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)

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)

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)

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)

The discrete–structural structure of the world is described. In comparison with the idea of Heraclitus about an indissoluble world, preference is given to the discrete world of Democritus. It is noted that if the discrete atoms of Democritus were simple and indivisible, the atoms of the modern world indicated in the article would possess, rather, a structural structure. The article proves the problem of how the mutual connection of mathematics and philosophy influences cognition, which creates a discrete–structural worldview. The (...) author notes that the appearance of writing, symbolic language and the depiction of the picture of the world through mathematics, led us into the sphere of discrete mathematical mathematics. (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)

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)

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)

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)

Marr’s seminal distinction between computational, algorithmic, and implementational levels of analysis has inspired research in cognitive science for more than 30 years. According to a widely-used paradigm, the modelling of cognitive processes should mainly operate on the computational level and be targeted at the idealised competence, rather than the actual performance of cognisers in a specific domain. In this paper, we explore how this paradigm can be adopted and revised to understand mathematical problem solving. The computational-level approach applies methods (...) from computational complexity theory and focuses on optimal strategies for completing cognitive tasks. However, human cognitive capacities in mathematical problem solving are essentially characterised by processes that are computationally sub-optimal, because they initially add to the computational complexity of the solutions. Yet, these solutions can be optimal for human cognisers given the acquisition and enactment of mathematical practices. Here we present diagrams and the spatial manipulation of symbols as two examples of problem solving strategies that can be computationally sub-optimal but humanly optimal. These aspects need to be taken into account when analysing competence in mathematical problem solving. Empirically informed considerations on enculturation can help identify, explore, and model the cognitive processes involved in problem solving tasks. The enculturation account of mathematical problem solving strongly suggests that computational-level analyses need to be complemented by considerations on the algorithmic and implementational levels. The emerging research strategy can help develop algorithms that model what we call enculturated cognitive optimality in an empirically plausible and ecologically valid way. (shrink)

Analytic semantics got its start when Frege pointed out differences in cognitive content between sentences that in some good sense “say the same.” Frege put cognitive content (in the form of sense) at the heart of semantic content. Most prefer nowadays to see cognitive contents as generated by semantic contents in context; a sentence's cognitive significance is an aspect rather of the information imparted by its use. I argue for a particular version of this idea. Semantic contents generate cognitive contents (...) by operating levers in a larger piece of machinery, involving also the presupposition P one is leaning on and the subject matter M under discussion. A sentence's cognitive content is what its literal-content-restricted-to-P-worlds says about M. It's by triggering different presuppositions and/or different subject matters that coreferential names make their distinctive mark on a sentence's felt information value. (shrink)

In the present article we attempt to show that Aristotle's syllogistic is an underlying logiC which includes a natural deductive system and that it isn't an axiomatic theory as had previously been thought. We construct a mathematical model which reflects certain structural aspects of Aristotle's logic. We examine the relation of the model to the system of logic envisaged in scattered parts of Prior and Posterior Analytics. Our interpretation restores Aristotle's reputation as a logician of consummate imagination and skill. (...) Several attributions of shortcomings and logical errors to Aristotle are shown to be without merit. Aristotle's logic is found to be self-sufficient in several senses: his theory of deduction is logically sound in every detail. (His indirect deductions have been criticized, but incorrectly on our account.) Aristotle's logic presupposes no other logical concepts, not even those of propositional logic. The Aristotelian system is seen to be complete in the sense that every valid argument expressible in his system admits of a deduction within his deductive system: every semantically valid argument is deducible. (shrink)

Mathematics is obviously important in the sciences. And so it is likely to be equally important in any effort that aims to understand God in a scientifically significant way or that aims to clarify the relations between science and theology. The degree to which God has any perfection is absolutely infinite. We use contemporary mathematics to precisely define that absolute infinity. For any perfection, we use transfinite recursion to define an endlessly ascending series of degrees of that perfection. That series (...) rises to an absolutely infinite degree of that perfection. God has that absolutely infinite degree. We focus on the perfections of knowledge, power, and benevolence. Our model of divine infinity thus builds a bridge between mathematics and theology. (shrink)

A reasoned argument or tarka is essential for a wholesome vāda that aims at establishing the truth. A strong tarka constitutes of a number of elements including an anumāna based on a valid hetu. Several scholars, such as Dharmakīrti, Vasubandhu and Dignāga, have worked on theories for the establishment of a valid hetu to distinguish it from an invalid one. This paper aims to interpret Dignāga’s hetu-cakra, called the wheel of grounds, from a modern philosophical perspective by deconstructing it into (...) a simple probabilistic mathematical model. The objective is to understand how and why a vāda based on a probabilistically weaker hetu can degrade into a Jalpa or vitaṇḍā. To do so, the paper maps the concept of ‘Bounded Rationality’ onto the hetu-cakra. Bounded Rationality, an idea coined by the management thinker Herbert Simon, is often employed in understanding decision-making processes of rational agents. In the context of this paper, the concept would state that the prativādin and ālocaka (debater) may not hold unbounded information to back their pratijñā (proposition). The paper argues that within the probabilistically deconstructed hetu-cakra model, most people argue in the ‘Zone of Bounded Rationality’, and thus, the probability of a debate degrading into Jalpa or vitaṇḍā is high. -/- . (shrink)

Cognition involves physical stimulation, neural coding, mental conception, and conscious perception. Beyond the neural coding of physical stimuli, it is not clear how exactly these component processes constitute cognition. Within mathematical sciences, category theory provides tools such as category, functor, and adjointness, which are indispensable in the explication of the mathematical calculations involved in acquiring mathematical knowledge. More speci cally, functorial semantics, in showing that theories and models can be construed as categories and functors, (...) respectively, and in establishing the adjointness between abstraction (of theories) and interpretation (to obtain models), mathematically accounts for knowing-within-mathematics. Here we show that mathematical knowing recapitulates--in an elementary form--ordinary cognition. The process of going from particulars (physical stimuli) to their concrete models (conscious percepts) via abstract theories (mental concepts) and measured properties (neural coding) is common to both mathematical knowing and ordinary cognition. Our investigation of the similarity between knowing-within-mathematics and knowing-in-general leads us to make a case for the development of the basic science of cognition in terms of the functorial semantics of mathematical knowing. (shrink)

A practical viewpoint links reality, representation, and language to calculation by the concept of Turing (1936) machine being the mathematical model of our computers. After the Gödel incompleteness theorems (1931) or the insolvability of the so-called halting problem (Turing 1936; Church 1936) as to a classical machine of Turing, one of the simplest hypotheses is completeness to be suggested for two ones. That is consistent with the provability of completeness by means of two independent Peano arithmetics discussed in Section (...) I. Many modifications of Turing machines cum quantum ones are researched in Section II for the Halting problem and completeness, and the model of two independent Turing machines seems to generalize them. Then, that pair can be postulated as the formal definition of reality therefore being complete unlike any of them standalone, remaining incomplete without its complementary counterpart. Representation is formal defined as a one-to-one mapping between the two Turing machines, and the set of all those mappings can be considered as “language” therefore including metaphors as mappings different than representation. Section III investigates that formal relation of “reality”, “representation”, and “language” modeled by (at least two) Turing machines. The independence of (two) Turing machines is interpreted by means of game theory and especially of the Nash equilibrium in Section IV. Choice and information as the quantity of choices are involved. That approach seems to be equivalent to that based on set theory and the concept of actual infinity in mathematics and allowing of practical implementations. (shrink)

Scientists use models to know the world. It i susually assumed that mathematicians doing pure mathematics do not. Mathematicians doing pure mathematics prove theorems about mathematical entities like sets, numbers, geometric figures, spaces, etc., they compute various functions and solve equations. In this paper, I want to exhibit models build by mathematicians to study the fundamental components of spaces and, more generally, of mathematical forms. I focus on one area of mathematics where models occupy a (...) central role, namely homotopy theory. I argue that mathematicians introduce genuine models and I offer a rough classification of these models. (shrink)

In computational complexity theory, decision problems are divided into complexity classes based on the amount of computational resources it takes for algorithms to solve them. In theoretical computer science, it is commonly accepted that only functions for solving problems in the complexity class P, solvable by a deterministic Turing machine in polynomial time, are considered to be tractable. In cognitive science and philosophy, this tractability result has been used to argue that only functions in P can feasibly work as computational (...) models of human cognitive capacities. One interesting area of computational complexity theory is descriptive complexity, which connects the expressive strength of systems of logic with the computational complexity classes. In descriptive complexity theory, it is established that only first-order systems are connected to P, or one of its subclasses. Consequently, second-order systems of logic are considered to be computationally intractable, and may therefore seem to be unfit to model human cognitive capacities. This would be problematic when we think of the role of logic as the foundations of mathematics. In order to express many important mathematical concepts and systematically prove theorems involving them, we need to have a system of logic stronger than classical first-order logic. But if such a system is considered to be intractable, it means that the logical foundation of mathematics can be prohibitively complex for human cognition. In this paper I will argue, however, that this problem is the result of an unjustified direct use of computational complexity classes in cognitive modelling. Placing my account in the recent literature on the topic, I argue that the problem can be solved by considering computational complexity for humanly relevant problem solving algorithms and input sizes. (shrink)

The mental rotation ability is an essential spatial reasoning skill in human cognition and has proven to be an essential predictor of mathematical and STEM skills, critical and computational thinking. Despite its importance, little is known about when and how mental rotation processes are activated in games explicitly targeting spatial reasoning tasks. In particular, the relationship between spatial abilities and TetrisTM has been analysed several times in the literature. However, these analyses have shown contrasting results between the effectiveness (...) of Tetris-based training activities to improve mental rotation skills. In this work, we studied whether, and under what conditions, such ability is used in the TetrisTM game by explicitly modelling mental rotation via an ACT-R based cognitive model controlling a virtual agent. The obtained results show meaningful insights into the activation of mental rotation during game dynamics. The study suggests the necessity to adapt game dynamics in order to force the activation of this process and, therefore, can be of inspiration to design learning activities based on TetrisTM or re-design the game itself to improve its educational effectiveness. (shrink)

Our manuscript addresses the foundational question of cognitive science: how do we know? Specifically, examination of the mathematics of acquiring mathematical knowledge revealed that knowing-within-mathematics is reflective of knowing-in-general. Based on the correspondence between ordinary cognition (involving physical stimuli, neural sensations, mental concepts, and conscious percepts) and mathematical knowing (involving objective particulars, measured properties, abstract theories, and concrete models), we put forward the functorial semantics of mathematical knowing as a formalization of cognition. Our investigation (...) of the similarity between mathematics and cognition led us to argue against the compartmentalization of scientific knowledge and ordinary cognition, and make a case for the understanding of the basic science of knowing in terms of functorial semantics. Recognizing functorial semantics as an elementary form of cognition can help advance cognitive science the way calculus helped in the advancement of physics. (shrink)

We propose a way to explain the diversification of branches of mathematics, distinguishing the different approaches by which mathematical objects can be studied. In our philosophy of mathematics, there is a base object, which is the abstract multiplicity that comes from our empirical experience. However, due to our human condition, the analysis of such multiplicity is covered by other empirical cognitive attitudes (approaches), diversifying the ways in which it can be conceived, and consequently giving rise to different mathematical (...) disciplines. This diversity of approaches is founded on the manifold categories that we find in physical reality. We also propose, grounded on this idea, the use of Aristotelian categories as a first model for this division, generating from it a classification of mathematical branches. Finally we make a history review to show that there is consistency between our classification, and the historical appearance of the different branches of mathematics. (shrink)

The essay is critically examines the conceptual problems with the influential modularity model of mind. We shall see that one of the essential characters of modules, namely informational encapsulation, is not only inessential, it ties a knot at a crucial place blocking the solution to the problem of understanding the formation of concepts from percepts (nodes of procedural knowledge). Subsequently I propose that concept formation takes place by modulation of modules leading to cross-representations, which were otherwise prevented by encapsulation. It (...) must be noted that the argument is not against modular architecture, but a variety of an architecture that prevents interaction among modules. This is followed by a brief argument demonstrating that module without modularization, i.e. without developmental history, is impossible. Finally the emerging picture of cognitive development is drawn in the form of the layers in the fabric of mind, with a brief statement of the possible implications. (shrink)

Artificial models of cognition serve different purposes, and their use determines the way they should be evaluated. There are also models that do not represent any particular biological agents, and there is controversy as to how they should be assessed. At the same time, modelers do evaluate such models as better or worse. There is also a widespread tendency to call for publicly available standards of replicability and benchmarking for such models. In this paper, I (...) argue that proper evaluation ofmodels does not depend on whether they target real biological agents or not; instead, the standards of evaluation depend on the use of models rather than on the reality of their targets. I discuss how models are validated depending on their use and argue that all-encompassing benchmarks for models may be well beyond reach. (shrink)

Models are the ultimate results of all (scientific, non-scientific, and anti-scientific) kinds of cognition. Therefore, philosophy of cognition should start with the following fundamental distinction: there are models, and there are means of model-building. Laws of nature and theories are useful only as a means of model-building. If it's true that models are the ultimate results of cognition, then shouldn't we try reordering the field, starting with the notion of model? In this way, couldn't (...) we obtain a unified and more productive picture - a model-based model of cognition? (shrink)

The 'Science of Knowing: Mathematics' textbook is the first book to put forward and substantiate the thesis that the mathematical understanding of mathematics, as exemplified in F. William Lawvere's Functorial Semantics, constitutes the science of knowing i.e. cognitive science. -/- This is a textbook, i.e. teaching material.

According to [Bayesian] models” in cognitive neuroscience, says a recent textbook, “the human mind behaves like a capable data scientist”. Do they? That is to say, do such model show we are rational? I argue that Bayesian models of cognition, perhaps surprisingly, do not and indeed cannot, show that we are Bayesian-rational. The key reason is that such models appeal to approximations, a fact that carries significant implications. After outlining the argument, I critique two responses, seen (...) in recent cognitive neuroscience. One says that the mind can be seen as approximately Bayes-rational, while the other reconceives norms of rationality. (shrink)

Bayesian cognitive science is a research programme that relies on modelling resources from Bayesian statistics for studying and understanding mind, brain, and behaviour. Conceiving of mental capacities as computing solutions to inductive problems, Bayesian cognitive scientists develop probabilistic models of mental capacities and evaluate their adequacy based on behavioural and neural data generated by humans (or other cognitive agents) performing a pertinent task. The overarching goal is to identify the mathematical principles, algorithmic procedures, and causal mechanisms that enable (...) cognitive agents to take uncertainty into account and weigh it appropriately in producing adaptive behaviour. -/- The appeal of Bayesian cognitive science derives from its transparency, normativity, and unifying power. Resources from Bayesian statistics allow cognitive scientists to characterise transparently, in terms of random variables and probabilistic dependencies between them, the inductive problems that many mental capacities are presumed to solve. Given this characterisation, researchers can define an upper bound on one’s performance on that problem and use it as a benchmark for interpreting experimental results, formulating hypotheses about why and when deviations from optimal performance should be expected, and seeking explanations of mental capacities within one encompassing theoretical framework. -/- Although critics question the testability, explanatory power, and plausibility of Bayesian models of mental capacities, Bayesian cognitive science has proved itself to be an incredibly fruitful research programme. Fuelled by empirical and theoretical results from psychology, neuroscience, philosophy, computer science, artificial intelligence, and evolution, Bayesian cognitive science has: advanced our understanding of why cognitive systems should keep track of uncertainty and use it to facilitate adaptive behaviour; clarified how the brain might encode information about the uncertainty of its stimuli and perform computations with probability distributions; and crystallised the insight that cognitive agents’ management of uncertainty might be grounded in predictive processes aimed at avoiding surprising exchanges with the environment. (shrink)

Recent advances in neuroimaging technologies have rendered multimodal analysis of operators’ cognitive processes in complex task settings and environments increasingly more practical. In this exploratory study, we utilized optical brain imaging and mobile eye tracking technologies to investigate the behavioral and neurophysiological differences among expert and novice operators while they operated a human-machine interface in normal and adverse conditions. In congruence with related work, we observed that experts tended to have lower prefrontal oxygenation and exhibit gaze patterns that are better (...) aligned with the optimal task sequence with shorter fixation durations as compared to novices. These trends reached statistical significance only in the adverse condition where the operators were prompted with an unexpected error message. Comparisons between hemodynamic and gaze measures before and after the error message indicated that experts’ neurophysiological response to the error involved a systematic increase in bilateral dorsolateral prefrontal cortex (dlPFC) activity accompanied with an increase in fixation durations, which suggests a shift in their attentional state, possibly from routine process execution to problem detection and resolution. The novices’ response was not as strong as that of experts, including a slight increase only in the left dlPFC with a decreasing trend in fixation durations, which is indicative of visual search behavior for possible cues to make sense of the unanticipated situation. A linear discriminant analysis model capitalizing on the covariance structure among hemodynamic and eye movement measures could distinguish experts from novices with 91% accuracy. Despite the small sample size, the performance of the linear discriminant analysis combining eye fixation and dorsolateral oxygenation measures before and after an unexpected event suggests that multimodal approaches may be fruitful for distinguishing novice and expert performance in similar neuroergonomic applications in the field. (shrink)

Recent advances in neuroimaging technologies have rendered multimodal analysis of operators’ cognitive processes in complex task settings and environments increasingly more practical. In this exploratory study, we utilized optical brain imaging and mobile eye tracking technologies to investigate the behavioral and neurophysiological differences among expert and novice operators while they operated a human-machine interface in normal and adverse conditions. In congruence with related work, we observed that experts tended to have lower prefrontal oxygenation and exhibit gaze patterns that are better (...) aligned with the optimal task sequence with shorter fixation durations as compared to novices. These trends reached statistical significance only in the adverse condition where the operators were prompted with an unexpected error message. Comparisons between hemodynamic and gaze measures before and after the error message indicated that experts’ neurophysiological response to the error involved a systematic increase in bilateral dorsolateral prefrontal cortex (dlPFC) activity accompanied with an increase in fixation durations, which suggests a shift in their attentional state, possibly from routine process execution to problem detection and resolution. The novices’ response was not as strong as that of experts, including a slight increase only in the left dlPFC with a decreasing trend in fixation durations, which is indicative of visual search behavior for possible cues to make sense of the unanticipated situation. A linear discriminant analysis model capitalizing on the covariance structure among hemodynamic and eye movement measures could distinguish experts from novices with 91% accuracy. Despite the small sample size, the performance of the linear discriminant analysis combining eye fixation and dorsolateral oxygenation measures before and after an unexpected event suggests that multimodal approaches may be fruitful for distinguishing novice and expert performance in similar neuroergonomic applications in the field. (shrink)

Evolutionary anthropologists and archaeologists have been considerably successful in modelling the cumulative evolution of culture, of technological skills and knowledge in particular. Recently, one of these models has been introduced in the philosophy of science by De Cruz and De Smedt (Philos Stud 157:411–429, 2012), in an attempt to demonstrate that scientists may collectively come to hold more truth-approximating beliefs, despite the cognitive biases which they individually are known to be subject to. Here we identify a major shortcoming in (...) that attempt: De Cruz & De Smedt’s mathematical model makes one particularly strong tractability assumption that causes the model to largely miss its target (namely, truth accumulation in science), and that moreover conflicts with empirical observations. The second, more constructive part of the paper presents an alternative, agent-based model, which allows one to much better examine the conditions for scientific progress and decline. (shrink)

Many biological processes and objects can be described by fractals. The paper uses a new type of objects – blinking fractals – that are not covered by traditional theories considering dynamics of self-similarity processes. It is shown that both traditional and blinking fractals can be successfully studied by a recent approach allowing one to work numerically with infinite and infinitesimal numbers. It is shown that blinking fractals can be applied for modeling complex processes of growth of biological systems including their (...) season changes. The new approach allows one to give various quantitative characteristics of the obtained blinking fractals models of biological systems. (shrink)

Investigation into the sequence structure of the genetic code by means of an informatic approach is a real success story. The features of human language are also the object of investigation within the realm of formal language theories. They focus on the common rules of a universal grammar that lies behind all languages and determine generation of syntactic structures. This universal grammar is a depiction of material reality, i.e., the hidden logical order of things and its relations determined by natural (...) laws. Therefore mathematics is viewed not only as an appropriate tool to investigate human language and genetic code structures through computer sciencebased formal language theory but is itself a depiction of material reality. This confusion between language as a scientific tool to describe observations/experiences within cognitive constructed models and formal language as a direct depiction of material reality occurs not only in current approaches but was the central focus of the philosophy of science debate in the twentieth century, with rather unexpected results. This article recalls these results and their implications for more recent mathematical approaches that also attempt to explain the evolution of human language. (shrink)

This article is an experiment. Consider a minimalist model of cognition (models, means of model-building and history of their evolution). In this model, explanation could be defined as a means allowing to advance: production of models and means of model-building (thus, yielding 1st class understanding), exploration and use of them (2nd class), and/or teaching (3rd class). At minimum, 3rd class understanding is necessary for an explanation to be respected.

3 Abstract This paper is about modeling morality, with a proposal as to the best 4 way to do it. There is the small problem, however, in continuing disagreements 5 over what morality actually is, and so what is worth modeling. This paper resolves 6 this problem around an understanding of the purpose of a moral model, and from 7 this purpose approaches the best way to model morality.

Cognitive Set Theory is a mathematical model of cognition which equates sets with concepts, and uses mereological elements. It has a holistic emphasis, as opposed to a reductionistic emphasis, and it therefore begins with a single universe (as opposed to an infinite collection of infinitesimal points).

The study investigated the effect of cognitive restructuring on junior secondary school mathematics test anxiety in Oshimili south L.G.A of Delta State. Two research questions and two hypotheses tested at 0.05 level of significance guided the study. Quasi-experimental research design was adopted for this study. The population for this study was a total of 1224 students. These comprised of all the JSS 2 students from Oshimili South Local Government Area of Delta State. Research sample consisted of 120 JSS 2 students (...) with high level of mathematics test anxiety selected through purposive sampling technique. The instrument adopted for this study was Maths anxiety rating scale-R. Data collected from the study were analyzed using mean and ANCOVA. Results obtained from the study indicated that Cognitive restructuring therapy (CRT) was effective on mathematics test anxiety of junior secondary school students. The results equally showed that Cognitive restructuring therapy was more effective on the female students’ mathematics anxiety than their male counterparts. Furthermore, the results indicated a significant difference in the post-test mathematics test anxiety mean scores of students treated with CRT and those in the control group. Also, there was significant difference in the post-test mathematics test anxiety mean scores of male and female students treated with CRT. Based on the findings of this study, some recommendations were noted. The researcher recommended, among others that Schools’ Guidance Counselors should be empowered to make good use of the cognitive restructuring technique, in counseling students who find it difficult to comprehend mathematics skills. (shrink)

Although the cognition is significant in strategic reasoning, its role has been weakly analyzed, because only the average intelligence is usually considered. For example, prisoner's dilemma in game theory, would have different outcomes for persons with different intelligence. I show how various levels of intelligence influence the quality of reasoning, decision, or the probability of psychosis. I explain my original methodology developed for my MA thesis in clinical psychology in 1998, and grant research in 1999, demonstrating the bias of (...) the classic IQ method, and how the intelligence limits thinking. Based on that I defined Personality Model, providing insight into understanding of psychosis (schizophrenia, bi-polar), which has not been explained yet by psychology or psychiatry. In addition, it enables to analyze and assess non-linear problems, utilizable in computer programming, visualization (animation) or other fields including Baduk game. I've already applied some principles in complex information system www.each.co.uk, and video-animations exhibited in London, Germany, Tokyo. I need to mention my experience in chess composition between 1994 and 2000, winning a few international prizes and inventing a special class of fairy rules redefining the mate. The chess composition principles or patterns show the way to organize logical series to higher advanced mechanisms (like calculus), applicable to other fields. One of such principles is a logical aesthetic innovation: new strategy, defined by Italian composers. Finally I show how the simple redefinition of the classic utility concept links economics and psychology to explain irrational / destructive behavior. All presented results (from the research) can be repeated. (shrink)

Current theories of the division of cognitive labor are confined to the “context of justification,” assuming exogenous theories. But new theories are made from the same labor that is used for developing existing theories, and if none of this labor is ever allocated to create new alternatives, then scientific progress is impossible. A unified model is proposed in which theories are no longer given but a function of the division of labor in the model itself. The interactions of individuals balancing (...) the exploitation of existing theories and the exploration of new theories results in a robust cyclical pattern. (shrink)

We argue the thesis that if (1) a physical process is mathematically representable by a Cauchy sequence; and (2) we accept that there can be no infinite processes, i.e., nothing corresponding to infinite sequences, in natural phenomena; then (a) in the absence of an extraneous, evidence-based, proof of `closure' which determines the behaviour of the physical process in the limit as corresponding to a `Cauchy' limit; (b) the physical process must tend to a discontinuity (singularity) which has not been reflected (...) in the Cauchy sequence that seeks to describe the behaviour of the physical process. We support our thesis by mathematical models of the putative behaviours of (i) a virus cluster; (ii) an elastic string; and (iii) a Universe that recycles from Big Bang to Ultimate Implosion, in which parity and local time reversal violation, and the existence of `dark energy' in a multiverse, need not violate Einstein's equations and quantum theory. We suggest that the barriers to modelling such processes in a mathematical language that seeks unambiguous communication are illusory; they merely reflect an attempt to ask of the language chosen for such representation more than it is designed to deliver. (shrink)

We present a model of the distribution of labour in science. Such models tend to rely on the mechanism of the invisible hand . Our analysis starts from the necessity of standards in distributed processes and the possibility of multiple standards in science. Invisible hand models turn out to have only limited scope because they are restricted to describing the atypical single-standard case. Our model is a generalisation of these models to J standards; single-standard models such (...) as Kitcher are a limiting case. We introduce and formalise this model, demonstrate its dynamics and conclude that the conclusions commonly derived from invisible hand models about the distribution of labour in science are not robust against changes in the number of standards. (shrink)

De Neys (2021) argues that the debate between single- and dual-process theorists of thought has become both empirically intractable and scientifically inconsequential. I argue that this is true only under the traditional framing of the debate—when single- and dual-process theories are understood as claims about whether thought processes share the same defining properties (e.g., making mathematical judgments) or have two different defining properties (e.g., making mathematical judgments autonomously versus via access to a central working memory capacity), respectively. But (...) if single- and dual-process theories are understood in cognitive modeling terms as claims about whether thought processes function to implement one or two broad types of algorithms, respectively, then the debate becomes scientifically consequential and, presumably, empirically tractable. So, I argue, the correct response to the current state of the debate is not to abandon it, as De Neys suggests, but to reframe it as a debate about cognitive models. (shrink)

Active materials are self-propelled non-living entities which, in some circumstances, exhibit a number of cognitively interesting behaviors such as gradient-following, avoiding obstacles, signaling and group coordination. This has led to scientific and philosophical discussion of whether this may make them useful as minimal models of cognition (Hanczyc, 2014; McGivern, 2019). Batterman and Rice (2014) have argued that what makes a minimal model explanatory is that the model is ultimately in the same universality class as the target system, which (...) underpins why it exhibits the same macrobehavior. We appeal to recent research in basal cognition (Lyon et al., 2021) to establish appropriate target systems and essential features of cognition as a target of modeling. Looking at self-propelled oil droplets, a type of active material, we do not find that organization alone indicates that these systems exhibit the essential features of cognition. We then examine the specific behaviors of oil droplets but also fail to find that these demonstrate the essential features of cognition. Without a universality class, Batterman & Rice’s account of the explanatory power of minimal models simply does not apply to cognition. However, we also want to stress that it is not intended to; cognition is not the same type of behavioral phenomena as those found in physics. We then look to the minimal cognition methodology of Beer (1996, 2020a, b) to show how active materials can be explanatorily valuable regardless of their cognitive status because they engage in specific behaviors that have traditionally been expected to involve internal representational dynamics, revealing misconceptions about the cognitive underpinnings of certain, specific behaviors in target systems where such behaviors are cognitive. Further, Beer’s models can also be genuinely explanatory by providing dynamical explanations. (shrink)

Autonomist accounts of cognitive science suggest that cognitive model building and theory construction (can or should) proceed independently of findings in neuroscience. Common functionalist justifications of autonomy rely on there being relatively few constraints between neural structure and cognitive function (e.g., Weiskopf, 2011). In contrast, an integrative mechanistic perspective stresses the mutual constraining of structure and function (e.g., Piccinini & Craver, 2011; Povich, 2015). In this paper, I show how model-based cognitive neuroscience (MBCN) epitomizes the integrative mechanistic perspective and concentrates (...) the most revolutionary elements of the cognitive neuroscience revolution (Boone & Piccinini, 2016). I also show how the prominent subset account of functional realization supports the integrative mechanistic perspective I take on MBCN and use it to clarify the intralevel and interlevel components of integration. (shrink)

A formal model of the processes of digestion in a hypothetical cell is developed and discussed as a case study of how the threefold logic of Peircean semiotics works within Rosen’s paradigm of relational ontology. The formal model is used to demonstrate several fundamental differences between a relational description of biological processes and a mechanistic description. The formal model produces a logic of embodied generalization that is mediated and determined by the cell through its interactions with the environment. Specifically, the (...) synchronization of the functions of pattern recognition and semantic attribution results in an open and adaptive learning system that is stabilized by a hermeneutical circle. The relational principles of biosemiotics demonstrated through this case study are applicable to other biological systems, as well as to the relational ontology of systems theory, artificial intelligence systems, and relativistic quantum theory. (shrink)

In the brain the relations between free neurons and the conditioned ones establish the constraints for the informational neural processes. These constraints reflect the systemenvironment state, i.e. the dynamics of homeocognitive activities. The constraints allow us to define the cost function in the phase space of free neurons so as to trace the trajectories of the possible configurations at minimal cost while respecting the constraints imposed. Since the space of the free states is a manifold or a non orthogonal space, (...) the minimum distance is not a straight line but a geodesic. The minimum condition is expressed by a set of ordinary differential equation ( ODE ) that in general are not linear. In the brain there is not an algorithm or a physical field that regulates the computation, then we must consider an emergent process coming out of the neural collective behavior triggered by synaptic variability. We define the neural computation as the study of the classes of trajectories on a manifold geometry defined under suitable constraints. The cost function supervises pseudo equilibrium thermodynamics effects that manage the computational activities from beginning to end and realizes an optimal control through constraints and geodetics. The task of this work is to establish a connection between the geometry of neural computation and cost functions. To illustrate the essential mathematical aspects we will use as toy model a Network Resistor with Adaptive Memory (Memristors).The information geometry here defined is an analog computation, therefore it does not suffer the limits of the Turing computation and it seems to respond to the demand for a greater biological plausibility. The model of brain optimal control proposed here can be a good foundation for implementing the concept of "intentionality",according to the suggestion of W. Freeman. Indeed, the geodesic in the brain states can produce suitable behavior to realize wanted functions and invariants as neural expressionsof cognitive intentions. (shrink)

Mathematical models provide explanations of limited power of specific aspects of phenomena. One way of articulating their limits here, without denying their essential powers, is in terms of contrastive explanation.

Much problem solving and learning research in math and science has focused on formal representations. Recently researchers have documented the use of unschooled strategies for solving daily problems -- informal strategies which can be as effective, and sometimes as sophisticated, as school-taught formalisms. Our research focuses on how formal and informal strategies interact in the process of doing and learning mathematics. We found that combining informal and formal strategies is more effective than single strategies. We provide a theoretical account of (...) this multiple strategy effect and have begun to formulate this theory in an ACT-R computer model. We show why students may reach common impasses in the use of written algebra, and how subsequent or concurrent use of informal strategies leads to better problem-solving performance. Formal strategies facilitate computation because of their abstract and syntactic nature; however, abstraction can lead to nonsensical interpretations and conceptual errors. Reapplying the formal strategy will not repair such errors; switching to an informal one may. We explain the multiple strategy effect as a complementary relationship between the computational efficiency of formal strategies and the sense-making function of informal strategies. (shrink)

On the question of whether gambling behavior can be changed as result of teaching gamblers the mathematics of gambling, past studies have yielded contradictory results, and a clear conclusion has not yet been drawn. In this paper, I bring some criticisms to the empirical studies that tended to answer no to this hypothesis, regarding the sampling and laboratory testing, and I argue that an optimal mathematical scholastic intervention with the objective of preventing problem gambling is possible, by providing the (...) principles that would optimize the structure and content of the teaching module. Given the ethical aspects of the exposure of mathematical facts behind games of chance, and starting from the slots case – where the parametric design is missing, we have to draw a line between ethical and optional information with respect to the mathematical content provided by a scholastic intervention. Arguing for the role of mathematics in problem-gambling prevention and treatment, interdisciplinary research directions are drawn toward implementing an optimal mathematical module in cognitive therapies. (shrink)

What is the class of possible semiotic systems? What kinds of systems could count as such systems? The human mind is naturally considered the prototypical semiotic system. During years of research in semiotics the class has been broadened to include i.e. living systems like animals, or even plants. It is suggested in the literature on artificial intelligence that artificial agents are typical examples of symbol-processing entities. It also seems that semiotic processes are in fact cognitive processes. In consequence, it is (...) natural to ask the question about the relation between semiotic studies and research on artificial cognitive systems within cognitive science. Consequently, my main question concerns the problem of inclusion or exclusion from the semiotic spectrum at least some artificial systems. I would like to consider some arguments against the possibility of artificial semiotic systems and I will try to repeal them. Then I will present an existing natural-language using agent of the SNePS system and interpret it in terms of Peircean theory of signs. I would like also to show that some properties of semiotic systems in Peircean sense could be also found in a discussed artificial system. Finally, I will have some remarks on the status of semiotics in general. (shrink)