtypes.


The ideal of the classifier is to build a tree in which a type at the bottom of the hierarchy has every feature of all the types above it. In such a strict or pure type hierarchy, each subtype “inherits” the features of all subtypes above it, and “extends” their features with additional ones.



Metalled road

Single carriage way


Dual carriage way




Plain railway


Electrified railway

Third line railway

Overhead line railway


A more famous generalization hierarchy is the classification of species. This wide and deep branching tree was built by comparing the morphology (structure or shape) of living and fossil organisms. It has been refined many times since the first version constructed by Linnaeus. Here is a tiny part of it.


Order Primates

Suborder Strepsirrhini: lemurs, etc.

Suborder Haplorhini: tarsiers, monkeys and apes

Infraorder Tarsiiformes

Infraorder Simiiformes 

Parvorder Platyrrhini: New World monkeys

Parvorder Catarrhini

Superfamily Cercopithecoidea

Family Cercopithecidae: Old World monkeys (138 species)

Superfamily Hominoidea

Family Hylobatidae: gibbons (18 species)

Family Hominidae: great apes, inc. humans (8 species)


This classification was first built with reference to the physical features of organisms preserved in the fossil record. However, studying jaws, bones, scales, teeth and fin spines on their own tends to deliver a confusing signal of evolutionary relationships. This is partly due to convergence. Evolution not only produced divergences between species. Convergent evolution leads some species in different branches to share the same structural and/or behavioral features (e.g. insects, birds, pterosaurs, and bats share wings and flight).

Ancestral hierarchies

Phylogeneticists look to construct a type hierarchy that reflects the divergence of species over time. A cladogram is a diagram, an ever-branching tree, drawn to represent a particular view of evolution. In a cladogram, the branch from fish to human usually runs more or less along these lines.


·       Earlier life form < Fish < Bony fish < Lobe-finned fish < Tetrapod (four-limbed animal) < Mammal < Ape < Human.


Note that tadpoles are on a different branch from humans.


To name a branch in the diagram (say, “fish”) is to name a type. Each subsequent branch is a new type, labelled either with a newly emerged property (say, scales) or a type name we already know (say, mammal).


However, a cladogram is not like a strict generalization hierarchy, because a type at the end of the hierarchy does not inherit all the features of its ancestral types. You may understand a little of the human type (ape, mammal) by reading the diagram backwards. But humans don’t breathe through gills or have scales. Through evolution, features are lost as well as gained.


How closely does a cladogram represent biological evolution? Real-world evolution is a well-nigh continuous process of tiny, incremental, nearly imperceptible changes. A cladogram shows it as a succession of major divisions, where species diverge. Different cladograms can show coarser or finer-grained divisions, but it usual to gloss over features gained and lost by one species over time.

Is there really no such thing as a fish?

After a lifetime of studying fish, the biologist Stephen Jay Gould concluded that there was no such thing as a fish. He reasoned that although there are many sea creatures, most are not closely related. For example, in the ever-branching tree of evolution, a salmon is more closely related to a camel than it is to a hagfish.


However, Gould’s conclusion can mislead people. He could define a fish set, by extension, by listing its members, but could not define a fish type without including several species he did not want to call fish, and perhaps saying that humans are fish.


The difficulty here is that people confuse three ways of answering the question Gould raised.


Can we enumerate a fish set that embraces all things called fish today?

“In days gone by, the word ‘fish’ was pinned to virtually any creature that lurked beneath the waterline. Gradually, as people paid more attention to the biology of those animals, it became clear that some ‘fish’ belonged to other groups, such as reptiles or mammals. This left a motley collection of aquatic vertebrates, including sharks, stingrays, hagfish, sturgeon, lungfish, goldfish and tuna.”


So yes, we may define a fish set by extension, by listing all the animals in that collection above, along with any others we want to call fish. The trouble is that people speak loosely, and other people may have a different fish set in mind.


How to avoid debate? How to agree the fish set without listing every species we like to call fish? Can we describe every member of our fish set by intension? Can we describe what all fish have in common?


Can we define a fish type that embraces all things called fish today?

Suppose we say fish are cold-bloodied animals that breathe through gills, have bodies covered in scales, and have limbs in the form of fins?


    Fishy features

Fishy things


“Gill breathing”



A haddock





A shark




A hagfish




A tadpole




A dolphin






As Gould observed there is no fish type that satisfactorily characterises every set member he wanted to call fish. No feature is present in all things most people call fish (hagfish have no scales or true fins) and not in things most people don’t call fish (tadpoles have gills).


On the other hand, we are free to construct whatever type we choose. We can define the fish type as “an animal with gills”, and so accept and allow tadpoles are fish. After all, a tadpole is indeed the fishy state or stage of an amphibian’s life.


Can we identity a common ancestor of all fish today?

“A stunningly preserved, soft-bodied fish that is more than 500 million years old could be the ancestor of almost all living vertebrates. The fossilized fish, called Metaspriggina, sports characteristic gill structures that later evolved into jawbones in jawed vertebrates, according to a new study. "For the first time, we are able to say this is really close to this hypothetical ancestor that was drawn based on a study of modern organisms in the 19th century," said study co-author Jean-Bernard Caron, a paleontologist at the Royal Ontario Museum in Toronto, Canada.”


Since all tetrapods (birds, frogs, lizards, turtles, mammals) evolved from this fishy ancestor, some cladists like to say humans are a kind of fish. That is contrary to common sense, since we are not cold-blooded, we don’t breathe through gills, and our limbs are significantly different from most fish fins in their form and function.


Discussions of the question Gould raised tend to confuse these two different concepts:


·       A common ancestor in an ancestral hierarchy

·       A supertype in a generalization hierarchy


The first hierarchy is a fact of history, which scientists uncover through painstaking research, evidence collection, hypothesis, trial and error. The second hierarchy is one we construct to classify things. So, in summary we might use the word “fish” to mean:


·       the ancestral type of all modern-day fish and four legged animals, including humans

·       a generic type (say) an animal with gills and without any human-specific features. 


It doesn’t matter, as long as we are consistent about which we mean when using the type name.

More about types

Observing and envisaging types

The table below defines a tiny universe with just three observable things, and four observable features. (For now, let us gloss over ambiguity or vagueness in the features.) Looking at the descriptions of the three things, how many types do you observe?



Things in reality




“Source of Light”

Our sun




Source of Light

The moon





My beach ball






In the table, you can see two things of the type “round and heavy”; and two things of the type “round and yellow”. So, you might observe there are two types in the table. But listen to this philosopher.


“The fact is that one cannot in language point to an object without describing it. If a sentence is to express a proposition, it cannot merely name a situation; it must say something about it. And in describing a situation, one is not merely ‘registering’ a sense-content; one is classifying it in some way or other, and this means going beyond what is immediately given.” Chapter 5 of “Language truth and logic” A J Ayer.


Ayer tells us that as soon as we describe one thing, we define a type. Even if we have observed only one sun, our description of it serves as a type we can compare other things with. And having described our sun by typifying it as a “round, heavy, yellow, source of light”, we can go on to find other suns in the universe that conform to the same type.


Observed and envisaged types: We can observe things and create types to describe their features. We can envisage types we have never observed.


Fantastic and realistic types: We can envisage fantastic types realizable only in fiction, such as “flying elephant”, and realistic types composed of features a thing may in fact instantiate, but we haven’t seen yet.


Simple and complex types: There are four simple types in the tiny universe above. We can envisage a set of “round” things, a set of “heavy” things, a set of “yellow” things, and a set of things that is each a “source of light”. Further, we can envisage 16 complex types by combining two or three of the features. Some combinations may describe nothing in the real word, so their corresponding sets are empty.

Types as states in the life of a thing

The conformance of a real-world thing to a type may change. Animals can change from one type to another by “morphogenesis”. E.g.


·       A tadpole changes into a frog

·       A caterpillar changes into a butterfly

·       You change from a child to an adult.


In each of these biological examples, the entity has a continuity of identity over time. It retains the same DNA and remains a member of the same species.


·       The tadpole-frog entity conforms throughout its life to the amphibian type.

·       The caterpillar-butterfly entity conforms throughout life to the insect type.

·       The child-adult entity conforms throughout life to the human type.


In biology, the type that defines an animal, found in its DNA, is called its genotype. What all members of species share is the locations in the DNA at which particular sets of genes occur, those being the locations and the sets that define the species.

Monothetic and polythetic types

The table is below is edited from one produced by Van Rijsbergen 1979; see also:


The table below defines a universe with eight things (1 to 8), and eight features (A to H). How many types of thing can you observe this table?



Things in reality


















































































To qualify as a thing of a monothetic type, a thing must have every feature. The quoted source observes there are three monothetic types in the second half of the table. However, we can observe several more monothetic types.



Monothetic type


Things in the set













and so on



To qualify as a thing of a polythetic type, a thing need not have every feature. The source observes that there two (four-part) polythetic types. But those two sets conform also to some two and three-part polythetic types.



Polythetic type


Things in the set















and so on



Combining the 8 features above in different ways give us 256 types that might be instantiated in this tiny universe. In the wider universe of human discourse, there are infinite possible types, even if most do not or cannot possibly represent a thing in the real world.

Family resemblances

The types discussed in mathematics (e.g. triangles) are often monothetic, meaning every set member exemplifies, exhibits or embodies every feature of the type.


The types we use in describing nature are often polythetic, meaning different set members exhibit different subsets of the features associated with the named type. E.g. a fish is “a cold blooded animal with gills, scales and fins”. A fish need not have all those features


The types we discuss in more general conversation are also often polythetic, with no particular feature required. A thing can qualify as an instance of the type by having any one of several features.


Wittgenstein suggested we recognize loose “family resemblances” rather than types. In such a family, A is similar to B is similar to C is similar to D; but A is not like D. For example, the things we call “games” don’t share any particular quality. We can formalise our sense of family resemblances by defining the polythetic verbal type below. Still, war games are very unlike games of solitaire.



Games in reality



“Rehearsal for action”








World series



Archery (middle ages)



Rehearsal for action

War game


Rehearsal for action


It is not clear whether or how far we recall family resemblances in our memory, as distinct from memories of particular things. But how the brain compares new perceptions to past memories doesn’t matter here; it only matters it evidently does.


Even what appears to be a monothetic type can turn out to be polythetic. Suppose we say a bird “is an animal with feathers, two legs, wings and a beak”. That type appears to be monothetic, but the question arises: If a bird, through disease, loses all its feathers, is it still a bird? Surely most people would say yes?

Mathematics as a tool based on typification

In a sense, we “discover” everything we “invent”. You might plausibly say tools like the axe and the wheel were always "out there" waiting to be discovered. You might say the same of the tool we call mathematics. But only in a useless metaphysical sense, which we can cut out of our philosophy using Occam's razor.


Moreover, you surely believe that different human societies, independently, were bound to invent the axe and the wheel. You might well say the same of mathematics, that people were bound to invent it. But to say “discover” rather than “invent” is sophistry.


Typification precedes enumeration. Mathematics is descriptive/modelling tool rooted in quantification. But we cannot count the instantiations of a general descriptive type (or rather count the things that near enough instantiate a type) until we have a type.


There were no shareable descriptive types before animals were able to a) recognise resemblances between discretely observable things b) group similar things in a category (e.g. "dangerous" and "mate-worthy") and c) communicate about them using symbols (e.g. alarm calls and mating calls), of which human language is a spectacular development.


The truth/usefulness of mathematical statements depends on our biologically evolved ability to differentiate types ("food" and "friend") and differentiate instances of those types (Flora, Fiona, Fred), and count the instances of types. As Maturana said "knowledge is a "biological phenomena".


In short, no life, no description, no types, no numbers, no invention of mathematics.

The evolution of formal types from fuzzy typification

In this chapter, fuzzy does not mean what you might find in more formal discussions of mathematics and logic. It embraces several ideas: a matter of degree, ambiguous, vague and context-dependent or subjective.

Fuzzy typification

Look again at the tiny universe described earlier.



Things in reality




“Source of Light”

Our sun




Source of Light

The moon





My beach ball






Here, the conformance of a thing to a type may be fuzzy for any of several reasons


·       The conformance may be a matter of degree. E.g. a circus ring is described as circular, though it is not a perfect circle. A play is said to be a performance of Hamlet, though it departs from the script.

·       The type may be ambiguous: Does round mean circular or spherical?

·       The type may be vague: How heavy is heavy?

·       The type may be context-dependent or subjective. Who decides whether the sun is rightly described as “yellow”? Might the moon be regarded as source of light?


As a result, the extent of the set associated with the type is debatable. Different observers may associate different sets with the same type, and vice-versa.  


In social conversation, the conformance of things to types tends to be fuzzy. We might well call a plant a “rose bush” if it has only two thorns and grows sturdily to the height of a tree.  The conformance of things to types may not be 100% true, yet be true enough for practical purposes. After all, it is for practical purposes that animals evolved the ability to characterize things using descriptive types.

Fuzzy matching of the new to the old

“In describing a situation, one is not merely registering a [perception], one is classifying it in some way, and this means going beyond what is immediately given.” Chapter 5 of “Language truth and logic” A J Ayer.


Biological evolution is parsimonious. The competition between individuals and between species favors the most efficient. Efficiency demands we cannot respond to every event as though it is new and unique, we must somehow classify events. Again, how the brain compares new perceptions to past memories doesn’t matter here; it only matters it evidently does.


To remember just one particular thing is to remember some of its features. In effect, that set of features forms a pattern or type that future things can be compared with. In other words, every description of particular thing can serve as a type for future reference.


If a man with a gun takes a pot shot at you, then remembering that particular situation as a type is valuable. The type is not the situation in reality, it is only the features describing that situation that you remember.


We can match a newly perceived thing to our memory of one particular thing. And then respond to it in the light of what we learned from the past. Thus, referring to our memory of just one particular thing can be a basis of learning by conditioning (or trial and error) in psychology.


In nature, the matching of newly-perceived things to past-remembered things must be fuzzy.  In science and business, the matching of perceived things to remembered things has to be more exact.

Formalizing descriptions of similar things into verbal types

In describing the locations of pollen sources, honey bees describe a set of things that resemble each other. But honey bees don’t discuss what those resemblances are.


By contrast, humans can and do discuss the resemblances between things, and create verbal types in doing so. We invent words (such as yellow, person, payment) to label similar things and qualities. Words enable us to can formalise our sense of family resemblances in the form of a polythetic verbal type.


The constructivist position is that those abstractions (descriptions, concepts, types) don’t exist until they are formed, conceived or defined.


The success of science demonstrates the effectiveness of describing phenomena using types and logic. And the success of business demonstrates the effective of designing systems using types and logic.


EA is about business roles and processes that create and use business data. We describe particular things by using abstract data types to classify or typify them. Say, order, invoice and payment. We define each entity and event type in terms of attributes (quantitative and qualitative variable types). At run-time, when a business records an entity or event as having values for those attribute values, it formally defines it as an instance of that type.


We also describe abstract process types (in data structures) which sequence atomic activities. Say, order value = order amount * unit price. At run-time, business processes remember things of the defined types the business needs to monitor and direct, and uses that memory to decide how to responds to events when they happen.

Our type theory

We have no direct knowledge of things in reality. We know them only in terms of types we construct (in the mind, in speech, writing, mathematics, whatever) to describe them. We (describers) can understand physical reality only in so far as it is correlated with a type we have access to, which we can correlate well enough with reality for practical use.


Types in a description theory

Descriptive types

<create and use>          <characterize>

Describers <observe and envisage > Described things


A described thing is any aspect or part of reality that a describer can observe or envisage. A type is a representation (in mind, speech, writing or other form) that is correlatable with a described thing. A describer is any animal or machine that can create or use (encode or decode) a type. AI software can now abstract types (patterns) from instances (things with similar features).


Note that each pair of concepts is related in a many-to-many way. A describer can apply several types to the same thing, which may be compatible or in conflict. Is light rightly described as waves or particles? Physicists do not say either model is the “true” model, they say only that each can be useful.


Types <represent> described things.

We usually discuss types as they are defined verbally or symbolically. However, the concept of type can be extended to embrace every kind of model an animal or machine can make of the world. Described things include everything that can be observed or envisaged, including types and describers.


Describers <create and use> types.

To create a symbolic type is to encode it. To use a type involves decoding it. Symbolic types are given meanings only in the acts of encoding or decoding. To create a type is to encode a model that represents some feature(s) of a described thing. To use a type is to decode it, then use it to respond to or manipulate whatever is described.


Describers <observe and envisage> described things.

Describers are animals or machines that can encode and decode descriptive types. A described thing is anything that can be observed or envisaged, including types and describers. In short, observing and envisaging are processes that involve encoding, decoding and comparing of types.


Describers and types are also typifiable.

Describers are physical animals or machines. Types are also physical matter/energy structures. Many copies of a type can be created and used. If all copies are deleted, then the type disappears from the cosmos. In other words, there is no ethereal type aside from what exists in one or more copies of it.


Describers can not only observe but also envisage things, which may be purely fantastic or might possibly exist. E.g. neither a unicorn, nor the next prime number beyond today’s largest known prime, exist in material reality today, but to envisage them is to typify them.

Every type is a description

A type is an intensional definition, a description of a set member. Here are two complex types that describe our sun in terms of simpler types. The second is more formal than the first.


·       A yellow, spherical, source of light, more than 100 times wider than the earth

·       A G-type, yellow-dwarf, main sequence, star.


Both types are abstractions, since we could never define every conceivable property of the sun (and it changes over time). Other stars conform to the same types. The two typified sets are not the same size.


In human evolution, surely, the induction of a generic type from features found in two or more descriptions was first and foremost fuzzy. Since in nature, there are usually exceptions to the rule and hybrid cases. So-called “monothetic” types - to which things conform perfectly and completely – in the world of mathematical description – are the exception rather than the norm.

Every description is a type

“The fact is that one cannot in language point to an object without describing it. If a sentence is to express a proposition, it cannot merely name a situation; it must say something about it. And in describing a situation, one is not merely ‘registering’ a sense-content; one is classifying it in some way or other, and this means going beyond what is immediately given.” Chapter 5 of “Language truth and logic” A J Ayer.


Physicists say that there is nothing in their description of the universe that prevents parallel universes from existing.


Think of any particular thing; a molecule, a game of chess, a galaxy, whatever. Write down a description of it. Perhaps the thing you have described is unique. But there is nothing to prevent your description being realized in more than one particular thing. To describe one thing is to create a type to which other things might conform.


In short, generalisation is not required to define a type. Since to remember or describe even one thing is to define a type. We can describe things we envisage just as well as things we observe. We can inductively draw common features from similar descriptions to create another description. A type that we create by generalization may have no practical use.


Remember: Every description we construct is a type, even if there is only one thing, or nothing, that conforms to it. This view of types underpins the simpler general description theory in the next chapter, which in turn underpins the system theory in chapter 1.

Types in description theory

A description is a structure to which some phenomena (observed or envisaged) may be correlated. Every description we construct is a type, even if there is only one phenomenon, or none, correlated with it.

·       Describers <observe and envisage> phenomena.

·       Describers <create and use> types to help them recognize and deal with phenomena.

·       Types <characterize> phenomena in terms of their structural and/or behavioral attributes.


Types in a description theory

Descriptive types

<create and use>          <characterize>

Describers <observe and envisage > Described things


Every description (however complex) serves as a type that may conceivably be embodied or realized in many physical instances, by many real-world things.

Types in systems theory

An abstract system (e.g. a musical score) is a description to which a physical system (e.g. a musical performance) may conform. Every abstract description we construct is a type, even if there is only one physical system, or none, that conforms to it.

·       Describers <observe and envisage > physical systems.

·       Describers <create and use> abstract systems to help them recognize and deal with physical systems.

·       Abstract systems <represent> physical systems in terms of their structural and/or behavioral attributes.





Abstract system

A type