Complexity science

Copyright 2020 Graham Berrisford. A chapter in “the book” at Last updated 10/07/2021 12:17


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Introduction. 1

What is a system?. 2

What is adaptation?. 2

What is complexity?. 3

How three sources define complex systems. 4

How three sources define a complex adaptive system.. 6

Conclusions and remarks. 8

Appendix 1: On Cillier’s idea of complexity. 9

Appendix 2: What does non-linear mean?. 10



"Complexity science" stretches from mathematics to sociology. This makes it difficult to define the whole field in a coherent way. This chapter analyses the terms and concepts of complexity science/theory with reference to many sources.


To begin, here are some examples of what different people have called complex systems.


·       A school of fish: nine fish swimming together.

·       A ridden bicycle: two actors, rider and bicycle, moving down the road. 

·       A melting ice cube.

·       Google’s software: reputedly two billion lines of code.

·       The weather

·       The biosphere


Questions to ponder. Do fish define or change the rules they follow? Can a rider change the rules that govern their bicycles motion? Is a melting ice cube complex? Is Google’s software complex? If all above are complex, can you give some examples of simple systems? 


And before we analyse complexity theory terms and concepts, below are several mind-bending points to bear in mind.


Complex activity systems?

We model a software system by explicitly relating components at multiple levels of abstraction (delegation from clients to servers, composition of larger from smaller, generalization of more universal from more particular). By contrast, we usually model our world using one level in the hierarchy of sciences (sociology, psychology, biology, chemistry, physics) rather than by including several levels in one model.


Does it make sense to speak of systems in higher-level sciences in terms of scaling up and/or increasing complexity? Which is more complex sun (1057 hydrogen atoms) or a virus (millions of atoms). Can a higher-level social system be smaller and simpler than lower-level biological systems it depends on?


Model or reality? In cybernetics and system dynamics, a system is a model - a selection of structural state variables (cf. stocks) and behavioral rules (cf. flows) that change the system state. The models are abstractions, idealizations or simplifications of material realities. Even the most complex model of an economy or a biosphere is still only a set of state variables and mathematical rules. Observers may abstract countless different systems from one material entity. So, the complexity of a "system of interest" is not the complexity of the entity regardless of any observer.


Prediction time? Given a system model, its complexity might plausibly be measured by how long a computer takes to predict the next state of the system. However, that complexity measure may differ for each event type a system responds to.


Chaos? Chaos theory tells us there are rule-bound systems whose state at some future date is in practice unpredictable. And yet, we do build complex system dynamics models, and do attempt to predict future realities using those models. That is how climate change scientists and epidemiologists do their job.


Using the term complex to mean unpredictable or chaotic is confusing, since even the simplest of systems (e.g. a double pendulum) can have lines of behavior that are unpredictable or chaotic.


Complex social entities?

When "systems thinkers" speak of complex systems, they are often talking about complex realities (social entities, cities or organizations) that evolve in unpredictable ways, rather than rule-bound systems in the sense of cybernetic or system dynamics.


Some refer to a group of cooperating people as a Complex, Adaptive and System (CAS), yet find it difficult to define what they mean by any of those terms. Each term may 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, both separately and together.


·       Complex can refer to disorder or order: the messiness of a disorderly entity in the real world, or the complexity in the orderliness of a system description. It can mean large, that there are many actors or entities in a real-world phenomenon (like viruses and people in an epidemic), even though those actors may interact in simple ways. Or it can mean that, even though a system may be small and simple (like a double pendulum), it produces a complicated line of behavior.


·       Adaptive can refer to the homeostatic behavior of an entity (e.g. shivering when it is cold), or to the evolution of an entity from one version or generation to another, or else to the design of a machine to be configurable. Sometimes it is equated to robust or resilient, two more terms used with various meanings.


·       The system in question is rarely an abstract system, a model of the kind drawn in cybernetics, system dynamics or soft systems thinking. It can be a social entity, a group of people, for which there is little or no definition of roles or rules. The membership of the social entity may clear, or not. So, a complex adaptive system (CAS) might better be called an evolving social entity (ESE) whose members may realize countless activity systems, and change them now and then.


Let us look at each term in more detail.

What is a system?

A thing can only rightly be called a "system" with respect to a description made by some observer/describer that bounds some aspect or part of the world in which some smaller parts interact. Say, a card school, or a rider on a bicycle.



A card school (social entity)

A ridden bicycle (activity system)

Interacting parts?

the card players

the rider and the bicycle


each part responds to feedback from what the others do

each part responds to feedback from what the other does


does what one part cannot do alone

does what one part cannot do alone


adapts to changing conditions

adapts to holes and bends in the road


no overarching director or controller

no overarching director or controller


exhibits non-linear behavior

exhibits non-linear behavior


Of course, despite sharing so many characteristics, a card school and a ridden bicycle are very different. The first is social entity (a community of actors), the second is an activity system (a coordinated set of actions).

What is adaptation?

An adaptation is a change made to a thing in response to a change in its environment. However, Ashby pointed out the word adaptation “is commonly used in two senses which refer to different kinds of processes.” There is the process kind whereby an entity (animal, mechanical or social) responds to external changes so as to maintains its state or integrity. And there is the process kind called re-configuration or mutation that produces a new version or generation of an entity.


In short, an entity can adapt or be adapted to events by rule-bound state change (whether to maintain homeostasis or to advance its state progressively) or by rule-changing mutation, from one version or generation to the next.


"Change the rules from those of football to those of basketball, and you’ve got, as they say, a whole new ball game.” Meadows


"Social systems are not just ‘complex adaptive systems’ bound by the fixed rules of interaction of their parts. Rather, they are ‘complex evolving systems’ that can change the rules of their development as they evolve over time." This book Jackson 2003


In discussing social entities, Jackson preferred the term evolving to adaptive. Again, what many call a complex adaptive system (CAS) may better be called an evolving social entity (ESE) whose members may realize countless activity systems, and change them now and then.

What is complexity?

A thing can only be called a "whole" or a "part" with respect to a scope defined by some observer/describer. E.g. consider the whole that is a fire, in the firebox, of a steam engine, at the front of train, carrying passengers, in the system defined by a train timetable.)


Then, the whole thing can only be called "simple" or "complex" with respect to the granularity of parts and activities regarded as atomic in the description of the thing.


Does complex mean messy? Given a disorganized and/or ever-changing entity, in which no pattern, regularity or repetition can be observed, there is no system. Surely, to call something a “complex system” is to imply the opposite, that it is organized or orderly in some way whose complexity can be agreed by some kind of objective assessment?


How to measure complexity? Google is two billion lines of code that interact to produce the search results you want. Is it simpler or more complex than a school of fish?


In practice, scores of complexity measures have been proposed, and using them is difficult. We do make subjective comparisons. However, to say the complexity of a thing is only in the eye of an observer is unhelpful. We do better to use a measure agreed by observers as useful for their purposes. And to agree how complexity is assessed or measured requires us to answer many questions.


How is complexity measured?

Do we measure reality or description? Cilliers (see appendix 1) didn't quite say so, but he implied systems thinkers tend to say "complex" when thinking of a physical entity (like a card school) and "complicated" when thinking of a description of what the entity is or does (such as the rules of poker).


We cannot measure the complexity of a card school per se, we can only measure it with respect to a description of what it is or does. We can count the roles, rules and variables in the definition of a game of poker. And we can measure the complexity of an actual poker game, in terms of the number of card players, the hands dealt, the money wagered and the decisions made by players.


So, do we count the types or the instances? Do we count the number of entity and event types in the abstract system description? Or the number of entity and event instances that exemplify those types in the operation of a physical activity system?


Do we measure structure or behavior? A simple structure can behave in complex ways. For example, look on the internet for a video of a double pendulum. [Links to videos to be included here.] Conversely, a simple system can produce a complex structure. For example, a Mandelbrot fractal.


(Aside: A fractal is an infinitely detailed two-dimensional image (a set of points). As you zoom in to look more closely at its border areas, you see the same pattern emerges, recursively, at lower levels of decomposition. By contrast, the process to generate or calculate this structure can be coded in less than 20 lines of code (based on a simple formula: z(n+1) = z(n)^2 + C). One systems thinker defines the term fractal very differently. "A fractal system is a complex, non-linear, interactive system which has the ability to adapt to a changing environment." Surely that describes every social entity you know of? No human organization is fractal. An organization's reporting hierarchy is not fractal. Zoom in, you divide one element into different elements. Zoom out, you group different elements into one.)


Which dimensions of system structure or behavior do we measure? There is complexity in memories actors maintain, messages actors exchange, activities actors perform, the resources actors need, the network in which actors connect, and the lines of behavior (trajectories of state variable value changes) over time.


At what level of abstraction do we measure? The atomic level of a description must be agreed. For a software application, do we count the modules, the operations, the lines of code, or the verbs and variables in the lines of code? Do we consider the internal complexity of human actors? The question is discussed in Brooks' no silver bullet article.


Appendix 1 contains a discussion of Cilliers’ ideas about complexity.

How three sources define complex systems

Source 1: “Systems innovation”

This section reviews this short guide to complexity theory:


In different posts/articles, the Systems Innovation source has defined complexity theory in terms of Emergence, Networks, Phase transitions, Non-linearity, Evolutionary dynamics, and Self-organization. Let us explore what these terms can mean.



Of several kinds of emergence, the kind most commonly discussed is the emergence of effects from the interaction of two or more things. All systems, including simple ones with only two actors, have emergent properties. Consider the holistic effects or results that emerge from interacting things in these examples:


·        the force produced by a wind passing over a sail

·        the progress of a rider on a bicycle

·        the V shape of three geese in flight

·        the shimmering of a school of fish

·        the price of fish that emerges when customers and suppliers strike a deal.


Judging by these examples, the emergence of effects or results does not require a system to have many actors, does not require a system to be complex in any normal sense of the term (we always ignore the internal complexity of what we see as atomic parts), and does not imply a system behaves in a surprising or unpredictable way. If it turns out that a designed system produces unexpected effects, we call them "unintended consequences".



Networks of actors? activities? physical connections? communication paths? logical communications? Simple systems (say, actors sending and receiving an SOS message) have such networks.


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


What does this mean? What “many networks” are found in a ridden bicycle or a melting ice cube? What way is there to interact other than dynamically? Does “overlapping networks” mean that two networks can share some members? Might that situation be better seen as one entity or network structure in which the members can play roles in two activity systems?


Phase transition?

As in physics? Water to steam? Or moving from one attractor state to another? Do all complex systems do that?


“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.” Ibid.


In related videos, the quoted source seems to merge the ideas of “phase transitions” and “far from equilibrium thermodynamics” with the more debatable idea called “the edge of chaos” - discussed in appendix 2 below.


Nonlinearity (emerging from feedback)?

Meaning what? What is the opposite, a simple, linear system? One with no feedback loops? Or only dampening feedback? Surely non-linear systems (e.g. a double pendulum) can be very simple in the everyday sense?


“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.” Ibid.


Which is necessary to the definition here: feedback, emergence or both? System dynamics models feature feedback loops that produce emergent effects, yet are entirely deterministic and mechanistic. See appendix 2 for discussion of what else “non-linear” might mean.


One of System Innovation’s videos speaks of complexity theory as “emerging, post-Newtonian, non-linear, system theory.” Yet almost all science is post-Newton. And given that systems with feedback loops that display non-linear lines of behavior, adapt and self-organize, were addressed by system theories in the 1960s, what new theory is emerging?


Evolutionary dynamics (as in CAS)?

As in homeostatic adaptation? Simple thermostatic devices can do that. As in phase transitions? Is a body of H2O a complex system? Or as in replacement by a different system/generation as in biological or business evolution?



Here, this ambiguous term means coordination by peer-to-peer choreography rather than overarching orchestration. Simple linear systems can work thus. And simple rules can generate both complex structures like fractals and simple structures like the V shape in a flight of geese.


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


The rules for how fish swim together, and ants build nests, emerge from evolution. Is that well-called “self-organizing”? Neither fish nor ants cannot define or change the rules of the activity systems they act in. Car traffic is organized in many ways, including but not only the way that individual car drivers avoid collisions, and look for the fastest route.

Source 2: source lost

This source defines a complex system as being “open, non-linear, chaotic, multi-dimensional and adaptive or self-organizing”. Yet a simple system can be open, and a closed system (e.g. an ecology modeled in causal loop diagram) can be complex.


Moreover, what most people would call a simple system can:

·       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).

·       adapt (e.g. consider a cooling system adapting to temperature change).

·       have the same dimensions as a complex system.

Source 3: "Understanding Complexity”

This source says complexity concepts include the following jumble of ideas.

·       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.


The simplest of system produce emergent properties. 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 towards the extreme and risky when acting as part of a group, compared with acting individually. This is one form of “group polarization”.

How three sources define a complex adaptive system

The current consensus seems to be that CAS does not have a strict definition. You might say CAS is a polythetic type - a collection of attributes, not all of which are necessary to be called a CAS. But then, confusingly, not only are some of the attributes ambiguous, but also many are shared by simple systems.

Source 1: The Merriam Webster dictionary

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

This means the complexity of the whole depends on how far you decompose your description of its parts. Other sources suggest complex systems have many actors or agents. However, this seems to confuse size with complexity. Is a school of 99 fish ten times more complex than a school of 9 fish?


“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."


Remember, an entity can respond to events by rule-bound state change or rule-changing mutation? The former is A; the latter is B.


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? And note that continual mutation undermines the concept of system in which there is some 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 interactions. In social entity thinking, might it instead mean merely frequent interactions?

Source 2: Axelrod and Cohen

Axelrod and Cohen ( defined a CAS thus.

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

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

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


Points 1 and 3 restrict complex systems to ones in which each agent has a “strategy” to “improve themselves” – which implies a human social entity of some kind. Aside from trying to improve themselves, what else makes people members of the system?


Point 2 (interaction) is the defining feature of all systems, be they simple or complex. The simplest of deterministic systems may have many actors/agents, which interact closely with each other, and produce emergent properties.


Again, a Messy Evolving Social Entity (MESE) may realize one or more activity systems, and revise/adapt 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 might be using those terms in very particular ways, but his definition entangles ideas that merit attention as distinct concepts.


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 intelligent animals, which narrows the range of systems of interest. C is the more general meaning of disposition used in philosophical discussion of causality, where things have dispositions to act or change in some way when triggered by a cause, by a describable event or condition. E.g.


·       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.


Of course, human dispositions are malleable, and human social entities display innovative causality, because people invent ad hoc responses to events and conditions. Whether an ever-changing social entity is well called a system is questionable, as discussed earlier.

Conclusions and remarks

Viewed as a whole, complexity science or complexity theory is incoherent. The words that people use to characterize complex systems are often ambiguous, ill-defined, misunderstood or abused. Briefly:


·       Adaptation can mean adaptation by state change or by inter-generational evolution

·       Agents can be actors who choose between actions in a role, or invent new actions.

·       Complex can mean messy/disorderly, or a measure of orderliness. It can refer to a real-world thing, or a model of it. It can be a measure of structure or of behavior.

·       Emergence occurs in the simplest of systems. 

·       Non-linear is used too variously to be discussed here! (See appendix 2).

·       Self-organization is used too variously to be discussed here!

·       Structures differ from the systems that create and use them.


It has been said that


“A complex system is one 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?


 “An adaptive system is disposed to act or change in response to events or conditions.” The trouble is that every activity system is disposed to act or change in response to events or conditions. If every system does that then what is a non-adaptive system?


 “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 talk of "adaptive" and "self-organizing" systems as in system dynamics, in which every agent strictly follows the rules imposed by "the system". Others mean the opposite, empowering each individual to define their own rules, which ends in chaos rather than complexity.


Some shift confusingly from one idea to another. E.g. from phase transition to far-from-equilibrium thermodynamics to the edge of chaos (a debatable idea) to chaos (a mathematical concept).


You might allow “complexity science” 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 uses the terminology with much more specific intent. Their CAS is a human social entity.


Again, what many people call a complex adaptive system (CAS) might better be called a messy evolving social entity (MESE) whose members may realize countless activity systems, and change them now and then.

Appendix 1: On Cillier’s idea of complexity

Cilliers is making a good try here … But because of some key omissions, his argument is confusing if not misleading. In particular, it is prone to be misinterpreted, as it does not deal with precision about the underlying technical material. I would not rely on it to teach students about complex systems and the limitations of modeling them. I would certainly not draw any clear, ontological lines between “complicated” and “complex” systems as Cilliers does not do this himself.” Will Harwood


Cilliers’ hedges against defining his terms

“I will not provide a detailed description of complexity here, but only summarize the general characteristics of complex systems as I see them.

1. Complex systems consist of a large number of elements that in themselves can be simple.
2. The elements interact dynamically by exchanging energy or information. These interactions are rich. Even if specific elements only interact with a few others, the effects of these interactions are propagated throughout the system. The interactions are nonlinear.
3. There are many direct and indirect feedback loops.
4. Complex systems are open systems—they exchange energy or information with their environment—and operate at conditions far from equilibrium.
5. Complex systems have memory, not located at a specific place, but distributed throughout the system. Any complex system thus has a history, and the history is of cardinal importance to the behavior of the system.
6. The behavior of the system is determined by the nature of the interactions, not by what is contained within the components. Since the interactions are rich, dynamic, fed back, and, above all, nonlinear, the behavior of the system as a whole cannot be predicted from an inspection of its components. The notion of “emergence” is used to describe this aspect. The presence of emergent properties does not provide an argument against causality, only against deterministic forms of prediction.
7. Complex systems are adaptive. They can (re)organize their internal structure without the intervention of an external agent.

Certain systems may display some of these characteristics more prominently than others.

These characteristics are not offered as a definition of complexity, but rather as a general, low-level, qualitative description. If we accept this description (which from the literature on complexity theory appears to be reasonable), we can investigate the implications it would have for social or organizational systems.”


The fact is, simple deterministic systems can and do produce emergent properties.


Cilliers’ distinction between complex and complicated

“complex is a term we use for something we cannot yet model. If there is nothing metaphysical about a complex system, and the notion of causality has to be retained,

then perhaps a complex system is ultimately nothing more than extremely complicated.”


Cilliers says a jumbo jet is complicated, a mayonnaise is complex. The issue is not the complex/complicated distinction. It is the entity/system distinction. Is every physical entity also a system?


IBM is a system of one kind to its owners, another kind to an employee, another to the taxman. A jar of mayonnaise is a system of one kind to a physicist, another to chemist, and another to a chef. A jumbo jet is a system of one kind to an engineer, another to a pilot, another to a passenger.


As Ashby said, we can find infinite systems in any physical or social entity. And its complexity can only be assessed wrt a description of it. You can change the level to which you (observer) choose to decompose an entity into "parts" - all the way down to quarks if you want.


So, which is easier to disassemble and reassemble – a jumbo jet or a jar of mayonnaise? It is far easier to join two half jars of mayonnaise into one than it is to join to halves of a jumbo jet into one. Unless, that is, you insist the full jar has all the same molecules in the same positions as it did before you divided it. And then, you turn the jar of mayonnaise from Cillier's complex system into his extremely complicated one.


The point is this: the entity (the jar or the jet) is not a system, it is as many systems as you choose to abstract from it.  And its complexity can only be assessed with respect to the model you choose to abstract from it.

Appendix 2: What does non-linear mean?

A term commonly associated with complex systems is non-linear. This term has many meanings, as discussed below.


Linear means in a straight line. 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 that is a straight line when drawn on a graph. Some use the term linear thinking as an insult, and without any clear meaning.


Generally, the term non-linear means not in a straight line. So, it could imply a line of behavior that is curved or jagged, perhaps an exponential increase or decrease, or a sine wave. Consider price movements in a stock market, or the population of a virus. However, the term non-linear is used with various other meanings discussed below.



Some systems thinkers call the system described in a casual loop diagram linear if it has no feedback loops; and call it non-linear if it shows at least two stocks connected by a feedback loop. Simple systems with feedback loops that display non-linear lines of behavior, can adapt and self-organize, were addressed by cybernetics 60 years ago. 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.



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.)



The term is ambiguous. In mathematics, a chaotic system is one whose state and line of behavior is highly sensitive to initial conditions. Given different initial values, the system can produce very different outcomes. 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.


Less formally, 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

The term is ambiguous. Using this phrase can give the impression of something profound when little has been said. 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. Some use it to describe a human social entity.


Human society

Social entity thinkers sometimes use the term non-linear to describe human behavior Obviously, people can perform both deterministic processes, and innovative (spontaneous) processes. Whether human thinking is deterministic or not at some level of biochemistry, we have no option but to treat people as having free will.


Non-linear systems in mathematics (after Will Harwood)

A function relating 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.


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