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The study of complexity has emerged out of a number of analytical trends in the physical and biological sciences in the last century, principally in the fields of computing and computer modelling, cybernetics, dynamical system theory (a branch of classical mechanics which studies the properties and interactions of many-bodied systems), 'organismic' biology (an approach to theoretical biology emphasizing an analytic approach to vitalistic concepts such as teleology) and thermodynamics. In an attempt to provide modern scientific foundations for vitalistic notions such as teleological behaviour, analytic biologists such as Cannon and Sommerhoff proposed analytic or behavioural analyses and definitions of biological notions. Once given a behavioural grounding, these models were able to migrate out of biology, to account for analogical features of non-biological systems: first to the study of machines and control systems in cybernetics, and thence to a wide range of physical and social processes, aided by developments in non-linear dynamics such as dynamical systems theory, the emergence of the statistical sciences, and the development of modern computer modelling. ‘Complexity’ (a term that can describe behaviour and function equally well as structure) has since become a trans-disciplinary umbrella term that is intended to denote that feature of entities which is claimed to be responsible or to account for such characteristics, in both living and non-living systems. Complexity, as a concept, thereby provides not only analysis, but also (and perhaps more crucially, yet contentiously) a uniform explanation for the structure and behaviour of a very extensive range of phenomena. Philosophical problems associated with complexity include clarifying the meanings of various concepts associated with complexity, such as emergence, non-linearity, feedback, adaptation, and self-organization, and the extent to which these terms can be given scientific meaning, that is, the extent to which these terms can be meaningfully used in the physical sciences themselves. The study of complexity also naturally intersects with more traditional problem areas in the philosophy of the sciences, such as the study of reductionism, modelling, supervenience, functionalism, and causality; however the focus of contemporary philosophy of complexity has largely tended towards the examination of (or in many cases, an attempt at the legitimization of) a scientific grounding of a particular set of approaches to these problem areas. Much of this focus is surely due to the fact that the study of complexity in the twentieth century has largely been driven by scientific practitioners themselves, and not by philosophers or philosophers of science. As such, contemporary complexity theory also makes assumptions about the relationship between scientific and philosophical theories, leading to one of its central problems: its essential ambiguity. Is complexity science a specific branch of physical science (for example, the study of 'complex adaptive systems'); a study of a widespread trans-disciplinary scientific phenomenon (leading to the study of, for example, various broad 'measures of complexity', not to speak of complexity in other divisions of science, including biological and social complexity); or even a general (and allegedly paradigmatic) approach to science itself (the source of many popularizations, and in some cases works bordering on pseudo-science)? This ambiguity (which is reflected in the bibliography) opens up further avenues for exploration, and has implications for the manner in which philosophers should attempt to approach the subject.

Key works Weaver 1948, Simon 1962, and Ashby 1962 are classic early works, generalizing concepts from cybernetics. Buckley 1968 is an early application to sociology and is likely the origin of the concept of a 'Complex Adaptive System', later explored in Holland 1992. Prigogine 1984 explores a model of complexity based on ideas from thermodynamics; Various proposed measures of complexity are explored in Bennett 1988, Lloyd & Pagels 1988 and Gell-Mann 1995. Kauffman 1969 and Bak 1996 are the origins of the influential models of Random Boolean Networks and Self-Organized Criticality, respectively.
Introductions A comprehensive introduction to many of the technical and philosophical issues of complexity can be found in Ladyman et al 2013. Book-length introductions to the diverse areas of research in complexity are Mitchell 2009 and Hooker 2011. Historical context is provided in Abraham 2011 and Francois 1999, as well as Keller 2008 and Keller 2009. There is a paucity of discussion of the subject in a manner that would be familiar to academic philosophers; in addition to Ladyman et al 2013, readers can consult Poser 2007Phelan 2001, and, on a more skeptical note, Taborsky 2014
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  1. H. D. I. Abarbanel (1992). Local Lyapunov Exponents Computed From Observed Data. Journal of Nonlinear Science 2 (3):343-365.
    We develop methods for determining local Lyapunov exponents from observations of a scalar data set. Using average mutual information and the method of false neighbors, we reconstruct a multivariate time series, and then use local polynomial neighborhood-to-neighborhood maps to determine the phase space partial derivatives required to compute Lyapunov exponents. In several examples we demonstrate that the methods allow one to accurately reproduce results determined when the dynamics is known beforehand. We present a new recursive QR decomposition method for finding (...)
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  2. Ralph H. Abraham (2011). The Genesis of Complexity. World Futures 67 (4-5):380 - 394.
    The theories of complexity comprise a system of great breadth. But what is included under this umbrella? Here we attempt a portrait of complexity theory, seen through the lens of complexity theory itself. That is, we portray the subject as an evolving complex dynamical system, or social network, with bifurcations, emergent properties, and so on. This is a capsule history covering the twentieth century. Extensive background data may be seen at www.visual-chaos.org/complexity.
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  3. Mohammad Pourmahmood Aghababa & Bijan Hashtarkhani (2015). A New Adaptive Observer Design for a Class of Nonautonomous Complex Chaotic Systems. Complexity 21 (2):145-153.
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  4. Michel Alhadeff-Jones (2008). Three Generations of Complexity Theories: Nuances and Ambiguities. Educational Philosophy and Theory 40 (1):66–82.
    The contemporary use of the term ‘complexity’ frequently indicates that it is considered a unified concept. This may lead to a neglect of the range of different theories that deal with the implications related to the notion of complexity. This paper, integrating both the English and the Latin traditions of research associated with this notion, suggests a more nuanced use of the term, thereby avoiding simplification of the concept to some of its dominant expressions only. The paper further explores the (...)
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  5. P. W. Anderson (1994). More is Different. In H. Gutfreund & G. Toulouse (eds.), Biology and Computation: A Physicist's Choice. World Scientific. pp. 3--21.
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  6. Philip W. Anderson (1999). The Eightfold Way to the Theory of Complexity: A Prologue. In G. Cowan, D. Pines & D. Meltzer (eds.), Complexity: metaphors, models, and reality. Perseus Books. pp. 7–16.
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  7. Philip Anderson & Jack Cohen (1999). Reviews: Coping with Uncertainty, Insights From the New Sciences of Chaos, Self-Organization, and Complexity, Uri Merry. [REVIEW] Emergence: Complexity and Organization 1 (2):106-108.
    (1999). Reviews: Coping with Uncertainty, Insights from the New Sciences of Chaos, Self-Organization, and Complexity, Uri Merry. Emergence: Vol. 1, No. 2, pp. 106-108.
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  8. W. Ross Ashby (1962). Principles of the Self-Organizing System. In H. Von Foerster & Zopf Jr (eds.), Principles of Self-Organization: Transactions of the University of Illinois Symposium. Pergamon Press. pp. 255–278.
  9. Fatihcan M. Atay & J.�Rgen Jost (2004). On the Emergence of Complex Systems on the Basis of the Coordination of Complex Behaviors of Their Elements: Synchronization and Complexity. Complexity 10 (1):17-22.
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  10. H. Atmanspacher & G. Demmel (2016). Methodological Issues in the Study of Complex Systems. In H. Atmanspacher & S. Maasen (eds.), Reproducibility: Principles, Problems, Practices, and Prospects. Wiley. pp. 233–250.
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  11. Harald Atmanspacher & Robert C. Bishop (2007). Stability Conditions in Contextual Emergence. Chaos and Complexity Letters 2:139-150.
    The concept of contextual emergence is proposed as a non-reductive, yet welldefined relation between different levels of description of physical and other systems. It is illustrated for the transition from statistical mechanics to thermodynamical properties such as temperature. Stability conditions are crucial for a rigorous implementation of contingent contexts that are required to understand temperature as an emergent property. It is proposed that such stability conditions are meaningful for contextual emergence beyond physics as well.
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  12. David Aubin (2008). 'The Memory of Life Itself': Bénard's Cells and the Cinematography of Self-Organization. Studies in History and Philosophy of Science Part A 39 (3):359-369.
    In 1900, the physicist Henri Bénard exhibited the spontaneous formation of cells in a layer of liquid heated from below. Six or seven decades later, drastic reinterpretations of this experiment formed an important component of ‘chaos theory’. This paper therefore is an attempt at writing the history of this experiment, its long neglect and its rediscovery. It examines Bénard’s experiments from three different perspectives. First, his results are viewed in the light of the relation between experimental and mathematical approaches in (...)
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  13. David Aubin (1998). A Cultural History of Catastrophes and Chaos: Around the Institut des Hautes Études Scientifiques. Princeton.
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  14. David Aubin & Amy Dalmedico (2002). Writing the History of Dynamics Systems and Chaos: Longue Durée and Revolution, Disciplines and Cultures. Historia Mathematica 29:1–67.
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  15. Sunny Auyang, Synthetic Analysis: How Science Combats Complexity.
    In the past two or three decades, complexity not only has been a hot research topic but has caught the popular imagination. Terms such as chaos and bifurcation become so common they find their way into Hollywood movies. What is complexity? What is the theory of complexity or the science of complexity? I do not think there is such a thing as the theory of complexity. Not even a rigid definition of complexity exists in the natural sciences. There are many (...)
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  16. Sunny Auyang (ed.) (1999). Foundations of Complex-System Theories In Economics, Evolutionary Biology, and Statistical Physics. Cambridge University Press.
  17. Nils Baas & Claus Emmeche (1997). On Emergence and Explanation. Intellectica 2 (25):67-83.
    Emergence is a universal phenomenon that can be defined mathematically in a very general way. This is useful for the study of scientifically legitimate explanations of complex systems, here defined as hyperstructures. A requirement is that the observation mechanisms are considered within the general framework. Two notions of emergence are defined, and specific examples of these are discussed.
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  18. R. Badii (1997). Complexity: Hierarchical Structures and Scaling in Physics. Cambridge University Press.
    This is a comprehensive discussion of complexity as it arises in physical, chemical, and biological systems, as well as in mathematical models of nature. Common features of these apparently unrelated fields are emphasised and incorporated into a uniform mathematical description, with the support of a large number of detailed examples and illustrations. The quantitative study of complexity is a rapidly developing subject with special impact in the fields of physics, mathematics, information science, and biology. Because of the variety of the (...)
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  19. Ion C. Baianu (2007). Categorical Ontology of Levels and Emergent Complexity: An Introduction. [REVIEW] Axiomathes 17 (3-4):209-222.
    An overview of the following three related papers in this issue presents the Emergence of Highly Complex Systems such as living organisms, man, society and the human mind from the viewpoint of the current Ontological Theory of Levels. The ontology of spacetime structures in the Universe is discussed beginning with the quantum level; then, the striking emergence of the higher levels of reality is examined from a categorical—relational and logical viewpoint. The ontological problems and methodology aspects discussed in the first (...)
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  20. P. Bak & K. Chen (1991). Self-Organized Criticality. Scientific American 264 (1):46–53.
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  21. Per Bak (1996). How Nature Works: The Science of Self-Organized Criticality. Copernicus.
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  22. Alan Baker (2013). Complexity, Networks, and Non-Uniqueness. Foundations of Science 18 (4):687-705.
    The aim of the paper is to introduce some of the history and key concepts of network science to a philosophical audience, and to highlight a crucial—and often problematic—presumption that underlies the network approach to complex systems. Network scientists often talk of “the structure” of a given complex system or phenomenon, which encourages the view that there is a unique and privileged structure inherent to the system, and that the aim of a network model is to delineate this structure. I (...)
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  23. Z. Bankovic, J. C. Vallejo, D. Fraga & J. M. Moya (2013). Detecting False Testimonies in Reputation Systems Using Self-Organizing Maps. Logic Journal of the IGPL 21 (4):549-559.
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  24. Anouk Barberousse & Cyrille Imbert (2013). New Mathematics for Old Physics: The Case of Lattice Fluids. Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 44 (3):231-241.
    We analyze the effects of the introduction of new mathematical tools on an old branch of physics by focusing on lattice fluids, which are cellular automata -based hydrodynamical models. We examine the nature of these discrete models, the type of novelty they bring about within scientific practice and the role they play in the field of fluid dynamics. We critically analyze Rohrlich's, Fox Keller's and Hughes' claims about CA-based models. We distinguish between different senses of the predicates “phenomenological” and “theoretical” (...)
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  25. Scott Barrton (1994). Chaos, Self-Organization, and Psychology. American Psychologist 49 (1):5–14.
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  26. Robert W. Batterman (1991). Randomness and Probability in Dynamical Theories: On the Proposals of the Prigogine School. Philosophy of Science 58 (2):241-263.
    I discuss recent work in ergodic theory and statistical mechanics, regarding the compatibility and origin of random and chaotic behavior in deterministic dynamical systems. A detailed critique of some quite radical proposals of the Prigogine school is given. I argue that their conclusion regarding the conceptual bankruptcy of the classical conceptions of an exact microstate and unique phase space trajectory is not completely justified. The analogy they want to draw with quantum mechanics is not sufficiently close to support their most (...)
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  27. Robert W. Batterman & Homer White (1996). Chaos and Algorithmic Complexity. Foundations of Physics 26 (3):307-336.
    Our aim is to discover whether the notion of algorithmic orbit-complexity can serve to define “chaos” in a dynamical system. We begin with a mostly expository discussion of algorithmic complexity and certain results of Brudno, Pesin, and Ruelle (BRP theorems) which relate the degree of exponential instability of a dynamical system to the average algorithmic complexity of its orbits. When one speaks of predicting the behavior of a dynamical system, one usually has in mind one or more variables in the (...)
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  28. William Bechtel & Robert C. Richardson (1993). Discovering Complexity Decomposition and Localization as Strategies in Scientific Research. Princeton.
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  29. A. Beckermann, H. Flohr & Jaegwon Kim (eds.) (1992). Emergence or Reduction? Essays on the Prospect of Nonreductive Physicalism. De Gruyter.
  30. Mark Bedau, Is Echo a Complex Adaptive System?
    We evaluate whether John Holland’s Echo model exemplifies his theory of complex adaptive systems. After reviewing Holland’s theory of complex adaptive systems and describing his Echo model, we describe and explain the characteristic evolutionary behavior observed in a series of Echo model runs. We conclude that Echo lacks the diversity of hierarchically organized aggregates that typify complex adaptive systems, and we explore possible explanations for this failure.
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  31. Michael J. Behe (2000). Self-Organization and Irreducibly Complex Systems: A Reply to Shanks and Joplin. Philosophy of Science 67 (1):155-162.
    Some biochemical systems require multiple, well-matched parts in order to function, and the removal of any of the parts eliminates the function. I have previously labeled such systems "irreducibly complex," and argued that they are stumbling blocks for Darwinian theory. Instead I proposed that they are best explained as the result of deliberate intelligent design. In a recent article Shanks and Joplin analyze and find wanting the use of irreducible complexity as a marker for intelligent design. Their primary counterexample is (...)
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  32. C. H. Bennett (1988). Logical Depth and Physical Complexity. In R. Herken (ed.), The universal Turing machine, a half century survey. Oxford University Press. pp. 227-257.
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  33. Charles H. Bennett (1986). On the Nature and Origin of Complexity in Discrete, Homogeneous, Locally-Interacting Systems. Foundations of Physics 16 (6):585-592.
    The observed complexity of nature is often attributed to an intrinsic propensity of matter to self-organize under certain (e.g., dissipative) conditions. In order better to understand and test this vague thesis, we define complexity as “logical depth,” a notion based on algorithmic information and computational time complexity. Informally, logical depth is the number of steps in the deductive or causal path connecting a thing with its plausible origin. We then assess the effects of dissipation, noise, and spatial and other symmetries (...)
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  34. Jane Bennett (2001). 5. Complexity and Enchantment. In The Enchantment of Modern Life: Attachments, Crossings, and Ethics. Princeton University Press. pp. 91-110.
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  35. P. Berge, Y. Pomeau & C. Vidal (1987). Order Within Chaos. Wiley.
  36. Bertuglia Cristoforo Sergio & Vaio Franco (2005). Nonlinearity, Chaos, and Complexity: The Dynamics of Natural and Social Systems. Oxford University Press.
    Covering a broad range of topics, this text provides a comprehensive survey of the modelling of chaotic dynamics and complexity in the natural and social sciences. Its attention to models in both the physical and social sciences and the detailed philosophical approach make this an unique text in the midst of many current books on chaos and complexity. Including an extensive index and bibliography along with numerous examples and simplified models, this is an ideal course text.
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  37. Robert Bishop (2015). Chaos. Stanford Encyclopedia of Philosophy.
    The big news about chaos is supposed to be that the smallest of changes in a system can result in very large differences in that system's behavior. The so-called butterfly effect has become one of the most popular images of chaos. The idea is that the flapping of a butterfly's wings in Argentina could cause a tornado in Texas three weeks later. By contrast, in an identical copy of the world sans the Argentinian butterfly, no such storm would have arisen (...)
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  38. Harold F. Blum (1963). Complexity and Organization. Synthese 15 (1):115 - 121.
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  39. Harold F. Blum (1957). On the Origin of Self-Replicating Systems. In D. Rudnick (ed.), Rhythmic and synthetic properties in growth. Princeton University Press. pp. 155–70.
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  40. S. Boccaletti, V. Latora, Y. Moreno, M. Chavez & D. U. Hwang (2006). Complex Networks: Structure and Dynamics. Physics Reports 424:175–308.
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  41. Kovas Boguta (2005). Complexity and the Paradigm of Wolfram's A New Kind of Science: From the Computational Sciences to the Science of Computation. Complexity 10 (4):15-21.
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  42. Fabio Boschetti (2012). Causality, Emergence, Computation and Unreasonable Expectations. Synthese 185 (2):187-194.
    I argue that much of current concern with the role of causality and strong emergence in natural processes is based upon an unreasonable expectation placed on our ability to formalize scientific knowledge. In most disciplines our formalization ability is an expectation rather than a scientific result. This calls for an empirical approach to the study of causation and emergence. Finally, I suggest that for advances in complexity research to occur, attention needs to be paid to understanding what role computation plays (...)
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  43. Fabio Boschetti & Randall Gray (2007). Emergence and Computability. Emergence: Complexity and Organization 9.
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  44. Fabio Boschetti, David McDonald & Randall Gray (2008). Complexity of a Modelling Exercise: A Discussion of the Role of Computer Simulation in Complex System Science. Complexity 13 (6):21-28.
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  45. Erez Braun & Shimon Marom (2015). Universality, Complexity and the Praxis of Biology: Two Case Studies. Studies in History and Philosophy of Science Part C: Studies in History and Philosophy of Biological and Biomedical Sciences 53:68-72.
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  46. D. R. Brooks (1988). Evolution as Entropy: Toward a Unified Theory of Biology. University of Chicago Press.
    "By combining recent advances in the physical sciences with some of the novel ideas, techniques, and data of modern biology, this book attempts to achieve a new and different kind of evolutionary synthesis. I found it to be challenging, fascinating, infuriating, and provocative, but certainly not dull."--James H, Brown, University of New Mexico "This book is unquestionably mandatory reading not only for every living biologist but for generations of biologists to come."--Jack P. Hailman, Animal Behaviour , review of the first (...)
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  47. Gregory G. Brunk (2000). Understanding Self‐Organized Criticality as a Statistical Process. Complexity 5 (3):26-33.
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  48. Walter Buckley (1968). Society as a Complex Adaptive System. In Modern Systems Research for the Behavioral Scientist. Aldine.
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  49. Walter Frederick Buckley (1998). Society-- A Complex Adaptive System Essays in Social Theory.
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  50. G. Caglioti, H. Haken & L. Lugiato (eds.) (1988). Synergetics and Dynamical Instabilities. North Holland.
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