Complexity science – is it a mirage?


Overviews of "complexity science" tend to present a jumble of ideas from mathematics to sociology.

This makes it difficult to define the whole field in a coherent way.

Writers who lean towards hard science present as "complex systems" ones that most people would describe as simple.

Writers who lean towards sociology speak of "complex adaptive systems", where most see nothing more or less than a group of people interacting as they see fit.

This article compares and contrasts definitions from several sources.


Introduction. 1

What is complexity science?. 1

How to assess the complexity of an entity or a system?. 1

What is a Complex Adaptive Systems (CAS)?. 1

On causality, catastrophe and chaos. 1

Conclusion. 1

Footnote on complexity as non-linearity. 1



This article explores ambiguities in the terminology of complexity science.

For example:

·       “Adapting” can mean changing the state of a thing and/or changing it into a different thing.

·       “Agency” can mean the freedom of an actor to choose between actions in a role, or the ability to invent new actions.

·       “Complex” can mean messy/disorderly, or the opposite, a measure of orderliness.

·       “Complex” can refer to a real-world thing, or a description or model of it.

·       “Complex” can be a measure of structure, or of behavior.

·       “Emergent properties” may arise from evolution of the universe, or from entities interacting produce results one cannot produce alone.

·       “Fractal” is abused to mean hierarchically composable and decomposable.

·       “Holism” is abused to mean wholeism.

·       “Linear” and “reductionist” are used misguidedly, and as insults.

·       “Non-linear” is used by people with various meanings, and no meaning.



A Mandelbrot fractal is an infinitely detailed two-dimensional image (set of points).

It looks interesting and complex as you zoom into its border areas.

You see the same pattern emerges, recursively, at lower levels of decomposition.

But the formula to generate or calculate this output is based on a simple formula: z(n+1) = z(n)^2 + C.

The process can be coded in less than 20 lines of code.

Is that well-called a complex system?

And by the way, no human organization is fractal.


Complex adaptive system?

Some refer to a group of cooperating people as a “complex adaptive system”.

But when pressed, find it difficult to define what they mean by complex, adaptive or even system.


Some say: “A system is a thing contained or connected in some coherent way.”

The trouble is that everything from an atom to a solar system is contained or connected in some sense.

If everything you can think of is a system, then, how to differentiate systems thinking from any other kind of thinking?


Some say: “An adaptive system is disposed to act or change in response to events or conditions.”

The trouble is that every dynamic system is disposed to act or change in response to events or conditions.

If every system changes over time, then what is a non-adaptive system?

The term adaptive is ambiguous.


Some say: “A complex system is that doesn’t behave in a simple linear way (cause A to effect B).”

The trouble is that simple structures can behave in ways describable as non-linear, complex, unpredictable, even chaotic.

If simple systems meet the definition of complex systems, then, how to differentiate complexity science from simplicity science?

The terms complex and linear are ambiguous.


This chapter addresses these ambiguities.

What is complexity science?

Overviews of "complexity science" tend to present a jumble of ideas from mathematics to sociology.

This makes it difficult to define the whole field in a coherent way.

A commonly mentioned feature of complex systems is “emergent properties”.

To begin with, let us dispel some myths about emergence.


On emergence

Consider the holistic effects or results that emerge from interacting things in these examples:


Emergence does not require a system to have many actors.


Emergence does not mean a system behaves in a surprising or unpredictable way.

Natural systems often produce results or effects that appear surprising or mysterious, until you know how they work.

Designed systems are intentionally designed to produce specified results or effects.

Given a business system to design, the designer must start with the desired effects of the whole.

And when a designed system produces unexpected effects, we call them "unintended consequences".


Emergence does not mean a system is complex in any normal sense of the term.

Because we ignore the internal complexity of whatever parts we select for discussion.


In this simple system

We ignore the complexity in

A rider riding a bicycle

the biology of the rider

Planets orbiting a star

the substance and surface of each planet, and life on earth.

A game of poker

the biology of the card players

Source 1:

This source presents a collection of disparate ideas, with no measure of complexity

It defines complexity as a polythetic type with one or more of four possible features.



“Complex organizations like schools of fish, ant colonies, or car traffic manage to organize themselves into emergent patterns without any form of global coordination.”

Hmm… consider the examples of emergence above.

In a school of nine fish, nine actors swim together following simple rules.

Q1) Is that well-called a complex system?

Q2) Given the rules emerge from evolution – is the school really “self-organizing”?

Q3) What about self-organization in the sense of defining or changing the rules of the system itself?



“Nonlinearity describes how when two things interact the output is more or less than the sum of their parts in isolation.

It arises out of the interdependency between elements within a system and interdependence over time through feedback loops.”


We can never study all that could ever be known about a phenomenon we perceive in reality.

We can only (and must necessarily) select those aspects of it relevant to our interest in it.

Only then can we consider how selected parts interact – holistically – to produce emergent properties.

Consider the examples of emergence above.

Q4) Are these well-called complex systems?


Phase Transition

“A phase transition may be defined as some smooth, small change in a quantitative input variable that results in an abrupt qualitative change in the system’s overall state.”


Consider the melting of ice into water

Q5) Is that well-called a complex system?



“It is important to be aware that real world complex systems are the product of many overlapping networks interacting dynamically.”


Q6) Does this exclude chains and hierarchies (which are subtypes of network)?

Q7) Does it really mean many networks with many-to-many connections?

Q8) Where are these many networks in a school of fish, a rider on a bicycle, or a melting ice cube?

Google is reported to have two billion lines of code.

Q9) Is that well-called a complex system?

Q10) Is any system not complex?

Source 2 (source lost)

This defines a complex system as being open, non-linear (see footnote), chaotic, multi-dimensional and adaptive or self-organizing.


Yet a simple system can

·       be open. And a closed system can be complex. E.g. an ecology as in modeled in causal loop diagram.

·       produce a non-linear line of behavior. E.g. a double pendulum.

·       have a non-linear disposition. E.g. a wine glass is disposed to one or other of two possible effects when struck.

·       be chaotic. E.g. a predator-prey system, given slightly different initial populations, can produce very different outcomes and population crashes.

·       have the same dimensions as a complex one.

·       adapt. E.g. consider a cooling system adapting to temperature change.

Source 3 on "Understanding Complexity”

This says complexity concepts include the following.

·       Tipping points. The sociological term used to describe moments when unique or rare phenomena become more commonplace.

·       The wisdom of crowds. The argument that certain types of groups harness information and make decisions in more effective ways than individuals.

·       Six degrees of separation. The idea that it takes no more than six steps to find some form of connection between two random individuals.

·       Emergence. The idea that new properties, processes, and structures can emerge unexpectedly from systems in operation.


These look like an arbitrary jumble of ideas.

And by the way, the wisdom of crowds is undermined by the "risky shift phenomena" identified by James Stoner in 1961.

People change their decisions or opinions to become more extreme and risky when acting as part of a group, compared with acting individually.

This is one form of “group polarization”.

How to assess the complexity of an entity or a system?

·       “complex is a term we use for something we cannot yet model.

·       If there is nothing metaphysical about a complex system... then perhaps a complex system is ultimately nothing more than extremely complicated.”

 Cilliers quoted in


Some in the field of complexity science presume a thing has an intrinsic amount of complexity - independent of any observer's perspective, description or model of that thing.

But as Ashby pointed out, the complexity of every substantial physical entity is well-nigh infinite, and certainly far beyond any model we can make of it.


The only complexity we can meaningfully discuss is that of a model or description of a thing.

And even that is measurable in a variety of ways.


For example: The most complex kind of model we produce is the code of a software system.

As a measure of complexity, you might count lines of code.

There are said to be 50 million lines in Windows, and 2 billion lines in Google.

As well or instead, you might measure the logical complexity of a system's component structure, or of it procedures, in various ways.


Still, there is even more complexity to be found in the physical entity that is a software system in operation.

Consider the number and variety of the states the system occupies over time.

Consider the structures and behaviors of the electronics and computing hardware, all the way down to subatomic particles and their movements.


Cilliers didn't quite say so, but he implied that systems thinkers say "complex" when thinking of a real-world thing.

And "complicated" when thinking of a thing in terms of a description or model of that thing.

So, one way to disambiguate the terminology is to say

·       "complex" when speaking of a real-world thing or situation and

·       "complicated" when speaking of a description or model of it.


The general issue in systems thinking is not so much the complex/complicated distinction as the entity/system distinction.


Is complexity found in entity or in a description of it?

In system theory, an entity is not a system, it is as many systems as observers choose to abstract from it.

And its complexity can only be assessed with respect to one such view.

How about a holistic view that encapsulates a system?

If it defines only the emergent properties of the system, then it hides some complexity of that system.

How about a holistic view that defines how the system's internal components interact to produce emergent properties?

Then, the complexity depends on the granularity at which you define those components.


Is a thing complex because it is messy, disorganized or ever-changing?

Surely, if there is no regularity or repetition, there is no system?

To call something complex is to imply it is organized or orderly in some way?


Is complexity found in the organization of a system's structure and/or its behavior?

Very simple structures can exhibit surprisingly complex/complicated behaviors.

See the two videos at the top of this article.


Is complexity found by counting a system's elements?

Do we count the number of entity and event types in a system's description?

Or the number of entity and event instances in the run-time operation of a system?


You cannot measure the complexity of a thing directly.

You can only measure it with respect to a particular description its properties.

The amount of complexity you find depends ow you answer these questions


What is the level of abstraction?

The atomic level of description must be agreed.

How far a description is abstracted (e.g. from sub-atomic particles) is discussed in Brooks' no silver bullet article.


Which dimensions are of interest?

There is complexity in memories actors maintain and messages actors exchange.

And the network in which actors connect

And the activities actors perform.

And the resources actors need

And the trajectories of state variable value changes over time.


Do we count variety or volume?

Do we count the types in a system's description.

Or do we count the instances in a system's operation?


How does complexity relate to entropy?

This question raises too many questions to address here.

Briefly, thermodynamics is largely irrelevant to social systems thinking.


In practice, scores of complexity measures have been proposed, and measuring every conceivable dimension of a system is too difficult.

We do make subjective comparisons.

However, to say the complexity of a thing is only in the eye of the observer seems unhelpful.

We should use a measure of complexity that is found in a description agreed by communicating observers, and agreed by those observers as useful for their purposes.

What is a Complex Adaptive Systems (CAS)?

Some refer to a group of cooperating people as a “complex adaptive system”.

But when pressed, find it difficult to define what they mean by complex, adaptive or even system.

Some have said:

·       Systems thinking is a way of thinking about selected aspects of the world and their interrelationships which is useful in relation to the individual’s concerns.

·       A system is a thing contained or connected in some coherent way.

·       An adaptive system is disposed to act or change in response to external events or internal conditions.

·       A complex system doesn’t behave in a simple linear way (cause A to effect B).


The trouble is that:

·       every way of thinking is about selected aspects of about the world, and relates them to each other and/or the concerns of the thinker.

·       everything from an atom to a solar system is contained or connected in some sense

·       everything is disposed to act or change in response to external events or internal conditions.

·       simple structures (especially ones with feedback loops) can behave in ways describable as non-linear, unpredictable, even chaotic.


If everything you can think of is a system, everything changes over time, and simple things meet the definition of complex things, then how to differentiate systems thinking from thinking?

And what useful meaning does the term "complex adaptive system" convey?


The terms Complex, Adaptive and System (CAS) can each be used for an idea that is interesting and valuable on its own.

However, you only have to skim the systems thinking literature to find different interpretations of the terms, separately and together.

It seems the current consensus is that CAS does not have a strict definition.

CAS is a polythetic type.

It is a collection of attributes, not all of which are necessary to be called a CAS.

Source 1: The Merriam Webster dictionary

“Complex: a whole made up of complicated or interrelated parts."

Note that different observers may divide the same thing into different parts.

So, the complexity of a thing necessarily relates which description you have in mind.


Other sources suggest complex systems have many actors or agents.

However, this seems to confuse size with complexity.

Is a school of 9 fish complex? Is a school of 99 fish ten times more complex?


“Adaptive: Marked by "adjustment to environmental conditions: such as

A.    adjustment of a sense organ to the intensity or quality of stimulation, and

B.    modification of an organism or its parts [to better fit] the conditions of its environment."


Note the dictionary examples illustrate two different kinds of change.

Whereas A is a state change in an organism's life time, B is an inter-generational mutation.


Generally, adaptation implies system elements adapt to events and conditions

But the word is used for two different kinds of change

·       system state change (as in homeostatic adaptation to temperature change)

·       system mutation (as in caterpillar to butterfly, or dinosaur to bird)


It seems some use CAS as pseudo-scientific term for a human social entity.

Surely, every human social entity may be called complex and adaptable?

Be it a small family business, or an army?

But note that continual mutation undermines the concept of system.

Since it can mean there is no pattern, regularity or repetition to be modelled.


System: "a regularly interacting or interdependent group of items forming a unified whole."

In activity system thinking, regular means rule bound,

In social entity thinking, might it instead mean frequent?

Source 2:

Axelrod and Cohen define a CAS in terms of characteristics found in a simple systems and human organizations.

·       "A “system” includes one or more populations of agents and all of the strategies that those agents employ."

·       "A “complex” system is one in which the actions of agents are tied very closely to the actions of other agents in the system."

·       "When the agents in a system are actively trying to improve themselves (“adapt”), then the system is a Complex Adaptive System."


Yet the simplest of deterministic systems may have

·       many component or actors/agents, which

·       interact closely with each other, and

·       produce emergent properties.


Which leaves is with adapt – meaning “learning and seeking to improve themselves”

This shows the authors are thinking of a human social entities in particular, rather than systems in general.


Again: where actors invent actions, a social entity is not well called a system.

A CAS with human actors may better be called a complex adaptive social entity (CASE).

CASE: a social entity that can realize activity systems, and revises/adapts those systems when needs arise.

Source 3: Snowden

Dave Snowden has distinguished complex adaptive systems from other systems by defining them as dispositional rather than linear casual 

He may be using those terms in very particular ways, but his definition entangles ideas that merit attention as distinct concepts.

See the next section on causality.

On causality, catastrophe and chaos

This work has distinguished, deterministic, probabilisitc, possibilistic and innovative causality.

Cybernetics accommodates the first three, but not the last.


The Merriam Webster dictionary defines a disposition as

A.    prevailing tendency, mood, or inclination.

B.    temperamental make up.

C.    the tendency to act in a certain manner under given circumstances.


A and B apply to people, but C is the more general meaning of disposition used in philosophical discussion of causality.

Things have dispositions to act or change in some way when triggered by a cause, by a describable event or condition.


·       A wine glass is disposed to ring or shatter when struck.

·       A cooling system is disposed to start up when the air gets hot.

·       A person is disposed to shiver when cold.

·       A species is disposed to acquire new characteristic(s) when a child is born.

·       A democracy is disposed to replace one government by another.


Consider the disposition of a wine glass to ring or to shatter when struck.

The cause-to-effect process is non-linear in the sense there are two possible outcomes.

The glass "chooses" which to do depending on the nature of the strike, and its own current state.

The outcome may be predictable in theory, but unless the strike is hard, it may be unpredictable in practice.


Human dispositions are malleable, and human social entities display innovative causality.

Human actors may change how an organization responds to events and conditions, and invent ad hoc responses.

Whether an ever-changing social entity is well called a system is questionable, as discussed earlier.


This is a discontinuity in a state change trajectory.

It appears as a sudden large jump or fold in a line-of-behavior graph.

If the graph is stretched over a very long time scale, then it might look more continuous.

Chaos? Ambiguous!

In mathematics, a chaotic system is one whose state and line of behavior is highly sensitive to initial conditions.

Or perhaps a system whose state is stuck in a "strange attractor".

The behavior of such a chaotic system is still deterministic at the level of a single event.

In activity systems thinking, chaos might mean a system's state change trajectory goes up or down in apparently random or irregular ways.

In social entity thinking, chaos might mean actors’ activities are messy, not rule-bound, and no pattern can be detected how actors interact, so there is no activity system.

The edge of chaos? Ambiguous!

Informally, the edge of chaos is a phase transition in a system from a predictable regime to chaotic (but still deterministic) regime.

But the phrase has no precise and general definition across domains where it is used.

It gives the impression of something profound when little has been said.

The phrase has been used (hijacked?) to describe a human social entity.

Obviously, people can perform both deterministic processes, and innovative (spontaneous and adaptive) processes.

Predictable, controllable and stable

These are different ideas.

A weather system is predicable but not controllable.

A rocket may be predictable and controllable, yet not be stable (it does not stay in or near some attractor state).


The more you read of "complexity science" the less coherent it looks.

You might allow the term to be just a label - a heading for a jumble of ideas from classical cybernetics and other hard science disciplines.

However, many in the humanities usually use term CAS with much more specific intent.

By "complex adaptive system", they really mean a human social entity.


In this case, it would be clearer to replace the term CAS by CAE.

A Complex Adaptive Entity is one that survives by responding to changes in its environment.

The change may be to adapt by changing state, or by mutating into a different entity.


A CAS with human actors may then be called a complex adaptive social entity (CASE).

A CASE is a social entity that can realize activity systems, and change those systems when needs arise.

Footnote on complexity as non-linearity

The term complexity is often defined in relation to non-linearity.

Which begs the question: What do linear and non-linear mean?


Linear means n a straight line.

That sounds simple, yet the term is applied by people in different situations with different meanings.

A linear cause is a cause that leads to one effect; it implies a straight line from cause A to effect B.

A linear line of behavior is a state change trajectory seen as a straight line when drawn on a graph of state change over time.

Linear thinking is a term used as an insult.


Non-linear is a term is applied by different people in different situations with many different meanings.

At its simplest, the term means not in a straight line.

So, in activity systems thinking, it could mean a line of behavior that is curved or jagged.

Perhaps an exponential increase or decrease, or a sine wave, or more complicated curve.

E.g. consider unit price movements in a stock market, or the population of a virus.

Non-linear systems in mathematics (after Will Harwood)

A relationship between x and y is linear if y can be expressed as ax+b.

A function f is said to be linear if f(ax) = af(x) and f(x+y) = f(x) + f(y).

A system is linear if it is described by a deterministic linear differential equation.

A system is non-linear if it is described by a deterministic non-linear differential equation.

And a system is non-linear if it is described by a stochastic differential equation.

Non-linear systems in system dynamics (after Will Harwood)

A causal loop diagram shows a set of flows between stocks, where the flows increase or decrease those stocks.

Some systems thinkers call the described system linear if it has no feedback loops; and call it non-linear if it shows at least two stocks connected by a feedback loop.

(That is a different distinction from the one drawn by mathematicians above.)


A causal loop diagram is a sketch of a system's dynamics.

The diagram is drawn with + and - signs on the flow arrows to show which flows (or activities) increase or decrease the stocks (or resources).

For examples, see this introductory book


A causal loop diagram can be a good way to tell a story, or express a hypothesis about the world.

However, it does not show all the quantitative features of stocks and flows needed to complete a system dynamics model (or the differential equations that define the dynamics of the system).

So, it is not enough to reveal all the behaviors the system could exhibit.


Proving such a diagram to be a usefully accurate model is very challenging.

You may be confident, from your experience of the world, that the + and - signs on the flow arrows are correct, but the strength of those of effects may be unclear.

You have no obvious way to judge whether critically important stocks or flows have been omitted from the model.

(e.g. the impact on the Covid virus population of vaccinations or social distancing policies).

And short of adding the mathematics, supplying some initial values to the stocks and running some simulations, it can be hard to be sure what effects the passage of time will produce.

Moreover, some would characterize a "complex system" as one that, given different initial values, can produce very different outcomes!

Non-linear information systems

Most business information systems are connected in a feedback loop with entities in their environment.

E.g. Outputting an order triggers an invoice to be input, which triggers a payment to be output, which triggers a receipt to be input.

Yet most systems thinkers would probably call the system linear.

Non-linear social entities

Social entity thinkers sometimes use the term non-linear to describe human, rather than demonstrably deterministic, behavior.

Whether human thinking is deterministic or not at some level of biochemistry, we have no option but to treat people as having free will.


For more, go to