I have tried to come to terms with the notion of "complex systems" of which I heard in one of the lessons at school though without too much depth. I grasp that a complex system is such that the behavior of the elements in the system do not yield wholly the behavior of the system as a whole - in some sense the system is not predicted and may be thought of as something between total order and total chaos. I also understand that practically everything around us might be of such nature - the internet, the infrastructures of electricity and water, the human society, the human brain, and so on. However I could not pick the practical value in keeping this view "all the way" while researching this or that.

Trying to better understand "complexity" I have ran into two wonders: is it the opposite of reductionism? Is it some sort of holism? What is the historical and philosophical context of complex systems?

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    To understand complex system is complicated due to it's internal structure. Actually for any computably generated object it may be such a complex system itself and there's a complexity description measure of any generated object called Solomonoff-Kolmogorov-Chaitin complexity and Algorithmic information theory principally studies complexity measures on strings (or other data structures). Because most mathematical objects can be described in terms of strings... it can be used to study a wide variety of mathematical objects... Feb 14, 2022 at 0:38
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    If restricted to understand complexity of the theoretical cases of any recursively axiomatized theory such as the common ZFC and PA, Chaitin proposed his famous “heuristic principle,” the theorems of a recursively axiomatized theory cannot be significantly more complex than the theory itself. Thus the complexity of such a theoretic system as a whole cannot exceed and very close to that of its most complicated theorem which is only part of the system. Also it can be shown the probability that a true sentence of length n is provable in the theory tends to zero when n tends to infinity... Feb 14, 2022 at 1:38
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    Related question on physics: What is the definition of "Complexity" in physics? Is it quantifiable?
    – Wrzlprmft
    Feb 16, 2022 at 17:35

5 Answers 5


A system is a complex system if its characteristic properties cannot be investigated by studying its components in isolation. A typical example of a complex system is the weather: One cannot study the weather by studying the path of each separate molecule within the atmosphere.

Hence a complex system resists a reductionist approach. But I would not throw out the baby with the bath water: Not each part of a complex system is connected to each other part in a holistic way. A possible approach to study complex systems is to decompose the system into layers. And then studying each layer by a suitable method adapted to the layer.

Many complex systems show chaotic behaviour though they are deterministic: A slight change in their initial conditions provides completely different developments, which cannot be predicted in the beginning. A typical example is the Mandelbrot set, which results from iterating again and again the simple function „z maps to z^2 +c“. The result depends in a sensible way on the value of „c“.

Hence one root of complexity is chaotic behaviour, it may result when the basic equations are non-linear. In a historical context non-linearity is one of the roots for studying complex systems.

  • This is just holism, nothing new here. Non-linearity is also a feature of classical systems. Nobody said that the classical systems theory study only linear systems or systems which parts have the same features as the whole. "division into lsyers" is the reductionist approach, which is not incompatible with holism. See my answer.
    – RodolfoAP
    Feb 14, 2022 at 5:01
  • @RodolfoAP 1. Is your objection: System Theory also comprises the theory of complex systems. Hence there is no need to coin a new term? 2. How and where do you establish a border between System Theory and other theories from natural science or sociology? Which topics do not belong to System Theory? 3. In your answer you „classify the Systems Theory in the category of metaphysics“. Why metaphysics?
    – Jo Wehler
    Feb 14, 2022 at 7:46
  • 1. Yes. 2. No relation, but anyway. In any case, the question is what is the demarcation between that allegedly "complex systems theory" and the traditional? 3. Instead, question yourself: are "complex systems" part of science (empirical truth, that is, knowledge dependent on the senses), or metaphysics (pure, ideal, related to mathematics)? If it depends on the senses, observation, it is science, so, near biology.
    – RodolfoAP
    Feb 14, 2022 at 8:31
  • This raises a further question: why can't a complex system be investigated by studying its components? Is it because it is practically in feasible for us, or because this will not work even in principle? In other words, is this a property of the system itself, or the people investigating the system?
    – gardenhead
    Feb 15, 2022 at 19:11
  • @gardenhead One reason: Because of the large number of components. There are about 6*10^^23 molecules of air within some cm^^3. One cannot solve Newtons equations for a system of about 10^^23 coupled differential equations. Here thermodynamics became the method of choice to investigate averaged quantities of the system like pressure, temperature and energy.
    – Jo Wehler
    Feb 15, 2022 at 19:27

Short Answer

Complex systems is a mathematical approach to studying certain objects of science, and is neither a science, nor a philosophy, but an approach that might be considered a combination of the mereological with mathematical modeling, which is a sub-discipline of mathematics. It is a topic frequently found in conjunction with chaos theory since real-world phenomena are complex, stochastic systems.

Long Answer


The paragon for the mathematization of the sciences is often considered Newtonian mechanics, since Isaac Newton took the philosophy of mathematical insights of Galileo Galilei and Rene Descartes and applied them to revolutionize physics, particularly with the laws of motion. By the time of Kepler, mathematical science was far more common, but after Newtonian mechanics became a shorthand for reductionism and determinism. Some scientists at the end of the 19th-century believed the human species was on the cusp of solving all important scientific problems, and it was just a question of how much computation was involved in modeling the universe, and the things in it. Of course, the revolution of quantum mechanics in physics and a set of incompleteness theorems in logic began to put the kibosh on that thinking by before WWII.


By 1990s, chaos and complexity theory began to enter the public consciousness outside of universities with books like Complexity: The Emerging Science at the Edge of Order and Chaos by Waldrop and Chaos: Making a New Science by Gleick. Today, there's a body of literature and can find manuals on programing those sorts of models. O'Reilly has a recent book in python Think Complexity that demonstrates how widely studied methods for constructing such models are.

At the root of complexity theory is the idea that behavior of systems can modeled from the very simple, such as a mathematical point on a plane, to having billions of data points in four dimensions, such as in weather systems. In fact, highly related to complex systems is the notion that to mathematically model them requires a science of its own, and in computer science, computational complexity deals with that. What is at play is a desire to understand 'when does a model require qualitative differences or when does it require quantitative ones?'

Scientific Application

As an example of modeling complex systems, one practical application is folding proteins. Stanford University many years ago launched a distributed client for utilizing unharnessed cycles on CPUs and their cores to do such research with their Folding@Home initiative. In the days of alchemy, alchemists would haphazardly combine substances through various processes and see what would happen, and a lot knowledge about substances like aqua fortis was accumulated. But as modern chemistry emerged out of these empirical activities and the evidence they accumulated, mathematical models emerged, simple ones at first, like Dalton's law or Avogadro's number.

These days, physical chemistry, affectionately known as p-chem, uses sophisticated mathematical models to predict the behavior of molecules. Since the human body can be seen as a large set of folded proteins, folding proteins is important to experimentation in medicine. In Edward Jenner's day, experimentation meant manually attenuating potential pathogens and injecting them into live people and seeing what happened. But there are several downsides to this sort of scientific practice. It's somewhat unethical; it's expensive to buy substances and build and operate laboratories; it's time-consuming to go through permutations of substance and process in the real world. So, chaos and complexity theory might be thought of as the intellectual substrate of an approach to use computation to model real-world phenomenon to narrow down what work to perform in the real world.


What makes chaos and complex theory a bit different than other contemporary academic disciplines is that it attempts to be interdisciplinary, and that's because all phenomena can be modeled in roughly the same way. Systems have parts, interact with other systems, and can be modeled with mathematics. In data science, big data models and machine learning techniques can be applied to these models which are often built on relational models. There are academic programs and institutions devoted to these sorts of techniques. One famous place where such ideas are studied is the Santa Fe Institute. From WP:

The Santa Fe Institute (SFI) is an independent, nonprofit theoretical research institute located in Santa Fe, New Mexico, United States and dedicated to the multidisciplinary study of the fundamental principles of complex adaptive systems, including physical, computational, biological, and social systems. The institute is ranked 24th among the world's "Top Science and Technology Think Tanks" and 24th among the world's "Best Transdisciplinary Research Think Tanks" according to the 2020 edition of the Global Go To Think Tank Index Reports, published annually by the University of Pennsylvania.2


So, these days, if you're looking to crush an epidemic, one of the best approaches is to build a series of sophisticated models: model the epidemiology, and the human body, and the biomolecular interactions. Then, search a problem-space for desirable outcomes. If you can fold a protein, and then use CRISPR to engineer a bioreactor to produce a substance, and then model the best way to economically distribute the substance, you have a tremendous opportunity to find solutions to problems quickly, cheaply, and effectively. But that requires a certain mastery of interdisciplinary thinking, an understanding of the probabilistic nature of physical phenomena, and an aptitude for modeling large, complex systems effectively on computers. And that's the hole complex systems seeks to fill.

  • And...? What is a complex system?
    – RodolfoAP
    Feb 13, 2022 at 21:26
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    I think that it's a rebranding to put the idea of interdisciplinary thinking at the forefront, and that like most definitions, there's no definition of sufficiency and necessity that is adequate. It might better be apprehended as a prototypical definition..
    – J D
    Feb 13, 2022 at 21:30
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    @RodolfoAP I'd argue you're engaged in a specious thinking of the sorites paradox, or the fallacy of the beard. The idea isn't that there is some sort of strict criterion that allows for a taxonomy from simple to complex, but rather an understanding that there are certain philosophical principles at play a la emergence/supervenience/mereology that form a fundamentally more sophisticated way to think of systems...
    – J D
    Feb 13, 2022 at 21:33
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    that's where the features section comes in. Complex systems in prototypical fashion exhibit some family resemblance of: cascading failures, openness, fuzzy boundaries, irregular transitions of state, have non-linear, system control loops, etc...
    – J D
    Feb 13, 2022 at 21:36
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    Complex systems in science are rooted in chaos theory and non-linear dynamics, where I'd say the definitions are quite clear: Have philosophers speculated on how chaotic forces meeting together can result in order?. Complex and emergent systems are subtypes of non-linear systems
    – CriglCragl
    Feb 15, 2022 at 17:11

I would suggest that a complex system is simply a system which is not a simple system; where a simple system is one where you can see with a (possibly long) glance at the components every way it's ever going to behave with its given rules and constraints.

In contrast, a complex system often displays dynamic, interesting emergent behaviour that you would never have predicted, even when having complete information about the static components and their rules and relationships in a snapshot in time.

Aside from that, I like to think that there is nothing particularly magic or, well, complex, about complex systems. Every "interesting" system is a complex system, unless one finds simplicity itself interesting.

Examples for complex systems made up from pretty simple and easily visible parts, rules and relationships (just to have a few alternatives to just the old "weather" example):

  • Traffic
  • The Internet
  • Stock markets (markets in general)
  • Factories (production chains in general)
  • Societies

To your questions:

Is the notion of "Complex System" a philosophy of science?

I don't exactly know what you mean with the term "philosophy of science" in this context. Calling something a complex system in science is a purely scientific event. Scientist can see if a system behaves chaotic or "interesting" pretty easily, and no particular philosophical approach is necessary.

Only because we then cannot predict what's happening does not put it into the purview of philosophy (in the sense that science is at its wits end). Only because a system shows complex behaviour does not mean that it is a particularly "special" system that needs to be explained from outside of science.

Is it the opposite of Reductionism?

The one does not seem to be related to the other. You can construct complex systems from very simple parts, and we humans do so all the time. While this is usually a one-way-street (it is very hard to deduce the simple parts from just witnessing the behaviour of the system if you only have little information), depending on which definition of Reductionism you subscribe to, complex systems can be perfectly and often easily reduced into their parts.

You usually cannot model the behaviour of a complex system easily (i.e., find a mathematical model which gives the same result as just "running" the system itself, whatever that means), but that does not mean that you cannot take the system apart just fine; and there is no "invisible magic" in-between the components which would require particularly intense philosophical attention.

Is it related to Holism?

Not really; only if you wish to treat the complex system as a black box for the sake of convenience and then insist that this is the only way ever possible. But in that respect it is not different from any other simple system.


This is a personal opinion. I do research related to the Systems Theory for long years, and it's clear for me that the alleged discipline of complex systems is just an academic fraud. Classifying a system as "complex" has a common cause: misunderstanding the classical Systems Theory, Epistemology and Philosophy.

I absolutely respect those who believe on it, but no debate has ever convinced me of the necessity of a "complexity" branch to the systems theory. The term complexity essentially means inability to understand, a feature that systems thinking is about; and systems thinking is a reduced pop subset of the Systems Theory, which already targets complexity. In fact, the classical Systems Theory was created to address complex problems, by splitting them into simpler ones (simple: easy to understand). A house system can be divided in multiple ways, but the proper division is that which responds better to a goal. For example, if my goal is to assign spaces to the members of my family, I can split the house system into room parts. That's the systemic way of solving the problem. etc. Removing complexity is what the Systems Theory is already about.

As you've found, there's not even a clear definition of this theory. Some associate the term "complex system" with classical systems (then, why is the name change necessary?), some associate it with some debatable features, allegedly not present in the classical theory, and in such case, either the features are part of it, they should be part of it without changing the name of the whole discipline, or they are (or should be) part of another discipline, commonly, biology.

The content of the "complex systems theory" is normally associated to biologic systems (Maturana et.al. target systemic features that should be part of biology, but without any rigor) or to classical systems (von Bertalanffy et.al).

... that a complex system is such that the behavior of the elements in the system do not yield wholly the behavior of the system as a whole

Association with non-linearity. Non-linearity is already a feature that the Systems Theory addresses, and it is a natural feature of systems (social organizations, for example; the central focus of the system theory). Non-linearity is common in nature, on non-living and living organizations. In addition: try finding a proper definition of systemic non-linearity. You fill, again, a lot of speculation based on subjectivity.

... in some sense the system is not predicted and may be thought of as something between total order and total chaos.

Chaos and emergence. The systemic evolution to chaos is simply described by thermodynamics, and the evolution towards order has no description ("complex systems" speculate about it, does not describe it). Naive ideas like emergence require more than speculation. For example, emergence is supposed to be the raise of new features, which are not present in the parts. Look this example: two dots. Alone they can describe two 0-dimensional sets. But together, in addition, they will describe a line. Wow. Emergence. And three dots can produce a plane. Loud wow. Emergence of a new feature, again. Is that excluded from the classical systems theory? No.

In addition, emergence describes a subjective perception. Life is a human subjective perception. And that is not mentioned in the "complex systems theory". The complex systems sellers should worry more about what is the content of the theory, instead of inflating superficial speculations. All the fuzz about emergence, non-linearity, self-organization ("self?" are those entities closed systems? closed systems are perfectly characterized by thermodynamics: no evolution towards order is possible in closed systems), spontaneous order (start defining order, in relation to thermodynamics, not as simple perceptions), cascading failures, openness (so, open systems are not addressed by the systems theory? what?), fuzzy boundaries (all physical systems have NO boundaries at all; all are open; on the contrary, ideal systems are essentially bounded), irregular transitions of state (a natural feature of systems), etc. should be part of some Theory of Biological Systems. The word "complex" is a very good seller, but not very scientific.

In addition, I tend to classify the Systems Theory in the category of metaphysics, as Kant classifies Arithmetic and Geometry (well, properly, Mathematics). Like numbers, systems are bounded ideals, not material things. Matter has no boundaries. There is no spoon out there.

I Repeat: this is a personal opinion.

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    Thanks for the lead on Maturana!
    – J D
    Feb 13, 2022 at 21:27
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    Despite your objections, I upvoted. It's good to be critical of any academic disciplines organization!
    – J D
    Feb 13, 2022 at 21:36
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    Great read, and well articulated. However I think you are stretching very hard to conflate the very real work being done in complex systems with some common embellishments that enthusiasts make. As for emergence there are many quantifiable examples outside in economics and biology.
    – Enforce
    Feb 14, 2022 at 14:48
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    -1 for the accusation of fraud
    – N. Virgo
    Feb 15, 2022 at 5:13
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    I have been in the vast and diffuse field of complex systems for years and I have never encountered: 1) Somebody who “believes” in complexity in any meaningful definition of believe. 2) Scientists who consider it worthwhile to rigorously define a complex system, complexity, etc., let alone try to do so. 3) A single complex-systems theory. 4) Somebody who cares about whether a given system is complex or not. — The main reason why there is something like “complex-systems science” in the first place is to have a name for a group of people working on similar problems using similar techniques.
    – Wrzlprmft
    Feb 16, 2022 at 18:03

There is a common phrase that illustrates exactly this:

The sum is greater than the parts

For example, a novel is just ink on paper. But if you examine both ink and paper separately, you will not be able to understand a novel. The novel is greater than its parts.

One could argue that here there is a third part that has been left out: the author, the man or woman that wrote the novel. This is true, but this is not apparent in the bare artifact. We only see this if we see the nocel in its composition, that is when it is being written.

Holism, is the idea that the whole has to be understood as a thing in itself and not reducible to its parts. Its a philosophy that is opposite to that of reductionism that states a whole can be understood from its parts. The latter philosophy has been the dominating influence in modern science.

Buddhism takes holism as its natural philosophy. In fact, it takes it to an extreme degree stating that parts have no independent reality. Only the All has a claim to reality.

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    A dot has no length. A system of dots (two dots in different places) has length (the distance between them). So, "the sum is greater than the parts" is evident here. And two dots would therefore be a complex system (?). This fallacious definition is quite common. All systems feature emergent (synergistic) qualia, but not all are, ideally, complex. Otherwise, the term "complex" exceeds. All systems feature emergent (synergistic) features.
    – RodolfoAP
    Mar 9, 2022 at 5:16

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