An ontological foundation
for systems
A type theory for system theory
Copyright 2019 Graham Berrisford. One of more than 100 papers on the “System Theory” page at http://avancier.website. Last updated 05/01/2020 12:42
A system theory should help us identity things in reality that
are describable as systems.
The systems of interest appear in two forms:
as a description of some systematic behaviour, and as a reality that behaves as
described.
The reality is only a system when, where and in so far as it realises a system description.
Ashby’s cybernetic view of systems is based on simple presumptions about how we describe reality.
We divide reality into discrete things, and describe them using types.
This paper discusses types under headings defined in an ISO standard for upper ontologies.
For a longer discussion of types in relation to sets, read A Philosophy of Systems.
Contents
Requirements
for a top-level ontology
Observing
actualities and envisaging possibilities
From
subjectivity to objectivity
Perceiving,
recognizing and describing things
Types,
qualities and other attributes
Scale and
granularity (and levels of description)
Parts,
wholes, unity and boundaries
Causality
(and levels of reality)
Footnote 1: on quantities and mathematical entities
Footnote 2: on 3D things in the BFO
Systems
The term “system” is used loosely and widely; it is often, in effect, a meaningless noise word.
Here, systems are defined in a way compatible with Ashby’s classical cybernetics and Forrester’s System Dynamics.
The systems of interest here contain actors that interact in regular ways.
Actors interact with each other (and with external entities) by sending and receiving stimuli or messages.
An actor plays a role in a system by responding to a stimulus or message, either unconditionally or as determined by its current state or a memory it has access to.
To describe a system is to bound some portion of the universe, then define its structures and behaviors.
That is, to typify the things that make it a system - including inputs, outputs, actors and activities.
Ontologies
Ontology is the branch of metaphysics dealing with the nature of things and their being.
An ontology is a set of concepts and categories (aka types) that shows their attributes and the relations between them.
An ontology may be constructed for a specific subject area or domain (say, banking).
An upper ontology is supposed to define and relate universal concepts, common to many or all domains of knowledge.
An upper ontology has to address some concepts that have troubled philosophers for millennia.
Here is a list of “Requirements for a Top-Level Ontology”, edited from ISO 21838-1.
1. Space and time
2. Actuality and possibility
3. Classes and types
4. Time and change
5. Parts, wholes, unity and boundaries
6. Space and place (locations, shape,
holes and a vacuum)
7. Scale and granularity (and levels of
reality)
8. Qualities and other attributes:
9. Quantities and mathematical entities
(quantitative data and with mathematical data and theories)
10. Processes and events
11. Constitution: (the relation between material entities and the material of which, at any given time, they are made)
12. Causality
13. Information and reference (information entities)
14. Artefacts and socially constructed entities (engineered and socially constructed items like money and laws)
This paper touches on most items in the list (others are addressed in related papers).
Ontology is said to be a branch of metaphysics – but the approach here is more physical and epistemological.
Epistemology is about what we know of reality, through observation, testing, reasoning and learning from others.
The triangle below identifies and relates three epistemological concepts.
|
Epistemology |
|
Descriptions <create and use>
<represent> Describers <observe and envisage> Realities |
This differs from Ogden and Richards’ "semiotic triangle" and Peirce’s triadic sign relation” discussed in other triangular philosophies.
In this triangle, describers are actors that create and use descriptions.
Descriptions are created to remember or convey meaningful information about some aspect or portion of reality that is observed or envisaged.
But social animals described things to each other for eons before verbal language.
And animals recognized things for eons before they attempted to communicate much more than mating intentions.
“Knowledge is a biological phenomenon” (Maturana, 1970).
“Knowledge "fits" but does not "match" the world of experience” (Glasersfeld, 1987).
Why did processes for encoding and decoding memories and messages evolve in animals? Because they proved helpful.
Animals remember things in the world so that
they can react appropriately to them.
And members of a social species find
communicating and cooperating helps them live better than they can do alone.
System definition?
Systems thinking discussions can become
confused when people don’t clearly distinguish:
· a description, an abstract system, which defines some systematic behavior – performable by one or more described realities.
· a described reality, a concrete system, which behaves systematically – in accord with one or more descriptions.
In practical applications of system theory, a reality is only a system when, where and in so far as it realises a system description.
For sure, stuff does actually exist out there.
We may observe a house, a horse, and the largest known prime number, which all exist in time and space, in material reality.
The types “prime number” and “largest known prime” exist in countless minds and records.
But the latest instance of “largest known prime” exists only in records, because it is too large to remember (in 2018 it had more than 23 million digits).
We can also envisage stuff that might possibly exist.
We can envisage a unicorn, and the next prime number beyond today’s largest known prime.
None of those exist in material reality today, but we can describe them, and infinite other possibilities.
We can describe the process for generating the next largest known prime, starting from the one known today.
The realities of interest to us here include whatever is
· observed (including observed describers and descriptions) or
· envisaged (including envisaged describers and descriptions).
Some portions or aspects of the universe that exist, in the everyday sense, may never be envisaged or observed by describers.
Conversely, some “realities” envisaged by describers will never actually be realized in the universe.
They will only “exist” in so far as describers envisage and create descriptions of them.
Yes, to call those “realities” is questionable, but to call them anything else would obscure the philosophy here.
System
definition?
We observe actual (current, baseline, as-is) systems; and envisage possible (required, target, to-be) systems.
We can describe a process for analysing and designing a required system.
That process may be seen as a meta system to the required system.
Certainly, what we know of the world is a biological phenomenon.
But that does not mean knowledge is entirely subjective or personal.
How do we find food to eat if we have no objective-enough knowledge of the world?
How many subjective interpretations can different observers make of Newton's f = ma or Einstein's e = mc2?
No biologist would accept the view expressed by one systems thinker that "internal cognitions do not reflect any external reality".
First, the neural systems evolved to represent things in their environment (food, friends and enemies) in bio-chemical memories.
This helps individual animals survive and thrive by recognizing and manipulating things in their environment.
Social animals evolved further to share knowledge of things in their world, by translating internal representations into external messages (like alarm calls).
Human animals evolved further to communicate information about the world (descriptions, directions and decisions) by using words.
We discuss what we observe and envisage using words; we propose theories and test them.
In short, to transcend our subjective experience, we use science, logic and domain-specific languages.
We turn the subjective into the objective by empirical, logical and social verification.
To deny that would be to deny the success of social animals and science.
System definition?
We test that things in the real world behave systematically, as expected from a system description.
The approach here is Darwinian; the premise is that there was no description of reality before life.
Consider some stages in the evolution of knowledge.
1. Molecular memory: organisms recognize molecular structures.
2. Neural memory: animals remember things they have seen, heard or otherwise perceived
3. Messaging: social animals share descriptions of things in fixed format messages (e.g. gestures and alarm calls)
4. Speech: humans encode descriptions in words
5. Writing: humans record descriptions in an external persistent form.
6. Science: humans learn to form theories, predict outcomes and test them.
Amoeba must recognize chemicals on the surfaces of things (food, predators, other) and react appropriately.
To do this, they have receptors that recognize instances of a chemical type that they need to react to.
To recognize a chemical structure, a receptor must be structured with a description of it.
Leap over some billennia from single-celled animals to social animals that communicate using a few fixed message types.
At first, communications were simple and imprecise (e.g. an alarm call meaning your situation is of the type “dangerous”).
Incrementally, communications become more complex and precise.
A honey bee gives values to the types we call the “direction” and “distance” of a pollen source from the hive.
Jump over many millennia to humans who communicate using the infinitely flexible tool of words.
We learn what a word means by hearing it used to refer to one thing several times, or to different things that resemble each other.
And by using words, we create extraordinarily rich descriptions of the world.
First, to make sense of the space-time continuum, we divide it up.
We carve the world we experience into discretely identifiable and recognizable individual things.
E.g. A person, a predator, a planet, a product, a process.
Newton divided the spectrum in a rainbow first into five colours, and then into seven.
Musicians divide sound into discrete notes, of discrete lengths and discrete pitches.
Having dividing the world into discrete things, we recognize resemblances between similar things.
Primitive animals surely don’t classify things into rigid “types”.
But they can recognize “family resemblances” between similar things.
And learn to respond to similar things in appropriate ways.
Body language, gestures and facial expressions are rarely ambiguous; because they evolved to convey specific meanings
By contrast, words are infinitely fluid and flexible, and we each develop a vocabulary that is unique to us
Natural language is so flexible and ambiguous that listeners cannot be sure they know what speakers mean.
This makes verbal communication a continual search for mutual understanding
In social situations, we reduce ambiguity by testing understanding (know what I mean?) and repeating our message in different words.
In scientific endeavours, and in business, we strive to eliminate ambiguity.
For communicating about things in a "bounded context”, we create and use a "domain-specific language" in which specific words represent specific types.
Science depends on creating and sharing types such as “molecule” that formalise family resemblances into defined categories.
Even within a domain of knowledge, scientists don’t presume a description is perfectly true in an absolute way.
Descriptions can be useful, yet inconsistent with each other.
E.g. Is light a stream of particles or waves? Physicists find each model has its practical uses.
A statement that is true in one domain-specific language may be untrue in another.
System
definition?
Ashby’s system is a complex type (composed of simpler types) that describes some behavior of a real machine.
Forrester’s system dynamics relates collections or quantities of different types to each other
“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.
We use types (classes, categories) to categorise and describe things.
The three essential relationships are:
· Describers <observe and envisage >
things.
· Describers <create and use> types to help them recognize and deal with things.
· Types <characterizes> things, be they entities or events, in terms of their structural and behavioral attributes.
|
Types |
|
Types <create and use>
<characterize> Describers <observe and envisage > Things |
To describe a thing (e.g. my pet) is to typify it (e.g. as a cat).
We may or may not share the types we use to describe things.
The types may be what linguists call ‘universals’, or cultural, or purely idiomatic.
To convey the meaning of a type to me, you may rely on cultural references we share.
Or strive to induce similar sensations in my brain to ones you experienced in creating or using the type.
You might point to things that instantiate the type, create a metaphor, or show me other uses of the type.
To reach a precisely shared understanding can be a very difficult process.
We create a type by describing a thing that instantiates it.
We can do this by adapting an old type, by adding, changing or removing attributes.
A common definition format is “genus + difference”, meaning a more generic type, extended by specific attributes.
|
Type name |
Type attributes |
|
Cat |
A
carnivore with soft fur, a short snout, and retractable claws. |
|
Unicorn |
A
mythical horse-like animal with a single straight horn projecting from its
forehead. |
In natural language, synonyms and homonyms abound.
People often use different type names for the same type, and give the same type name to different types.
But when it matters, in a system or domain-specific language, we fix one type name to one meaning, and vice-versa.
Type-to-thing is a description-to-reality relationship.
Every type is a description - of a thing that instantiates the type.
Conversely, every description is a type
As soon as you describe one observed thing, you can envisage more things of that same type.
Many things can realise the attributes of one type. E.g. Many people are “young”; many are “tall”.
Conversely, one thing can instantiate (realise, manifest, or embody) many types. E.g. I am “tall” and “blue eyed”.
And one thing may be described by alternative types. E.g. scientists describe a beam of light as stream of either “waves” or “particles”.
Structural and behavioral attributes
You may notice the lack of operations, functions or rules in the example type attribute lists above and below.
But let us begin with simple structural examples, much as Ashby started by defining a system as a set of variables.
We’ll come later to the rules or processes that relate and change the values of those variables.
System
definition?
An ontology is composed of types that are defined by their attributes and the relations between them.
A system description is, in effect, an ontology.
We often make sense of things by arranging them in a hierarchy.
We use two very different kinds of hierarchy to organise our descriptions of the world.
Composition-decomposition in whole-part hierarchies
Describers
can describe a thing of any scale they choose, from very large to very small
(from the universe to subatomic particles).
They
can successively decompose a larger thing into smaller things, as many times as
they choose (though not often more than 4 times).
They
can stop decomposing at any level of granularity they choose (their atomic
things might be as large as galaxies or as small as quarks).
Enterprise
architects describe business using various kinds of hierarchy.
Notably,
an organization (management) hierarchy and a function/capability hierarchy.
Generalisation-specialisation in type hierarchies
A type is commonly defined in the form called “genus plus difference”, as a subtype of a more generic type with some additional attributes.
By incremental, successive generalisation of types, you reach a top level, where the types are considered to be axiomatic.
|
Type name |
Type qualities or
attributes |
|
Thing |
A
discrete unit or division of the world or phenomena in it. |
|
Entity |
A
thing whose identity persists over lifetime, and has a structure at
each moment in time. |
|
Organism |
An
entity that is autopoietic, its components perform processes that
consume simple chemicals and sustain its components. |
|
Animal |
An
organism that feeds on other organisms. |
|
Carnivore |
An
animal that eats a diet composed mainly of meat. |
|
Cat |
A
carnivore with soft fur, a short snout, and retractable claws. |
|
Lion |
A
cat that is muscular, deep-chested, with a rounded head and ears, a
reduced neck, and a hairy tuft at the end of its tail. |
At a lower level, types may be particular to one domain or system, or to a narrow range of interests.
At a higher level, types are more generic.
An “upper ontology” is supposedly universal, but even so, several different ones have been published.
A type hierarchy is a very limited kind of ontology.
Infinite different type hierarchies are definable.
One type can appear in several type hierarchies (which is to say, there can be multiple inheritance).
One thing can change from one type to another over time (e.g. from egg to caterpillar to pupa to adult butterfly).
There is no fixed ontology, or one upper ontology.
In analysing and designing a system we usually build a domain-specific language from the ground up.
System definition?
Both composition and generalisation are employed in system definition.
We naturally classify things into two broad kinds: discrete structures (in space) and behaviors (in time).
· Some philosophers divide things into “continuants” and “occurrents”.
· Linguists divide words into “nouns” and “verbs”.
· The space-time dichotomy can be seen in other pairs of terms: actors and activities, entities and events, forms and functions.
· Software engineers know the object/operation dichotomy underpins much of software engineering.
One upper level ontology, the Basic Formal Ontology (BFO) divides the space-time continuum into two disjoint types.
· Continuants: substantial entities that endure through time, while maintaining their identity. E.g. the sun, the moon, and the solar system.
· Occurrents: things that happen, both events and processes that unfold and develop in time. E.g. an orbit, a football match, and goal scored, and a heartbeat cycle.
Nevertheless, the division is questionable; is “order” a noun or verb, an entity or an event?
Structures in space: substantial things
Every material thing, at a moment in time, can be located in the three dimensions of space.
A thing can be solid or holed; the holes may contain other things, nothing we know of or care about, or a vacuum.
A structural thing can be of any shape, regular or irregular; its shape may endure or change over time.
Two shapes often discussed by system theorists are hierarchical structures and network structures.
These and other shapes may be used as “design patterns”.
A line of behavior is a graph showing the trajectory over time of changes to a structural state variable quantity.
Behaviors
over time: processes and events
A process runs over time, either from start to end (like a mating ritual) or cyclically (like your heart beat).
A process can be continuous or divided into discrete steps, each triggered by an event.
Later, like Ashby, we will model continuous behavior as discrete event-driven behavior.
Because event-driven behavior is what we observe and envisage in the social and business systems of interest.
An event triggers a process (or is a process) that occurs in what we regard as a single moment in time.
The process/event distinction is a matter of perspective.
On a geological time-scale, world war one might be regarded as an event that changed countries’ boundaries.
To the user of a database, a transaction is an event that moves the database from one consistent state to the next.
To the programmer, the transaction is an extensive process.
Four dimensional things (continuants as
side effects of occurrents)
The distinction above, though we all use it in describing things, is blurred if not illusory.
Can you classify each of the following as either a structure
in space or a behavior over time?
· The sun
· The solar system
· A cloud
· A cyclone
· A rainbow
· The Amazon river
· A queen bee
· Your lungs
· A conveyor belt
· A project
· A banker’s draft
· IBM
The contrary view here is that continuants and
occurrents are structural and behavioral aspects of
the same reality.
Continuants (substantial entities) are the
structural facets (results or side effects) of occurrents (events and
processes).
E.g. the sun is both a substantial entity that endures and a process.
E.g. an order (a structural entity) is the side
effect or result of an order (a behavioral event or
process).
All actor-performed processes have a substantial or material face located in or distributed across space at each moment in time.
·
A solar system is a process (orbiting planets)
whose state at each moment is observed in the positions of planets.
·
A person is an unfolding process whose state at
each moment is a material entity (every decade, every cell in
your body is replaced by a new cell).
·
A butterfly is an unfolding process (egg > larva
> pupa > adult > dead > decayed) whose state at each moment is a
material entity.
·
A football match is an unfolding process that
depends on (and manifest in) changes to the states of material entities (players, football and scoreboard).
·
A football is a structural side effect of
processes: it is manufactured and used; it deflates and is re-inflated; its valve is replaced; it is dumped and decays.
Note: continuity of identity (assignable to a thing by an observer, or by its DNA) does not imply continuity of the substance or material.
See Footnote 2 for more on the continuant/occurrent distinction.
System definition?
A system’s structures and their behaviors can be modelled in object-oriented and event-oriented ways.
Object-oriented approach
The whole system is seen as a set of parts – objects - which interact when higher-level processes occur.
This approach defines types in a model that relates classes of objects. (Some differentiate types and classes, but we don’t do that here.)
It define each class of objects in terms of structural attributes and behavioural operations allowed on an object of that class.
Then defines for each event, the communication required to access and modify all affected entities.
Event-oriented approach
The whole system is seen as a set of parts - state machines – which are coordinated by discrete events
This approach defines types as entities in an entity-attribute-relationship model that relates entity types.
It defines the life history of each entity as a state machine (or several parallel state machines).
It defines system behavior as event-triggered state changes allowed on each entity type.
Then defines for each event, the process (cf. transaction) required to access and modify all affected entities.
What is the difference?
The object-oriented approach tends to encourage reuse by subtypes of operations inherited from more general supertypes.
The event-oriented approach factors out processes shared by different events into “superevents” (think subroutines).
(“Object-Oriented SSADM” Prentice Hall, Berrisford G. and Robinson K. 1994).
We found a way to resolve the paradigm clash between the approaches, if an obscure one.
(“Reconciling OO with Turing Machines” in Computing. J. 37(10): 888-906 Berrisford G & Burrows M. 1995).
Imagine observing a universe in which nothing changes, you would think time had stopped.
Time is observed and measured by change.
The lifetimes of discrete things
Every persistent thing has a lifetime, even celestial bodies.
Mars was created, has competed billions of orbits, has changed in various ways, and will eventually decay or die in some way.
Note: before the type "planet" was named and defined, Mars was not and could not be called an instance of that type.
The lifetimes of types
A descriptive type cannot exist until a life form (or artificial intelligence) creates it.
Once a type has been created, we may share
and use that type to describe things for millennia.
E.g. The type "planet" was created in ancient times, and since then has been held in minds and other records.
A type can be changed, or replaced by a new type.
E.g. In the late 20th century, astronomers redefined the attributes of a thing conforming to the type “planet”.
And since then, they no longer call Pluto a “planet”.
Eventually, when the universe is too cold to support life, even the types “planet” and “even number” will disappear.
The lifetimes of things’ instantiations of
types
There are not two but three general concepts of interest here:
|
Type theory |
Social systems |
|
Types |
Roles
(in abstract system descriptions) |
|
Instantiations
of types |
Performances
(in concrete system realizations) |
|
Things
that instantiate types |
Actors
who perform roles. |
As instantiation has a life time, as does the performance of a role.
|
Type theory |
Social systems |
|
Instantiations
are transient. |
Performances
are transient. |
|
A
thing may instantiate a type for a period of time, or intermittently. |
An
actor may play a role for a period of time, or intermittently. |
|
A
thing may instantiate several types in parallel, or swap between them. |
An
actor may play several roles in parallel, or swap between them. |
System definition?
In analysing and designing a discrete system we usually build a domain-specific ontology from the ground up.
A common practice is to define and relate long-lived business entity types in some kind of data model.
Domain experts must agree axiomatic definitions of a few kernel or core types (like, customer, supplier, product, asset, employee, project).
Other types further characterise those kernel types and relate them to each other.
The enterprise data architect’s challenge is to address commonalities and communications between discretely defined systems.
The presumption in all system definition is that a type has a lifetime.
It is created when the system is created; it may survive unchanged through several system generations.
It may be changed in, or removed from, a new system generation.
Thinking about the lifetimes of types exposes the limitations of a type hierarchy.
The Linnean hierarchy of biological species/types does not show evolution over time – a cladogram does that better.
A type hierarchy of the BFO kind is also undermined by changes to things and types.
Are Patient and Doctor subtypes of Person-in-Hospital? Or different roles that can be associated with a Person over time?
Are Lecturer and Professor subtypes of University Employee? Or states in the life history of a University Employee?
The passage of time tends to turn subtypes of a type into roles played by a thing, or states in the life history of a thing.
Another effect of time can be to weaken the degree to which a thing conforms to a type.
(“How the Fuzziness of the Real-World Limits Reuse by Inheritance Between Business Objects.” in OOIS 1995: 3-18 Berrisford G.)
Mereology, the study of wholes and parts, is bedevilled by the ambiguity of its basic terms.
Generally, a whole is a complete, entire, bounded thing; it may be atomic or composed of other things.
It could be a collection of assorted things, with little or no relationship between them.
Here, the whole is a system of interest.
And what unifies the parts is that they contribute to the functioning of the whole system.
The boundary of a whole is a decision made by the observer or envisager of the thing interest.
A boundary may be drawn in space, time or logic
· A whole clock bounds its parts in space.
· A whole process bounds its parts (steps) in time.
· A whole software application bounds its component parts in logic.
· A social system bounds the roles played in the system, by actors.
The boundary of the whole may be crystallized in the form of an interface definition.
The whole-part association is a very loose relationship.
One thing can be composed (in space, time or logic) of many other things.
One thing can be a part (in space, time or logic) of many other things.
The boundaries of things can be discrete, overlapping, cooperating or competing.
Bear in mind that a part-whole membership (like each part and the whole) has a life time.
System definition?
The whole-part relation is especially loose in social systems, since you may join and leave many social systems as you choose.
Consider actors as parts, and social systems as wholes.
|
The mereology of social systems |
|
Memberships
are transient |
|
An
actor may belong to a social system for a period of time, or intermittently. |
|
An
actor may belong to several social systems in parallel, or swap between them. |
You may flip from minute-to-minute (even second-to-second) between roles in different social systems.
The boundaries of those social systems may be discrete, overlapping, cooperating or competing.
Causal
connections
One thing may create, change or destroy another thing
Which is to say, they are connected in a cause-effect relationship.
These causal effects may be typical of how things one type (say wolves) affect things of other types (say sheep).
Emergent
properties and levels of reality
The behavior of a whole system emerges from interactions between its parts.
The behavior of a higher-level system depends on the actions of lower level systems
In other words, there are different levels of reality and of causality.
System definition?
We can draw “causal loop diagrams” to show cause-effect relationships between types of thing (say wolves and sheep)..
And use “system dynamics” to model how these relationships change a system’s state over time.
Nobody really understands how higher-level knowledge and consciousness emerge from our DNA.
But we do have a basis for that understanding, in principles articulated by Darwin (evolution) and Ashby (cybernetics).
Ashby’s cybernetics is based on simple presumptions about how we describe reality.
We divide the world into things, describable using types, with some interest or requirement in mind.
|
Epistemology |
A type theory |
|
Descriptions <create and
use> <represent> Describers <observe and
envisage> Realities |
Types <create and
use> <characterize> Describers <observe and
envisage> Things |
Ashby’s real machine is a thing that behaves regularly, according to rules.
E.g. a clock, a pair of sticklebacks in their mating ritual, a flight of geese, a game of poker.
Internally, the machine contains actors interacting, by observing and reacting to each other.
Externally, the machine can be encapsulated and observed as though it were a single actor.
The externally observable behavior emerges from rule-bound interactions between actors “inside” the system.
Ashby’s system is a complex type (composed of simpler types) that describes some behavior of a real machine.
The triangular view runs through much discussion of systems that follows.
|
A system theory |
|
Systems <create and use> <represent> Systems thinkers <observe and envisage> Real machines |
Descriptions, types and systems all occupy the same position in the triangle, as abstractions.
Remember, at the same time, they are also realities/things that might themselves be described/typified at a higher level of abstraction.
Emergent properties
“Behavioral synthesis” is finding a set of actors and interactions to produce a given behavior.
“Behavioral analysis” is determining what behavior emerges from a given set of actors and interactions.
Emergent properties can be calculated and predicted from actors’ interactions, only up to the limits set by probability, chaos and computation.
A system, despite being rule-bound, may have surprising “emergent properties” and “chaotic” outcomes.
The premise here is that there were no concepts before conceivers, no descriptive types before life.
Type are instruments that life forms
developed as side effects of biological evolution.
There
is no type outside of a description encoded in a matter and/or energy
structure.
And
when all descriptions of a “rock”, “plant” or “circle” destroyed, that type
will disappear from the universe.
The
idea of an ethereal type is useless,
redundant, and better cut out using Occam’s razor.
Just as natural language is an instrument for description and prediction - useful in so far as it works.
So, mathematics is also an instrument for
description and prediction - useful in so far as it works.
Where
do numbers and other mathematical types and concepts come from?
Like words, numbers are side effects of biological evolution, have evolved over time, and only exist in material forms.
This instrumentalist and materialistic view may seem radical or strange to many – including some mathematicians
But it seems enough for the purpose of describing “real machines” as “systems”.
For a longer discussion of types in
relation to sets, read A
Philosophy of Systems.
The Basic Formal Ontology (BFO) divides time and space into two disjoint types.
Continuants: substantial entities that endure through time, while maintaining their identity. E.g. the sun, the moon, and the solar system
Occurrents: things that happen, both events and processes that unfold and develop in time. E.g. a goal, a football match, and a heartbeat cycle.
This BFO paper refers to occurrents as 4D objects and continuants as 3D objects.
The notion of 3D objects as highly questionable.
Think of a 4D occurrent as a sausage through time, or a movie.
Now think of a 3D continuant as a slice of the sausage, or a still in the movie.
If a 3D thing endures, then it has a life history, for which there are three possibilities.
First, it moves in time, the slice along the sausage, the still along the movie.
But a slice or a still does not do that; it is replaced by the next slice or still.
Second, a 3D thing has no time dimension, is infinitely thin in time, is undetectable and indescribable.
If so, then there are infinite 3D things for one 4D thing, and the notion that a continuant “endures” or is "substantial" is odd.
Third, a 3D thing has a finite width, that is, the time span between one state change in the life of a 4D thing and the next.
Then there are many 3D things and the notion that a continuant "endures" makes no sense.
The view here is that continuants
(substantial entities) are the structural facets (or side effects) of
occurrents (events and processes).
In other words, continuants and
occurrents are structural and behavioral aspects of
the same reality.