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)

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)

Advances in cognitive science are relevant to the debate between moral pluralism and absolutism. Parametric structure, which plausibly underlies syntax, gives some idea of how pluralism might be true. The cognitive mechanisms underlying mathematical intelligence give some idea of how far absolutism is right. Advances in cognitive science should help us better understand the extent to which we are divided and how far we are potentially harmonious in our values.

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)

This book is meant as a part of the larger contemporary philosophical project of naturalizing logico-mathematical knowledge, and addresses the key question that motivates most of the work in this field: What is philosophically relevant about the nature of logico-mathematical knowledge in recent research in psychology and cognitive science? The question about this distinctive kind of knowledge is rooted in Plato’s dialogues, and virtually all major philosophers have expressed interest in it. The essays in this collection tackle this (...) important philosophical query from the perspective of the modern sciences of cognition, namely cognitive psychology and neuroscience. _Naturalizing Logico-Mathematical Knowledge _contributes to consolidating a new, emerging direction in the philosophy of mathematics, which, while keeping the traditional concerns of this sub-discipline in sight, aims to engage with them in a scientifically-informed manner. A subsequent aim is to signal the philosophers’ willingness to enter into a fruitful dialogue with the community of cognitive scientists and psychologists by examining their methods and interpretive strategies. (shrink)

The aim of the paper is to argue for the cognitive unity of the mathematical results ascribed by ancient authors to Thales. These results are late ascriptions and so it is difficult to say anything certain about them on philological grounds. I will seek characteristic features of the cognitive unity of the mathematical results ascribed to Thales by comparing them with Galilean physics. This might seem at a first sight a rather unusual move. Nevertheless, I suggest viewing the (...) process of turning geometry into an axiomatic-deductive science as a process of idealization in mathematics that is parallel to the process of idealization in physics. In Kvasz I offered an epistemological reconstruction of the process of idealization in physics during the scientific revolution of the seventeenth century. In the present paper I try to employ these epistemological insights in the process of idealization in physics and propose a reconstruction of the cognitive unity of the mathematical results ascribed to Thales, who can, on the basis of these ascriptions, be seen as one of the initiators of idealization in mathematics. (shrink)

The problem of semantic content is the problem of explicating those features of brain processes by virtue of which they may properly be thought to possess meaning or reference. This paper criticizes the account of semantic content associated with fodor's version of cognitive science, And offers an alternative account based on mathematical communication theory. Its key concept is that of a neuronal representation maintaining a high-Level of mutual information with a designated external state of affairs under changing conditions (...) of perceptual presentation. (shrink)

For all of recorded history prior to the second half of the twentieth century, there has been but one realm in which the cognitive processes of reasoning and problem solving, learning and discovery, language and mathematics took place. The realm of human intellect no longer has an exclusive claim on these cognitive processes--artificial intelligence represents a parallel claim. Wagman compares the two realms, focusing on each of the major components of cognition: logic, reasoning, problem-solving, language, memory, learning, (...) and discovery. He identifies consonant and disparate modes of cognition, and develops a general theory of human and artificial intelligence. (shrink)

The problem of semantic content is the problem of explicating those features of brain processes by virtue of which they may properly be thought to possess meaning or reference. This paper criticizes the account of semantic content associated with fodor's version of cognitive science, And offers an alternative account based on mathematical communication theory. Its key concept is that of a neuronal representation maintaining a high-Level of mutual information with a designated external state of affairs under changing conditions (...) of perceptual presentation. (shrink)

Philosophical analysis of mathematical knowledge are commonly conducted within the realist/antirealist dichotomy. Nevertheless, philosophers working within this dichotomy pay little attention to the way in which mathematics evolves and structures itself. Focusing on mathematical practice, I propose a weak notion of objectivity of mathematical knowledge that preserves the intersubjective character of mathematical knowledge but does not bear on a view of mathematics as a body of mind-independent necessary truths. Furthermore, I show how that the successful application of (...) class='Hi'>mathematics in science is an important trigger for the objectivity of mathematical knowledge. (shrink)

According to a grand narrative that long ago ceased to be told, there was a seventeenth century Scientific Revolution, during which a few heroes conquered nature thanks to mathematics. This grand narrative began with the exhibition of quantitative laws that these heroes, Galileo and Newton for example, had disclosed: the law of falling bodies, according to which the speed of a falling body is proportional to the square of the time that has elapsed since the beginning of its fall; (...) the law of gravitation, according to which two bodies are attracted to one another in proportion to the sum of their masses and in inverse proportion to the square of the distance separating them -- according to his own preferences, each narrator added one or two quantitative laws of this kind. The essential feature was not so much the examples that were chosen, but, rather, the more or less explicit theses that accompanied them. First, mathematization would be taken as the criterion for distinguishing between a qualitative Aristotelian philosophy and the new quantitative physics. Secondly, mathematization was founded on the metaphysical conviction that the world was created pondere, numero et mensura, or that the ultimate components of natural things are triangles, circles, and other geometrical objects. This metaphysical conviction had two immediate consequences: that all the phenomena of nature can be in principle submitted to mathematics and that mathematical language is transparent; it is the language of nature itself and has simply to be picked up at the surface of phenomena. Finally, it goes without saying that, from a social point of view, the evolution of the sciences was apprehended through what has been aptly called the "relay runner model," according to which science progresses as a result of individual discoveries. Grand narratives such as this are perhaps simply fictions doomed to ruin as soon as they are clearly expressed. In any case, the very assumption on which this grand narrative relies can be brought into question: even in the canonical domain of mechanics, the relevant epistemological units crucial to understanding the dynamics of the Scientific Revolution are perhaps not a few laws of motion, but a complex set of problems embodied in mundane objects. Moreover, each of the theses just mentioned was actually challenged during the long period of historiographical reappraisal, out of which we have probably not yet stepped. Against the sharp distinction between a qualitative Aristotelian philosophy and the new quantitative physics, numerous studies insist that Rome wasn't built in a day, so to speak. Since Antiquity, there have always been mixed sciences; the emergence of pre-classical mechanics depends on both medieval treatises and the practical challenges met by Renaissance engineers. It is indeed true that, for Aristotle, mathematics merely captures the superficial properties of things, but the Aristotelianisms were many during the Renaissance and the Early Modern period, with some of them being compatible with the introduction of mathematics in natural philosophy. In addition, the gap between the alleged program of mathematizing nature and its effective realization was underlined as most natural phenomena actually escaped mathematization; at best they were enrolled in what Thomas Kuhn began to rehabilitate under the appellation of the "Baconian sciences," i.e., empirical investigations aiming at establishing isolated facts, without relating them to any overarching theory. Hence, mathematization of nature cannot pretend to capture a historical fact: at most, it expresses an indeterminate task for generations to come. On top of these first two considerations, and against the thesis of the neutrality of the mathematical language, it was urged that mathematics is not "only a language" and that, exactly as other symbolic means or cognitive tools, it has its own constraints. For example, it has been thoroughly explained that the Euclidean theory of proportions both guides and frustrates the Galilean analysis of motion; its shortages were particularly clear with respect to the expression of continuity, which is crucial in the case of motion. Consequently, when calculus was invented and applied to the analysis of motion, it was not a transposition that left things as they stood. Even more clearly than in the case of a translation from one natural language to another, the shift from one symbolic language to another entails that certain possibilities are opened while others are closed. The cognitive constraints imposed by established mathematical theories, as seen in the theory of proportions or calculus, were not the only ones to be studied in relation to mathematization. Certain schemes dependent on the grammar of natural languages, e.g., the scheme of contrariety, or certain symbolic means of representation, e.g. geometrical diagrams and numerical tables, were also subject to such scrutiny. Lastly, it was insisted that, even if we concede the existence of scientific geniuses, mathematics is largely produced by intellectual communities and embedded within social practices. More attention was consequently paid to the forms of communication in given mathematical networks, or to the teaching of the discipline in, for example, Jesuit colleges and universities. The set of mathematical practices specific to specialized craftsmen, highly-qualified experts and engineers began to be studied in its own right. All these reflections may have helped us change our perspectives on the question of mathematization. It seems, however, that they were instead set aside, both because of a general distrust towards sweeping narratives that are always subject to the suspicion that they overlook the unyielding complexity of real history, and because of a shift in our interests. The more obscure and idiosyncratic they are, an alchemist, a patron of the sciences or a lunatic collector is nowadays honored in journals of the history of sciences. As for the general issues involved in the question of mathematization, they are rejected as obsolete, or reserved for specialized journals in the history of mathematics. Consequently, before presenting the essays of this fascicle, I would like to say a few words in favor of a renewed study of the forms of mathematization in the history of the early sciences. (shrink)

In the past 10 years, contemporary philosophy of mathematics has seen the development of a trend that conceives mathematics as first and foremost a human activity and in particular as a kind of practice. However, only recently the need for a general framework to account for the target of the so-called philosophy of mathematical practice has emerged. The purpose of the present article is to make progress towards the definition of a more precise general framework for the philosophy (...) of mathematical practice by exploring two strategies. A first strategy is to turn to philosophy of mind and Edwin Hutchins' view of distributed cognition in order to better understand the cognitive issues at play when considering a community of mathematicians; a second strategy is to refer to philosophy of language and focus on Robert Brandom's inferentialism and mathematical conceptual content. A possible combination of these two views, called enhanced material inferentialism, is then put forward as a promising framework to account for the philosophy of mathematical practice. (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)

All normal human beings alive in the last fifty thousand years appear to have possessed, in Mark Turner's phrase, 'impressively atful minds'. Cognitively modern minds produced a staggering list of behavioural singularities - science, religion, mathematics, language, advanced tool use, decorative dress, dance, culture, art - that seems to indicate a mysterious and unexplained discontinuity between us and all other living things. This brute fact gives rise to some tantalizing questions: How did the artful mind emerge? What are (...) the basic mental operations that make art possible for us now, and how do they operate? These are the questions that occupy the distinguished contributors to this book, which emerged from a year-long Getty-funded research project hosted by the Centre for Advanced Study in the Behavioural Sciences at Stanford. (shrink)

The interdisciplinary field of cognitive science brings together elements of cognitive psychology, mathematics, perception, and linguistics. Focusing on the main areas of exploration in this field today, Cognitive Science presents comprehensive overviews of research findings and discusses new cross-over areas of interest. Contributors represent the most senior and well-established names in the field. This volume serves as a high-level introduction, with sufficient breadth to be a graduate-level text, and enough depth to be a valued (...) reference source to researchers. (shrink)

This article looks at recent work in cognitive science on mathematical cognition from the perspective of history and philosophy of mathematical practice. The discussion is focused on the work of Lakoff and Núñez, because this is the first comprehensive account of mathematical cognition that also addresses advanced mathematics and its history. Building on a distinction between mathematics as it is presented in textbooks and as it presents itself to the researcher, it is argued that the focus (...) of cognitive analyses of historical developments of mathematics has been primarily on the former, even if they claim to be about the latter. (shrink)

In this paper, I shall argue that both cognitivism and liberal contractualism defend a pre-moral conception of human desire that has its origin in the Hobbesian and Humean tradition that both theories share. Moreover, the computational and syntactic themes in cognitive science support the notion, which Gauthier evidently shares, that the human mind ¿ or, in Gauthier¿s case, the mind of ¿economic man¿ ¿ is a purely formal mechanism, characterized by logical and mathematical operations. I shall conclude that (...) a single conception of human behaviour runs through the various dominant psychological, moral and political theories of analytic inspiration. (shrink)

This book reports on cutting-edge concepts related to Bourbaki’s notion of structures mères. It merges perspectives from logic, philosophy, linguistics and cognitive science, suggesting how they can be combined with Bourbaki’s mathematical structuralism in order to solve foundational, ontological and epistemological problems using a novel category-theoretic approach. By offering a comprehensive account of Bourbaki’s structuralism and answers to several important questions that have arisen in connection with it, the book provides readers with a unique source of information and (...) inspiration for future research on this topic. (shrink)

In recent years philosophers have used results from cognitive science to formulate epistemologies of arithmetic :5–18, 2001). Such epistemologies have, however, been criticised, e.g. by Azzouni, for interpreting the capacities found by cognitive science in an overly numerical way. I offer an alternative framework for the way these psychological processes can be combined, forming the basis for an epistemology for arithmetic. The resulting framework avoids assigning numerical content to the Approximate Number System and Object Tracking System, (...) two systems that have so far been the basis of epistemologies of arithmetic informed by cognitive science. The resulting account is, however, only a framework for an epistemology: in the final part of the paper I argue that it is compatible with both platonist and nominalist views of numbers by fitting it into an epistemology for ante rem structuralism and one for fictionalism. Unsurprisingly, cognitive science does not settle the debate between these positions in the philosophy of mathematics, but I it can be used to refine existing epistemologies and restrict our focus to the capacities that cognitive science has found to underly our mathematical knowledge. (shrink)

In recent years philosophers have used results from cognitive science to formulate epistemologies of arithmetic :5–18, 2001). Such epistemologies have, however, been criticised, e.g. by Azzouni, for interpreting the capacities found by cognitive science in an overly numerical way. I offer an alternative framework for the way these psychological processes can be combined, forming the basis for an epistemology for arithmetic. The resulting framework avoids assigning numerical content to the Approximate Number System and Object Tracking System, (...) two systems that have so far been the basis of epistemologies of arithmetic informed by cognitive science. The resulting account is, however, only a framework for an epistemology: in the final part of the paper I argue that it is compatible with both platonist and nominalist views of numbers by fitting it into an epistemology for ante rem structuralism and one for fictionalism. Unsurprisingly, cognitive science does not settle the debate between these positions in the philosophy of mathematics, but I it can be used to refine existing epistemologies and restrict our focus to the capacities that cognitive science has found to underly our mathematical knowledge. (shrink)

This collection of readings shows how cognitive science can influence most of the primary branches of philosophy, as well as how philosophy critically examines the foundations of cognitive science. Its broad coverage extends beyond current texts that focus mainly on the impact of cognitive science on philosophy of mind and philosophy of psychology, to include materials that are relevant to five other branches of philosophy: epistemology, philosophy of science (and mathematics), metaphysics, language, (...) and ethics. The readings are organized by philosophical fields, with selections evenly divided between philosophers and cognitive scientists. They draw on research in numerous areas of cognitive science, including cognitive psychology, developmental psychology, social psychology, psychology of reasoning and judgment, artificial intelligence, linguistics, and neuropsychology. There are timely treatments of current topics and debates such as the innate understanding of number, children's theory of mind, self-knowledge, consciousness, connectionism, and ethics and cognitive science. (shrink)

Prior research has investigated the recruitment of inhibition in the use of science/mathematics concepts in tasks that require the rejection of a conflicting, nonscientific initial concept. The present research examines if inhibition is the only EF skill recruited in such tasks and investigates whether shifting is also involved. It also investigates whether inhibition and/or shifting are recruited in tasks in which the use of science/mathematics concepts does not require the rejection of an initial concept, or which (...) require only the use of initial concepts. One hundred and thirty‐three third‐ and fifth‐grade children participated in two inhibition and shifting tasks and two science and mathematics conceptual understanding and conceptual change (CU&C) tasks. All the tasks were on‐line, and performance was measured in accuracy and RTs. The CU&C tasks involved the use of initial concepts and of science/mathematics concepts which required conceptual changes for their initial formation. Only in one of the tasks the use of the science/mathematics concepts required the concurrent rejection of an initial concept. The results confirmed that in this task inhibition was recruited and also showed that the speed of shifting was a significant predictor of performance. Shifting was a significant predictor of performance in all the tasks, regardless of whether they involved science/mathematics or initial concepts. It is argued that shifting is likely to be recruited in complex tasks that require multiple comparisons of stimuli and the entertainment of different perspectives. Inhibition seems to be a more selective cognitive skill likely to be recruited when the use of science/mathematics concepts requires the rejection of a conflicting initial concept. (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)

In the late summer of 1998, the authors, a cognitive scientist and a logician, started talking about the relevance of modern mathematical logic to the study of human reasoning, and we have been talking ever since. This book is an interim report of that conversation. It argues that results such as those on the Wason selection task, purportedly showing the irrelevance of formal logic to actual human reasoning, have been widely misinterpreted, mainly because the picture of logic current in (...) psychology and cognitive science is completely mistaken. We aim to give the reader a more accurate picture of mathematical logic and, in doing so, hope to show that logic, properly conceived, is still a very helpful tool in cognitive science. The main thrust of the book is therefore constructive. We give a number of examples in which logical theorizing helps in understanding and modeling observed behavior in reasoning tasks, deviations of that behavior in a psychiatric disorder (autism), and even the roots of that behavior in the evolution of the brain. (shrink)

David Marr provided a useful framework for theorizing about cognition within classical, AI-style cognitive science, in terms of three levels of description: the levels of (i) cognitive function, (ii) algorithm and (iii) physical implementation. We generalize this framework: (i) cognitive state transitions, (ii) mathematical/functional design and (iii) physical implementation or realization. Specifying the middle, design level to be the theory of dynamical systems yields a nonclassical, alternative framework that suits (but is not committed to) connectionism. We (...) consider how a brain's (or a network's) being a dynamical system might be the key both to its realizing various essential features of cognition — productivity, systematicity, structure-sensitive processing, syntax — and also to a non-classical solution of (frame-type) problems plaguing classical cognitive science. (shrink)

Sociology originated in the mid-nineteenth century from a new confidence in the power of science to explain the world on a mathematical foundation. Both mathematics and sociology transformed over the ensuing century, inverting the hierarchical relationship from sociology as a mathematics-based science of complex human configurations to mathematics as a complex science based on social institutions. That is, where sociology began as the hard case for mathematics, it became possible to see mathematics (...) as the hard case for sociology. In this light, we examine a number of motivations and provocations for the sociology of mathematics, from language and social cognition to Marxist and materialist skepticism of ideology, and sample the evidence and arguments brought to bear on the sociology of mathematics at a range of scales. (shrink)

The Phenomenological Mind, by Shaun Gallagher and Dan Zahavi, is part of a recent initiative to show that phenomenology, classically conceived as the tradition inaugurated by Edmund Husserl and not as mere introspection, contributes something important to cognitive science. (For other examples, see “References” below.) Phenomenology, of course, has been a part of cognitive science for a long time. It implicitly informs the works of Andy Clark (e.g. 1997) and John Haugeland (e.g. 1998), and Hubert Dreyfus (...) explicitly uses it (e.g. 1992). But where the former use phenomenology in the background as broad context and Dreyfus uses it primarily (though not exclusively) as a critique of conventional AI, Gallagher and Zahavi wish to indicate a positive and constructive place for it within cognitive science. They do not recommend that we simply accept pronouncements of thinkers like Husserl, Heidegger, Sartre and Merleau‐Ponty and apply them to questions of cognition, but that we use revised forms of phenomenology to illuminate dimensions of cognitive experience that are missing in current research. The book is presented as an “introduction to philosophy of mind and cognitive science” written from a phenomenological perspective. It seeks to justify the use of phenomenology in cognitive science by showing what kinds of questions it asks and answers, the variety of uses to which it has recently been put and the fruitfulness of some of its findings. The catalog of topics, for the most part, matches other introductions to the philosophy of mind, such as questions of method, consciousness, perception, intentionality, embodiment, action, agency and other minds. One issue presented here that is not generally dealt with in existing philosophy of mind and cognitive science texts is temporality, a mainstay of the continental tradition. After an introductory chapter that places phenomenology in the context of other approaches, the book lays out the main tenets of phenomenological method. Here, one encounters expected components of phenomenology: the epoché (described below), phenomenological reduction, eidetic variation, and so on. This traditional fare is soon followed by some potential surprises, namely, attempts to “naturalize” phenomenology, a few attempts to formalize it, and the emergence of ‘neurophenomenology’. Each of these is a bit surprising because Husserl was a vocal critic of naturalism, seeing transcendental phenomenology as an alternative to the empirical study of consciousness. He was also skeptical about the possibilities of mathematizing phenomenology. Gallagher and Zahavi acknowledge these points, but since they are not repeating history or undertaking exegesis, strict adherence to canonical phenomenology is not required. Naturalizing phenomenology means recognizing that “the phenomena it studies are part of nature and are therefore also open to empirical investigation” (p.. (shrink)

This commentary gives a personal perspective on modeling and modeling developments in cognitive science, starting in the 1950s, but focusing on the author’s personal views of modeling since training in the late 1960s, and particularly focusing on advances since the official founding of the Cognitive Science Society. The range and variety of modeling approaches in use today are remarkable, and for many, bewildering. Yet to come to anything approaching adequate insights into the infinitely complex fields of (...) mind, brain, and intelligent systems, an extremely wide array of modeling approaches is vital and necessary. (shrink)

Complexity science is an investigative framework that stems from a number of tried and tested disciplines—including systems theory, nonlinear dynamical systems theory, and synergetics—and extends a common set of concepts, methods, and principles to understand how natural systems operate. By quantitatively employing concepts, such as emergence, nonlinearity, and self‐organization, complexity science offers a way to understand the structures and operations of natural cognitive systems in a manner that is conceptually compelling and mathematically rigorous. Thus, complexity science (...) both transforms understandings of cognition and reframes more traditional approaches. Consequently, if cognitive systems are indeed complex systems, then cognitive science ought to consider complexity science as a centerpiece of the discipline. (shrink)

Offering a collection of fifteen essays that deal with issues at the intersection of phenomenology, logic, and the philosophy of mathematics, this 2005 book is divided into three parts. Part I contains a general essay on Husserl's conception of science and logic, an essay of mathematics and transcendental phenomenology, and an essay on phenomenology and modern pure geometry. Part II is focused on Kurt Godel's interest in phenomenology. It explores Godel's ideas and also some work of Quine, (...) Penelope Maddy and Roger Penrose. Part III deals with elementary, constructive areas of mathematics. These are areas of mathematics that are closer to their origins in simple cognitive activities and in everyday experience. This part of the book contains essays on intuitionism, Hermann Weyl, the notion of constructive proof, Poincaré and Frege. (shrink)

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

October 14, 2007: Studying how a broker's brain works. swissinfo. "To help maintain its competitive edge, the Swiss banking industry is investing heavily in financial engineering. Its latest recruit is economist Peter Bossaerts. swissinfo talked to Bossaerts, a leading expert in neuroeconomics – the study of how we make financial choices - about his recent appointment as professor at the Federal Institute of Technology in Lausanne.... swissinfo: So what exactly is neuroeconomics? Peter Bossaerts: It's a mixture of decisional theory - (...) mathematical theories applied in risk-based decision-making - and neuroscience.... Neurofinance, therefore, tries to understand how choices are made in a risky world. It looks closely at the workings of the brain while taking into account human emotions.... swissinfo: What is the aim of your work? P.B.: Firstly, to make progress on how people make choices when dealing with risk.... Neuroeconomics should also help improve decisional theory, which doesn't work in the real world where rules are vague and probabilities are unknown. The aim is to build up artificial intelligence based on a theory where decision-making is repeated." >>> Neuroscience, Cognitive Science, Finance & Investing, Applications. (shrink)

It is sometimes suggested that the history of computation in cognitive science is one in which the formal apparatus of Turing-equivalent computation, or effective computability, was exported from mathematical logic to ever wider areas of cognitive science and its environs. This paper, however, indicates some respects in which this suggestion is inaccurate. Computability theory has not been focused exclusively on Turing-equivalent computation. Many essential features of Turing-equivalent computation are not captured in definitions of computation as symbol (...) manipulation. Turing-equivalent computation did not play the role in McCulloch and Pitts’s early cybernetic work that is sometimes attributed to it. Finally, various segments of the neuroscientific community invoke a notion of computation that differs from the Turing-equivalent notion.Keywords: Circular causality; Computation; Cortical maps; Neural networks; Symbol manipulation; Turing-equivalent computation. (shrink)

A new theory of time; Events and observability; Multimedial units and language: verbal and nonverbal communication; Judgment formation and problems of description; Memory and perception: some new models; Stochastic models of expertise formation, opinion change, and learning.

The important role of mathematical representations in scientific thinking has received little attention from cognitive scientists. This study argues that neglect of this issue is unwarranted, given existing cognitive theories and laws, together with promising results from the cognitive historical analysis of several important scientists. In particular, while the mathematical wizardry of James Clerk Maxwell differed dramatically from the experimental approaches favored by Michael Faraday, Maxwell himself recognized Faraday as “in reality a mathematician of a very high (...) order,” and his own work as in some respects a re‐representation of Faraday’s field theory in analytic terms. The implications of the similarities and differences between the two figures open new perspectives on the cognitive role of mathematics as a learned mode of representation in science. (shrink)

Dynamical systems theory (DST) is a branch of mathematics that assesses abstract or physical systems that change over time. It has a quantitative part (mathematical equations) and a related qualitative part (plotting equations in a state space). Nonlinear dynamical systems theory applies the same tools in research involving phenomena such as chaos and hysteresis. These approaches have provided different ways of investigating and understanding cognitive systems in cognitive science and neuroscience. The ‘dynamical hypothesis’ claims that cognition (...) is and can be understood as dynamical systems. Common consequences for such an approach include rejecting understanding cognition as information processing in nature, including eschewing explanatory roles for computation or representation. Contemporary applications of DST include mouse‐tracking studies in cognitive science and nonrepresentational perspectives on motor control in neuroscience. Such work has philosophical implications concerning the boundaries of cognition, explanation, and representations. DST offers powerful methodology and theories that raise many topics of philosophical significance. (shrink)

Attempts to apply the mathematical tools of dynamical systems theory to cognition in a systematic way has been well under way since the early 90s and has been recognised as a “third contender” to computationalist and connectionist approaches :441–463, 1996). Nevertheless, it was also realised that such an application will not lead to a solid paradigm as straightforwardly as was initially hoped. In this paper I explicate a method for assessing such proposals by drawing upon Lakatos’s Criticism and the growth (...) of knowledge, Cambridge University Press, London, pp 91–195, 1970) methodology of scientific research programs. MSRP focuses on the heuristics of a particular field and gauges the model/theory building stratagems by reference to theoretical and empirical progress, on the one hand, and the continuity and the autonomy of the way the field’s heuristic generates its series of models/theories, on the other. The requirement of continuity and autonomy afford distinct senses of ad hoc-ness, which serve as an effective tool to detect various subtleties which may otherwise be missed: the present approach identifies shortcomings missed by Chemero’s radical embodied cognitive science and falsifies Chemero’s claim that the methodological powers of his model-based account is on a par with computationalism. In general, I claim that MSRP is relevant to current methodological issues in cognitive science and can supplement debates regarding “local” assessments of methodologies, such as that between mechanical versus covering-law explanations. MSRP must at least be viewed as a necessary constraint for any methodological considerations in cognitive science. (shrink)

This paper reviews 30 years of progress in U.S. cognitive science research related to education and training, as seen from the perspective of a research manager who was personally involved in many of these developments.

Bayesian reverse-engineering is a research strategy for developing three-level explanations of behavior and cognition. Starting from a computational-level analysis of behavior and cognition as optimal probabilistic inference, Bayesian reverse-engineers apply numerous tweaks and heuristics to formulate testable hypotheses at the algorithmic and implementational levels. In so doing, they exploit recent technological advances in Bayesian artificial intelligence, machine learning, and statistics, but also consider established principles from cognitive psychology and neuroscience. Although these tweaks and heuristics are highly pragmatic in character (...) and are often deployed unsystematically, Bayesian reverse-engineering avoids several important worries that have been raised about the explanatory credentials of Bayesian cognitive science: the worry that the lower levels of analysis are being ignored altogether; the challenge that the mathematical models being developed are unfalsifiable; and the charge that the terms ‘optimal’ and ‘rational’ have lost their customary normative force. But while Bayesian reverse-engineering is therefore a viable and productive research strategy, it is also no fool-proof recipe for explanatory success. (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)

In this book, Earl Mac Cormac presents an original and unified cognitive theory of metaphor using philosophical arguments which draw upon evidence from psychological experiments and theories. He notes that implications of this theory for meaning and truth with specific attention to metaphor as a speech act, the iconic meaning of metaphor, and the development of a four-valued system of truth. Numerous examples of metaphor from poetry and science are presented and analyzed to support Mac Cormac's theory. A (...) Cognitive Theory of Metaphortakes up three levels of explanation—metaphor as expressed in surface language, the semantics of metaphor, and metaphor as a cogitive process—and unifies these by interpreting metaphor as an evolutionary knowledge process in which metaphors mediate between minds and culture. Mac Cormac considers, and rejects, the radical theory that all use of language is metaphorical; however, this argument also recognizes that the theoryof metaphor may itself be metaphorical. The book first considers the computational metaphor often adopted by cognitive psychology as an example of metaphor requiring analysis. In contrast to three well-known philosophical theories of metaphor—the tension theory, the controversion theory, and the grammatical deviance theory—it develops a semantical anomaly theory of metaphor based on a quasi-mathematical hierarchy of words. In developing the theory, Mac Cormac makes much-needed connections between theories of metaphor and more orthodox analytic philosophy of meaning, including discussions of speech acts and the logic of fuzzy sets. This semantical theory of explanation is then shown to be compatible with contemporary psychological theories of memory. (shrink)

In this article, I review recent findings in cognitive neuroscience in learning, particularly in the learning of mathematics and of reading. I argue that while cognitive neuroscience is in its infancy as a field, theories of learning will need to incorporate and account for this growing body of empirical data.

In this article, it is suggested a possible profile for the historical and philosophical migration of several issues from the metamathematical domain to the domain of functionalist neuro-computational Cognitive Science. The description of such a transition is accomplished by an analysis of the ideas of Post, Church, Gödel, and, in particular, Turing on the possibility of formalization of creative thinking in Mathematics.Neste artigo, propõe-se uma configuração possível para a transição histórico-filosófica de temas investigados no domínio da metamatemática (...) para o domínio da Ciência Cognitiva funcionalista neurocomputacional. A descrição de tal transição é feita por meio de uma breve análise das idéias de Post, Church, Gödel e Turing sobre a possibilidade de formalização do pensamento criador na matemática, enfatizando as contribuições deste último. (shrink)

The cognitive foundations of geometry have puzzled academics for a long time, and even today are mostly unknown to many scholars, including mathematical cognition researchers. -/- Foundations of Geometric Cognition shows that basic geometric skills are deeply hardwired in the visuospatial cognitive capacities of our brains, namely spatial navigation and object recognition. These capacities, shared with non-human animals and appearing in early stages of the human ontogeny, cannot, however, fully explain a uniquely human form of geometric cognition. In (...) the book, Hohol argues that Euclidean geometry would not be possible without the human capacity to create and use abstract concepts, demonstrating how language and diagrams provide cognitive scaffolding for abstract geometric thinking, within a context of a Euclidean system of thought. -/- Taking an interdisciplinary approach and drawing on research from diverse fields including psychology, cognitive science, and mathematics, this book is a must-read for cognitive psychologists and cognitive scientists of mathematics, alongside anyone interested in mathematical education or the philosophical and historical aspects of geometry. (shrink)

According to the traditional nomological-deductive methodology of physics and chemistry [Hempel and Oppenheim, 1948], explaining a phenomenon means subsuming it under a law. Logic becomes then the glue of explanation and laws the primary explainers. Thus, the scientific study of a system would consist in the development of a logically sound model of it, once the relevant observables (state variables) are identified and the general laws governing their change (expressed as differential equations, state transition rules, maximization/minimization principles,. . . ) (...) are well determined, together with the initial or boundary conditions for each particular case. Often this also involves making a set of assumptions about the elementary components of the system (e.g., their structural and dynamic properties) and modes of local interaction. In this framework, predictability becomes the main goal and that is why research is carried out through the construction of accurate mathematical models. Thus, physics and chemistry have made most progress so far by focusing on systems that, either due to their intrinsic properties or to the conditions in which they are investigated, allow for very strong simplifying assumptions, under which, nevertheless, those highly idealized models of reality are deemed to be in good correspondence with reality itself. Despite the enormous success that this methodology has had, the study of living and cognitive phenomena had to follow a very different road, because these phenomena are produced by systems whose underlying material structure and organization do not permit such crude approximations. Seen from the perspective of physics or chemistry, biological and cognitive systems are made of an enormous number of parts or elements interacting in non-linear and selective ways, which makes very difficult their tractability through mathematical models. In addition, many of those interacting elements are hierarchically organized, in a way that the “macroscopic” (observable) parts behave according to rules that cannot be, in practice, derived from simple principles at the level of their “microscopic” dynamics.. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:The Philosophic Foundations of Mimetic Theory and Cognitive Science(Including Artificial Intelligence)Jean-Pierre Dupuy (bio)In the mid 1970s I discovered at the same time cognitive science and mimetic theory. Being a philosopher with a scientific background, I immediately brought them together and tried to reconceptualize the latter in terms of the former. In a sense, I haven't stopped doing that in the last 45 years. That is (...) why I feel fully at home with the goals pursued by the organizers of this conference.I do not believe it is feasible or even desirable to operate the rapprochement between the two disciplines at a global level. The history of cognitive science is exceedingly involved, and the part played by artificial intelligence in it compared to the part played by the neurosciences has varied greatly over time.1 Only a piecemeal approach makes sense. I present here four short case studies that cover the whole period, from the beginnings in cybernetics to today's artificial intelligence (AI).In order to better situate these studies within a broader frame, I propose the diagram in Figure 1. To comment on it would require more time than available for my whole presentation. Let me just say the following. [End Page 1]It is essential to distinguish between two kinds of AI. The first to bear this name was launched by economist Herbert Simon and computer scientist Allan Newell at a Dartmouth conference in 1956. It was based on the premise that thinking is computing, with this computation bearing on sentences of a "language of thought" implementable in a digital computer. It turns out that this AI(1) was progressively substituted from the 1980s onward by an AI(2), which, to some extent, was a return to a fundamental result obtained by cybernetics in 1943. Neurophysiologist Warren McCulloch and mathematician Walter Pitts had constructed a mathematical model of a network of elementary calculators called "neurons" and explored its properties. The capacity to "think" was one of these. It was not situated at the elementary level, as was the case in AI(1), but it emerged at the collective level from the interactions between the neurons.This model was heavily criticized by the supporters of AI(1),2 and its first implementations, in the notorious "Perceptron" in particular (1957), were pitiful failures. However, the emergence of "second-order cybernetics" under the guidance of cybernetician Heinz von Foerster and the theory of complex, self-organizing systems that it helped develop, in cooperation and rivalry with similar efforts in the physical sciences, enabled the neural network model to come to a second life under different names: formal neurons, deep learning, big data, and the like. Together, they constitute AI(2).The following network of influences wouldn't have been possible without its stem, that is, the boxes containing the names of Kurt Gödel and Alan Turing. The former revolutionized logic by showing that reasoning about arithmetic and arithmetic are one and the same, insofar as it is possible to code in a 1-to-1 fashion propositions about integers (for instance, "7 is a prime number") by means of integers. The latter demonstrated that some mechanisms can self-transcend inasmuch as they generate an ensemble of outputs that cannot be defined mechanistically. Later, John von Neumann, who is credited along with Alan Turing with having invented the digital computer, would use this logical possibility to define complexity: A mechanism is complex if it is capable of generating one more complex than itself. The gate was open for the theories of complex, self-organizing systems to thrive and, by way of consequence, for AI(2) to emerge.fascination with modelsHerbert Simon (1916–2001), winner of the Nobel Prize in economics for his work on what he called "bounded rationality," is generally considered to be one [End Page 2] Click for larger view View full resolutionFigure 1.of the founders of artificial intelligence. Some cognitive scientists still rely on the computer program that he devised with Alan Newell to simulate the creative processes of human thought—a program aptly named "General Problem Solver." The title of one of his principal works, The Sciences of the... (shrink)

The published works of scientists often conceal the cognitive processes that led to their results. Scholars of mathematical practice must therefore seek out less obvious sources. This article analyzes a widely circulated mathematical joke, comprising a list of spurious proof types. An account is proposed in terms of argumentation schemes: stereotypical patterns of reasoning, which may be accompanied by critical questions itemizing possible lines of defeat. It is argued that humor is associated with risky forms of inference, which are (...) essential to creative mathematics. The components of the joke are explicated by argumentation schemes devised for application to topic-neutral reasoning. These in turn are classified under seven headings: retroduction, citation, intuition, meta-argument, closure, generalization, and definition. Finally, the wider significance of this account for the cognitive science of mathematics is discussed. (shrink)