Computational complexity theory is a subfield of computer science originating in computability theory and the study of algorithms for solving practical mathematical problems. Amongst its aims is classifying problems by their degree of difficulty — i.e., how hard they are to solve computationally. This paper highlights the significance of complexity theory relative to questions traditionally asked by philosophers of mathematics while also attempting to isolate some new ones — e.g., about the notion of feasibility in (...) mathematics, the $\mathbf{P} \neq \mathbf{NP}$ problem and why it has proven hard to resolve, and the role of non-classical modes of computation and proof. (shrink)

This volume is both a state-of-the-art display of current thinking on second language development as a complex system. It is also a tribute to Diane Larsen-Freeman for her decades of intellectual leadership in the academic disciplines of applied linguistics and second language acquisition. The chapters therein range from theoretical expositions to methodological analyses, pedagogical proposals, and conceptual frameworks for future research. In a balanced and in-depth manner, the authors provide a comprehensive and interdisciplinary understanding of second language development, with a (...) wealth of insights that promise to break the status-quo of current research and take it to exciting new territory. The book will appeal to both seasoned and novice researchers in applied linguistics, second language acquisition, bilingualism, cognitive psychology, and education, as well as to practitioners in second or foreign language teaching of any language. (shrink)

In a spirit of collegial support, this paper argues that Beckett and Hager’s theoretical justification and empirical exemplifications do not do full justice to the complexity of group or team learning. We firstly reaffirm our support for the theoretical argument Becket and Hager make, though expressing some reservations about Complexity Theory, to explain the taken-for-granted assumptions that learning by an individual is the paradigm case of learning and that context plays a minimal role in this process. Drawing (...) on our joint and separate work, we demonstrate that Becket and Hager’s argument is less radical than it may initially seem because it is predicated on: (i) cognitive-bounded rather than “distributed” or “extended” conception of mind; (ii) material rather than a “immaterial” conception of activity; and (iii) co-present rather than a “fractional” or “connective” conception of ontology. Despite making this critique, we conclude by making the case that we are adding further substance to Becket and Hager’s overarching argument and, in doing so, encouraging them to be more radical about how they conceptualise the complexity of learning. (shrink)

This article contributes to the revision of the concept of system in social theory using complexity theory. The old concept of social system is widely discredited; a new concept of social system can more adequately constitute an explanatory framework. Complexity theory offers the toolkit needed for this paradigm shift in social theory. The route taken is not via Luhmann, but rather the insights of complexity theorists in the sciences are applied to the tradition (...) of social theory inspired by Marx, Weber, and Simmel. The article contributes to the theorization of intersectionality in social theory as well as to the philosophy of social science. It addresses the challenge of theorizing the intersection of multiple complex social inequalities, exploring the various alternative approaches, before rethinking the concept of social system. It investigates and applies, for the first time, the implications of complexity theory for the analysis of multiple intersecting social inequalities. Key Words: complexity theory • inequality • intersectionality • social theory. (shrink)

Complexity theory attempts to explain, at the most general possible level, the interesting behaviors of complex systems. Two such behaviors are the emergence of simple or stable high-level behavior from relatively complex low-level behavior, and the emergence of sophisticated high-level behavior from relatively simple low-level behavior; they are often found nested in the same system. Concerning the emergence of simplicity, this essay examines Herbert Simon's explanation from near-decomposability and a stochastic explanation that generalizes the approach of statistical physics. (...) A more general notion of an abstract difference-making structure is introduced with examples, and a discussion of evolvability follows. Concerning the emergence of sophistication, this essay focuses on, first, the energetics approach associated with dissipative structures and the fourth law of thermodynamics, and second, the notion of a complex adaptive system. (shrink)

A collection of scholarly essays, __Complexity Theory and the Philosophy of Education__ provides an accessible theoretical introduction to the topic of complexity theory while considering its broader implications for educational change. Explains the contributions of complexity theory to philosophy of education, curriculum, and educational research Brings together new research by an international team of contributors Debates issues ranging from the culture of curriculum, to the implications of work of key philosophers such as Foucault and John (...) Dewey for educational change Demonstrates how social scientists and social and education policy makers are drawing on complexity theory to answer questions such as: why is it that education decision-makers are so resistant to change; how does change in education happen; and what does it take to make these changes sustainable? Considers changes in use of complexity theory; developed principally in the fields of physics, biology, chemistry, and economics, and now being applied more broadly to the social sciences and to the study of education. (shrink)

In the post-Apartheid era South Africa’s universities have faced serious questions about the quality of their student learning in the face of near impossible challenges. The University of Cape Town, widely seen as the country’s leading higher education institution, has shown remarkable resilience, however, in the range of initiatives it has launched to support and enhance student learning. These initiatives, designed with a common purpose, are of course intended to work together so that their effects might be compounded and realized (...) in enhanced student learning outcomes. Drawing substantially on the power of compounding (which is itself redolent of the claim made by complexity theory that relations and emergence are crucial concepts), complexity theory offers unique insights into how and why things change – and also into how and why things remain largely the same or inexorably grind towards failure. The networked initiatives undertaken by the University of Cape Town constitute a case of learning refurbishment that is also well explained and understood in terms of complexity theory. This paper draws on concepts from complexity theory both to understand how learning might be enhanced in higher education institutions in severely straitened circumstances and to offer insights for education leaders and policy-makers in this domain. (shrink)

This volume provides an accessible theoretical introduction to the topic of complexity theory while considering its broader implications for educational change.

This chapter contains sections titled: Complexity Theory and the Philosophy of Education Complexity Theory and Educational Research Complexity Theory and the Curriculum Concluding, and Simultaneously Introductory, Remarks References.

In this article, the author argues that complexity theories have limited use in the study of society, and that social processes are too complex and particular to be rigorously modeled in complexity terms. Theories of social complexity are shown to be inadequately developed, and typical weaknesses in the literature on social complexity are discussed. Two stronger analyses, of Luhmann and of Harvey and Reed, are also critically considered. New considerations regarding social complexity are advanced, on (...) the lines that simplicity, complexity that can be modeled, and incondensible complexity permeate society simultaneously. The difficulty of establishing complexity models for processes involving ongoing interpretation is discussed. It is argued that the notions of system and environment need recasting in social studies. Existing social studies and literature, it is argued, reflect a polymorphous, contextual, contingent, labyrinthine, dramatic and political face to social complexity. Students of social complexity must be literate in such studies. (shrink)

This research draws on complexity theory to provide an alternative conceptualization of issue management. We use six dynamics of complexity drawn from complex adaptive systems—equipoise, turbulence, sensitive conditions, bifurcation, attractor emergence, and symmetry breaking—to develop a metaphorical framework that describes what occurs during various periods of issue activity and what propels issues from one period of activity to another. We illustrate the framework with a case study of the pharmaceutical industry response to the HIV/aids pandemic in Sub-Saharan (...) Africa. The article concludes with a discussion of the additional affordances this framework provides to extend and compliment issue life cycle models. (shrink)

This paper reflects on the relationship between theory and practice in bioethics, by using various concepts drawn from debates on innovation in healthcare research—in particular debates around how best to connect up blue skies ‘basic’ research with practical innovations that can improve human lives. It argues that it is a mistake to assume that the most difficult and important questions in bioethics are the most abstract ones, and also a mistake to assume that getting clear about abstract cases will (...) automatically be of much help in getting clear about more complex cases. It replaces this implicitly linear model with a more complex one that draws on the idea of translational research in healthcare. On the translational model, there is a continuum of cases from the most simple and abstract to the most concrete and complex. Insights need to travel in both directions along this continuum—from the more abstract to the more concrete and from the more concrete to the more abstract. The paper maps out some difficulties in moving from simpler to more complex cases, and in doing so makes recommendations about the future of bioethics. (shrink)

We show how Abstract Complexity Theory is related to the degrees of unsolvability and develop machinery by which computability theoretic hierarchies with a complexity theoretic flavor can be defined and investigated. This machinery is used to prove results both on hierarchies of Δ 2 0 sets and hierarchies of Δ 2 0 degrees. We prove a near-optimal lower bound on the effectivity of the Low Basis Theorem and a result showing that array computable c.e. degrees are, in (...) some sense, the simplest possible Δ 2 0 degrees. We also examine the growth rates of iterates of m K . Finally, we indicate how complexity theory can be used to analyze notions of genericity intermediate between 1 -genericity and 2 -genericity, and produce a hierarchy of such notions. (shrink)

It has been claimed that quantum mechanics, unlike classical mechanics, allows for free will. In this paper I articulate that claim and explain how a complex physical system possessing fractal-like self similarity could exhibitboth self consciousness and self determination. I use complexity theory to show how quantum mechanical indeterminacies at the neural level could “percolate up” to the levels of scale within the brain at which sensory-motor information transformations occur. Finally, I explain how macro level indeterminacy could be (...) coupled with self determination to provide a physical system with the capacity for radically free willing. (shrink)

In this article we review some of the main results of descriptive complexity theory in order to make the reader familiar with the nature of the investigations in this area. We start by presenting the characterization of automata recognizable languages by monadic second-order logic. Afterwards we explain the characterization of various logics by fIxed-point logics. We assume familiarity with logic but try to keep knowledge of complexity theory to aminimum.

A growing body of literature focuses on practice as a central aspect of teacher education. Whereas this approach emerged mainly from teacher preparation programs in specific content areas such as math, science, and literacy studies, socially related educational fields have served as a peripheral player alone. Recently, however, scholars have suggested incorporating a practice-based approach to teacher education into the social studies. In this article, we draw on complexity theory to reexamine this proposal, evaluating the connections between teaching (...) practices and democratic citizenship. First, we introduce complexity theory. Second, we offer a critical examination of key publications discussing practice-based teacher education, using the components of complexity theory as an organizing scheme. We then highlight practice-based teacher education’s reductive aspects, leading to an argument that promoting democratic citizenship through civics teacher education should be seen as a complex endeavor where borders are blurred, interactions are nonlinear, and reality changes frequently. Finally, we offer to supplement the practice approach with aspects of teacher education that stem from complexity theory. (shrink)

This Special Issue addresses a range of educational issues linked to main themes from our 2019 book The Emergence of Complexity: Rethinking Education as a Social Science. This book elaborated two major theses that raise fundamental questions for philosophy of education. First, that learning by groups is typically a distinctive kind of learning that is not reducible to learning by individuals. Second, that a degree of holism, as against a focus on individuals, is essential for achieving a convincing understanding (...) of this distinctive type of group learning. These two theses are of direct interest to philosophy of education since they challenges the resources of received theories of learning. Yet this kind of group learning characterises the vast bulk of human learning situations that occur outside of formal education systems. In this Special Issue introductory article we, firstly, introduce some basic principles of complexity theory, and demonstrate that, together with the concept of a ‘co-present group’ (between 2 and about 12 individuals), these ideas offer develop novel understandings of the distinctive learning that occurs within such groups. Secondly, we outline and illustrate with further examples the main features of co-present groups. Thirdly, we consolidate our conceptual work by showing five ways in which our account of group learning serves to refurbish the concept of ‘learning’. (shrink)

Enlightened experience (i.e. awakened to the truth) is the most valuable one in most religions including Christianity and Buddhism. As well-known cases of such experience are Apocalypse St. Paul and many Grand Zen masters in Zen Buddhism, it is natural for us to believe that the enlighten is for very talented or speciallytrained ones. However, applying the complexity theory on the structure of enlightenment, based on the power law function, selforganized criticality, phase transition, and emergence, it is clear (...) that the full awakening can be experienced by everyone in daily life. Furthermore, such experiences are not necessary to be limited in religious one. Rediscovering the path to full awakening by complexity theory leads to the demythologization of the enlightenment in Korean Zen Buddhism, which emphasizes a hard apprenticeship in temple for long periods for the purpose. (shrink)

This article provides new perspectives on navigating complex social identity in the letter to Philemon by means of the heuristic use of social identity complexity theory in combination with socio-rhetorical analysis. The application of SICT as a heuristic tool in New Testament studies is relatively new, but it is positioned within the novel research being carried out on social identity theory in the NT. CONTRIBUTION: This article wants to make a new contribution by illustrating how SICT can (...) help us to think in more nuanced ways about nested identity in Philemon. (shrink)

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 etymology of ‘complexity’ and offers a chronological presentation of three generations of theories that have shaped its uses; the epistemic and socio‐cultural roots of these theories are also introduced. From an epistemological point of view, this reflection sheds light on the competing interpretations underlying the definition of what is considered as complex. Also, from an anthropological perspective it considers both the emancipatory as well as the alienating dimensions of complexity. Based on the highlighted ambiguities, the paper suggests in conclusion that contributions grounded in contemporary theories related to complexity, as well as critical appraisals of their epistemological and ethical legitimacy, need to follow the recursive feedback loops and dynamics that they constitute. In doing so, researchers and practitioners in education should consider their own practice as a learning process that does not require the reduction of the antagonisms and the complementarities that shape its own complexity. (shrink)

Recent years have seen an increasing interest in the study of continuous-time computational models. However, not so much has been done with respect to setting up a complexity theoretic framework for such models. The present paper intends to go a step into this direction. We consider problems over the real numbers which we try to relate to Lyapunov theory for dynamical systems: The global minimizers of particular energy functions are supposed to give solutions of the problem. The structure (...) of such energy functions leads to the introduction of problem classes U and NU; for the systems we are considering they parallel the classical complexity classes P and NP. We then introduce a notion of reducibility among problems and show existence of complete problems for NU and for PU, a polynomial hierarchy of continuous-time problems.For previous work on the computational capabilities of continuous-time systems see the surveys by Cris Moore [9] and by Pekka Orponen [10]. Our paper presents a step into the direction of creating a general framework for a complexity theory of continuous-time systems as outlined in [10]. It is closely related to work done by A. Ben-Hur, H. Siegelmann, and S. Fishman [12, 11]. (shrink)

This book explores the theories of chaos and complexity as applied to a variety of disciplines including political science, organizational and management science, economics, and education.

In this paper we interpret a characterization of the Gödel speed-up phenomenon as providing support for the ‘Nagel-Newman thesis’ that human theorem recognizers differ from mechanical theorem recognizers in that the former do not seem to be limited by Gödel's incompleteness theorems whereas the latter do seem to be thus limited.However, we also maintain that (currently non-existent) programs which are open systems in that they continuously interact with, and are thus inseparable from, their environment, are not covered by the above (...) (or probably any other recursion-theoretic) argument. (shrink)

This chapter contains sections titled: Introduction Complexity Theory Complexity Theory and Educational Change Conclusion: The ‘Conditions of Emergence’ References.

In this paper we interpret a characterization of the Gödel speed-up phenomenon as providing support for the ‘Nagel-Newman thesis’ that human theorem recognizers differ from mechanical theorem recognizers in that the former do not seem to be limited by Gödel's incompleteness theorems whereas the latter do seem to be thus limited. However, we also maintain that (currently non-existent) programs which are open systems in that they continuously interact with, and are thus inseparable from, their environment, are not covered by the (...) above (or probably any other recursion-theoretic) argument. (shrink)

This paper considers questions of continuity and change in education from the perspective of complexity theory, introducing the field to educationists who might not be familiar with it. Given a significant degree of complexity in a particular environment , new properties and behaviours, which are not necessarily contained in the essence of the constituent elements or able to be predicted from a knowledge of initial conditions, will emerge. These concepts of emergent phenomena from a critical mass, associated (...) with notions of lock‐in, path dependence, and inertial momentum, suggest that it is in the dynamic interactions and adaptive orientation of a system that new phenomena, new properties and behaviours, emerge. The focus thus shifts from a concern with decontextualised and universalised essence to contextualised and contingent complex wholes. This is where complexity theory seeks the levers of history. The paper posits the notion of inertial momentum as the conceptual link between the principle of emergent phenomena as developed principally in the natural sciences and the notion of socio‐historical change in human society. It is argued that educational and institutional change is less a consequence of effecting change in one particular factor or variable, and more a case of generating momentum in a new direction by attention to as many factors as possible. Complexity theory suggests, in other words, that what it might take to change a school's inertial momentum from an ethos of failure is massive and sustained intervention at every possible level until the phenomenon of learning excellence emerges from this new set of interactions among these new factors, and sustains itself autocatalytically. The paper concludes with a summary consideration of the conditions that contribute to the emergence of new properties and behaviours in a system. (shrink)

This chapter contains sections titled: Introduction: Complexity versus Complexities Etymological Roots Contemporary Genesis First Generation Second Generation Third Generation Keeping Complexity Complex Towards a New Form of Critique? Notes References.

The basic concept of Type-2 Theory of Effectivity to define computability on topological spaces or limit spaces are representations, i. e. surjection functions from the Baire space onto X. Representations having the topological property of admissibility are known to provide a reasonable computability theory. In this article, we investigate several additional properties of representations which guarantee that such representations induce a reasonable Type-2 Complexity Theory on the represented spaces. For each of these properties, we give a (...) nice characterization of the class of spaces that are equipped with a representation having the respective property. (shrink)

This paper considers questions of continuity and change in education from the perspective of complexity theory, introducing the field to educationists who might not be familiar with it. Given a significant degree of complexity in a particular environment (or ‘dynamical system’), new properties and behaviours, which are not necessarily contained in the essence of the constituent elements or able to be predicted from a knowledge of initial conditions, will emerge. These concepts of emergent phenomena from a critical (...) mass, associated with notions of lock‐in, path dependence, and inertial momentum, suggest that it is in the dynamic interactions and adaptive orientation of a system that new phenomena, new properties and behaviours, emerge. The focus thus shifts from a concern with decontextualised and universalised essence to contextualised and contingent complex wholes. This is where complexity theory seeks the levers of history. The paper posits the notion of inertial momentum as the conceptual link between the principle of emergent phenomena as developed principally in the natural sciences and the notion of socio‐historical change in human society. It is argued that educational and institutional change is less a consequence of effecting change in one particular factor or variable, and more a case of generating momentum in a new direction by attention to as many factors as possible. Complexity theory suggests, in other words, that what it might take to change a school's inertial momentum from an ethos of failure is massive and sustained intervention at every possible level until the phenomenon of learning excellence emerges from this new set of interactions among these new factors, and sustains itself autocatalytically. The paper concludes with a summary consideration of the conditions that contribute to the emergence of new properties and behaviours in a system. (shrink)

The correct solution to complex problems, such as those involved in international relations, can generally be discovered ex post but not predicted ex ante. Economics and game theory attempt to model such complexity, but have difficulty taking into account psychological subtleties, the myriad factors that each agent considers when making a decision, and cultural differences. And understanding that one is dealing with a system—that is, with interacting factors instead of with insulated monads—may not make the questions any more (...) amenable to prediction, particularly because the more unique an event, the less likely it is to be foreseen. Jervis's analysis of complex systems may therefore be more of a contribution to the historical sciences than to predictive social science. (shrink)

The disavowal of positivist science by many educational researchers has resulted in a deepening polarization of research agendas and an epistemological divide that appears increasingly difficult to span. Despite a turning away from science altogether by some, and thus toward various forms of poststructuralist inquiry, this has not held back the renewed entrenchment of more narrow definitions by policy elites of what constitutes scientific educational research. The new sciences of complexity signal the emergence of a new scientific paradigm that (...) challenges some of the core assumptions of positivism, while offering the potential to develop a new kind of social science that demands both rigour and imagination in coming to understand the emergence and behaviours of social systems and the subsystems that comprise them. The language, concepts and principles of complexity are central to the development of a new science of qualities to complement the science of quantities that has shaped our understanding of the physical and social worlds. Accomplishing this task promises to 1) open up new investigations that have thus far been beyond the purview of scientific study, 2) allow the study of social phenomena as fully embodied, or at least as more robust models than those represented in the abstracted empiricism upon which the sciences of quantities are predicated, and 3) allow for more coarse‐grained explanations and predictions of social phenomena to be legitimated as scientific. Both educational research and educational practice stand to gain from this expansion of the scientific repertoire to include rigorous and imaginative investigations of phenomena characterized by change and transformation. (shrink)

[I] Divine Simplicity: God and His Existence Types of Divine Simplicity Of the properties ascribed to God in Aquinas' natural theology, we may call one sort ...

Reductionism and complexity theory are two paradigms frequently found in language research. There exist a number of conflicts in terms of concepts and methodologies between reductionism and complexity theory, which are not conducive to creating a unified language research framework. This paper starts by discussing the adaptability of complex dynamic systems and combines cognitive processing model and artificial neural networks to construct and verify an adaptive weight model, showing that the study of reductionism is induction of (...) high-weight elements and the study of complexity theory is a discussion of system complexity from adaptability, meaning that there is a good fit between the two frameworks. The adaptive weight model is conducive to developing a unified interpretation of language research results. (shrink)

A central aim of cognitive science is to understand the fundamental mechanisms that enable humans to navigate and make sense of complex environments. In this letter, we argue that computational complexity theory, a foundational framework for evaluating computational resource requirements, holds significant potential in addressing this challenge. As humans possess limited cognitive resources for processing vast amounts of information, understanding how humans perform complex cognitive tasks requires comprehending the underlying factors that drive information processing demands. Computational complexity (...) theory provides a comprehensive theoretical framework to achieve this goal. By adopting this framework, we can gain new insights into how cognitive systems work and develop a more nuanced understanding of the relation between task complexity and human behavior. We provide empirical evidence supporting our argument and identify several open research questions and challenges in applying computational complexity theory to human decision‐making and cognitive science at large. (shrink)

This chapter contains sections titled: Introduction What Complexity Is Toward a Qualitative Approach to a Science of Qualities From Simplicity to Complexity Researching the ‘Edge of Chaos’ Ethics and Complexity Research References.

Complexity theory challenges educational philosophy to reconsider accepted paradigms of teaching, learning and educational research. However, though attractive, not least because of its critique of positivism, its affinity to Dewey and Habermas, and its arguments for openness, diversity, relationships, agency and creativity, the theory is not without its difficulties. These are seen to lie in terms of complexity theory's nature, status, methodology, utility and contribution to the philosophy of education, being a descriptive theory that (...) is easily misunderstood as a prescriptive theory, silent on key issues of values and ethics that educational philosophy should embrace, of questionable internal consistency, and of limited ‘added value’ in educational philosophy. The paper sets out key tenets of complexity theory and argues that, though it is useful for educational philosophy, it requires new conceptual tools and mind‐sets to comprehend fully its significance. (shrink)

Around 1989, a striking letter written in March 1956 from Kurt Gödel to John von Neumann came to light. It poses some problems about the complexity of algorithms; in particular, it asks a question that can be seen as the first formulation of the P=?NP question. This paper discusses some of the background to this letter, including von Neumann's own ideas on complexity theory. Von Neumann had already raised explicit questions about the complexity of Tarski's decision (...) procedure for elementary algebra and geometry in a letter of 1949 to J. C. C. McKinsey. The paper concludes with a discussion of why theoretical computer science did not emerge as a separate discipline until the 1960s. (shrink)