Mathematical cognition is one of the most important cognitive functions of human beings. The latest brain and cognitive research have shown that mathematical cognition is a system with multiple components and subsystems. It has phylogenetic root, also is related to ontogenetic development and learning, relying on a large-scale cerebral network including parietal, frontal and temporal regions. Especially, the parietal cortex plays an important role during mathematical cognitive processes. This indicates that language and visuospatial functions are (...) both key to mathematical cognition. All of those advances have important implications for basic mathematical education. (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)

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)

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)

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)

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)

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)

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)

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)

This study determined the cognitive skills achievement in mathematics of elementary pre-service teachers as a basis for improving problem-solving and critical thinking which was analyzed using Piaget's seven logical operations namely: classification, seriation, logical multiplication, compensation, ratio and proportional thinking, probability thinking, and correlational thinking. This study utilized an adopted Test on Logical Operations (TLO) and descriptive research design to describe the cognitive skills achievement and to determine the affecting factors. Overall, elementary pre-service teachers performed with sufficient understanding in dealing (...) with the TLO. However, in each logical operation, students were found to have insufficient understanding of probability thinking and correlational thinking. These insufficiencies in other logical operations were attributed to their misconceptions on some mathematical terms, misinterpretation of the problems, poor comprehension skills, poor problem-solving skills, and poor performance in mathematics in general. (shrink)

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

Many students consider mathematics as the most dreaded subject in their curriculum, so much so that the term “math phobia” or “math anxiety” is practically a part of clinical psychological literature. This symptom is widespread and students suffer mental disturbances when facing mathematical activity because understanding mathematics is a great task for them. This paper described the students’ cognitive skills performance in Basic Mathematics based on the following logical operations: Classification, Seriation, Logical Multiplication, Compensation, Ratio and Proportional Thinking, Probability (...) Thinking and Correlation Thinking as it serves a critical element of teaching and learning, that is determining the current position of the learners’ mathematical capacity. Its implications for mathematics education were also revealed. A descriptive quantitative design was used in this study. The study included 1011 first-year college students from six state universities in the Philippines who were enrolled in the Bachelor of Science of Secondary Education Program during the first semester of the Academic Year 2019-2020. This study made use of the teacher-made test called the Test on Logical Operations. Findings revealed that the cognitive skills achievement of the first year BSE students from different state universities in the Philippines falls under the category of late concrete operational stage. As a result, students are unable to perform the logical operational skills expected of their age. At their age level, they are expected to be under the formal operational stage based on Piaget’s stages of cognitive development. These findings revealed a weak mathematical education foundation among students which requires immediate attention. This reality should be recognized by educational planners and implementers when making curricular and other instructional decisions. (shrink)

That the state-of-affairs of cognitive science is not good is brought into figural salience in "What happened to cognitive science?" (Núñez et al., 2019). We extend their objective description of 'what's wrong' to a prescription of 'how to correct'. Cognitive science, in its quest to elucidate 'how we know', embraces a long list of subjects, while ignoring Mathematics (Fig. 1a, Núñez et al., 2019). Mathematics is known for making the unknown to be known (cf. solving for unknowns). This acknowledgement naturally (...) raises the question: does mathematical knowing inform knowing in general? Here we show that drawing parallels to mathematical knowing can facilitate the advancement of cognitive science (Lawvere, 1994). (shrink)

In this paper I study the development of arithmetical cognition with the focus on metaphorical thinking. In an approach developing on Lakoff and Núñez, I propose one particular conceptual metaphor, the Process → Object Metaphor, as a key element in understanding the development of mathematical thinking.

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

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

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)

Recent experimental evidence from developmental psychology and cognitive neuroscience indicates that humans are equipped with unlearned elementary mathematical skills. However, formal mathematics has properties that cannot be reduced to these elementary cognitive capacities. The question then arises how human beings cognitively deal with more advanced mathematical ideas. This paper draws on the extended mind thesis to suggest that mathematical symbols enable us to delegate some mathematical operations to the external environment. In this view, mathematical symbols (...) are not only used to express mathematical concepts—they are constitutive of the mathematical concepts themselves. Mathematical symbols are epistemic actions, because they enable us to represent concepts that are literally unthinkable with our bare brains. Using case-studies from the history of mathematics and from educational psychology, we argue for an intimate relationship between mathematical symbols and mathematical cognition. (shrink)

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

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)

We describe (essential features and an axiomatization of) a new metamathematical (cognitive) ability, i.e., functional conceptual substratum, used implicitly in the generation of several mathematical proofs and definitions, and playing a fundamental role in Artificial Mathematical Intelligence (or Cognitive-computational metamathematics). Furthermore, we present an initial (first-order) formalization of this mechanism together with its characterizing relation with classic notions like primitive positive definability and Diophantiveness. Additionally, we analyze the semantic variability of functional conceptual substratum when small syntactic modifications are (...) done. Finally, we describe cognitively natural inference rules for (mathematical) definitions inspired by functional conceptual substratum and we show that they are sound and complete w.r.t.\ standard calculi. (shrink)

This paper aims to clarify Merleau-Ponty’s contribution to an embodied-enactive account of mathematical cognition. I first identify the main points of interest in the current discussions of embodied higher cognition and explain how they relate to Merleau-Ponty and his sources, in particular Husserl’s late works. Subsequently, I explain these convergences in greater detail by more specifically discussing the domains of geometry and algebra and by clarifying the role of gestalt psychology in Merleau-Ponty’s account. Beyond that, I explain (...) how, for Merleau-Ponty, mathematical cognition requires not only the presence and actual manipulation of some concrete perceptible symbols but, more strongly, how it is fundamentally linked to the structural transformation of the concrete configurations of symbolic systems to which these symbols appertain. Furthemore, I fill a gap in the literature by explaining Merleau-Ponty’s claim that these structural transformations are operated through motor intentionality. This makes it possible, in turn, to contrast Merleau-Ponty’s approach to ontologically idealistic and realistic views on mathematical objects. On Merleau-Ponty’s account, mathematical objects are relational entities, that is, gestalts that necessarily imply situated cognizers to whom they afford a specific type of engagement in the world and on whom they depend in their eventual structural transformations. I argue that, by attributing a strongly constitutive role to phenomenal configurations and their motor transformation in mathematical thinking, Merleau-Ponty contributes to clarifying the worldly, historical, and socio-cultural aspects of mathematical truths without compromising what we perceive as their universality, certainty, and necessity. (shrink)

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

The paper discusses Hilbert mathematics, a kind of Pythagorean mathematics, to which the physical world is a particular case. The parameter of the “distance between finiteness and infinity” is crucial. Any nonzero finite value of it features the particular case in the frameworks of Hilbert mathematics where the physical world appears “ex nihilo” by virtue of an only mathematical necessity or quantum information conservation physically. One does not need the mythical Big Bang which serves to concentrate all the violations (...) of energy conservation in a “safe”, maximally remote point in the alleged “beginning of the universe”. On the contrary, an omnipresent and omnitemporal medium obeying quantum information conservation rather than energy conservation permanently generates action and thus the physical world. The utilization of that creation “ex nihilo” is accessible to humankind, at least theoretically, as long as one observes the physical laws, which admit it in their new and wider generalization. One can oppose Hilbert mathematics to Gödel mathematics, which can be identified as all the standard mathematics until now featureable by the Gödel dichotomy of arithmetic to set theory: and then, “dialectic”, “intuitionistic”, and “Gödelian” mathematics within the former, according to a negative, positive, or zero value of the distance between finiteness and infinity. A mapping of Hilbert mathematics into pseudo-Riemannian space corresponds, therefore allowing for gravitation to be interpreted purely mathematically and ontologically in a Pythagorean sense. Information and quantum information can be involved in the foundations of mathematics and linked to the axiom of choice or alternatively, to the field of all rational numbers, from which the pair of both dual and anti-isometric Peano arithmetics featuring Hilbert arithmetic are immediately inferable. Noether’s theorems (1918) imply quantum information conservation as the maximally possible generalization of the pair of the conservation of a physical quantity and the corresponding Lie group of its conjugate. Hilbert mathematics can be interpreted from their viewpoint also after an algebraic generalization of them. Following the ideas of Noether’s theorem (1918), locality and nonlocality can be realized both physically and mathematically. The “light phase of the universe” can be linked to the gap of mathematics and physics in the Cartesian organization of cognition in Modernity and then opposed to its “dark phase”, in which physics and mathematics are merged. All physical quantities can be deduced from only mathematical premises by the mediation of the most fundamental physical constants such as the speed of light in a vacuum, the Planck and gravitational constants once they have been interpreted by the relation of locality and nonlocality. (shrink)

One of the important challenges in the philosophy of mathematics is to account for the semantics of sentences that express mathematical propositions while simultaneously explaining our access to their contents. This is Benacerraf’s Dilemma. In this dissertation, I argue that cognitive science furnishes new tools by means of which we can make progress on this problem. The foundation of the solution, I argue, must be an ontologically realist, albeit non-platonist, conception of mathematical reality. The semantic portion of the (...) problem can be addressed by accepting a Chomskyan conception of natural languages and a matching internalist, mentalist and nativist view of semantics. A helpful perspective on the epistemic aspect of the puzzle can be gained by translating Kurt G ̈odel’s neo-Kantian conception of the nature of mathematics and its objects into modern, cognitive terms. (shrink)

This chapter tries to answer the following question: How should we conceive of the method of mathematics, if we take a naturalist stance? The problem arises since mathematical knowledge is regarded as the paradigm of certain knowledge, because mathematics is based on the axiomatic method. Moreover, natural science is deeply mathematized, and science is crucial for any naturalist perspective. But mathematics seems to provide a counterexample both to methodological and ontological naturalism. To face this problem, some authors tried to (...) naturalize mathematics by relying on evolutionism. But several difficulties arise when we try to do this. This chapter suggests that, in order to naturalize mathematics, it is better to take the method of mathematics to be the analytic method, rather than the axiomatic method, and thus conceive of mathematical knowledge as plausible knowledge. (shrink)

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

There is a noticeable gap between results of cognitive neuroscientific research into basic mathematical abilities and philosophical and empirical investigations of mathematics as a distinct intellectual activity. The paper explores the relevance of a Wittgensteinian framework for dealing with this discrepancy.

In light of the observations and research design presented in the previous reports, the current technical report is focused on the relationship between the quality and specificity of the content of the gambling sites and the site’s SEO and marketing policy. This relationship is dependent upon the category of the gambling site and the difference in content quality, and the degree to which the mathematical dimension of gambling is reflected in this content is explained by this dependence.

Elijah Chudnoff’s case for irreducible cognitive phenomenology hinges on seeming to see the truth of a mathematical proposition (Chudnoff 2015). In the following, I develop an augmented version of Chudnoff’s case, not based on seeming to see, or intuition, but based on being in a state with presentational phenomenology of high-level content. In contrast to other cases for cognitive phenomenology, those based on Strawson’s case (Strawson 2011), I argue that the case presented here is able to withstand counterarguments, which (...) attempt to reduce cognitive phenomenology to sensory phenomenology. To support my argument, I present findings from Bowden and Jung-Beeman’s experiments with the Aha! Experience (Bowden & Jung-Beeman 2004), and argue that the Aha! Experience is a species of the experience of understanding presented here. I interpret the results of these experiments to provide further evidence for irreducible cognitive phenomenology. (shrink)

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

In this paper I discuss Mark Steiner’s view of the contribution of mathematics to physics and take up some of the questions it raises. In particular, I take up the question of discovery and explore two aspects of this question – a metaphysical aspect and a related epistemic aspect. The metaphysical aspect concerns the formal structure of the physical world. Does the physical world have mathematical or formal features or constituents, and what is the nature of these constituents? The (...) related epistemic question concerns humans’ cognitive ability to reach the formal structure of the physical world. Among other things, I explore the interaction of mathematical and non-mathematical cognition in physical discovery. (shrink)

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

The main purpose of the study is to understand the mathematical problem-solving strategies among student teachers. This study used both quantitative and qualitative type of research. Aside from the semi-structured interviews, data were gathered through participant's actual mathematical problem-solving outputs and the videotaped interviews. Findings revealed that the problem-solving strategies among student teachers in the Problem-Solving subject are cognitive, metacognitive and other strategies. The cognitive strategies used by the student teachers are rehearsal, elaboration, and organization. The metacognitive strategies (...) are critical thinking and self-regulation. Other strategies are overlapping the cognitive and metacognitive strategies. These are prediction/orientation, planning, monitoring, and evaluating. The findings also suggest a significant influence of the strategies to the academic performance of the student teachers. (shrink)

2nd edition. Many-valued logics are those logics that have more than the two classical truth values, to wit, true and false; in fact, they can have from three to infinitely many truth values. This property, together with truth-functionality, provides a powerful formalism to reason in settings where classical logic—as well as other non-classical logics—is of no avail. Indeed, originally motivated by philosophical concerns, these logics soon proved relevant for a plethora of applications ranging from switching theory to cognitive modeling, and (...) they are today in more demand than ever, due to the realization that inconsistency and vagueness in knowledge bases and information processes are not only inevitable and acceptable, but also perhaps welcome. The main modern applications of (any) logic are to be found in the digital computer, and we thus require the practical knowledge how to computerize—which also means automate—decisions (i.e. reasoning) in many-valued logics. This, in turn, necessitates a mathematical foundation for these logics. This book provides both these mathematical foundation and practical knowledge in a rigorous, yet accessible, text, while at the same time situating these logics in the context of the satisfiability problem (SAT) and automated deduction. The main text is complemented with a large selection of exercises, a plus for the reader wishing to not only learn about, but also do something with, many-valued logics. (shrink)

Siyaves Azeri (2020) quite well shows that arithmetical thinking emerges on the basis of specific social practices and material engagement (clay tokens for economic exchange practices beget number concepts, e.g.). But his discussion here is relegated mostly to Neolithic and Bronze Age practices. While surely such practices produced revolutions in the cognitive abilities of many humans, much of the cognitive architecture that allows normative conceptual thought was already in place long before this time. This response, then, is an attempt to (...) sketch the deep prehistory of human cognition in order to show the inter-social bases of normative thought in general. To do this, I will look first to the work of Vygotsky and Leontiev, two often neglected psychologists whose combined efforts culminate in a developmental account of human cognitive origins. Then, I will review some key insights from the contemporary comparative psychologist Michael Tomasello—whose project is admittedly a Vygotskian one—in order to further shed light on the social-practical basis of abstract thought, of which mathematical cognition is surely a part. (shrink)

The Computational Theory of Mind (CTM) holds that cognitive processes are essentially computational, and hence computation provides the scientific key to explaining mentality. The Representational Theory of Mind (RTM) holds that representational content is the key feature in distinguishing mental from non-mental systems. I argue that there is a deep incompatibility between these two theoretical frameworks, and that the acceptance of CTM provides strong grounds for rejecting RTM. The focal point of the incompatibility is the fact that representational content is (...) extrinsic to formal procedures as such, and the intended interpretation of syntax makes no difference to the execution of an algorithm. So the unique 'content' postulated by RTM is superfluous to the formal procedures of CTM. And once these procedures are implemented in a physical mechanism, it is exclusively the causal properties of the physical mechanism that are responsible for all aspects of the system's behaviour. So once again, postulated content is rendered superfluous. To the extent that semantic content may appear to play a role in behaviour, it must be syntactically encoded within the system, and just as in a standard computational artefact, so too with the human mind/brain - it's pure syntax all the way down to the level of physical implementation. Hence 'content' is at most a convenient meta-level gloss, projected from the outside by human theorists, which itself can play no role in cognitive processing. (shrink)

If there is an area of discourse in which disagreement is virtually absent, it is mathematics. After all, mathematicians justify their claims with deductive proofs: arguments that entail their conclusions. But is mathematics really exceptional in this respect? Looking at the history and practice of mathematics, we soon realize that it is not. First, deductive arguments must start somewhere. How should we choose the starting points (i.e., the axioms)? Second, mathematicians, like the rest of us, are fallible. Their ability to (...) recognize whether a putative proof is correct is not infallible. In most cases, disagreement over the correctness of a putative proof is, however, evanescent. Once an error is spotted and communicated, the disagreement disappears. But this is not always the case. Sometimes it is recalcitrant; that is, it persists over time. In order to zoom in on this type of disagreement and explain its very possibility, we focus on a single case study: a decades-long (1921-1949) controversy between Federigo Enriques and Francesco Severi, two prominent exponents of the Italian school of algebraic geometry. We suggest that the instability of the mathematical community to which they belonged can be explained by the gap between an abstract criterion of rigor and local criteria of acceptability. It is this instability that made the existence of recalcitrant disagreement over putative proofs possible. We do not condemn speculative mathematics but rather its pretense of being rigorous mathematics. In this respect, we show that the overly self-confident Severi and the more intuitive, visionary Enriques had a completely different attitude. (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)

In the course of daily life we solve problems often enough that there is a special term to characterize the activity and the right to expect a scientific theory to explain its dynamics. The classical view in psychology is that to solve a problem a subject must frame it by creating an internal representation of the problem’s structure, usually called a problem space. This space is an internally generable representation that is mathematically identical to a graph structure with nodes and (...) links. The nodes can be annotated with useful information, and the whole representation can be distributed over internal and external structures such as symbolic notations on paper or diagrams. If the representation is distributed across internal and external structures the subject must be able to keep track of activity in the distributed structure. Problem solving proceeds as the subject works from an initial state in mentally supported space, actively constructing possible solution paths, evaluating them and heuristically choosing the best. Control of this exploratory process is not well understood, as it is not always systematic, but various heuristic search algorithms have been proposed and some experimental support has been provided for them. (shrink)

The dominant approach to analyzing the meaning of natural language sentences that express mathematical knowl- edge relies on a referential, formal semantics. Below, I discuss an argument against this approach and in favour of an internalist, conceptual, intensional alternative. The proposed shift in analytic method offers several benefits, including a novel perspective on what is required to track mathematical content, and hence on the Benacerraf dilemma. The new perspective also promises to facilitate discussion between philosophers of mathematics and (...) cognitive scientists working on topics of common interest. (shrink)

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

I identify two reasons for believing in the objectivity of mathematical knowledge: apparent objectivity and applications in science. Focusing on arithmetic, I analyze platonism and cognitive nativism in terms of explaining these two reasons. After establishing that both theories run into difficulties, I present an alternative epistemological account that combines the theoretical frameworks of enculturation and cumulative cultural evolution. I show that this account can explain why arithmetical knowledge appears to be objective and has scientific applications. Finally, I will (...) argue that, while this account is compatible with platonist metaphysics, it does not require postulating mind-independent mathematical objects. (shrink)

According to the received view, genuine mathematical justification derives from proofs. In this article, I challenge this view. First, I sketch a notion of proof that cannot be reduced to deduction from the axioms but rather is tailored to human agents. Secondly, I identify a tension between the received view and mathematical practice. In some cases, cognitively diligent, well-functioning mathematicians go wrong. In these cases, it is plausible to think that proof sets the bar for justification too high. (...) I then propose a fallibilist account of mathematical justification. I show that the main function of mathematical justification is to guarantee that the mathematical community can correct the errors that inevitably arise from our fallible practices. (shrink)

Guided by key insights of the four great philosophers mentioned in the title, here, in review of and expanding on our earlier work (Burchard, 2005, 2011), we present an exposition of the role played by language, & in the broader sense, λογοζ, the Logos, in how the CNS, the brain, is running the human being. Evolution by neural Darwinism has been forcing the linguistic nature of mind, enabling it to overcome & exploit the cognitive gap between an animal and its (...) world by recognizing environmental structures. Our work was greatly influenced by Heidegger’s lecture notes on metaphysics (Heidegger, 1935). We found agreement with recent progress in neuroscience, but also mathematical foundations of language theory, equating Logos with the mathematical concept of structure. The mystery of perception across the gap is analyzed as radiation and molecules impinging on sensory neurons that carry linguistic information about gross environmental structures, and only remotely about the physical reality of elementary particles. The most important logical brain function is Ego or Self, guiding the workings of the brain as a logos machine. Ego or Self operates from neurons in frontopolar cortex with global receptive fields. The logos machine can function only by availing itself of global context, its internally stored noumenal cosmos NK, and the categorical-conceptual apparatus CCA, updated continually through the neural default mode network (Raichle, 2005). In the Transcendental Deduction, Immanuel Kant discovered that Ego or Self is responsible for conscious control in perception relying on concepts & categories for a fitting percept to be incorporated intoNK. The entire CNS runs as a “movie-in-the-brain” (Parvizi & Damasio, 2001), at peak speed processing simultaneously in a series of cortical centers a stack of up to twelve frames in gamma rhythm of 25 ms intervals. We equate global context, or NK, with our human world, Heidegger’s Dasein being-in-the-world, and are able to demonstrate that the great philosopher in EM parallels neuro-science concerning the human mind. (shrink)

Why study notations, diagrams, or more broadly the variety of nonverbal “representations” or “signs” that are used in mathematical practice? This chapter maps out recent work on the topic by distinguishing three main philosophical motivations for doing so. First, some work (like that on diagrammatic reasoning) studies signs to recover norms of informal or historical mathematical practices that would get lost if the particular signs that these practices rely on were translated away; work in this vein has the (...) potential to broaden our view of the specific normativity of mathematics beyond logical proof. Second, some work focuses on signs to highlight the open-ended and “nonrigorous” aspects of mathematical practice, and the role of these aspects in the historical development of mathematics. The third motivation is based on the following observation: the reason differences in signs matter in mathematics is that humans are finite agents, with cognitive and computational limitations. Accordingly, studying signs can be a way to study how human computational constraints shape the mathematics that we do, and to tackle the topic of mathematical understanding. (shrink)

To account for the explanatory role representations play in cognitive science, Egan’s deflationary account introduces a distinction between cognitive and mathematical contents. According to that account, only the latter are genuine explanatory posits of cognitive-scientific theories, as they represent the arguments and values cognitive devices need to represent to compute. Here, I argue that the deflationary account suffers from two important problems, whose roots trace back to the introduction of mathematical contents. First, I will argue that mathematical (...) contents do not satisfy important and widely accepted desiderata all theories of content are called to satisfy, such as content determinacy and naturalism. Secondly, I will claim that there are cases in which mathematical contents cannot play the explanatory role the deflationary account claims they play, proposing an empirical counterexample. Lastly, I will conclude the paper highlighting two important implications of my arguments, concerning recent theoretical proposals to naturalize representations via physical computation, and the popular predictive processing theory of cognition. (shrink)

The essential role of language in rational cognition is analysed. The approach is functional: only the results of the connection between language, reality, and thinking are considered. Scientific language is analysed as an extension and improvement of everyday language. The analysis gives a uniform view of language and rational cognition. The consequences for the nature of ontology, truth, logic, thinking, scientific theories, and mathematics are derived.