Despite the importance of mathematics in our educational systems little is known about how abstract mathematical thinking emerges. Under the uniting thread of mathematical development, we hope to connect researchers from various backgrounds to provide an integrated view of abstract mathematical cognition. Much progress has been made in the last 20 years on how numeracy is acquired. Experimental psychology has brought to light the fact that numerical cognition stems from spatial cognition. The findings from (...) neuroimaging and single cell recording experiments converge to show that numerical representations take place in the intraparietal sulcus. Further research has demonstrated that supplementary neural networks might be recruited to carry out subtasks; for example, the retrieval of arithmetic facts is done by the angular gyrus. Now that the neural networks in charge of basic mathematical cognition are identified, we can move onto the stage where we seek to understand how these basics skills are used to support the acquisition and use of abstract mathematical concepts. (shrink)

Vehicle externalism maintains that the vehicles of our mental representations can be located outside of the head, that is, they need not be instantiated by neurons located inside the brain of the cogniser. But some disagree, insisting that ‘non-derived’, or ‘original’, content is the mark of the cognitive and that only biologically instantiated representational vehicles can have non-derived content, while the contents of all extra-neural representational vehicles are derived and thus lie outside the scope of the cognitive. In this paper (...) we develop one aspect of Menary’s vehicle externalist theory of cognitive integration—the process of enculturation—to respond to this longstanding objection. We offer examples of how expert mathematicians introduce new symbols to represent new mathematical possibilities that are not yet understood, and we argue that these new symbols have genuine non-derived content, that is, content that is not dependent on an act of interpretation by a cognitive agent and that does not derive from conventional associations, as many linguistic representations do. (shrink)

Kant's Mathematical World aims to transform our understanding of Kant's philosophy of mathematics and his account of the mathematical character of the world. Daniel Sutherland reconstructs Kant's project of explaining both mathematical cognition and our cognition of the world in terms of our most basic cognitive capacities. He situates Kant in a long mathematical tradition with roots in Euclid's Elements, and thereby recovers the very different way of thinking about mathematics which existed prior to (...) its 'arithmetization' in the nineteenth century. He shows that Kant thought of mathematics as a science of magnitudes and their measurement, and all objects of experience as extensive magnitudes whose real properties have intensive magnitudes, thus tying mathematics directly to the world. His book will appeal to anyone interested in Kant's critical philosophy -- either his account of the world of experience, or his philosophy of mathematics, or how the two inform each other. (shrink)

Mathematics is a powerful tool for describing and developing our knowledge of the physical world. It informs our understanding of subjects as diverse as music, games, science, economics, communications protocols, and visual arts. Mathematical thinking has its roots in the adaptive behavior of living creatures: animals must employ judgments about quantities and magnitudes in the assessment of both threats (how many foes) and opportunities (how much food) in order to make effective decisions, and use geometric information in the environment (...) for recognizing landmarks and navigating environments. Correspondingly, cognitive systems that are dedicated to the processing of distinctly mathematical information have developed. In particular, there is evidence that certain core systems for understanding different aspects of arithmetic as well as geometry are employed by humans and many other animals. They become active early in life and, particularly in the case of humans, develop through maturation. Although these core systems individually appear to be quite limited in application, in combination they allow for the recognition of mathematical properties and the formation of appropriate inferences based upon those properties. In this overview, the core systems, their roles, their limitations, and their interaction with external representations are discussed, as well as possibilities for how they can be employed together to allow us to reason about more complex mathematical domains. (shrink)

The theory suggests that quantum computations in brain neuronal dendritic-somatic microtubules regulate axonal firings to control conscious behavior. Within microtubule subunit proteins, collective dipoles in arrays of contiguous amino acid electron clouds enable suitable for topological dipole able to physically represent cognitive values, for example, those portrayed by Pothos & Busemeyer (P&B) as projections in abstract Hilbert space.

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)

In their recent paper on “Challenges in mathematical cognition”, Alcock and colleagues (Alcock et al. [2016]. Challenges in mathematical cognition: A collaboratively-derived research agenda. Journal of Numerical Cognition, 2, 20-41) defined a research agenda through 26 specific research questions. An important dimension of mathematical cognition almost completely absent from their discussion is the cultural constitution of mathematical cognition. Spanning work from a broad range of disciplines – including anthropology, archaeology, cognitive science, (...) history of science, linguistics, philosophy, and psychology – we argue that for any research agenda on mathematical cognition the cultural dimension is indispensable, and we propose a set of exemplary research questions related to it. (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)

The purpose of this book is to initiate a new discipline, namely a formalized epistemological method drawn from the cognitive strategies practised in the most effective among the modern scientific disciplines, as well as from general philosophical thinking. Indeed, what is lacking in order to improve our knowledge and our domination of the modes which nowadays are available for the generation and communication of knowledge, thoroughly and rapidly and with precision and detail? It is a systematic explication of the epistemological (...) essence encrypted in the specialized languages and algorithms of the major modern scientific approaches, a systematic cross-referencing of the explicated results, and a final elaboration of a new coherent whole. Quantum mechanics, like a diver, can take us down to the level of the very first actions of our conceptualization of reality. And starting from there, it can induce an explicit understanding of certain fundamental features of the new scientific thinking. A formalized epistemology should not be mistaken for a crossdisciplinary or a multidisciplinary project. The latter projects are designed to offer to nonspecialists access to information, to results obtained inside specialized disciplines, as well as a certain understanding of these results; whereas a formalized epistemology should equip anyone with a framework for conceptualizing himself in whatever domain and direction he or she might choose. A formalized epistemology should not be mistaken either for an approach belonging to the modern cognitive sciences. These try to establish as neutrally as possible descriptions of how the human body-and-mind work spontaneously when knowledge is generated; whereas a method of conceptualization should establish what conceptual-operational deliberate procedures have to be applied in order to represent and to achieve processes of generation of knowledge optimized accordingly to any definite aims. This book addresses philosophers of science, physicists, mathematicians, logicians, computer scientists, researchers in cognitive sciences, and biologists, as well as any intellectual who is interested in scientific and philosophical thinking. (shrink)

In this lengthy book, Daniel Sutherland proposes to rectify our long neglect of Kant's theory of mathematics by means of both historical and systematic analyses. This is a worthy undertaking, since the scope and significance of that theory were lost from view during the twentieth century.

The premise of this article is that cognitive development involves both conceptual and semiotic achievements. From this perspective, the authors emphasize the distinctness of the semiotic issues and develop a differentiated appreciation of the semiotic aspects of cognition, particularly in the field of elementary mathematical cognition. The authors provide semiotic analyses of the differences between counting, adding, and multiplying and of the conventional place-value system. The authors introduce the concept of the field of reference of a sign, (...) the differentiation of the field into foreground and background, and the dynamics within the field of reference. Finally, the authors relate these ideas to the dynamics between two dimensions of semiotic relations: the sign-referent dimension and the sign-sign dimension. (shrink)

Rips et al. appear to discuss, and then dismiss with counterexamples, the brain-based theory of mathematical cognition given in Lakoff and Nez (2000). Instead, they present another theory of their own that they correctly dismiss. Our theory is based on neural learning. Rips et al. misrepresent our theory as being directly about real-world experience and mappings directly from that experience.

Educational interventions are necessary to develop mathematical competence at early ages and prevent widespread mathematics learning failure in the education system as indicated by the results of European reports. Numerous studies agree that domain-specific predictors related to mathematics are symbolic and non-symbolic magnitude comparison, as well as, number line estimation. The goal of this study was to design 4 digital learning app games to train specific cognitive bases of mathematical learning in order to create resources and promote the (...) use of these technologies in the educational community and to promote effective scientific transfer and increase the research visibility. This study involved 193 preschoolers aged 57–79 months. A quasi-experimental design was carried out with 3 groups created after scores were obtained in a standardised mathematical competence assessment test, i.e., low-performance group, high-performance group, and control group. The results show that training with the 4 digital learning app games focusing on magnitude, subitizing, number facts, and estimation tasks improved the numerical skills of the experimental groups, compared to the control group. The implications of the study were, on the one hand, provided verified technological tools for teaching early mathematical competence. On the other hand, this study supports other studies on the importance of cognitive precursors in mathematics performance. (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)

Humans and other animals have an evolved ability to detect discrete magnitudes in their environment. Does this observation support evolutionary debunking arguments against mathematical realism, as has been recently argued by Clarke-Doane, or does it bolster mathematical realism, as authors such as Joyce and Sinnott-Armstrong have assumed? To find out, we need to pay closer attention to the features of evolved numerical cognition. I provide a detailed examination of the functional properties of evolved numerical cognition, and (...) propose that they prima facie favor a realist account of numbers. (shrink)

In his most comprehensive book on the subject , Roger Penrose provides arguments to demonstrate that there are aspects of human understanding which could not, in principle, be attained by any purely computational system. His central argument relies crucially on oft-cited theorems proven by Gödel and Turing. However, that key argument has been the subject of numerous trenchant critiques, which is unfortunate if one believes Penrose's conclusions to be plausible. In the present article, alternative arguments are offered in support of (...) Penrose-like conclusions . It is argued here that a purely computational agent, which lacked conscious awareness, would be incapable of possessing crucial concepts and of understanding certain kinds of geometrically-based proofs. Specifically, it is argued that the acquisition of human-like concepts of countable and non-denumerable infinities, and human-like comprehension of a particular geometrically motivated proof does require conscious apprehension of the subject matter involved. This does not preclude the possibility that a computational agent might come to possess the requisite consciousness, but it is argued that if this consciousness does arise within the agent, it does so, at best, as an emergent, contingent side-effect of the underlying processes involved. (shrink)

We demonstrate how real progress can be made in the debate surrounding the enhanced indispensability argument. Drawing on a counterfactual theory of explanation, well-motivated independently of the debate, we provide a novel analysis of ‘explanatory generality’ and how mathematics is involved in its procurement. On our analysis, mathematics’ sole explanatory contribution to the procurement of explanatory generality is to make counterfactual information about physical dependencies easier to grasp and reason with for creatures like us. This gives precise content to key (...) intuitions traded in the debate, regarding mathematics’ procurement of explanatory generality, and adjudicates unambiguously in favour of the nominalist, at least as far as explanatory generality is concerned. (shrink)

Most previous research on human cognition has focused on problem-solving, and has confined its investigations to the laboratory. As a result, it has been difficult to account for complex mental processes and their place in culture and history. In this startling - indeed, disco in forting - study, Jean Lave moves the analysis of one particular form of cognitive activity, - arithmetic problem-solving - out of the laboratory into the domain of everyday life. In so doing, she shows how (...) mathematics in the 'real world', like all thinking, is shaped by the dynamic encounter between the culturally endowed mind and its total context, a subtle interaction that shapes 1) Both tile human subject and the world within which it acts. The study is focused on mundane daily, activities, such as grocery shopping for 'best buys' in the supermarket, dieting, and so on. Innovative in its method, fascinating in its findings, the research is above all significant in its theoretical contributions. Have offers a cogent critique of conventional cognitive theory, turning for an alternative to recent social theory, and weaving a compelling synthesis from elements of culture theory, theories of practice, and Marxist discourse. The result is a new way of understanding human thought processes, a vision of cognition as the dialectic between persons-acting, and the settings in which their activity is constituted. The book will appeal to anthropologists, for its novel theory of the relation of cognition to culture and context; to cognitive scientists and educational theorists; and to the 'plain folks' who form its subject, and who will recognize themselves in it, a rare accomplishment in the modern social sciences. (shrink)

Mathematics is a highly specialised arena of human endeavour, one in which complex notations are invented and are subjected to complex and involved manipulations in the course of everyday work. What part do these writing practices play in mathematical communication, and how can we understand their use in the mathematical world in relation to theories of communication and cognition? To answer this, I examine in detail an excerpt from a research meeting in which communicative board-writing practices can (...) be observed, and attempt to explain the observed exchanges with reference to relevance theory (a cognitivist theory of pragmatics), in combination with the situated cognition paradigm. Successful communication in the excerpt appears well described by the notion of metacognitive acquaintance, as described in Sperber & Wilson (2015), especially when a situated view of cognition is adopted. Though this example is taken from a relatively informal face-to-face communicative situation, characteristics of such situations extend even into formal written mathematics, since even the technical terminology developed in the course of the meeting includes details that may provide readers with a kind of access to the mathematicians’ cognitive processes. (shrink)

This book is devoted to the study of human thought, its systemic structure, and the historical development of mathematics both as a product of thought and as a fascinating case analysis. After demonstrating that systems research constitutes the second dimension of modern science, the monograph discusses the yoyo model, a recent ground-breaking development of systems research, which has brought forward revolutionary applications of systems research in various areas of the traditional disciplines, the first dimension of science. After the systemic structure (...) of thought is factually revealed, mathematics, as a product of thought, is analyzed by using the age-old concepts of actual and potential infinities. In an attempt to rebuild the system of mathematics, this volume first provides a new look at some of the most important paradoxes, which have played a crucial role in the development of mathematics, in proving what these paradoxes really entail. Attention is then turned to constructing the logical foundation of two different systems of mathematics, one assuming that actual infinity is different than potential infinity, and the other that these infinities are the same. This volume will be of interest to academic researchers, students and professionals in the areas of systems science, mathematics, philosophy of mathematics, and philosophy of science. (shrink)

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)

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) (...) class='Hi'>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)

In the field of mathematics education, one of the main questions remaining under debate is whether students’ development of mathematical reasoning and problem-solving is aided more by solving tasks with given instructions or by solving them without instructions. It has been argued, that providing little or no instruction for a mathematical task generates a mathematical struggle, which can facilitate learning. This view in contrast, tasks in which routine procedures can be applied can lead to mechanical repetition with (...) little or no conceptual understanding. This study contrasts Creative Mathematical Reasoning (CMR), in which students must construct the mathematical method, with Algorithmic Reasoning (AR), in which predetermined methods and procedures on how to solve the task are given. Moreover, measures of fluid intelligence and working memory capacity are included in the analyses alongside the students’ math tracks. The results show that practicing with CMR tasks was superior to practicing with AR tasks in terms of students’ performance onpracticed test tasksandtransfer test tasks. Cognitive proficiency was shown to have an effect on students’ learning for both CMR and AR learning conditions. However, math tracks (advanced versus a more basic level) showed no significant effect. It is argued that going beyond step-by-step textbook solutions is essential and that students need to be presented with mathematical activities involving a struggle. In the CMR approach, students must focus on the relevant information in order to solve the task, and the characteristics of CMR tasks can guide students to the structural features that are critical for aiding comprehension. (shrink)

What can we infer from numerical cognition about mathematical realism? In this paper, I will consider one aspect of numerical cognition that has received little attention in the literature: the remarkable similarities of numerical cognitive capacities across many animal species. This Invariantism in Numerical Cognition (INC) indicates that mathematics and morality are disanalogous in an important respect: proto-moral beliefs differ substantially between animal species, whereas proto-mathematical beliefs (at least in the animals studied) seem to show (...) more similarities. This makes moral beliefs more susceptible to a contingency challenge from evolution compared to mathematical beliefs, and indicates that mathematical beliefs might be less vulnerable to evolutionary debunking arguments. I will then examine to what extent INC can be used to flesh out a positive case for mathematical realism. Finally, I will review two forms of mathematical realism that are promising in the light of the evolutionary evidence about numerical cognition, ante rem structuralism and Millean empiricism. (shrink)

The foundation of Mathematics is both a logico-formal issue and an epistemological one. By the first, we mean the explicitation and analysis of formal proof principles, which, largely a posteriori, ground proof on general deduction rules and schemata. By the second, we mean the investigation of the constitutive genesis of concepts and structures, the aim of this paper. This “genealogy of concepts”, so dear to Riemann, Poincaré and Enriques among others, is necessary both in order to enrich the foundational analysis (...) with an often disregarded aspect (the cognitive and historical constitution of mathematical structures) and because of the provable incompleteness of proof principles also in the analysis of deduction. For the purposes of our investigation, we will hint here to a philosophical frame as well as to some recent experimental studies on numerical cognition that support our claim on the cognitive origin and the constitutive role of mathematical intuition. (shrink)

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

An increasing number of learning goals refer to the acquisition of cognitive skills that can be described as ‘resource-based,’ as they require the availability, coordination, and integration of multiple underlying resources such as skills and knowledge facets. However, research on the support of cognitive skills rarely takes this resource-based nature explicitly into account. This is mirrored in prior research on mathematical argumentation and proof skills: Although repeatedly highlighted as resource-based, for example relying on mathematical topic knowledge, methodological knowledge, (...) mathematical strategic knowledge, and problem-solving skills, little evidence exists on how to support mathematical argumentation and proof skills based on its resources. To address this gap, a quasi-experimental intervention study with undergraduate mathematics students examined the effectiveness of different approaches to support both mathematical argumentation and proof skills and four of its resources. Based on the part-/whole-task debate from instructional design, two approaches were implemented during students’ work on proof construction tasks: (i) asequential approachfocusing and supporting each resource of mathematical argumentation and proof skills sequentially after each other and (ii) aconcurrent approachfocusing and supporting multiple resources concurrently. Empirical analyses show pronounced effects of both approaches regarding the resources underlying mathematical argumentation and proof skills. However, the effects of both approaches are mostly comparable, and only mathematical strategic knowledge benefits significantly more from the concurrent approach. Regarding mathematical argumentation and proof skills, short-term effects of both approaches are at best mixed and show differing effects based on prior attainment, possibly indicating an expertise reversal effect of the relatively short intervention. Data suggests that students with low prior attainment benefited most from the intervention, specifically from the concurrent approach. A supplementary qualitative analysis showcases how supporting multiple resources concurrently alongside mathematical argumentation and proof skills can lead to a synergistic integration of these during proof construction and can be beneficial yet demanding for students. Although results require further empirical underpinning, both approaches appear promising to support the resources underlying mathematical argumentation and proof skills and likely also show positive long-term effects on mathematical argumentation and proof skills, especially for initially weaker students. (shrink)

This paper is concerned with the genesis of mathematical knowledge. While some philosophers might argue that mathematics has no real subject matter and thus is not a body of knowledge, I will not try to dissuade them directly. I shall not attempt such a refutation because it seems clear to me that mathematicians do know such things as the Mean Value Theorem, The Fundamental Theorem of Arithmetic, Godel's Theorems, etc. Moreover, this is much more evident to me than any (...) philosophical view of mathematics I know of — including my own. So I am going to take mathematics as my starting point. (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)

This paper advances the view that some explanations in cognitive science are extra-mathematical explanations. Demonstrating the plausibility of this interpretation centers around certain efficient coding cases which ineliminably enlist information theoretic laws, facts and theorems to identify in-principle, mathematical constraints on neuronal information processing capacities. The explanatory structure in these cases is shown to parallel other putative instances of mathematical explanation. The upshot for cognitive mathematical explanations is thus two-fold: first, the view capably rebuts standard mechanistic (...) objections to non-mechanistic explanation; second, this view serves to importantly widen the possibility space for non-mechanistic forms of explanation in cognitive science. (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)