The creation of paraconsistent logics have expanded the boundaries of formal logic by introducing coherent systems that tolerate contradictions without triviality. Thanks to their novel approach and rigorous formalization they have already found many applications in computer science, linguistics and mathematics. As a natural next step, some philosophers have also tried to answer the question if human everyday reasoning could be accurately modelled with paraconsistent logics. The purpose of this article is to argue against the notion that human reasoning (...) is paraconsistent. Numerous findings in the area of cognitive psychology and cognitive neuroscience go against the hypothesis that humans tolerate contradictions in their inferences. Humans experience severe stress and confusion when confronted with contradictions. Experiments on the ways in which humans process contradictions point out that the first thing humans do is remove or modify one of the contradictory statements. From an evolutionary perspective, contradiction is useless and even more dangerous than lack of information because it takes up resources to process. Furthermore, it appears that when logicians, anthropologists or psychologists provide examples of contradictions in human culture and behaviour, their examples very rarely take the form of: \\). Instead, they are often conditional statements, probabilistic judgments, metaphors or seemingly incompatible beliefs. At different points in time humans are definitely able to hold contradictory beliefs, but within one reasoning leading to a particular behaviour, contradiction is never tolerated. (shrink)

This book offers an innovative introduction to the psychological basis of mathematics and the nature of mathematical thinking and learning, using an approach that empowers students by fostering their own construction of mathematical structures. Through accessible and engaging writing, award-winning mathematician and educator Anderson Norton reframes mathematics as something that exists first in the minds of students, rather than something that exists first in a textbook. By exploring the psychological basis for mathematics at every level (...) - including geometry, algebra, calculus, complex analysis and more - Norton unlocks students' personal power to construct mathematical objects based on their own mental activity, and illustrates the power of mathematics in organizing the world as we know it. Including reflections and activities designed to inspire awareness of the mental actions and processes coordinated in practicing mathematics, the book is geared towards current and future secondary and elementary mathematics teachers who will empower the next generation of mathematicians and STEM majors. Those interested in the history and philosophy that underpins mathematics will also benefit from this book, as well as those informed and curious minds attentive to the human experience more generally. Anderson Norton is Professor in the Department of Mathematics at Virginia Tech, USA, where he has been teaching mathematics courses for future teachers for 15 years. He is the editor of Constructing Number, and the co-author of Developing Fractions Knowledge and Numeracy for All Learners. (shrink)

The contemplation of the heavenly vault suggests to any generalizing intelligence the idea of the world suspended within this concavity. During rude barbaric times such as the High Middle Ages – the records of which are precious in this respect – it was thought that this vault rested on the surface of the Earth like a glass bell. And when experience gained firstly by terrestrial travels and afterwards by circumnavigation showed that this was a delusory phenomenon and, at the same (...) time, that the Earth was an autonomous sphere floating within space, the [heavenly] vault became in turn a hollow and translucent sphere containing the concentric world, like the white of an egg surrounding the yolk. Cosmological experience then revealed that the stars were situated at different heights in the supposed vault, which compelled imagining new concentric spheres. Finally, it was discovered that there are neither such spheres nor such a vault; that the amplification and multiplicity of the latter were illusions – as was the concave aspect of the sky – and that the space enclosing the universe is an abyss. (shrink)

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)

The fifteenth Journal for Research in Mathematics Education monograph had its origins in a conference titled An Interdisciplinary Conference on Assessment in K--12 Mathematics: Collaborations Between Mathematics Education and Psychometrics, which was held in 2011 in Atlanta, Georgia. The basis for the conference was the renaissance in the field of psychometrics in which an increasing variety of psychometric models are becoming available through advances in computer hardware and software. This is opening new avenues for studying the mathematical (...) knowledge of teachers and students. The overarching purpose of the monograph is to guide further interdisciplinary collaborations between mathematics education researchers and psychometricians by examining theoretical and conceptual issues that have arisen in recent efforts to apply contemporary psychometric models to mathematics education research. Specifically, its chapters are intended to (a) illustrate for mathematics education researchers the two main categories of psychometric models-models that locate individuals along continua and models that place individuals into discrete groups, as well as hybrids of these approaches; (b) provide examples that apply these different categories of psychometric models to mathematics education research; (c) illustrate how researchers have selected different psychometric models on the basis of the researchers' goals and (d) demonstrate issues related to item development. With these goals in mind, the monograph will enhance an awareness among mathematics education researchers that it is increasingly possible to select from a variety of model options when pursuing particular research goals and that choosing among models involves trade-offs. (shrink)

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)

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)

Mathematicians and physical scientists depend heavily on the formal symbolism of mathematics in order to express and develop their theories. For this and other reasons the last hundred years has seen a growing interest in the nature of formal language and the way it expresses meaning; particularly the objective, shared aspect of meaning as opposed to subjective, personal aspects. This dichotomy suggests the question: do the objective philosophical theories of meaning offer concepts which can be applied in psychological (...) theories of meaning? In recent years cognitive scientists such as Chomsky [1980], Fodor [1981] and MacNamara [1982] have used philosophical approaches to the meaning of formal language expressions as the basis for their psychological theories. Following this lead it seems appropriate to review some of the main treatments of meaning with a view to their transferability. (shrink)

I. Prelude -- About this Book -- II. School Mathematics and Its Consequences -- The Foundations of Mathematical Thinking -- Compression, Connection and Blending of Mathematical Ideas -- Set-befores, Met-befores and Long-term Learning -- Mathematics and the Emotions -- The Three Worlds of Mathematics -- Journeys through Embodiment and Symbolism -- Problem-Solving and Proof -- III. Interlude -- The Historical Evolution of Mathematics -- IV. University Mathematics and Beyond -- The Transition to Formal Knowledge -- (...) Blending Knowledge Structures in the Calculus -- Expert Thinking and Structure Theorems -- Contemplating the Infinitely Large and the Infinitely Small -- Expanding the Frontiers through Mathematical Research -- Reflections -- Appendix: Where the Ideas Came From. (shrink)

If mathematics is the purest form of knowledge, the perfect foundation of all the hard sciences, and a uniquely precise discipline, then how can the human brain, an imperfect and imprecise organ, process mathematical ideas? Is mathematics made up of eternal, universal truths? Or, as some have claimed, could mathematics simply be a human invention, a kind of tool or metaphor? These questions are among the greatest enigmas of science and epistemology, discussed at length by mathematicians, physicians, (...) and philosophers. But, curiously enough, neuroscientists have been absent in the debate, even though it is precisely the field of neuroscience—which studies the brain’s mechanisms for thinking and reasoning—that ought to be at the very center of these discussions. How our Emotions and Bodies are Vital for Abstract Thought explores the unique mechanisms of cooperation between the body, emotions, and the cortex, based on fundamental physical principles. It is these mechanisms that help us to overcome the limitations of our physiology and allow our imperfect, human brains to make transcendent mathematical discoveries. This book is written for anyone who is interested in the nature of abstract thought, including mathematicians, physicists, computer scientists, psychologists, and psychiatrists. (shrink)

Professor Michael Edgeworth McIntyre is an eminent scientist who has also had a part-time career as a musician. From a lifetime's thinking, he offers this extraordinary synthesis exposing the deepest connections between science, music, and mathematics, while avoiding equations and technical jargon. He begins with perception psychology and the dichotomization instinct and then takes us through biological evolution, human language, and acausality illusions all the way to the climate crisis and the weaponization of the social media, and beyond that (...) into the deepest parts of theoretical physics - demonstrating our unconscious mathematical abilities. He also has an important message of hope for the future. Contrary to popular belief, biological evolution has given us not only the nastiest, but also the most compassionate and cooperative parts of human nature. This insight comes from recognizing that biological evolution is more than a simple competition between selfish genes. Rather, he suggests, in some ways it is more like turbulent fluid flow, a complex process spanning a vast range of timescales. Professor McIntyre is a Fellow of the Royal Society of London (FRS) and has worked on problems as diverse as the Sun's magnetic interior, the Antarctic ozone hole, jet streams in the atmosphere, and the psychophysics of violin sound. He has long been interested in how different branches of science can better communicate with each other and with the public, harnessing aspects of neuroscience and psychology that point toward the deep 'lucidity principles' that underlie skilful communication. (shrink)

Contemporary psychological research that studies how people apply mathematics has largely viewed mathematics as a computational tool for deriving an answer. The tacit assumption has been that people first understand a situation, and then choose which computations to apply. We examine an alternative assumption that mathematics can also serve as a tool that helps one to construct an understanding of a situation in the first place. Three studies were conducted with 6th‐grade children in the context of (...) proportional situations because early proportional reasoning is a premier example of where mathematics may provide new understanding of the world. The children predicted whether two differently‐sized glasses of orange juice would taste the same when they were filled from a single carton of juice made from concentrate and water. To examine the relative contributions and interactions of situational and mathematical knowledge, we manipulated the formal features of the problem display (e.g., diagram vs. photograph) and the numerical complexity (e.g., divisibility) of the containers and the ingredient ratios. When the problem was presented as a diagram with complex numbers, or “realistically” with easy numbers, the children predicted the glasses would taste different because one glass had more juice than the other. But, when the problem was presented realistically with complex numbers, the children predicted the glasses would taste the same on the basis of empirical knowledge (e.g., “Juice can't change by itself”). And finally, when the problem was presented as a diagram with easy numbers, the children predicted the glasses would taste the same on the basis of proportional relations. These complex interactions illuminate how mathematical and empirical knowledge can jointly constrain the construction of a new understanding of the world. We propose that mathematics helped in the case of successful proportional reasoning because it made a complex empirical situation cognitively tractable, and thereby helped the children construct mental models of that situation. We sketch one aspect of the mental models that are constructed in the domain of quantity—a preference for specificity—that helps explain the current findings. (shrink)

Studies within mathematics education have shown the existence of a so-called “mathematics anxiety” among secondary and high school students as a global phenomenon and have related it to various social and psychological aspects of the processes of teaching and learning mathematics, which are responsible for the understanding of mathematics. Current research project aims at crystallizing a primary conceptual and theoretical framework necessary for the application of contributions of philosophy of mathematics and of science and (...) epistemology of mathematics in mathematics education from an epistemological perspective. It argues for and discusses applicability of the general principle that teaching about mathematics along with traditionally teaching formal mathematics and ‘doing’ mathematics does not render the process more difficult, but rather, this approach enhances mathematics education, with effects on the mathematics anxiety. -/- . (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)

Mathematics is one of the core STEM (science, technology, engineering, and mathematics) subject disciplines. Engaging students in learning mathematics helps retain students in STEM fields and thus contributes to the sustainable development of society. To increase student engagement, some mathematics instructors have redesigned their courses using the flipped classroom approach. In this review, we examined the results of comparative studies published between 2011 and 2020 to summarize the effects of this instructional approach (vs. traditional lecturing) on (...) students’ behavioral, emotional, and cognitive engagement with mathematics courses. Thirty-three articles in K–12 and higher education contexts were included for analysis. The results suggest that the use of the flipped classroom approach may increase some aspects of behavioral engagement (e.g., interaction and attention/participation), emotional engagement (e.g., course satisfaction), and cognitive engagement (e.g., understanding of mathematics). However, we discovered that several aspects (e.g., students’ attendance, mathematics anxiety, and self-regulation) of student engagement have not been thoroughly explored and are worthy of further study. The results of this review have important implications for future flipped classroom practice (e.g., engaging students in solving real-world problems), and for research on student engagement (e.g., using more objective measures, such as classroom observation) in mathematics education. (shrink)

Seeing-as and Novelty brings together new essays that consider Wittgenstein’s treatment of the phenomenon of aspect perception in relation to the broader idea of conceptual novelty; that is, the acquisition or creation of new concepts, and the application of an acquired understanding in unfamiliar or novel situations. Over the last twenty years, aspect perception has received increasing philosophical attention, largely related to applying Wittgenstein’s remarks on the phenomena of seeing-as, found in Part II of Philosophical Investigations , to issues within (...) philosophical aesthetics. Seeing-as, however, has come to occupy a broader conceptual category, particularly in philosophy of mind and philosophical psychology. The essays in this volume examine the exegetical issues arising within Wittgenstein studies, while also considering the broader utility and implications of the phenomenon of seeing-as in the fields of aesthetics, philosophical psychology, and philosophy of mathematics, with a thematic focus on questions of novelty and creativity. The collection constitutes a fruitful interpretative engagement with the later Wittgenstein, as well as a unique contribution to considerations of philosophical methodology. (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)

Teaching mathematics and improving mathematics competence are pending subjects within our educational system. The PEIM (Programa Evolutivo Instruccional para Matemáticas), a constructivist intervention program for the improvement of mathematical performance, affects the different agents involved in math learning, guaranteeing a significant improvement in students’ performance. The program is based on the following pillars: (a) students become the main agents of their learning by constructing their own knowledge; (b) the teacher must be the guide to facilitate and guarantee such (...) a construction by being a great connoisseur of the fundamental aspects of the development of the child’s mathematical thinking; (c) the mathematical contents must be sequenced in terms of the complexity and significance for the student as well as contextualized at all times; and (d) the classroom must have a constructivist climate highlighting cooperative work among students. The implementation of PEIM along with the empirical evaluation conducted in several centers in Madrid and Zaragoza (Spain) confirm how students improve their mathematical competence. Both first- and second-grade students in elementary education were far more effective in solving problems, highlighting the use of more advanced strategies in their resolution and a lower incidence of conceptual errors. Moreover, it was possible to verify how the students proving greater difficulty, experienced an evolution in learning similarly to those who did not present it. The program provides customized education to allow the teacher to know at all times how he should be more influential on the students’ learning through mathematical profiles. Both teaching practice and teachers were observed, being that of the experimental group more prone to analyzing processes and allowing the construction of knowledge by students, due to their psycho-developmental training. As a result, we found several improvements through the implementation of the program that may serve, for upcoming years, as a basis for the necessary changes in the teaching of mathematics. (shrink)

This volume, edited by three leading proponents and practitioners of human science psychology, serves as an invitation to readers new to this approach while also renewing that invitation to those who have long embraced and advanced research in the field from this perspective. It is a timely and important invitation. In 2009, the American Psychological Association declared psychology to be a core STEM (science, technology, engineering, mathematics) discipline and advocated the teaching and practice of psychology with this natural (...) science understanding in mind, but in 2014 further reaffirmed alternative methods by adding a new journal, Qualitative Psychology. The varied essays in this volume, certainly, bolster the view that a purely STEM-centered vision would ignore much about the very experience of being human.In fact, it would be dangerous to rely solely on the methods of the natural sciences to study human beings, who operate in the realm of meanings, lived experience, and complex and complicated relationships with self and others. We create societies and belief systems, orient ourselves in time, experience beauty and pain. The Qualitative Vision for Psychology: An Invitation to a Human Science Approach argues that because we have aspects that are distinctly and uniquely human -- we are not rats, hydrogen, or rocks, for example -- this necessitates a distinctly human science, one that regards persons as humans rather than objects of study. The laws and formulas of the natural sciences simply do not take into account that particularly human way of being in the world.There are few comprehensive books on psychology conceived as a human science, even though it has a long history with roots in phenomenology, existentialism, psychoanalysis, humanistic psychology, and hermeneutics. In recent years, as these essays discuss, the field has been transformed through its contact with feminism, critical historical analysis, and deconstruction, and it has continued to examine new challenges. Further, we see here its specific applicability to issues as diverse as empathy, cultural history, apartheid, sexual assault, fetishes, and our natural environment. (shrink)

This book unites the worlds of physics and depth psychology through analysis of carefully selected existing and new dream materials. Their interpretation by Matthews provides fertile ground for the unifying of the extreme opposites of psyche and matter and forms a continuation of the deep dialogue between acclaimed psychologist Carl Jung and Nobel physicist Wolfgang Pauli. What emerges is an individuation process where inner and outer worlds are intertwined through a succession of dream images, culminating with that of the ring (...) i, the mathematical function at the heart of quantum physics. This mysterious function unites wave and particle and symbolically carries the quality of paradox. The occurrence of the ring i in Pauli's and the author's dreams suggests paradox is a necessary psychological state to experience a living union between psyche and matter. Analysis of accompanying materials further indicates the arising of a new world view where inner and outer, mind and matter, may again be seen as a unified whole. This book is an engaging read for academics and researchers in the field of Jungian psychology and will appeal to those interested in the novel application of quantum physics to philosophy, psychology and spirituality. (shrink)

It is a well-known fact that mathematics plays a crucial role in physics; in fact, it is virtually impossible to imagine contemporary physics without it. But it is questionable whether mathematical concepts could ever play such a role in psychology or philosophy. In this paper, we set out to examine a rather unobvious example of the application of topology, in the form of the theory of persons proposed by Kurt Lewin in his Principles of Topological Psychology. Our aim is (...) to show that this branch of mathematics can furnish a natural conceptual system for Gestalt psychology, in that it provides effective tools for describing global qualitative aspects of the latter’s object of investigation. We distinguish three possible ways in which mathematics can contribute to this: explanation, explication and metaphor. We hold that all three of these can be usefully characterized as throwing light on their subject matter, and argue that in each case this contrasts with the role of explanations in physics. Mathematics itself, we argue, provides something different from such explanations when applied in the field of psychology, and this is nevertheless still cognitively fruitful. (shrink)

The aim of this paper is to show that (elementary) topology may be useful for dealing with problems of epistemology and metaphysics. More precisely, I want to show that the introduction of topological structures may elucidate the role of the spatial structures (in a broad sense) that underly logic and cognition. In some detail I’ll deal with “Cassirer’s problem” that may be characterized as an early forrunner of Goodman’s “grue-bleen” problem. On a larger scale, topology turns out to be useful (...) in elaborating the approach of conceptual spaces that in the last twenty years or so has found quite a few applications in cognitive science, psychology, and linguistics. In particular, topology may help distinguish “natural” from “not-so-natural” concepts. This classical problem that up to now has withstood all efforts to solve (or dissolve) it by purely logical methods. Finally, in order to show that a topological perspective may also offer a fresh look on classical metaphysical problems, it is shown that Leibniz’s famous principle of the identity of indiscernibles is closely related to some well-known topological separation axioms. More precisely, the topological perspective gives rise in a natural way to some novel variations of Leibniz’s principle. (shrink)

Technology-based assessment offers unique possibilities for collecting data about students’ cognitive development and using this data to provide students and teachers with feedback to improve learning. The aim of this study was to show how the psychological dimension of learning can be assessed in everyday educational practice through technology-based assessment in reading, mathematics and science. We analyzed three related aspects of the assessments: cognitive development, gender differences and vertical scaling. The sample for the study was drawn from primary (...) school students in Grades 1–8 (aged 7 to 14) in Hungary. There were 1500 to 2000 students in each grade cohort. Online tests were constructed from 1638 items from the reading, mathematics and science domains in the eDia system. The results confirmed empirically that the disciplinary, application and psychological dimensions of learning can be empirically distinguished. Students’ cognitive development was the most steady (and effective) in mathematics where the greatest development happened in the first years of schooling. Path models suggested that the psychological dimension of learning can be predicted at a moderate level based on students’ level of school knowledge consisting of the disciplinary and application dimensions of learning as latent constructs. The predictive power was almost the same in both of the dimensions. Generally, girls developed faster in the psychological dimension of reading, mathematics and science learning; however, the size of gender differences varied by age and domain. This study (1) provides evidence that the psychological dimension of learning can be made visible even in an educational context and (2) highlights the importance of the explicit development of the psychological dimension of learning during school lessons, and (3) shows that there are gender differences in the developmental level of the psychological dimension of learning in favour of girls, but it varies by grade and domain. (shrink)

The most puzzling and intriguing aspect of intuitionism as a philosophy of mathematics is its claim that classical deductive reasoning in mathematics is illegitimate. The two most well-known proponents of this position are L. E. J. Brouwer and Michael Dummett. Both of their criticisms of the use of classical logic in mathematics have, by and large, been taken to depend on the thesis that the principle of bivalence does not apply to mathematical statements; and the difference between (...) these criticisms is taken to consist in their grounds for this thesis. Brouwer bases this thesis on an account of the nature of mathematics: it is an activity of mental construction essentially independent of language. Dummett, in contrast, bases the thesis on his anti-realist view of the nature of meaning. ;My first principal argument in this essay shows that the conception of anti-realism just outlined is problematic as a reading of Dummett. To begin with, Dummett explicitly rejects, not only the evidentiary restrictions of this conception, but also, more seriously, the use of specific epistemological theses in a philosophical account of meaning. Moreover, I give an alternative account of anti-realism, consisting of alternative interpretations of the manifestation requirement and of the rejection of realist truth conditions, that do not rely on an epistemology of meaning, and hence, a fortiori, not on evidential restrictions. ;My second principal argument in this essay consists of drawing a consequence of this conception of anti-realism, namely, that the failure of bivalence cannot, by itself, support a cogent criticism of classical logic. The basic reason is that, given my view of anti-realism, the claim that the legitimacy of applying classical reasoning to a class of statements depends on whether bivalence applies to that class already implies that the system of deductive reasoning applying to these statements is non-classical. Hence an argument against classical logic based on this claim is question-begging. ;Finally, I conclude by showing that Dummett's criticism of classical reasoning, as reconstructed by my third argument, fails. (shrink)

Quoi de plus simple que l'idée d'objets dont le mathématicien se proposerait de connaître les propriétés ?

Ontology, the ideas we have about the nature of reality, and epistemology, our concepts about how to gain knowledge about the world, are interdependent. Currently, the dominant ontology in science is a materialist model, and associated with it an empiricist epistemology. Historically speaking, there was a more comprehensive notion at the cradle of modern science in the middle ages. Then “experience” meant both inner, or first person, and outer, or third person, experience. With the historical development, experience has come to (...) mean only sense experience of outer reality. This has become associated with the ontology that matter is the most important substance in the universe, everything else—consciousness, mind, values, etc., —being derived thereof or reducible to it. This ontology is insufficient to explain the phenomena we are living with—consciousness, as a precondition of this idea, or anomalous cognitions. These have a robust empirical grounding, although we do not understand them sufficiently. The phenomenology, though, demands some sort of non-local model of the world and one in which consciousness is not derivative of, but coprimary with matter. I propose such a complementarist dual aspect model of consciousness and brain, or mind and matter. This then also entails a different epistemology. For if consciousness is coprimary with matter, then we can also use a deeper exploration of consciousness as happens in contemplative practice to reach an understanding of the deep structure of the world, for instance in mathematical or theoretical intuition, and perhaps also in other areas such as in ethics. This would entail a kind of contemplative science that would also complement our current experiential mode that is exclusively directed to the outside aspect of our world. Such an epistemology might help us with various issues, such as good theoretical and other intuitions. (shrink)

It is the purpose of this paper to present a theory of probability derived from two-valued logic—the logic of which an aspect is given in Part I, Section A, of Principia Mathematica. The symbolic system of Mr. Keynes, given in his Treatise on Probability, will be shown to be a part of our system. We have, however, little if anything in common with his philosophical analysis; a definition of Keynes’ fundamental probability relation, free from psychological or material reference, will (...) be given, enabling us to offset some of the objections to his theory. (shrink)

Author demonstrates that Russell’s conception of space and time diverges from Kant’s transcendental aesthetics and leans towards logical and mathematical topology. Russell’s approach is grounded in analytical rather than synthetic judgments, contrasting with Kant’s perspective. The British philosopher develops a subjective-psychological model of space and time that complements the logical-mathematical model, serving as the foundation for human experience and cognition. This Russellian model considers the psychological aspects of perceptual and tactile space and time, highlighting their intersection in human (...) perception, which facilitates the experience. It represents a form of «return» to Kant’s subjective theory, albeit on psychological rather than transcendental principles. It is argued that Russell incorporates physical topology into subjective-psychological space and time because he views things as physical objects rather than a complex of sensory associations. This incorporation is indicative of Russell's commitment to neorealism. (shrink)

The article examines the development of Husserl’s early philosophy from his Habilitationsschrift (1887) to the Philosophie der Arithmetik (1891). -/- An attempt will be made at reconstructing the lost Habilitationsschrift (of which only the first chapter survives, which we know as Über den Begriff der Zahl). The examined sources show that the original version of the Habilitationsschrift was by far broader than the printed version, and included most topics of the PA. -/- The article contains an extensive and detailed comparison (...) of these texts to illustrate the changes in Husserl’s position before and after February 1890. This date is taken as a turning point in his development, because of Husserl’s announcement in a letter to Carl Stumpf that he was mistaken in his basic assumption, i.e. that the (psychological) analysis of the concept of Anzahl would yield a foundation for arithmetic. Some interesting conclusions in this respect can also be drawn from an unpublished lecture that Husserl held in the WS 1889/90, in which he anticipates aspects of his position hitherto deemed to belong to the last phases of the PA. (shrink)

In the paper, a detailed analysis of some new logical aspects of Cantor’s diagonal proof of the uncountability of continuum is presented. For the first time, strict formal, axiomatic, and algorithmic definitions of the notions of potential and actual infinities are presented. It is shown that the actualization of infinite sets and sequences used in Cantor’s proof is a necessary, but hidden, condition of the proof. The explication of the necessary condition and its factual usage within the framework of Cantor’s (...) proof makes, Cantor’s proof invalid. It’s shown that traditional Cantor’s proof has a second necessary, but hidden as well, condition which is teleological by its nature, i.e., is not mathematical. The explication of the second necessary condition makes Cantor’s statement on the uncountability of continuum unprovable from the point of view of classical logic.One of the most dramatic facts in history of science is connected just with the notion of actual infinity and consists in the following. On the one hand, Aristotle, Berkeley, Locke, Descartes, Spinoza, Gauss, Kant, Cauchy, Kronecker, Hermite, Poincaré, Bair, Borel, Brouwer, Quine, Wittgenstein, Weyl, Luzin, and a lot of other outstanding creators of classical logic and classical mathematics, during millenniums, stated categorically and insistently warned about that the actual infinity is a self-contradictory notion and its usage in mathematics is inadmissible (e.g., according to Poincare, “There is no actual infinity. - Cantorians forgot that and fell into contradictions. Later generations will regard Mengenlehre (set theory) as a disease from which one has recovered”). On the other hand, a lot of outstanding scientists of the XX c.c., such as Hilbert, Church, Turing, Gödel, etc., and modern axiomatic set theory as a whole ignored this (intuitive though) opinion of genii and accepted Cantor’s “transfinite paradise”, based on the actual infinity. In the paper some logical, philosophical, and psychological aspects of this historical “favorable opposition” are analyzed, and the fact that the scientific intuition of genii has won the historical battle against the mytho-’logic’ of Cantor’s transfinite ‘paradise’ is explained and logically justified. (shrink)

The entrepreneurial element is one of the aspects emphasized in the primary school mathematics education curriculum in Malaysia. However, previous studies have found that application of entrepreneurial elements in mathematics teaching is still lacking. This study was therefore conducted to identify the real challenges that mathematics teachers face in applying the entrepreneurial element in mathematics teaching. This study is qualitative case study which involved six primary school mathematics teachers. Semi-structured interviews, observation, document analysis and field (...) notes were used for the data collection process in this study. The data obtained was analyzed using the constant comparative analysis method to determine themes and sub-themes. The findings were that a teacher’s knowledge, teaching style, limited time of teaching, and attitudes, and in-service courses and training are among the identified challenges in applying entrepreneurial elements in mathematics teaching. This study expands knowledge and the literature on the challenges faced by school mathematics teachers in applying entrepreneurial elements in mathematics teaching. Various initiatives need to be taken to ensure that all the challenges can be overcome, and ultimately enable the entrepreneurial elements to be applied to students more widely. (shrink)

The paper presents an overview of research on working memory as a predictor of early school achievements. We contrast two main areas of research on the role of working memory in school achievements: the first concerns the structural model of working memory and the second focuses on executive functions. Then, we discuss the facet model of working memory as a promising approach merging the two research branches on working memory tasks as predictors of early school achievements. At the end we (...) present exemplary results of the research conducted on a national sample of six- and seven-year-olds in Poland, which indicates strong relation of working memory functions with the measures of competences in mathematics, reading, and writing. Additionally, the mediation analyses, with parents’ education as a covariate, indicate that the influence of age on achievements in math, reading, and writing in six- and seven-year olds is mediated by working memory functions. (shrink)

This is the most comprehensive book ever published on philosophical methodology. A team of thirty-eight of the world's leading philosophers present original essays on various aspects of how philosophy should be and is done. The first part is devoted to broad traditions and approaches to philosophical methodology. The entries in the second part address topics in philosophical methodology, such as intuitions, conceptual analysis, and transcendental arguments. The third part of the book is devoted to essays about the interconnections between philosophy (...) and neighbouring fields, including those of mathematics, psychology, literature and film, and neuroscience. (shrink)

Chacun d'entre nous fait partie du monde, certes. Mais, au-delà de son évidence, quelle est la signification d'une telle formule? Comment faisons-nous partie du monde? Comment sommes-nous reliés à tout ce qui n'est pas nous : les autres, le reste de la nature, le cosmos? Lionel Naccache répond à cette énigme intemporelle à partir d'une approche mathématique fondée sur la distinction entre ensembles discrets et ensembles continus : alors qu'un élément, qui fait partie d'un ensemble discret, reste séparé des autres (...) éléments par des frontières tranchées, ces frontières disparaissent dans un ensemble continu. La question devient alors : sommes-nous reliés au reste du monde par des liens continus ou par des liens discrets? Muni de cette boussole, Lionel Naccache nous propose une odyssée passionnante et pleine de gai savoir éclectique qui nous entraîne de la psychologie de la subjectivité vers les sciences du cerveau et de la conscience, les origines et l'évolution des idées mathématiques et la philosophie politique. Au bout du voyage, Lionel Naccache formule une nouvelle éthique pour le XXIe siècle"--Page 4 of cover. (shrink)

"Edmund Husserl is widely regarded as the principal founder of phenomenology, one of the most important movements in twentieth-century philosophy. His work inspired subsequent figures such as Martin Heidegger, his most renowned pupil, as well as Jean-Paul Sartre and Maurice Merleau-Ponty, all of whom engaged with and developed his insights in significant ways. He also made important contributions to logic and philosophy of mathematics and his work on fundamental problems such as intentionality, consciousness and subjectivity continues to animate philosophical (...) research and argument. The Husserlian Mind is an outstanding reference source to the full range of Husserl's philosophy. Forty chapters by a team of international contributors are divided into seven clear parts covering the following areas: Husserl's major works, Husserl's phenomenological method, Phenomenology of consciousness, Epistemology, Ethics and social and political philosophy, Science, Metaphysics. Contained in these sections are chapters on many of the key aspects of Husserl's thought, including intentionality, transcendental philosophy, reduction, perception, time, self and subjectivity, personhood, logic and mathematics, psychology, ontology and idealism. Offering an unparalleled guide to the enormous range of his thought, The Husserlian Mind is essential reading for students and scholars of Husserl, phenomenology, and the history of twentieth-century philosophy. It will also be of interest to those in related fields, such as religion and sociology"--. (shrink)

Excerpt from The Mathematical Psychology of Gratry and Boole: Translated From the Language of the Higher Calculus Into That of Elementary Geometry Dear Dr. Maudsley, - You have often asked me to explain, for students unaquainted with the Infinitesimal Calculus, certain doctrines expressed in terms of that Calculus by P. Gratry and my late husband. That you permit me to dedicate my attempt to you will, at least, be a guarantee that the main ideas of mathematical psychology are based, not (...) on mystic dreams, but on scientific induction. I am glad of this opportunity of expressing to you my gratitude, both for your published works, and for much personal kindness to myself. About the Publisher Forgotten Books publishes hundreds of thousands of rare and classic books. Find more at www.forgottenbooks.com This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully; any imperfections that remain are intentionally left to preserve the state of such historical works. (shrink)

_Despite advances in standard of living of the population, the condition of widows and divorced women remains deplorable in society. The situation is worse in developing nations with their unique social, cultural and economic milieu, which at times ignores the basic human rights of this vulnerable section of society. A gap exists in life expectancies of men and women in both developing and developed nations. This, coupled with greater remarriage rates in men, ensures that the number of widows continues to (...) exceed that of widowers. Moreover, with women becoming more educated, economically independent and aware of their rights, divorce rates are increasing along with associated psychological ramifications. The fact that widowed/divorced women suffer from varying psychological stressors is often ignored. It has been concluded in various studies that such stressors could be harbingers of psychiatric illnesses (e.g., depression, anxiety, substance dependence), and hence should be taken into account by treating physicians, social workers and others who come to the aid of such women. A change in mindset of the society is required before these women get their rightful place, for which a strong will is needed in the minds of the people, and in law-governing bodies._. (shrink)

Psychological Aspects of Risk and Aggression among Motorcyclists - "Mad Max" Syndrome The primary objective of this study was the psychological examination of a group of Polish motorcyclists against a group of students and graduates of Technical Universities. This work poses a question regarding the differences in temperament, aggression and the level of risk between motorcyclists and the control group. The second question was whether it was possible to create a typology of Polish motorcyclists taking into account the (...) variables describing risk, temperament and aggression. This study used the Pavlovian Temperamental Scale, Stimulating-Instrumental Risk Inventory, SIRI 2001, Multifactor Risky Behavior Scale, Unhealthy behavior Inventory and the Buss and Perry Aggression Questionnaire. The tests were performed on 267 motorcyclists and 188 students and graduates of Technical Universities. A number of important differences was observed between the motorcyclists and the control group. Motorcyclists had a higher level of mobility of nervous processes and higher degree of excitation at the lower level of inhibitory processes. In terms of the measured level of risk they also had a higher level of stimulating, instrumental and unhealthy risk, at the lower levels of physical, social, ethical and financial risk in relation to the control group. Cluster and confirmation analyses showed the existence of three sub-groups of motorcyclists: prevaricators, ones with a Mad Max syndrome, and adventurers. The confirmation analysis was carried out on an additional group of 243 motorcyclists. (shrink)

Excerpt from The Psychological Aspects of Christian Experience Continued observation of the various methods of religious instruction has brought upon me a conviction that grows stronger as the years go by, that there must be, as Francis Peabody says, a new expansion of the range of studies appropri ate to the teachers of religion. There ought to be a far deeper study of the psychology of reli gion. The appeal to the impulses and emotions, so prevalent in the methods (...) of some teachers, is far behind twentieth century enlightenment. It was foreign to the methods of our Saviour. He taught men. He did not frighten them into ao tion, neither entice them. He instructed their minds in truth. Ye shall know the truth and the truth shall make you free. About the Publisher Forgotten Books publishes hundreds of thousands of rare and classic books. Find more at www.forgottenbooks.com This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully; any imperfections that remain are intentionally left to preserve the state of such historical works. (shrink)

Probably the main purpose of reproductive technologies is to enable people who choose to do so to avoid the birth of a baby with a disabling condition. However the conditions women want information about and the ‘price’ they are willing to pay for obtaining that information vary enormously. Individual women have to arrive at their own prenatal testing choices by ‘trading off’ means and ends in order to resolve the dilemmas facing them. We know very little about how individuals make (...) these trade-offs, so it is difficult to predict how new technologies will affect their choices and preferences. Uptake decisions can be expected to change, especially in the group of women who now are put off by some aspect of the current screening approach, where the avoidance of miscarriage risk may have provided a kind of ‘psychological shelter’, protecting a lot of people from having to make other decisions. Technologies such as Pre-implantation Genetic Diagnosis may remove a second ‘psychological shelter’ because they offer the means of avoiding the birth of an affected child without terminating a pregnancy. Even if new technologies will make some decisions easier in terms of their cognitive demands, they will also create new dilemmas and decision making will not necessarily become less stressful in emotional terms. Key challenges concern information and decision-making. (shrink)

This is the most comprehensive book ever published on philosophical methodology. A team of thirty-eight of the world's leading philosophers present original essays on various aspects of how philosophy should be and is done. The first part is devoted to broad traditions and approaches to philosophical methodology. The entries in the second part address topics in philosophical methodology, such as intuitions, conceptual analysis, and transcendental arguments. The third part of the book is devoted to essays about the interconnections between philosophy (...) and neighbouring fields, including those of mathematics, psychology, literature and film, and neuroscience. (shrink)

Mathematical models and their modeling frameworks developed to advance knowledge in one discipline are sometimes sourced to answer questions or solve problems in another discipline. Studying this aspect of cross-disciplinary transfer of knowledge objects, philosophers of science have weighed in on the question of whether knowledge about how a mathematical model is previously applied in one discipline is necessary for the success of reapplying said model in a different discipline. However, not much has been said about whether the answer to (...) that epistemological question applies to the reapplication of a modeling framework. More generally, regarding the nature of the production of knowledge in science, a metaphysical question remains to be explored whether historical contingencies associated with a mathematical construct have a genuine impact on the nature—as opposed to sociological practices or individual psychology—of advancing scientific knowledge with said construct. Focusing on this metaphysical question, this paper analyzes the use of mathematical logic in the development of the Chomsky hierarchy and subsequent reapplications of said hierarchy; with these examples, this paper develops the notion of “spillovers” as a way to detect cross-disciplinary justifications for better understanding the relations between reapplications of the same mathematical construct across disciplines. (shrink)