Complex Systems

Edited by Jon Lawhead (University of Southern California)
About this topic
Summary The study of complex systems is an interdisciplinary field that examines how the interaction of many parts can give rise to holistic collective behavior at the system level.  Contemporary complex systems science is a synthesis of many different areas of inquiry, including non-linear dynamical systems theory, chaos theory, cybernetics, control theory, information theory, multiscale modeling, and non-equilibrium statistical mechanics.  There is as-yet no widely accepted general definition of "complex system," but a few common themes or properties can be observed.  Complex systems frequently tend to display self-organization, autopoiesis, non-linearity in their dynamics, chaotic behavior, emergent properties, adaptation or some combination of these traits.  In the natural sciences, the global climate, the economy, neural networks, and living organisms are among the systems generally regarded as "complex," and the methods or tools of complex systems theory are frequently applied to their study.  A holistic understanding of complex systems frequently involves contributions from many areas, including both the social and physical sciences as well as the humanities.  Given the challenges associated with coordinating this kind of vast interdisciplinary collaboration, philosophy--with its emphasis on what Wilfrid Sellars famously called "bridge-building" between disparate disciplines--has a clear and obvious role to play.  The study of complex systems also overlaps with a number of traditional problems in the philosophy of science and metaphysics, including mereology, the nature of laws and explanations, supervenience, emergence and reduction, the scale-relativity of ontology, and functionalism.  Applied philosophical issues raised by complex systems include: how do we understand causation and explanation in systems that require analysis from multiple perspectives, and which resist hierarchical organizational schemes?  Can computer simulations and multi-scale modeling provide a new way to explore strong emergence and self-organization?  How can we design organizational systems to most effectively engage in collaborative decision making while still mitigating the risks associated with large-scale collective action problems?  These questions are of extremely general importance as we move forward into the 21st century, and how we choose to address them will have implications for a diverse set of topics: challenges like how to meet the problems posed by anthropogenic climate change, how the digital revolution stands to impact our social organizations, how human society will cope with increasingly autonomous artificially intelligent agents, and how to design or manage the behavior of novel complex adaptive organisms all involve coming to grips with complexity theoretic concepts to some degree.
Key works Work in the fields from which modern complexity theory emerged, including information theory (Weaver 1948), chaos theory (Lorenz 1963; Prigogine 1984), statistical physics (Anderson 1994), and cybernetics (Simon 1962) are important for a foundational understanding of the relevant concepts.  Important early works in complexity theory include Lloyd & Pagels 1988 and Gell-Mann 1995.  More contemporary contributions have been made by Bar-Yam 2004 (which explores the mathematical foundations of strong emergence), Ladyman et al 2013 (which offers a taxonomy of definitions of 'complexity'), and Hooker 2013 (which explores the physical underpinnings of complex dynamics).
Introductions Mitchell 2009Auyang 1998Hooker msMitchell 2009Dennett 1991
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  1. Failures of Methodological Individualism: The Materiality of Social Systems.Sally Haslanger - 2020 - Journal of Social Philosophy.
  2. Failures of Methodological Individualism: The Materiality of Social Systems.Sally Haslanger - 2020 - Journal of Social Philosophy.
  3. Teaching for Complex Systems Thinking.Rosemary Hipkins - 2021 - Wellington, New Zealand: NZCER Press.
    What do a short car trip, a pandemic, the wood-wide fungal web, a challenging learning experience, a storm, transport logistics, and the language(s) we speak have in common? All of them are systems, or multiple sets of systems within systems. What happens in any set of circumstances will depend on a mix of initial conditions, complexity dynamics, and the odd wild card (e.g., a chance event). While it is possible to model and predict what might or perhaps should happen, it (...)
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  4. Why Machines Will Never Rule the World: Artificial Intelligence Without Fear.Jobst Landgrebe & Barry Smith - 2022 - Abingdon, England: Routledge.
    The book’s core argument is that an artificial intelligence that could equal or exceed human intelligence—sometimes called artificial general intelligence (AGI)—is for mathematical reasons impossible. It offers two specific reasons for this claim: Human intelligence is a capability of a complex dynamic system—the human brain and central nervous system. Systems of this sort cannot be modelled mathematically in a way that allows them to operate inside a computer. In supporting their claim, the authors, Jobst Landgrebe and Barry Smith, marshal evidence (...)
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  5. The Complex 'I'. The Formation of Identity in Complex Systems.Paul Cilliers & Tanya De Villiers-Botha - 2010 - In F. P. Cilliers & R. Preiser (eds.), Complexity, Difference and Identity. Issues in Business Ethics. Dordrecht: Springer. pp. 19–38.
    When we deal with complex things, like human subjects or organizations, we deal with identity – that which makes a person or an organization what it is and distinguishes him/her/it from other persons or organizations, a kind of “self”. Our identity determines how we think about and interact with others. It will be argued in this chapter that the self is constituted relationally. Moreover, when we are in the realm of the self, we are always already in the realm of (...)
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  6. Drift Into Failure: From Hunting Broken Components to Understanding Complex Systems.Sidney Dekker - 2011 - Crc Press.
    Explores complexity theory and systems thinking to better understand how complex systems drift into failure. This book develops a vocabulary that allows us to harness complexity and find different ways of managing drift.
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  7. The Methodological Challenges of Complex Systems.Stephan Hartmann - 2013 - In Ulrich Gähde, Stephan Hartmann & Jörn Henning Wolf (eds.), Models, Simulations, and the Reduction of Complexity. De Gruyter. pp. 81-86.
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  8. Pluralistic Modeling of Complex Systems.Dirk Helbing - 2013 - In Ulrich Gähde, Stephan Hartmann & Jörn Henning Wolf (eds.), Models, Simulations, and the Reduction of Complexity. De Gruyter. pp. 53-80.
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  9. Sizing Up Free Will: The Scale of Compatibilism.Stuart Doyle - 2021 - Journal of Mind and Behavior 42 (3 & 4):271-289.
    Is human free will compatible with the natural laws of the universe? To “compatibilists” who see free actions as emanating from the wants and reasons of human agents, free will looks perfectly plausible. However, “incompatibilists” claim to see the more ultimate sources of human action. The wants and reasons of agents are said to be caused by physical processes which are themselves mere natural results of the previous state of the world and the natural laws which govern it. This paper (...)
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  10. An Epistemological and Bio-Physical Point of View on Complex Systems.Stefano Polizzi - 2021 - Science and Philosophy 9 (2):115-127.
    In this article, after a historical introduction, we give an epistemological point of view of the physics of complex systems. Complex systems are epistemologically interesting because of the fundamental interaction experiment/observer and physicists in their everyday life can experience the paradoxes given by this interaction. Here we describe some of these paradoxes, we make a parallel with quantum mechanics and give a possible philosophical solution, based on notorious physicists/philosopher from the past, transposing and reinterpreting their ideas to modern times. In (...)
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  11. Assyrian Merchants Meet Nuclear Physicists: History of the Early Contributions From Social Sciences to Computer Science. The Case of Automatic Pattern Detection in Graphs (1950s-1970s).Sébastien Plutniak - 2021 - Interdisciplinary Science Reviews 46 (4):547-568.
    Community detection is a major issue in network analysis. This paper combines a socio-historical approach with an experimental reconstruction of programs to investigate the early automation of clique detection algorithms, which remains one of the unsolved NP-complete problems today. The research led by the archaeologist Jean-Claude Gardin from the 1950s on non-numerical information and graph analysis is retraced to demonstrate the early contributions of social sciences and humanities. The limited recognition and reception of Gardin's innovative computer application to the humanities (...)
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  12. Fair Machine Learning Under Partial Compliance.Jessica Dai, Sina Fazelpour & Zachary Lipton - 2021 - In Proceedings of the 2021 AAAI/ACM Conference on AI, Ethics, and Society. pp. 55–65.
    Typically, fair machine learning research focuses on a single decision maker and assumes that the underlying population is stationary. However, many of the critical domains motivating this work are characterized by competitive marketplaces with many decision makers. Realistically, we might expect only a subset of them to adopt any non-compulsory fairness-conscious policy, a situation that political philosophers call partial compliance. This possibility raises important questions: how does partial compliance and the consequent strategic behavior of decision subjects affect the allocation outcomes? (...)
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  13. Not All Structure and Dynamics Are Equal.Garrett Mindt - 2021 - Entropy 23 (9).
    The hard problem of consciousness has been a perennially vexing issue for the study of consciousness, particularly in giving a scientific and naturalized account of phenomenal experience. At the heart of the hard problem is an often-overlooked argument, which is at the core of the hard problem, and that is the structure and dynamics (S&D) argument. In this essay, I will argue that we have good reason to suspect that the S&D argument given by David Chalmers rests on a limited (...)
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  14. Maria Burguete, Lui Lam (eds.), Science matters: humanities as complex Systems. [REVIEW]Ma Aránzazu Serantes López & Martin Pereira Farina - 2010 - Philosophica: International Journal for the History of Philosophy 18 (36):196-200.
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  15. Econophysics: Making Sense of a Chimera.Adrian K. Yee - 2021 - European Journal for Philosophy of Science 11 (4):1-34.
    The history of economic thought witnessed several prominent economists who took seriously models and concepts in physics for the elucidation and prediction of economic phenomena. Econophysics is an emerging discipline at the intersection of heterodox economics and the physics of complex systems, with practitioners typically engaged in two overlapping but distinct methodological programs. The first is to export mathematical methods used in physics for the purposes of studying economic phenomena. The second is to export mechanisms in physics into economics. A (...)
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  16. Autonomous Systems and the Place of Biology Among Sciences. Perspectives for an Epistemology of Complex Systems.Leonardo Bich - 2021 - In Gianfranco Minati (ed.), Multiplicity and Interdisciplinarity. Essays in Honor of Eliano Pessa. Springer. pp. 41-57.
    This paper discusses the epistemic status of biology from the standpoint of the systemic approach to living systems based on the notion of biological autonomy. This approach aims to provide an understanding of the distinctive character of biological systems and this paper analyses its theoretical and epistemological dimensions. The paper argues that, considered from this perspective, biological systems are examples of emergent phenomena, that the biological domain exhibits special features with respect to other domains, and that biology as a discipline (...)
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  17. Philosophie de la simulation et finitude.Franck Varenne - 2021 - Revue Philosophique de la France Et de l'Etranger 2 (146):183-201.
    This study shows firstly that it is necessary to characterize a computer simulation at a finer level than that of formal models: that of symbols and their various modes of reference. This is particularly true for those that integrate models and formalisms of a heterogeneous nature. This study then examines the ontological causes that, consequently, could explain their epistemic success. It is argued that they can be conveniently explained if one adopts a conception of nature that is both discontinuous and (...)
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  18. A Multidisciplinary Understanding of Polarization.Jiin Jung, Patrick Grim, Daniel J. Singer, Aaron Bramson, William J. Berger, Bennett Holman & Karen Kovaka - 2019 - American Psychologist 74:301-314.
    This article aims to describe the last 10 years of the collaborative scientific endeavors on polarization in particular and collective problem-solving in general by our multidisciplinary research team. We describe the team’s disciplinary composition—social psychology, political science, social philosophy/epistemology, and complex systems science— highlighting the shared and unique skill sets of our group members and how each discipline contributes to studying polarization and collective problem-solving. With an eye to the literature on team dynamics, we describe team logistics and processes that (...)
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  19. The Biosemiotic Implications of 'Bacterial Wisdom'.Felipe-Andres Piedra & Donald R. Frohlich - manuscript
    Eshel Ben-Jacob’s manuscript entitled ‘Bacterial wisdom, Gödel’s theorem and creative genomic webs’ summarizes decades of work demonstrating adaptive mutagenesis in bacterial genomes. Bacterial genomes, each an essential part of a Kantian whole that is a single bacterium, are thus not independent of the environment as sensed; and a single bacterium is therefore a semiotic entity. Ben-Jacob suggests this but errs in 1) assigning autonomy to the genome, and 2) analogizing through computation without making clear whether he is doing so for (...)
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  20. Life, the Universe and Consciousness: An Introduction to the Theory of Universal Life.A. T. Bollands - 2020 - Oxford, UK: Bollands Publishing.
    We live in a world full of mysteries. How do our brains create consciousness? Which animals are conscious, and which are not? How can we have free-will in a deterministic universe? What are the fundamental Laws of Nature? What caused the Big Bang? How can we make sense of Quantum Mechanics? Why is the universe so finely-tuned for Life? And how did Life begin? Despite investigating such mysteries for decades or more, scientists and philosophers are no closer to finding clear (...)
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  21. Encountering Complexity: In Need For A Self-Reflecting (Pre)Epistemology.Vasileios Basios - 2007 - In Avshalom C. Elitzur, Metod Saniga & Rosolino Buccheri (eds.), Endophysics, Time, Quantum and the Subjective. Singapore: World Scientific Publishing. pp. 547-566.
    We have recently started to understand that fundamental aspects of complex systems such as emergence, the measurement problem, inherent uncertainty, complex causality in connection with unpredictable determinism, time­irreversibility and non­locality all highlight the observer's participatory role in determining their workings. In addition, the principle of 'limited universality' in complex systems, which prompts us to search for the appropriate 'level of description in which unification and universality can be expected', looks like a version of Bohr's 'complementarity principle'. It is more or (...)
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  22. Introduction to the Neoclassical Interpretation: Quantum Steampunk.Shiva Meucci - 2020 - Cosmos and History: The Journal of Natural and Social Philosophy (1):406-451.
    In a previous paper we outlined a series of historical touchpoints between classical aether theories and modern theoretical physics which showed a shared conceptual lineage for the modern tools and methods of the most common interpretations and fluid based “Hydrodynamic” treatments of an electromagnetic medium. It was proposed that, though the weight of modern experimentation leaves an extremely narrow and convoluted window for even a reconceptualization of a medium, all of modern physics recognizes a plethora of behaviors and attributes for (...)
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  23. Pluralidad de conocimientos y sistemas complejos.Mónica Gómez - 2018 - Ludus Vitalis 26 (49):43-59.
    In this paper, I defend an epistemological pluralism that moves away from universal and absolute rationality, as well as from a radical and arbitrary relativism, where any criterion is valid for decision making. Such epistemological pluralism maintains that subjects know the world in which they live according to different conceptual schemes. We posit a notion of truth linked to the justification process and practical effectiveness. Then we present the importance of traditional knowledge in the socio-ecological field and relative to complex (...)
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  24. Climate Change and Spiritual Transformation.David John Midgley - 2007 - In Mary Midgley (ed.), Earthy Realism: The Meaning of Gaia. Exeter, UK: pp. 95-101.
    The continued failure of our civilisation to mobilise an adequate response to the crisis of climate change is traced to a pathological condition of culture analogous to addiction in the case of an individual. The exponential increase in the use of fossil fuel energy has both fuelled, and been driven by, an increasingly mechanistic and materialistic world-outlook that is inimical to acceptance of the measures needed to prevent catastrophic anthropogenic climate change. A holistic view of nature, drawn from such disciplines (...)
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  25. There is No General AI.Jobst Landgrebe & Barry Smith - 2020 - arXiv.
    The goal of creating Artificial General Intelligence (AGI) – or in other words of creating Turing machines (modern computers) that can behave in a way that mimics human intelligence – has occupied AI researchers ever since the idea of AI was first proposed. One common theme in these discussions is the thesis that the ability of a machine to conduct convincing dialogues with human beings can serve as at least a sufficient criterion of AGI. We argue that this very ability (...)
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  26. Emergent Quasiparticles. Or How to Get a Rich Physics From a Sober Metaphysics.Alexandre Guay & Olivier Sartenaer - 2018 - In Melinda Fagan, Otávio Bueno & Ruey-Lin Chen (eds.), Individuation, Process and Scientific Practices. New York, USA: Oxford University Press. pp. 214-235.
    Among the very architects of the recent re-emergence of emergentism in the physical sciences, Robert B. Laughlin certainly occupies a prominent place. Through a series of works beginning as early as his Nobel lecture in 1998, a lecture given after having been awarded, together with Störmer and Tsui, the Nobel prize in physics for its contribution in the elucidation of the fractional quantum Hall effect, Laughlin openly and relentlessly advocated a strongly anti-reductionistic view of physics – and, more particularly, of (...)
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  27. Compressibility, Laws of Nature, Initial Conditions and Complexity.Sergio Chibbaro & Angelo Vulpiani - 2017 - Foundations of Physics 47 (10):1368-1386.
    We critically analyse the point of view for which laws of nature are just a mean to compress data. Discussing some basic notions of dynamical systems and information theory, we show that the idea that the analysis of large amount of data by means of an algorithm of compression is equivalent to the knowledge one can have from scientific laws, is rather naive. In particular we discuss the subtle conceptual topic of the initial conditions of phenomena which are generally incompressible. (...)
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  28. The Value of Weather Event Science for Pending UN Climate Policy Decisions.Justin Donhauser - 2017 - Ethics, Policy and Environment (3):263-278.
    This essay furthers debate about the burgeoning science of Probabilistic Event Attribution (PEA) and its relevance to imminent climate policy decisions. It critically examines Allen Thompson and Friederike Otto’s recent arguments concerning the implications of PEA studies for how the United Nations Framework Convention on Climate Change (UNFCCC) policy framework should be revised during the 2016 ‘review and decision.’ I show that their contention that PEA studies cannot usefully inform decision-making about adaptation policies and strategies is misguided and argue that (...)
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  29. Optimal Synchronization of Two Different in-Commensurate Fractional-Order Chaotic Systems with Fractional Cost Function.Reza Behinfaraz & Mohammadali Badamchizadeh - 2016 - Complexity 21 (S1):401-416.
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  30. Synchronization Analysis of Complex-Variable Chaotic Systems with Discontinuous Unidirectional Coupling.Song Zheng - 2016 - Complexity 21 (6):343-355.
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  31. On Two Mathematical Definitions of Observational Equivalence: Manifest Isomorphism and Ε - Congruence Reconsidered.Christopher Belanger - 2013 - Studies in History and Philosophy of Modern Physics 44 (2):69-76.
  32. The Ergodic Hierarchy, Randomness and Hamiltonian Chaos.Joseph Berkovitz, Roman Frigg & Fred Kronz - 2006 - Studies in History and Philosophy of Modern Physics 37 (4):661-691.
    Various processes are often classified as both deterministic and random or chaotic. The main difficulty in analysing the randomness of such processes is the apparent tension between the notions of randomness and determinism: what type of randomness could exist in a deterministic process? Ergodic theory seems to offer a particularly promising theoretical tool for tackling this problem by positing a hierarchy, the so-called ‘ergodic hierarchy’, which is commonly assumed to provide a hierarchy of increasing degrees of randomness. However, that notion (...)
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  33. Notes From the Existential Underground: The Universe as a Complex Emergent System.Michelle Kathryn McGee - 2016 - Cosmos and History 12 (2):172-183.
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  34. An Inhomogeneous Space–Time Patching Model Based on a Nonlocal and Nonlinear Schrödinger Equation.Christine C. Dantas - 2016 - Foundations of Physics 46 (10):1269-1292.
    We consider an integrable, nonlocal and nonlinear, Schrödinger equation as a model for building space–time patchings in inhomogeneous loop quantum cosmology. We briefly review exact solutions of the NNSE, specially those obtained through “geometric equivalence” methods. Furthemore, we argue that the integrability of the NNSE could be linked to consistency conditions derived from LQC, under the assumption that the patchwork dynamics behaves as an integrable many-body system.
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  35. The Crack-Branching Velocity.S. R. Anthony, J. P. Chubb & J. Congleton - 1970 - Philosophical Magazine 22 (180):1201-1216.
  36. Non-Linear Lattice Models: Complex Dynamics, Pattern Formation and Aspects of Chaos.J. Pouget - 2005 - Philosophical Magazine 85 (33-35):4067-4094.
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  37. Further Results on the Impulsive Synchronization of Uncertain Complex-Variable Chaotic Delayed Systems.Song Zheng - 2016 - Complexity 21 (5):131-142.
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  38. Adaptive Impulsive Synchronization of Uncertain Delayed Chaotic System with Full Unknown Parameters Via Discrete-Time Drive Signals.Xiaojing Gao, Mengfan Cheng & Hanping Hu - 2016 - Complexity 21 (5):43-51.
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  39. Adaptive-Impulsive Function Projective Synchronization for a Class of Time-Delay Chaotic Systems.Song Zheng - 2016 - Complexity 21 (2):333-341.
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  40. Fractional Modeling and Control of a Complex Nonlinear Energy Supply-Demand System.Mohammad Pourmahmood Aghababa - 2015 - Complexity 20 (6):74-86.
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  41. In What Sense is the Kolmogorov-Sinai Entropy a Measure for Chaotic Behaviour?—Bridging the Gap Between Dynamical Systems Theory and Communication Theory.Roman Frigg - 2004 - British Journal for the Philosophy of Science 55 (3):411-434.
    On an influential account, chaos is explained in terms of random behaviour; and random behaviour in turn is explained in terms of having positive Kolmogorov-Sinai entropy (KSE). Though intuitively plausible, the association of the KSE with random behaviour needs justification since the definition of the KSE does not make reference to any notion that is connected to randomness. I provide this justification for the case of Hamiltonian systems by proving that the KSE is equivalent to a generalized version of Shannon's (...)
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  42. Identifying Complex Causal Dependencies in Configurational Data with Coincidence Analysis.Michael Baumgartner & Alrik Thiem - 2015 - R Journal 7 (1):176-184.
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  43. Explanation as Condition Satisfaction.Paul Humphreys - 2014 - Philosophy of Science 81 (5):1103-1116.
    It is shown that three common conditions for scientific explanations are violated by a widely used class of domain-independent explanations. These explanations can accommodate both complex and noncomplex systems and do not require the use of detailed models of system-specific processes for their effectiveness, although they are compatible with such model-based explanations. The approach also shows how a clean separation can be maintained between mathematical representations and empirical content.
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  44. Worlds of Flow: A History of Hydrodynamics From the Bernoullis to Prandtl.Olivier Darrigol - 2005 - Oxford University Press UK.
    The first of its kind, this book is an in-depth history of hydrodynamics from its eighteenth-century foundations to its first major successes in twentieth-century hydraulics and aeronautics. It documents the foundational role of fluid mechanics in developing a new mathematical physics. It gives full and clear accounts of the conceptual breakthroughs of physicists and engineers who tried to meet challenges in the practical worlds of hydraulics, navigation, blood circulation, meteorology, and aeronautics, and it shows how hydrodynamics at last began to (...)
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  45. Between Rigor and Reality: Many-Body Models in Condensed Matter Physics.Axel Gelfert - 2015 - In Brigitte Falkenburg & Margaret Morrison (eds.), Why More Is Different: Philosophical Issues in Condensed Matter Physics and Complex Systems. Springer. pp. 201-226.
    The present paper focuses on a particular class of models intended to describe and explain the physical behaviour of systems that consist of a large number of interacting particles. Such many-body models are characterized by a specific Hamiltonian (energy operator) and are frequently employed in condensed matter physics in order to account for such phenomena as magnetism, superconductivity, and other phase transitions. Because of the dual role of many-body models as models of physical sys-tems (with specific physical phenomena as their (...)
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  46. Toward 'Complexics' as a Transdiscipline.Albert Bastardas I. Boada - unknown
    The proposed transdisciplinary field of ‘complexics’ would bring together allcontemporary efforts in any specific disciplines or by any researchersspecifically devoted to constructing tools, procedures, models and conceptsintended for transversal application that are aimed at understanding andexplaining the most interwoven and dynamic phenomena of reality. Our aimneeds to be, as Morin says, not “to reduce complexity to simplicity, [but] totranslate complexity into theory”.New tools for the conception, apprehension and treatment of the data ofexperience will need to be devised to complement existing (...)
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  47. Continuability in Time of Smooth Solutions of Strong-Nonlinear Nondiagonal Parabolic Systems.Arina Arkhipova - 2002 - Annali della Scuola Normale Superiore di Pisa- Classe di Scienze 1 (1):153-167.
    A class of quasilinear parabolic systems with quadratic nonlinearities in the gradient is considered. It is assumed that the elliptic operator of a system has variational structure. In the multidimensional case, the behavior of solutions of the Cauchy-Dirichlet problem smooth on a time interval $[0,T)$ is studied. Smooth extendibility of the solution up to $t=T$ is proved, provided that “normilized local energies” of the solution are uniformly bounded on $[0,T)$. For the case where $[0,T)$ determines the maximal interval of existence (...)
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  48. Periodic Orbits Close to Elliptic Tori and Applications to the Three-Body Problem.Massimiliano Berti, Luca Biasco & Enrico Valdinoci - 2004 - Annali della Scuola Normale Superiore di Pisa- Classe di Scienze 3 (1):87-138.
    We prove, under suitable non-resonance and non-degeneracy “twist” conditions, a Birkhoff-Lewis type result showing the existence of infinitely many periodic solutions, with larger and larger minimal period, accumulating onto elliptic invariant tori. We prove the applicability of this result to the spatial planetary three-body problem in the small eccentricity-inclination regime. Furthermore, we find other periodic orbits under some restrictions on the period and the masses of the “planets”. The proofs are based on averaging theory, KAM theory and variational methods.
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  49. Periodic Solutions of Forced Kirchhoff Equations.Pietro Baldi - 2009 - Annali della Scuola Normale Superiore di Pisa- Classe di Scienze 8 (1):117-141.
  50. Nonlinear Potentials, Local Solutions to Elliptic Equations and Rearrangements.Andrea Cianchi - 2011 - Annali della Scuola Normale Superiore di Pisa- Classe di Scienze 10 (2):335-361.
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