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A system is something that can be separated from its surrounds, but this definition leaves much scope for refinement. Starting with the notion of measurement, we explore increasingly contextual system behaviour and identify three major forms of contextuality that might be exhibited by a system: (1) between components; (2) between system and experimental method; and (3) between a system and its environment. Quantum theory is shown to provide a highly useful formalism from which all three forms of contextuality can be analysed, offering numerous tests for contextual behaviour, as well as modelling possibilities for systems that do indeed display it. I conclude with the introduction of a contextualised general systems theory based on an extension of this formalism.

Systems theory has reached something of an impasse. While early work yielded a range of insights about general system behaviour [

We will start with a consideration of the very notion of a system. What allows us to model some part of the world as a system, separate from its surroundings? This will lead to a discussion of the idea of a measurement that can be performed upon a system. At this point, we shall see that only a very small subset of systems displays objective behaviour upon measurement. This observation allowed entire fields to progress in their attempts to understand reality; however, the systems modelled by fields, such as physics and chemistry, are not all there is to reality; we must be very careful as to how deeply we hold the idea that objective responses to measurement are essential to the very idea of science. Many systems do not display such behaviour, and it is essential that we understand the different ways in which they can violate what might otherwise be regarded as a core tenet of science.

The attempt to understand this violation of objectivity will form the core contribution of this work. We will consider the notion of context and the way in which contextual behaviour leads to the manifestation of behaviour that is often denoted as complex. Contextual behaviour has been witnessed in physics, too, and quantum theory (QT) was the result. QT has developed a whole range of formalisms to describe contextuality, but there is no

However, stepping back from this general modelling paradigm, it is possible to realise that QT did not cease with the standard formulation of quantum mechanics. The description of structures appearing and disappearing required a move to quantum field theory (QFT), historically through the mechanism of second quantization, but this is not the only path available, as demonstrated by the functional methods of QFT [

We begin with the notion of a system.

At its simplest, a system

Some examples will serve as guiding illustrations for the arguments that we will develop here.

Of course, many different behaviours can be described by the tools developed in physics, which suggests that the framework of a system plus the environment,

General systems theory (GST) attempts to specify a set of principles that can apply to all types of systems [

However, formalising these notions has proven to be difficult. Instead of a generalised formal model, we have seen a range of different approaches and methods proposed, especially as GST was gradually replaced by complex systems science (CSS). Network theory, agent-based modelling, statistical models, spin glass models and evolutionary approaches have all been applied to the modelling of systems well beyond the physical [

The complexity required of a model depends not just upon the system to be described. The requirements of the modeller also play a role.

Even for the case of the tennis ball, we saw that varying levels of model complexity were possible. However, in that case, it was possible to choose between them according to the accuracy required of the model. This is because tennis balls exhibit a clear boundary separating the ball from its environment, which means that it is only necessary to consider those factors in the environment that are required for an accurate enough description of its dynamics. This relatively straightforward scenario can be contrasted with other systems that do not display such a clear separation. In particular, it is frequently the case that different levels of description are possible within the one system. Thus, in the case of an ecosystem, we see that:

Of course, this emphasis on levels of description is by no means new. For example, in cognitive science, Marr proposed that three complementary levels of analysis should be used to understand information processing systems: (i) computational; (ii) representational and algorithmic; and (iii) implementational [

Such levels of description pose a challenge to the more standard definitions of complexity, which often list a series of properties that are displayed by complex systems. For example, Mitchell [

But this raises an interesting question; how are we to analyse such contextual complexity?

For many systems, the decision as to which level of description should be adopted depends upon the observer, who is required to make an epistemic cut between the system and the environment [

However, our understanding of the role of the observer suggests that understanding measurement is key to understanding contextuality. For this reason, we shall now turn to a consideration of measurement. In what follows, we shall gradually increase the contextuality of the behaviour revealed by measurement, asking what this tells us about the formalism required to understand and model the resulting behaviour.

In order to describe measurement, we will have to enlarge our notion of a system. Measurements form a boundary between the system and the environment (in which the observer traditionally resides). Introducing both the observer

At its simplest, a system

The classical ball, moving according to the laws of Newtonian physics, must hit a device of some form to be measured. This could be a wall, a basketball hoop, a net, a tennis racket,

Such assumptions, while sometimes correct, can be markedly dangerous. For example, assuming that a photon can be described in the same way as the tennis ball results in surprise. Consider, for example, the modern quantum version of the Young’s double slit experiment, which reveals interference patterns even when single electrons are sent into the apparatus [

Further afield, the concept of a ‘gene’ in biology has moved from protein coding sequences to a more modern understanding that genes are comprised of many interdependent elements and that this makes it very difficult to delimit even the boundaries of a gene. Even when classically conceived, it is not clear where the boundary between selective units and their environments lie. A gene is subject to its environment, but this includes not only its cellular and extracellular environment; other genes and regulatory elements, homologues,

Similarly, in psychology and sociology, we find that the way we ask questions inevitably affects the response that we obtain. For example, framing a question in a positive or negative light can result in statistically significant preference changes, even if the same question is being asked [

Systems, such as these, pose problems for any theory that assumes that a system ‘has’ objectively defined properties, without reference to the context in which these occur (which could be a measurement scenario, an environment, another confounding measurement,

This variety of different uses often results in category errors. Much of what we call contextual can be reduced to an extra parameter in a model (e.g., the location of a mobile device). Such data exist out there; we do not know this, but these have a well-defined value. This calls into question the original designation of such a factor as contextual. If this data were relevant to the description of the system of interest, then we have drawn our boundary incorrectly (in this case, around the mobile phone, opting for a model that is primarily influenced by the boundaries of objects, instead of the more abstract model that described the object in a location). Note that the biotic system that is placed in a different environment is not of the same form. It is impossible to be sure what form the phenotype will have as it depends upon an interaction between the system and the environment, and here, we find an important clue. Contextuality must be ontological. That is, it must refer to fundamental uncertainty that is displayed by the system once a level of analysis has been designated.

Very few tools have been developed for the description of contextual behaviour. This problem is compounded when we consider the manner in which this behaviour manifests; it is often the case that our very attempts to measure, model or interact with a contextually-dependent system result in a change to the very results obtained from our measurements of that system (not just its future dynamics). Consider the way in which the assumption that the global financial market was crash-free led to the Black–Scholes model, which itself precipitated the 1987 stock market crash [

There appear to be a number of different classes of contextuality that might be displayed by a system of interest. We might initially expect that different effects would result from contextual responses:

These different contextual responses may require different modelling methodologies, but it might also be possible to generalise our understanding of context from a proper consideration of their similarities. While contextual dependencies between components are the focus of much current complex systems science (CSS) research, less work looks at contextual dependencies of Types (2) and (3). Given the apparent prevalence of contextually-dependent complex systems in our modern world, one might wonder at the apparent lack of theories capable of modelling such behaviour. One way forward would involve taking contextuality seriously. Intriguingly, one formalism already exists that takes contextuality between a system and experimental method very seriously: quantum theory.

Context matters in the formalism of quantum theory (QT). From von Neumann’s measurement theory and the Heisenberg uncertainty relations [

This implicit recognition of context in standard quantum measurement theory can be quickly demonstrated. According to the formalism (see below for more details), the probability of some measurement outcome can be extracted by representing the state of the system (

A state

This is a geometrical account of probability, given by a straightforward application of Pythagoras theorem [

This intuition can be formalised, as follows. First,

This simple approach to the modelling of measurement means that the quantum formalism incorporates the experimental context of a system into its description of that system, and this context can profoundly affect the experimental result obtained. Thus, QT can describe the system contextuality of Type (2). This is a highly unusual state of affairs in scientific modelling, which almost by definition assumes that a system of interest can be separated from the models that are used to analyse its behaviour.

The generalisation of QT beyond the physical realm would make it possible to model similar contextual effects in systems not presently well modelled by the scientific method. Indeed, this process has begun already; quantum inspired models have already been used to model a wide range of non-physical systems [

In addition to this immediate application to the description of contextual measurements, the phenomenon of entanglement [

Finally, contextual dependencies of Type (3) can sometimes be modelled using dissipative, or open, quantum models [

Thus, we see that the three forms of contextuality introduced in

There are a wide range of contextuality effects in QT. In this section, we will start with a class of tests that can be used on systems with one component and then gradually increase the system complexity, by adding more components, and the possibility of interactions between those components. We shall see that QT has provided an entire class of tests that can be used to determine the validity of the assumption that a system is non-contextual. Violations of the Law of Total Probability, Bell’s theorem, the Clauser–Horne–Shimony–Holt theorem, the Kochen–Specker theorem and Fine’s theorems all generate strong restrictions on the possible form that a separable system can have, and their violation frequently entails contextual behaviour. Here, we shall only explore some of the more accessible examples. The interested reader is encouraged to consult the many different references cited here and elsewhere for further details.

Quantum measurements are frequently shown to behave in a contextual manner using a violation of the Law of Total Probability as a test [

For example, Busemeyer

This basic law of probability is frequently violated by humans across a wide range of demographics (including educational ones). Such violations are commonly generated via a story that proceeds something like the following:

A large majority of people (85% in the original case presented by Tversky and Kahneman [

It might be claimed that this form of contextuality is not particularly interesting; the context of a stimulus is affecting a behavioural response (

Another form of contextuality upon measurement is exhibited by systems that display order effects (meaning that the order in which two measurements are performed affects the outcome). Wang and Busemeyer [

A further experiment, discussed by White

This section has discussed a number of different types of context, along with a consistent model that can be used to describe them. Furthermore, this model provides natural tests that can be used to determine whether contextuality is being displayed by the system under consideration or not. However, this is only a first step towards what we might call a contextualised generalised systems theory. A second set of models and tests come from the quantum formalism when it is applied to systems that consist of multi-component states. These tests can be used to discover whether the system should be considered using a reductive model or not, and while they are related to the tests and models discussed in this section (as they still involve measurement), they add to the formalism by providing further information about the internal behaviour of the system (

What happens when we combine two component systems into one joint system? The quantum formalism has a well-developed mechanism for combining systems, using the tensor product and time evolution operations [

While it is generally assumed that systems are by definition separable in a well-defined manner from their environments, a similar separation between

For example, suppose that two different experiments can be carried out upon each of the presumed subsystems, which will answer a set of ‘questions’ with binary outcomes. We shall represent these questions using four possible measurement settings, consisting of two alternative questions asked of either sub-component. Thus, a choice of two experimental settings

A potentially compositional system

As with many systems, the outcomes of our experiments will have a statistical distribution over all available outcomes, and this can be used to determine whether the sub-components can be considered as isolated, influencing one another or in some sense irreducible. Frequently, joint probability distributions, such as

It is possible to derive a number of restrictions on the probability distributions that must be satisfied by a separable system. For example, we could define such a system as one for which experiments performed at

This is a very general statement about the possibility of separating a system into objective components that interact only via the proposed variable

This class of tests and their more advanced forms have been applied to both language [

There are a number of extensions of this test. Many of them either use multi-partite systems or describe two-partite systems with a larger number of operators [

Standard QT (or quantum mechanics) does not describe the emergence of novelty. Indeed, physicists found it necessary to extend the formalism of QT with quantum field theory (QFT) when they started to model situations, such as the creation and annihilation of particles within physics. In contrast to standard QT, which preserves the number of particles in a system and, hence, cannot describe the complex interactions occurring in much of the physical world (such as, for example, the behaviour of atomic nuclei), QFT allows for the description of a number of inequivalent representations of the same physical system [

While those approaches that use standard QT are modelling the context of the system in terms of different observations, it seems likely that QFT can provide science with a genuine theory of ontological emergence [

Stepping back, it is necessary to briefly consider where exactly a quantum field theoretic model of ontological emergence could fit into the larger understanding of emergence and into systems theory as a whole. One overarching understanding is provided by Atmanspacher [

This brings us to our final discussion: how do all of the concepts that we have discussed so far fit together into an overarching framework that could drive a new contextualised general systems theory?

At the beginning of this discussion, we started by examining the notion of a system. We saw that while a system in an environment is easy to imagine, the reality is not so simple. We quickly found examples of systems that exhibit a variety of contextual responses to measurement and their environment, which makes it markedly difficult to draw a clean line between the system and the environment. Many of these systems are termed complex; however, the designation of a system as complex often depends upon what aspect of it we are interested in modelling; it is necessary to specify the level of description before we can make statements as to a system’s complexity. We then sought to examine the manner in which system contextuality affects measurement results, and the advantage of the quantum formalism became apparent. It is a formalism that recognises the manner in which our interactions with a contextually-dependent system will affect the results of measurements that we perform upon it. The geometric probabilities used in QT link experimental manipulations made by an observer with the behaviour exhibited by the system, which provides an invaluable modelling tool. A number of different proposals have been made as to why QT should be used to model systems that are not traditionally deemed physical (see e.g., [

One particularly important clarification will require a much more sophisticated understanding of quantum operators for non-physical systems. QT arose in physics only after a notion of classical measurement was defined. This meant that a dynamics had been defined classically, which mapped naturally into QT via Hamiltonians, energy, Pauli matrices,

Referring to

If direct paths could be found, then they would allow for models of contextual systems to be constructed without following the standard reductive modelling methodology, where we assume a set of objects and then gradually relax our assumptions about their behaviour (through first and second quantization).

A contextual general systems theory would explain all of the relationships between the below four types of systems. Importantly, it would provide natural explanations of how Paths (4) and (5) could be followed without assuming a classical (

One extant theory almost follows Path (5). Modern path integral forms of quantum field theory [

It is worth noting that the schema proposed here is not a hierarchy of theories (such as the one proposed by Marr; see

Formalising the notion of context is not impossible, but it will require sophisticated new mathematical techniques. It also requires a thorough re-examination of the assumptions that we make when attempting a scientific explanation. The fact that a system does not display objective responses to measurement need not imply that it is beyond the realms of science; QT provides a direct counterexample, and much can be learned from a careful consideration of the system of techniques that make up this approach.

Here, I have attempted to show that a unified approach to contextuality is possible. Taking inspiration from the quantum formalism, we can start to understand many different types of contextual systems. We can also start to utilise a range of tests that can be used to determine whether a system is contextual, and a formalism that can help us to model its behaviour if it is. While I hope that the ideas discussed in this paper have sketched out a range of possibilities, much work remains to be done. However, it seems like a contextualised general systems theory is possible and not too far away on the horizon.

This paper benefited substantially from the comments of its anonymous reviewers, who assisted greatly in improving its arguments and focus. I would also like to acknowledge the ongoing support and influence of many figures in the quantum interaction community. In particular, this paper would not have been possible without years of shared work and discussions with Peter Bruza, Fabio Boschetti and Daniel Kortschak, to name just some of the most direct influences. Much of the work towards the contextualised general systems theory that this paper discusses was completed under the Australian Research Council Discovery Grant DP1094974.

The author declares no conflicts of interest.