Much of our understanding of human thinking is based on probabilistic models. This innovative book by Jerome R. Busemeyer and Peter D. Bruza argues that, actually, the underlying mathematical structures from quantum theory provide a much better account of human thinking than traditional models. They introduce the foundations for modelling probabilistic-dynamic systems using two aspects of quantum theory. The first, 'contextuality', is a way to understand interference effects found with inferences and decisions under conditions of uncertainty. The (...) second, 'quantum entanglement', allows cognitive phenomena to be modeled in non-reductionist ways. Employing these principles drawn from quantum theory allows us to view human cognition and decision in a totally new light. Introducing the basic principles in an easy-to-follow way, this book does not assume a physics background or a quantum brain and comes complete with a tutorial and fully worked-out applications in important areas of cognition and decision. (shrink)

Games of chance are developed in their physical consumer-ready form on the basis of mathematical models, which stand as the premises of their existence and represent their physical processes. There is a prevalence of statistical and probabilistic models in the interest of all parties involved in the study of gambling – researchers, game producers and operators, and players – while functional models are of interest more to math-inclined players than problem-gambling researchers. In this paper I present (...) a structural analysis of the knowledge attached to mathematical models of games of chance and the act of modeling, arguing that such knowledge holds potential in the prevention and cognitive treatment of excessive gambling, and I propose further research in this direction. (shrink)

Remarkable progress in the mathematics and computer science of probability has led to a revolution in the scope of probabilistic models. In particular, ‘sophisticated’ probabilistic methods apply to structured relational systems such as graphs and grammars, of immediate relevance to the cognitive sciences. This Special Issue outlines progress in this rapidly developing field, which provides a potentially unifying perspective across a wide range of domains and levels of explanation. Here, we introduce the historical and conceptual foundations of the approach, (...) explore how the approach relates to studies of explicit probabilistic reasoning, and give a brief overview of the field as it stands today. (shrink)

This study tested a structural model of cognitive-emotional explanatory variables to explain performance in mathematics. The predictor variables assessed were related to students’ level of development of early mathematical competencies (EMCs), specifically, relational and numerical competencies, predisposition toward mathematics, and the level of logical intelligence in a population of primary school Chilean students (n = 634). This longitudinal study also included the academic performance of the students during a period of four years as a variable. The sampled students were (...) initially assessed by means of an Early Numeracy Test (ENT), and, subsequently, they were administered a Likert-type scale to measure their predisposition toward mathematics (EPMAT) and a basic test of logical intelligence (TILE). The results of these tests were used to analyse the interaction of all the aforementioned variables by means of a structural equations model. This combined interaction model was able to predict 64.3% of the variability of observed performance. Preschool students' performance in EMCs was a strong predictor for achievement in mathematics for students between 8 to 11 years of age. Therefore, this paper highlights the importance of EMCs and the modulating role of predisposition toward mathematics. Also, this paper discusses the educational role of these findings, as well as possible ways to improve negative predispositions toward mathematical tasks in the school domain. (shrink)

Current understanding of the development of quantity representations is based primarily on performance in the number‐line task. We posit that the data from number‐line tasks reflect the observer's underlying representation of quantity, together with the cognitive strategies and skills required to equate line length and quantity. Here, we specify a unified theory linking the underlying psychological representation of quantity and the associated strategies in four variations of the number‐line task: the production and estimation variations of the bounded and unbounded number‐line (...) tasks. Comparison of performance in the bounded and unbounded number‐line tasks provides a unique and direct way to assess the role of strategy in number‐line completion. Each task produces a distinct pattern of data, yet each pattern is hypothesized to arise, at least in part, from the same underlying psychological representation of quantity. Our model predicts that the estimated biases from each task should be equivalent if the different completion strategies are modeled appropriately and no other influences are at play. We test this equivalence hypothesis in two experiments. The data reveal all variations of the number‐line task produce equivalent biases except for one: the estimation variation of the bounded number‐line task. We discuss the important implications of these findings. (shrink)

This paper sets out to show how mathematical modelling can serve as a way of ampliating knowledge. To this end, I discuss the mathematical modelling of time in theoretical physics. In particular I examine the construction of the formal treatment of time in classical physics, based on Barrow’s analogy between time and the real number line, and the modelling of time resulting from the Wheeler-DeWitt equation. I will show how mathematics shapes physical concepts, like time, acting as a (...) heuristic means—a discovery tool—, which enables us to construct hypotheses on certain problems that would be hard, and in some cases impossible, to understand otherwise. (shrink)

There are presently two leading foreign policy decision-making paradigms in vogue. The first is based on the classical or rational model originally posited by von Neumann and Morgenstern to explain microeconomic decisions. The second is based on the cybernetic perspective whose groundwork was laid by Herbert Simon in his early research on bounded rationality. In this paper we introduce a third perspective — thepoliheuristic theory of decision-making — as an alternative to the rational actor and cybernetic paradigms in international relations. (...) This theory is drawn in large part from research on heuristics done in experimental cognitive psychology. According to the poliheuristic theory, policy makers use poly (many) heuristics while focusing on a very narrow range of options and dimensions when making decisions. Among them, the political dimension is noncompensatory. The paper also delineates the mathematical formulations of the three decision-making models. (shrink)

There are presently two leading foreign policy decision-making paradigms in vogue. The first is based on the classical or rational model originally posited by von Neumann and Morgenstern to explain microeconomic decisions. The second is based on the cybernetic perspective whose groundwork was laid by Herbert Simon in his early research on bounded rationality. In this paper we introduce a third perspective -- the poliheuristic theory of decision-making -- as an alternative to the rational actor and cybernetic paradigms in international (...) relations. This theory is drawn in large part from research on heuristics done in experimental cognitive psychology. According to the poliheuristic theory, policy makers use poly (many) heuristics while focusing on a very narrow range of options and dimensions when making decisions. Among them, the political dimension is noncompensatory. The paper also delineates the mathematical formulations of the three decision-making models. (shrink)

Statistical learning refers to the ability to identify structure in the input based on its statistical properties. For many linguistic structures, the relevant statistical features are distributional: They are related to the frequency and variability of exemplars in the input. These distributional regularities have been suggested to play a role in many different aspects of language learning, including phonetic categories, using phonemic distinctions in word learning, and discovering non-adjacent relations. On the surface, these different aspects share few commonalities. Despite this, (...) we demonstrate that the same computational framework can account for learning in all of these tasks. These results support two conclusions. The first is that much, and perhaps all, of distributional statistical learning can be explained by the same underlying set of processes. The second is that some aspects of language can be learned due to domain-general characteristics of memory. (shrink)

Cognitive function certainly poses the biggest challenge for computational neuroscience. As we argue, past efforts to build neural models of cognition (the target article included) had too narrow a focus on implementing rule-based language processing. The problem with these models is that they sacrifice the advantages of connectionism rather than building on them. Recent and more promising approaches for modeling cognition build on the mathematical properties of distributed neural representations. These approaches truly exploit the key (...) advantages of connectionism, that is, the high representational power of distributed neural codes and similarity-based pattern recognition. The architectures for cognitive computing that emerge from these approaches are neural associative memories endowed with additional mapping operations to handle invariances and to form reduced representations of combinatorial structures. (shrink)

The prominence of Bayesian modeling of cognition has increased recently largely because of mathematical advances in specifying and deriving predictions from complex probabilistic models. Much of this research aims to demonstrate that cognitive behavior can be explained from rational principles alone, without recourse to psychological or neurological processes and representations. We note commonalities between this rational approach and other movements in psychology – namely, Behaviorism and evolutionary psychology – that set aside mechanistic explanations or make use of (...) optimality assumptions. Through these comparisons, we identify a number of challenges that limit the rational program's potential contribution to psychological theory. Specifically, rational Bayesian models are significantly unconstrained, both because they are uninformed by a wide range of process-level data and because their assumptions about the environment are generally not grounded in empirical measurement. The psychological implications of most Bayesian models are also unclear. Bayesian inference itself is conceptually trivial, but strong assumptions are often embedded in the hypothesis sets and the approximation algorithms used to derive model predictions, without a clear delineation between psychological commitments and implementational details. Comparing multiple Bayesian models of the same task is rare, as is the realization that many Bayesian models recapitulate existing (mechanistic level) theories. Despite the expressive power of current Bayesian models, we argue they must be developed in conjunction with mechanistic considerations to offer substantive explanations of cognition. We lay out several means for such an integration, which take into account the representations on which Bayesian inference operates, as well as the algorithms and heuristics that carry it out. We argue this unification will better facilitate lasting contributions to psychological theory, avoiding the pitfalls that have plagued previous theoretical movements. (shrink)

By the end of his life Plato had rearranged the theory of ideas into his teaching about ideal numbers, but no written records have been left. The Ideal mathematics of Plato is present in all his dialogues. It can be clearly grasped in relation to the effective use of mathematical modelling. Many problems of mathematical modelling were laid in the foundation of the method by cutting the three-level idealism of Plato to the single-level “ideism” of Aristotle. For a (...) long time, the real, ideal numbers of Plato’s Ideal mathematics eliminates many mathematical problems, extends the capabilities of modelling, and improves mathematics. (shrink)

Quantum cognition research applies abstract, mathematical principles of quantum theory to inquiries in cognitive science. It differs fundamentally from alternative speculations about quantum brain processes. This topic presents new developments within this research program. In the introduction to this topic, we try to answer three questions: Why apply quantum concepts to human cognition? How is quantum cognitive modeling different from traditional cognitive modeling? What cognitive processes have been modeled using a quantum account? In addition, a brief introduction (...) to quantum probability theory and a concrete example is provided to illustrate how a quantum cognitive model can be developed to explain paradoxical empirical findings in psychological literature. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Handbook of Cognitive Mathematics ed. by Marcel DanesiNathan HaydonMarcel Danesi (Ed) Handbook of Cognitive Mathematics Cham, Switzerland: Springer International, 2022, vii + 1383, including indexFor one acquainted with C.S. Peirce, it is hard to see Springer's recent Handbook of Cognitive Mathematics (editor: Marcel Danesi) through none other than a Peircean lens. Short for the cognitive science of mathematics, such a modern, scientific pursuit into the nature and study (...) of mathematical practice would no doubt be found agreeable to Peirce. The fact that references to Peirce appear often throughout the Handbook is a welcome find, with Peirce's ideas being a key subject of half a dozen chapters, and where a reader of any other chapter may well find further connections to Peirce's ideas. After spending time with the Handbook, it is clear that cognitive mathematics has not only embraced some of Pierce's ideas but may be at an important forefront of Peirce studies. In the end, the field may well be an area a Peircean should pay attention to.The book itself is pitched as a reference volume to the field (p. v) with the necessary background to familiarize oneself with the aims and results. The connection to cognitive science may call to mind detailed cognitive models [Ch. 10–14], theories on the origins of numeracy and other theories behind the biological and evolutionary requirements of mathematical thought [Ch.15–18], and the like. These are all present. But a key theme of the Handbook is to situate mathematics not just within more traditional 'cognitive' faculties and the more formal, i.e. algebraic, presentations of mathematics, but also to place mathematics within other human faculties and practices, from the arts to language [Ch. 19–22], within education and learning more broadly [Ch. 23–26], and in relation to the significant, though at first perhaps less quantitative parts of mathematics, like the use of metaphor, gesture, analogy, [End Page 243] abstraction, as well as further cultural and ethnographic considerations [Ch. 5–8]. The Handbook has explicitly taken a broader, more interdisciplinary approach (p. vi–vii) towards the scientific aspects of mathematical practice—choosing to regulate the study not by antecedently drawn opinions about what mathematics is (or has traditionally been taken to be) but by what future quantifiable and diverse study may come to bear on the practice and have to say about those engaging in it.This broad interdisciplinarity has a pragmatist ring, where theory cannot so easily be separated from the normative, social, and, more generally, the more thoroughly human aspects that we encounter and employ when we engage in it. Two further commitments of cognitive mathematics steer us even closer towards Peirce's views. The first—going back, for example, to Lakoff and Núñez's Where Mathematics Comes From (2000), which is taken to be a key early work in shaping the field—is that mathematics is taken to be a language like any other and as such must be learned and situated within our other cognitive faculties (p. vi, Ch. 4). The second—and largely, one imagines, following from a concern for seeing how mathematics is actually practiced—is that mathematics is essentially diagrammatic, involving as it does experimenting in diagrams and employing other signs (one may think here even of the mathematician sketching equations or geometric figures on a sheet of paper). What is perhaps initially most noteworthy about Peirce's philosophy of mathematics is his early concern for both these aspects of the practice.The book (more like a tome of over 1300 pages) is too large to be covered here in all its aspects, hence the review will focus on those aspects a Peircean or Pragmatist may find of interest. In particular, this review focuses on several of the themes above—situating mathematics more broadly within our everyday (cognitive) lives, the necessarily diagrammatic, fallible, and creative aspects of the practice, and, finally, what these aspects may have to say about Peirce's philosophy more generally, particularly with respect to (scientific) inquiry.A Peircean reader may well begin with Pietarinen's "Pragmaticism as Philosophy of Cognitive Mathematics" [Ch. 39] and "Peirce on Mathematical... (shrink)

In this paper, we propose a new mathematical model of retrieval of intentions from long-term memory. We model retrieval as a stochastic race between a plurality of potentially relevant intentions stored in long-term memory. Psychological theories are dominated by two opposing conceptions of the role of memory in temporally extended agency – as when a person has to remember to make a phone call in the afternoon because, in the morning, she promised she would do so. According to the (...) Working Memory conception, remembering to make the phone call is explained in terms of the construction and maintenance of intentions in working-memory. According to the Long-Term Memory conception, we should explain the episode in terms of an ability to store intentions in long-term memory. The two conceptions predict different processing profiles. The aim of this paper is to present a new mathematical model of the type of memory mechanism that could realise the long-term memory representations of intentions necessary for the Long-Term Memory conception. We present and illustrate the formal model and propose a new type of experimental paradigm that could allow us to test which of the two conceptions provides the best explanation of the role of memory in temporally extended agency. (shrink)

This paper describes model building and manipulation in the solution of problems in mechanics. An automatic problem solver, MECHO, solving problems in several areas of mechanics, employs (1) a knowledge base representing the semantic content of the particular problem area, (2) a means-ends search strategy similar to GPS to produce sets of simultaneous equations and (3) a “focusing” technique, based on the data within the knowledge base, to guide the GSP-like search through possible equation instantiations. Sets of predicate logic statements (...) are employed to describe this model building activity. These clauses are used to give information content to the knowledge base and provide both basis and guidance for the goal driven search.It is hypothesized that humon subjects solving mechanics problems employ similar model building techniques. Protocols of several subjects are presented and comparisons are drawn with the traces of the automated problem solver. Adjustments are made to the program to provide a better fit of traces to protocols. Some implications are presented for using a rule based model for describing human problem solving performance in solving mechanics problems. (shrink)

Children learn and use various strategies to solve math problems. One way children's math learning can be supported is through their use of and exposure to hand gestures. Children's self-produced gestures can reveal unique, math-relevant knowledge that is not contained in their speech. Additionally, these gestures can assist with their math learning and problem solving by supporting their cognitive processes, such as executive function. The gestures that children observe during math instructions are also linked to supporting cognition. Specifically, children (...) are better able to learn, retain, and generalize knowledge about math when that information is presented within the gestures that accompany an instructor's speech. To date, no conceptual model provides an outline regarding how these gestures and the math environment are connected, nor how they may interact with children's underlying cognitive capacities such as their executive function. In this review, we propose a new model based on an integration of the information processing approach and theory of embodied cognition. We provide an in-depth review of the related literature and consider how prior research aligns with each link within the proposed model. Finally, we discuss the utility of the proposed model as it pertains to future research endeavors. (shrink)

Theories of grounded and embodied cognition offer a range of accounts of how reasoning and body‐based processes are related to each other. To advance theories of grounded and embodied cognition, we explore the cognitive relevance of particular body states to associated math concepts. We test competing models of action‐cognition transduction to investigate the cognitive relevance of directed actions to students’ mathematical reasoning in the area of geometry. The hypotheses we test include (1) that cognitively relevant (...) directed actions have a direct effect on performance (direct cognitive relevance hypothesis), (2) that cognitively relevant directed actions lead to more frequent production of gestures during explanations, which leads to improved performance (mediated cognitive relevance hypothesis), and (3) that performance effects of directed actions are influenced by the presence or absence of gesture production during mathematical explanations (moderated cognitive relevance hypothesis). We explore these hypotheses in an experiment where high school students (N = 85) evaluated the truth of geometry conjectures after performing cognitively relevant or cognitively irrelevant directed actions while playing a movement‐based video game. Contrary to the direct and mediated cognitive relevance hypotheses, we found no overall differences in performance or gesture production between relevant and irrelevant conditions. Consistent with the moderated cognitive relevance hypothesis, cognitive relevance influenced mathematical performance, as measured by the accuracy of students’ intuitions, insights, and the validity of their proofs, provided that students produced certain kinds of gestures during mathematical explanations (i.e., with explanatory gestures as the moderator). Implications for theories of grounded and embodied cognition and the design of embodied forms of educational interventions are discussed. (shrink)

We argue that dynamical and mathematical models in systems and cognitive neuro- science explain (rather than redescribe) a phenomenon only if there is a plausible mapping between elements in the model and elements in the mechanism for the phe- nomenon. We demonstrate how this model-to-mechanism-mapping constraint, when satisfied, endows a model with explanatory force with respect to the phenomenon to be explained. Several paradigmatic models including the Haken-Kelso-Bunz model of bimanual coordination and the difference-of-Gaussians model of visual (...) receptive fields are explored. (shrink)

This paper is devoted to some generalizations of explanatory potential of connectionist approaches to theoretical problems of the philosophy of mind. Are considered both strong, and weaknesses of neural network models. Connectionism has close methodological ties with modern neurosciences and neurophilosophy. And this fact strengthens its positions, in terms of empirical naturalistic approaches. However, at the same time this direction inherits weaknesses of computational approach, and in this case all system of anticomputational critical arguments becomes applicable to the connectionst (...) models of mind. The last developments in the field of deep learning gave rich empirical material for cognitive sciences. Multilayered networks, mathematical models of associative dynamics of learning, self-organizing neuronets and all that allow to explain the principles of human conceptual organizing and after this to emulate these processes in computer systems. At all engineering achievements of this technology there is a traditional criticism from representatives of cognitive psychology who cannot accept a thesis about learning ability of a neuronet on the basis of redistribution of scales. Process of learning of natural intelligence, according to cognitive models, happens due to attraction of knowledge broadcast in a symbolical form (mental representations, concepts) at the expense of the systems of output knowledge expressed in the propositional contents. Some philosophical aspects of «neural metaphor» in modern cognitive sciences create the problem field which demands comprehensive understanding, the first step towards which is taken in this work. (shrink)

Stemming from Uexkull’s fundamental concepts of Umwelt and Innenwelt as developed in the biosemiotic approach of Ferreira 2010, 2011, the present work models mathematically the semiosis of cognition and proposes an artificial cognitive architecture to be deployed in a robotic structure.

Computational models of memory presented in this issue reflect varied empirical data and levels of representation. From mathematical models to neural and cognitive architectures, all aim to converge on a unified theory of the mind.

Open peer commentary on the article “Info-computational Constructivism and Cognition” by Gordana Dodig-Crnkovic. Upshot: I propose a mathematical approach to the framework developed in Dodig-Crnkovic’s target article. It points to an important property of natural computation, called the multiplicity principle (MP), which allows the development of increasingly complex cognitive processes and knowledge. While local dynamics are classically computable, a consequence of the MP is that the global dynamics is not, thus raising the problem of developing more elaborate computations, (...) perhaps with the help of Turing oracles. (shrink)

The ability to improve in speed and accuracy as a result of repeating some task is an important hallmark of intelligent biological systems. Although gradual behavioral improvements from practice have been modeled in spiking neural networks, few such models have attempted to explain cognitive development of a task as complex as addition. In this work, we model the progression from a counting-based strategy for addition to a recall-based strategy. The model consists of two networks working in parallel: a slower (...) basal ganglia loop and a faster cortical network. The slow network methodically computes the count from one digit given another, corresponding to the addition of two digits, whereas the fast network gradually “memorizes” the output from the slow network. The faster network eventually learns how to add the same digits that initially drove the behavior of the slower network. Performance of this model is demonstrated by simulating a fully spiking neural network that includes basal ganglia, thalamus, and various cortical areas. Consequently, the model incorporates various neuroanatomical data, in terms of brain areas used for calculation and makes psychologically testable predictions related to frequency of rehearsal. Furthermore, the model replicates developmental progression through addition strategies in terms of reaction times and accuracy, and naturally explains observed symptoms of dyscalculia. (shrink)

A case study of quantum mechanics is investigated in the framework of the philosophical opposition “mathematical model – reality”. All classical science obeys the postulate about the fundamental difference of model and reality, and thus distinguishing epistemology from ontology fundamentally. The theorems about the absence of hidden variables in quantum mechanics imply for it to be “complete” (versus Einstein’s opinion). That consistent completeness (unlike arithmetic to set theory in the foundations of mathematics in Gödel’s opinion) can be interpreted furthermore (...) as the coincidence of model and reality. The paper discusses the option and fact of that coincidence it its base: the fundamental postulate formulated by Niels Bohr about what quantum mechanics studies (unlike all classical science). Quantum mechanics involves and develops further both identification and disjunctive distinction of the global space of the apparatus and the local space of the investigated quantum entity as complementary to each other. This results into the analogical complementarity of model and reality in quantum mechanics. The apparatus turns out to be both absolutely “transparent” and identically coinciding simultaneously with the reflected quantum reality. Thus, the coincidence of model and reality is postulated as necessary condition for cognition in quantum mechanics by Bohr’s postulate and further, embodied in its formalism of the separable complex Hilbert space, in turn, implying the theorems of the absence of hidden variables (or the equivalent to them “conservation of energy conservation” in quantum mechanics). What the apparatus and measured entity exchange cannot be energy (for the different exponents of energy), but quantum information (as a certain, unambiguously determined wave function) therefore a generalized law of conservation, from which the conservation of energy conservation is a corollary. Particularly, the local and global space (rigorously justified in the Standard model) share the complementarity isomorphic to that of model and reality in the foundation of quantum mechanics. On that background, one can think of the troubles of “quantum gravity” as fundamental, direct corollaries from the postulates of quantum mechanics. Gravity can be defined only as a relation or by a pair of non-orthogonal separable complex Hilbert space attachable whether to two “parts” or to a whole and its parts. On the contrary, all the three fundamental interactions in the Standard model are “flat” and only “properties”: they need only a single separable complex Hilbert space to be defined. (shrink)

This paper intends to further the understanding of the formal properties of (higher-order) vagueness by connecting theories of (higher-order) vagueness with more recent work in topology. First, we provide a “translation” of Bobzien's account of columnar higher-order vagueness into the logic of topological spaces. Since columnar vagueness is an essential ingredient of her solution to the Sorites paradox, a central problem of any theory of vagueness comes into contact with the modern mathematical theory of topology. Second, Rumfitt’s recent topological (...) reconstruction of Sainsbury’s theory of prototypically defined concepts is shown to lead to the same class of spaces that characterize Bobzien’s account of columnar vagueness, namely, weakly scattered spaces. Rumfitt calls these spaces polar spaces. They turn out to be closely related to Gärdenfors’ conceptual spaces, which have come to play an ever more important role in cognitive science and related disciplines. Finally, Williamson’s “logic of clarity” is explicated in terms of a generalized topology (“locology”) that can be considered an alternative to standard topology. Arguably, locology has some conceptual advantages over topology with respect to the conceptualization of a boundary and a borderline. Moreover, in Williamson’s logic of clarity, vague concepts with respect to a notion of a locologically inspired notion of a “slim boundary” are (stably) columnar. Thus, Williamson’s logic of clarity also exhibits a certain affinity for columnar vagueness. In sum, a topological perspective is useful for a conceptual elucidation and unification of central aspects of a variety of contemporary accounts of vagueness. (shrink)

Analysing several characteristic mathematical models: natural and real numbers, Euclidean geometry, group theory, and set theory, I argue that a mathematical model in its final form is a junction of a set of axioms and an internal partial interpretation of the corresponding language. It follows from the analysis that (i) mathematical objects do not exist in the external world: they are our internally imagined objects, some of which, at least approximately, we can realize or represent; (ii) (...) mathematical truths are not truths about the external world but specifications (formulations) of mathematical conceptions; (iii) mathematics is first and foremost our imagined tool by which, with certain assumptions about its applicability, we explore nature and synthesize our rational cognition of it. (shrink)

BackgroundAdolescents’ engagement in daily physical activity brings multiple benefits, including reduction in obesity, improvement of mental health, and enhancement of cognitive function. While prior studies have examined the link between physical activity and cognitive function, little is known regarding the extent to which this relationship is shaped by health and wellbeing factors. This study examines how subjective wellbeing and general health mediate the relationship between adolescents’ physical activity and cognitive function.MethodsThis study estimates a parallel structural equation model using the Program (...) for International Student Assessment 2018 dataset. Specifically, a total of 63,228 15-year-old subjects in nine countries/economies satisfied the study inclusion criteria, including in Bulgaria, Georgia, Hong Kong, Ireland, Mexico, Panama, Serbia, Spain, and United Arab Emirates. Frequency of moderate physical activity was reported weekly; SWB and GH were assessed using an internationally validated multi-item standardized questionnaire. SWB was measured by students’ self-evaluated satisfaction with their health, life, and schooling. GH was measured by students’ physical health and mental health status. Cognitive function was modeled as a latent function consisting of plausible values derived using item response theory on reading, mathematics, and science achievement tests.ResultsFindings indicated that increase in weekly MPA was positively associated with higher levels of SWB, GH, and CF among the study subjects. Parallel mediation analyses revealed that more frequent weekly MPA had relatively large direct effects on CF, and indirect effects channeling through improvements in SWB and GH were non-trivial. Heterogeneity results showed that boosts to CF, associated with MPA, were larger for mathematics and science than for reading.ConclusionThis study used a large-scale international dataset to show that the positive relationship observed between MPA and CF among adolescents was robust, and that SWB and GH were two critical mediators through which physical activity positively bolster CF. (shrink)

Effective conditioning requires a correlation between the experimenter's definition of a response and an organism's, but an animal's perception of its behavior differs from ours. These experiments explore various definitions of the response, using the slopes of learning curves to infer which comes closest to the organism's definition. The resulting exponentially weighted moving average provides a model of memory that is used to ground a quantitative theory of reinforcement. The theory assumes that: incentives excite behavior and focus the excitement on (...) responses that are contemporaneous in memory. The correlation between the organism's memory and the behavior measured by the experimenter is given by coupling coefficients, which are derived for various schedules of reinforcement. The coupling coefficients for simple schedules may be concatenated to predict the effects of complex schedules. The coefficients are inserted into a generic model of arousal and temporal constraint to predict response rates under any scheduling arrangement. The theory posits a response-indexed decay of memory, not a time-indexed one. It requires that incentives displace memory for the responses that occur before them, and may truncate the representation of the response that brings them about. As a contiguity-weighted correlation model, it bridges opposing views of the reinforcement process. By placing the short-term memory of behavior in so central a role, it provides a behavioral account of a key cognitive process. (shrink)

Understanding arithmetic word problems involves a complex interaction of text comprehension and mathematical processes. This article presents a computer simulation designed to capture the working memory demands required in “bottomup” comprehension of arithmetic word problems. The simulation's sentence‐level parser and text integration component reflect the importance of processing the problem from its original natural language presentation. Children's probability of solution was analyzed in exploratory regression analyses as a function of the simulation's sentence‐level and text integration processes. Working memory variables (...) measuring the combined effects of concepts to remember and text integration inferences account for a significant proportion of variance in children's solution probabilities across the first four grade levels (K‐3). Consistent with previous results from others, which highlighted the significance of small changes in problem wording, the simulation offers a process‐oriented perspective as to why natural language presentation constrains the comprehension of mathematical relationships. (shrink)

Since Darwin, the idea of psychological continuity between humans and other animals has dominated theory and research in investigating the minds of other species. Indeed, the field of comparative psychology was founded on two assumptions. First, it was assumed that introspection could provide humans with reliable knowledge about the causal connection between specific mental states and specific behaviors. Second, it was assumed that in those cases in which other species exhibited behaviors similar to our own, similar psychological causes were at (...) work. In this paper. we show how this argument by analogy is flowed with respect to the case of second-order mental states. As a test case, we focus on the question of how other species conceive of visual attention, and in particular whether chimpanzees interpret seeing as a mentalistic event involving internal states of perception, attention, and belief. We conclude that chimpanzees do not reason about seeing in this manner, and indeed, there is considerable reason to suppose that they do not harbor representations of mental states in general. We propose a reinterpretation model in which the majority of the rich social behaviors that humans and other primates share in common emerged long before the human lineage evolved the psychological means of interpreting those behaviors in mentalistic terms. Although humans, chimpanzees, and most other species may be said to possess mental states, humans alone may have evolved a cognitive specialization for reasoning about such states. (shrink)

This article responds to two unresolved and crucial problems of cognitive science: (1) What is actually accomplished by functions of the nervous system that we ordinarily describe in the intentional idiom? and (2) What makes the information processing involved in these functions semantic? It is argued that, contrary to the assumptions of many cognitive theorists, the computational approach does not provide coherent answers to these problems, and that a more promising start would be to fall back on mathematical communication (...) theory and, with the help of evolutionary biology and neurophysiology, to attempt a characterization of the adaptive processes involved in visual perception. Visual representations are explained as patterns of cortical activity that are enabled to focus on objects in the changing visual environment by constantly adjusting to maintain levels of mutual information between pattern and object that are adequate for continuing perceptual control. In these terms, the answer proposed to (1) is that the intentional functions of vision are those involved in the establishment and maintenance of such representations, and to (2) that semantic features are added to the information processes of vision with the focus on objects that these representations accomplish. The article concludes with proposals for extending this account of intentionality to the higher domains of conceptualization and reason, and with speculation about how semantic information-processing might be achieved in mechanical systems. (shrink)

How do minds emerge from developing brains? According to the representational features of cortex are built from the dynamic interaction between neural growth mechanisms and environmentally derived neural activity. Contrary to popular selectionist models that emphasize regressive mechanisms, the neurobiological evidence suggests that this growth is a progressive increase in the representational properties of cortex. The interaction between the environment and neural growth results in a flexible type of learning: minimizes the need for prespecification in accordance with recent neurobiological (...) evidence that the developing cerebral cortex is largely free of domain-specific structure. Instead, the representational properties of cortex are built by the nature of the problem domain confronting it. This uniquely powerful and general learning strategy undermines the central assumption of classical learnability theory, that the learning properties of a system can be deduced from a fixed computational architecture. Neural constructivism suggests that the evolutionary emergence of neocortex in mammals is a progression toward more flexible representational structures, in contrast to the popular view of cortical evolution as an increase in innate, specialized circuits. Human cortical postnatal development is also more extensive and protracted than generally supposed, suggesting that cortex has evolved so as to maximize the capacity of environmental structure to shape its structure and function through constructive learning. (shrink)

In recent years, several remarkable results have shown that certain theorems of finite combinatorics are unprovable in certain logical systems. These developments have been instrumental in stimulating research in both areas, with the interface between logic and combinatorics being especially important because of its relation to crucial issues in the foundations of mathematics which were raised by the work of Kurt Godel. Because of the diversity of the lines of research that have begun to shed light on these issues, there (...) was a need for a comprehensive overview which would tie the lines together. This volume fills that need by presenting a balanced mixture of high quality expository and research articles that were presented at the August 1985 AMS-IMS-SIAM Joint Summer Research Conference, held at Humboldt State University in Arcata, California.With an introductory survey to put the works into an appropriate context, the collection consists of papers dealing with various aspects of 'unprovable theorems and fast-growing functions'. Among the topics addressed are: ordinal notations, the dynamical systems approach to Ramsey theory, Hindman's finite sums theorem and related ultrafilters, well quasiordering theory, uncountable combinatorics, nonstandard models of set theory, and a length-of-proof analysis of Godel's incompleteness theorem. Many of the articles bring the reader to the frontiers of research in this area, and most assume familiarity with combinatorics and/or mathematical logic only at the senior undergraduate or first-year graduate level. (shrink)

The main problem discussed in this paper is: Why and how did animal cognition abilities arise? It is argued that investigations of the evolution of animal cognition abilities are very important from an epistemological point of view. A new direction for interdisciplinary researches – the creation and development of the theory of human logic origin – is proposed. The approaches to the origination of such a theory (mathematical models of ``intelligent invention'' of biological evolution, the cybernetic (...) schemes of evolutionary progress and purposeful adaptive behavior) as well as potential interdisciplinary links of the theory are described and analyzed. (shrink)

In the paper I discuss the problem of the nature of the relationship between objects and their properties. There are three contexts of the problem: of comparison, of change and of interaction. Philosophical explanations of facts indicated in the three contexts need reference to properties and to a proper understanding of a relationship between them and their bearers. My aim is to get closer to this understanding with the use of some models but previously I present the substantialist theory (...) of object and shortly argue for its main theses. The two models enabling us the understanding of the subject–properties structure are: the plastic stuff model and the functional model. On the ground of the first a subject is compared to a piece of plastic stuff which is informed by different shapes. Properties are ways how a subject is, they “give” some “figure” to a subject. The core idea of the second model is that essences (performing the role of subjects) are immanent functional laws governing correlations of properties. As such they are similar to mathematical functions which are saturated by values. The relationship between a subject and properties can be grasped by analogy to such a saturation. (shrink)

Open peer commentary on the article “Ethics: A Radical-constructivist Approach” by Andreas Quale. Upshot: The first of my two main goals in this commentary is to establish thinking of ethics as concepts rather than as non-cognitive knowledge. The second is to argue that establishing models of individuals’ ethical concepts is a scientific enterprise that is quite similar to establishing models of individuals’ mathematical concepts. To accomplish these two primary goals, I draw from my experience of working scientifically (...) with von Glasersfeld for 25 years while he was developing radical constructivism as a coherent model of knowing, and appeal to several of his basic insights to establish constructing models of ethical concepts as a scientific enterprise. (shrink)

Proof requires a dialogue between agents to clarify obscure inference steps, fill gaps, or reveal implicit assumptions in a purported proof. Hence, argumentation is an integral component of the discovery process for mathematical proofs. This work presents how argumentation theories can be applied to describe specific informal features in the development of proof-events. The concept of proof-event was coined by Goguen who described mathematical proof as a public social event that takes place in space and time. This new (...) meta-methodological concept is designed to cover not only “traditional” formal proofs but all kinds of proofs and inference steps, including incomplete or purported proofs. Our approach attempts to make proof-events more comprehensive to express the complete trajectory of a mathematical proof-event until the ultimate validation of the proving outcome. Thus, we advance an extended version of proof-event calculus which is built on argumentation theories designed to capture the internal and external structure of collaborative mathematical practice and highlight the relationship between proof, human reasoning, and cognitive processes. In addition, another area in which argumentation can make a significant contribution is dealing with the defeasible knowledge of the Web which is a product of its open and ubiquitous nature. This approach seems to be sufficient for the presentation of Web-based proving processes as manifested in the case of the Mini-Polymath 4 project. (shrink)

A formal model of metaphor is introduced. It models metaphor, first, as an interaction of “frames” according to the frame semantics, and then, as a wave function in Hilbert space. The practical way for a probability distribution and a corresponding wave function to be assigned to a given metaphor in a given language is considered. A series of formal definitions is deduced from this for: “representation”, “reality”, “language”, “ontology”, etc. All are based on Hilbert space. A few statements about (...) a quantum computer are implied: The sodefined reality is inherent and internal to it. It can report a result only “metaphorically”. It will demolish transmitting the result “literally”, i.e. absolutely exactly. A new and different formal definition of metaphor is introduced as a few entangled wave functions corresponding to different “signs” in different language formally defined as above. The change of frames as the change from the one to the other formal definition of metaphor is interpreted as a formal definition of thought. Four areas of cognition are unified as different but isomorphic interpretations of the mathematical model based on Hilbert space. These are: quantum mechanics, frame semantics, formal semantics by means of quantum computer, and the theory of metaphor in linguistics. (shrink)

A given but otherwise random environmental time series impinging on the input of a certain biological processor passes through with overwhelming probability practically undetected. A very small percentage of environmental stimuli, though, is ‘captured’ by the processor's nonlinear dissipative operator as initial conditions, and is ‘processed’ as solutions of its dynamics. The processor, then, is in such cases instrumental in compressing or abstracting those stimuli, thereby making the external world to collapse from a previous regime of a ‘pure state’ of (...) suspended animation into a set of stable complementary and mutually exclusive eigenfunctions or ‘categories'. The characteristics of this cognitive set depend on the operator involved and the hierarchical level where the abstraction takes place. Depending on the context, the transition from one state to another occurs in such a cognitive operator. The chaotic itinerancy may play a crucial role for this process. In this paper we model the dynamics which may underlie such a cognitive process and the role of the thalamo-cortical pacemaker of the brain. In order to model them, conceptualization by the notion of ‘attractor ruin’ in high-dimensional dynamical systems is necessary. (shrink)

We define a mathematical formalism based on the concept of an ‘‘open dynamical system” and show how it can be used to model embodied cognition. This formalism extends classical dynamical systems theory by distinguishing a ‘‘total system’’ (which models an agent in an environment) and an ‘‘agent system’’ (which models an agent by itself), and it includes tools for analyzing the collections of overlapping paths that occur in an embedded agent's state space. To illustrate the way (...) this formalism can be applied, several neural network models are embedded in a simple model environment. Such phenomena as masking, perceptual ambiguity, and priming are then observed. We also use this formalism to reinterpret examples from the embodiment literature, arguing that it provides for a more thorough analysis of the relevant phenomena. (shrink)

This paper considers two models of embodied mathematical cognition, and analyses the image of mathematics that they support.

Following Marr’s famous three-level distinction between explanations in cognitive science, it is often accepted that focus on modeling cognitive tasks should be on the computational level rather than the algorithmic level. When it comes to mathematical problem solving, this approach suggests that the complexity of the task of solving a problem can be characterized by the computational complexity of that problem. In this paper, I argue that human cognizers use heuristic and didactic tools and thus engage in cognitive processes (...) that make their problem solving algorithms computationally suboptimal, in contrast with the optimal algorithms studied in the computational approach. Therefore, in order to accurately model the human cognitive tasks involved in mathematical problem solving, we need to expand our methodology to also include aspects relevant to the algorithmic level. This allows us to study algorithms that are cognitively optimal for human problem solvers. Since problem solving methods are not universal, I propose that they should be studied in the framework of enculturation, which can explain the expected cultural variance in the humanly optimal algorithms. While mathematical problem solving is used as the case study, the considerations in this paper concern modeling of cognitive tasks in general. (shrink)

This article argues that many, if not most, behavior descriptions and sequencing are in essence an interpretation of signs, and are evaluated as sequences of signs by researchers. Thus, narrative analysis, as developed by Barthes and others, seems best suited to be used in behavioral/biosemiotic studies rather than mathematical modeling, and is very similar to some classic ethology methods. As our brain interprets behaviors as signs and attributes meaning to them, narrative analysis seems more suitable than mathematical modeling (...) to describe and study behavior. Mathematical models are, on many occasions, extremely reductionist and simplifying because of computational and/or numerical representation limitations that lead to errors and straitjacketing interpretations of reality. Since actual animals are not as optimal in real life as our mathematical models, here it is proposed that we should consider logical/verbal models and semantic interpretations as equally or even better suited for behavioral analysis, and refrain from enforcing mathematical modeling as the only way to study and understand biological problems, especially those of a behavioral and biosemiotic nature. (shrink)