The present study has two goals: to explore elementary students’ understanding of evidence and the ways they deploy it to construct arguments, and to examine whether eliciting their concept of evidence during argumentation improves students’ evidence-based reasoning. Individual semi-structured interviews were conducted with 4th and 6th graders in a public school in Mexico. We found significant differences between groups regarding the concept of evidence, with better performance in the older group. A positive correlation between the concept of evidence and (...) the quality of evidence-based reasoning was found. Also, three performance profiles were observed after eliciting the concept of evidence when grade was excluded as a factor. Results suggest that the concept of evidence plays an essential role in developing argumentative competence in pre-adolescence. (shrink)

The aim of the paper is to show that topoi are useful in the categorial analysis of the constructive logic with strong negation. In any topos ϵ we can distinguish an object Λ and its truth-arrows such that sets ϵ have a Nelson algebra structure. The object Λ is defined by the categorial counterpart of the algebraic FIDEL-VAKARELOV construction. Then it is possible to define the universal quantifier morphism which permits us to make the first order predicate calculus. The (...) completeness theorem is proved using the Kripke-type semantic defined by THOMASON. (shrink)

This book explores and articulates the concepts of the continuous and the infinitesimal from two points of view: the philosophical and the mathematical. The first section covers the history of these ideas in philosophy. Chapter one, entitled ‘The continuous and the discrete in Ancient Greece, the Orient and the European Middle Ages,’ reviews the work of Plato, Aristotle, Epicurus, and other Ancient Greeks; the elements of early Chinese, Indian and Islamic thought; and early Europeans including Henry of Harclay, Nicholas of (...) Autrecourt, Duns Scotus, William of Ockham, Thomas Bradwardine and Nicolas Oreme. The second chapter of the book covers European thinkers of the sixteenth and seventeenth centuries: Galileo, Newton, Leibniz, Descartes, Arnauld, Fermat, and more. Chapter three, 'The age of continuity,’ discusses eighteenth century mathematicians including Euler and Carnot, and philosophers, among them Hume, Kant and Hegel. Examining the nineteenth and early twentieth centuries, the fourth chapter describes the reduction of the continuous to the discrete, citing the contributions of Bolzano, Cauchy and Reimann. Part one of the book concludes with a chapter on divergent conceptions of the continuum, with the work of nineteenth and early twentieth century philosophers and mathematicians, including Veronese, Poincaré, Brouwer, and Weyl. Part two of this book covers contemporary mathematics, discussing topology and manifolds, categories, and functors, Grothendieck topologies, sheaves, and elementary topoi. Among the theories presented in detail are non-standard analysis, constructive and intuitionist analysis, and smooth infinitesimal analysis/synthetic differential geometry. No other book so thoroughly covers the history and development of the concepts of the continuous and the infinitesimal. (shrink)

This paper considers a topos-theoretic structure for the interpretation of co-constructive logic for proofs and refutations following Trafford :22–40, 2015). It is notoriously tricky to define a proof-theoretic semantics for logics that adequately represent constructivity over proofs and refutations. By developing abstractions of elementary topoi, we consider an elementary topos as structure for proofs, and complement topos as structure for refutation. In doing so, it is possible to consider a dialogue structure between these topoi, and also (...) control their relation such that classical logic is simulated where proofs and refutations are conclusive. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Journal of the History of Ideas 61.2 (2000) 305-326 [Access article in PDF] Logos and Kratos: Gramsci and the Ancients on Hegemony Benedetto Fontana * The purpose of this paper is to locate Gramsci's concept of hegemony, and its related ideas of civil society, the national-popular and the people-nation, within the political thought of classical antiquity. 1 In so doing, the paper seeks to identify strands or elements within (...) aspects of ancient political thought which may usefully be seen as conceptual prefigurations or as political anticipations of Gramsci's hegemony. It will show that Gramsci shares with the ancients specific theoretical, linguistic, and intellectual topoi. Hegemony has hitherto been located within the context, both theoretical and historical, of debates and controversies arising out of the Marxist and later Leninist revolutionary tradition. It is rightly seen as a notion developed by Gramsci to explain revolutionary failure in Italy and in the West generally, and consequently its antecedents are traced to the problems attendant upon the collapse of the Second International and the rise of Bolshevism in Russia. 2 While such an approach has been useful in revealing the immediate (both political and tactical) constraints acting upon Gramsci's thinking, it has overlooked passages in his writings which reveal an interest in, and [End Page 305] familiarity with, philosophical and theoretical themes originally formulated and elaborated by classical political thought.I. In a note entitled "Passage from Knowing to Understanding and to Feeling and vice versa from Feeling to Understanding and to Knowing" Gramsci establishes a distinction between intellectuals who "know" and the "people-nation" that "feels." The former may know but do not always understand or feel, while the latter may feel but does not always know. The intellectual, in order to know something politically and socially, not merely abstractly or philosophically, must understand it with feeling and passion. As Gramsci writes: The intellectual's error consists in believing that one can know without understanding and even more without feeling and being impassioned (not only for knowledge in itself but also for the object of knowledge); in other words that the intellectual can be an intellectual (and not a pure pedant) if distinct and separate from the people-nation, that is, without feeling the elementary passions of the people, understanding them and therefore explaining and justifying them in the particular historical situation and connecting them dialectically to the laws of history and to a superior conception of the world, scientifically and coherently elaborated--i.e. knowledge. One cannot make politics-history without this passion, without this sentimental connection between intellectuals and people-nation. 3The merely abstract knowledge of the intellectual becomes life and politics when linked to the experiential and passionate feelings of the people. At the same time, the feeling-passion of the people acquires the character of knowledge. Of course such a formulation is reminiscent of Marx's comments in the Theses on Feuerbach. 4 The dyadic relation between intellectual and people-nation, and between knowledge and feeling-passion, both parallels and informs the relation between common sense and good sense. Common sense is opinion which is incoherent and ambiguous 5 but which may nevertheless contain elements of truth to the extent that they are proliferated throughout a people. Good sense, on the other hand, is the common sense of the people as their passion and experience are imbued with knowledge and reason--that is, as the people begin to "think" coherently by producing their own intellectuals, the organic intellectual, or the democratic philosopher. 6 [End Page 306]Gramsci recalls the problem originally posed by Plato and Aristotle and running through the entire history of Western political thought up to thinkers as diverse as Hegel, Nietzsche, Weber, and Croce: namely, the relation between knowledge and politics, philosophy and rhetoric, ruler and people, reason and desire/appetite. 7 This relation poses the question regarding the role and status of reason. It is the way reason (the logos as transcendent reason or as speech and language) is perceived that moves the theory in either an ossified (absolutist... (shrink)

In historical claims for nativism, mathematics is a paradigmatic example of innate knowledge. Claims by contemporary developmental psychologists of elementary mathematical skills in human infants are a legacy of this. However, the connection between these skills and more formal mathematical concepts and methods remains unclear. This paper assesses the current debates surrounding nativism and mathematical knowledge by teasing them apart into two distinct claims. First, in what way does the experimental evidence from infants, nonhuman animals and neuropsychology support the (...) nativist hypothesis? Second, granting that infants have some elementary mathematical skills, does this mean that such skills play an important role in the development of mathematical knowledge? (shrink)

The infinite regress of Carroll’s ‘What the Tortoise said to Achilles’ is interpreted as a problem in the epistemology of mathematical proof. An approach to the problem that is both diagrammatic and non-logical is presented with respect to a specific inference of elementary geometry.

In historical claims for nativism, mathematics is a paradigmatic example of innate knowledge. Claims by contemporary developmental psychologists of elementary mathematical skills in human infants are a legacy of this. However, the connection between these skills and more formal mathematical concepts and methods remains unclear. This paper assesses the current debates surrounding nativism and mathematical knowledge by teasing them apart into two distinct claims. First, in what way does the experimental evidence from infants, nonhuman animals and neuropsychology support the (...) nativist hypothesis? Second, granting that infants have some elementary mathematical skills, does this mean that such skills play an important role in the development of mathematical knowledge? (shrink)

A down-to-earth admission of abstract objects can be based on detailed explanation of where the objectivity of mathematics comes from, and how a ‘thin’ notion of object emerges from objective mathematical discourse or practices. We offer a sketch of arguments concerning both points, as a basis for critical scrutiny of the idea that mathematical and social objects are essentially of the same kind—which is criticized. Some authors have proposed that mathematical entities are indeed institutional objects, a product of our collective (...) imposition of function onto reality (the phrase comes from Searle) and of surrogation or hypostasis. Yet there are significant disanalogies between the typical social objects and mathemata, on which basis I argue that one should make a clear distinction between both. The comparison of mathematical with social objects helps understanding how non-physical objects can figure prominently in our explanations of reality. Yet mathematical objects have a different kind of cognitive grounding, and the more elementary of them emerge under relatively very simple sociocultural conditions. The differences are also reflected in the wide scope of use of mathematical concepts, and the much higher degree of variation found among social objects. On the basis of all of these features, I defend the thesis that one can significantly distinguish degrees of objectivity, and I use the distinction to articulate a graded ontology where one can locate the different kinds of mathematical and social objects. (shrink)

One of Kant’s famous questions about being human asks, “What may I hope?” This question places individual life within an encompassing horizon of human history and speculates on the possibility of perspectives beyond death. In our time mortality is generally repressed, though the development of personal consciousness is closely linked to realization of one’s finitude. This raises especially urgent questions for children, and they are left to deal with them alone. From the time awareness begins, knowledge that death can occur (...) at any moment is one of the a priori determinants of being alive; that is, life is structured in advance by its future pastness. . According to Max Scheler, death reveals itself as a necessary and manifest constituent of all possible inner experiences in the life process. The cultural and individual interpretations of death’s meaning and consequences derived from this insight opened up an ample space for imagined possibilities of “continued existence,” which affected approaches to life. In our research project “Inochi – The Concept of Life after Death in Children’s Construction of the World. A German-Japanese Comparison,” carried out with German and Japanese research support, we examine concepts developed within the community of inquiry concerning “the individual’s afterlife” as soul, angel, animal, star, etc. In this we want to examine whether a globalized media environment leads to a cultural convergence in children’s ideas, and whether there are differences between views of girls and boys. Relevance of ideas about death is seen in the example of Japanese children who, believing in reincarnation chose “killing oneself” with relative frequency as a problem-solving strategy. Our contribution will present children’s imaginings using the drawings they created within this framework, since these can be interpreted as expressions of “narrative philosophizing,” especially for the Japanese children. Here we follow Mark Johnson who says” Human beings are imaginative synthesizing animals” . These imaginations make up a large part of our understanding, not just our beliefs, but rather our socially constructed way of being in and inhabiting a world. (shrink)

Most children experience some form of grouping in the classroom every day. Understanding how teachers make grouping decisions and their impacts on children’s social development can shed light on effective teacher practices for promoting positive social dynamics in the classroom. This study examined the influence of teachers’ grouping strategies on changes in young children’s social experiences with peers across an academic year. A total of 1,463 children and 79 teachers from kindergarten to third-grade classrooms participated in this study. Teachers rated (...) children’s behavioral problems as the most important consideration when creating seating charts or assigning children to small groups. Promoting existing or new friendships was rated as the least important consideration. Heterogeneous ability grouping, rated as somewhat important by the teachers, was associated with a decrease in children’s friendships and yet also a decrease in girls’ experience with peer conflicts. Our findings begin to fill in the gaps in the literature on the social impacts of ability grouping for young children. (shrink)

Topoi in Critical Discourse Analysis Topos is one of the most widely-used concepts from classical argumentation theory. It found its way not only in philosophy, sociology, anthropology, and linguistics; it found its way in everyday life and everyday conversation as well.In this article, I will examine the role that topoi play in Critical Discourse Analysis. Starting with definitions from Aristotle and Cicero, contrasting them with new conceptualisations by Perelman and Toulmin, and examining the superficial use of topoi (...) in everyday conversation, I will try to show that Critical Discourse Analysis relies mostly on simplified, unreflected use of topoi as found in everyday use, thus neglecting much more productive, theoretical elaborations of the concept. (shrink)

In partial fulfilment of the requirements for the degree of Master of Education, Department of Educational Administration.

To represent extension of objects in particle physics, a modified Weyl theory is used by gauging the curvature radius of the local fibers in a soldered bundle over space-time possessing a homogeneous space G/H of the (4, 1)-de Sitter group G as fiber. Objects with extension determined by a fundamental length parameter R0 appear as islands D(i) in space-time characterized by a geometry of the Cartan-Weyl type (i.e., involving torsion and modified Weyl degrees of freedom). Farther away from the domains (...) D(i), space-time is identified with the pseudo-Riemannian space of general relativity. Extension and symmetry breaking are described by a set of additional fields ( $(\tilde \xi ^a (x),R(x)) \in {G \mathord{\left/ {\vphantom {G H}} \right. \kern-0em} H}x R^ +$ , given as a section on an associated bundle $\tilde E(B,{G \mathord{\left/ {\vphantom {G H}} \right. \kern-0em} H}x R^ + ,\tilde G)$ over space-time B with structural group $\tilde G$ = G ⊗ D(1), where D(1) is the dilation group. Field equations for the quantities defining the underlying bundle geometry and for the fields $\tilde \xi ^a (x)$ are established involving matter source currents derived from a generalized spinor wave function. Einstein's equations for the metric are regarded as the part of the $\tilde G$ -gauge theory related to the Lorentz subgroup H of G exhibiting thereby the broken nature of the $\tilde G$ -symmetry for regions outside the domains D(i). (shrink)

As it is currently used, "foundations of arithmetic" can be a misleading expression. It is not always, as the name might indicate, being used as a plural term meaning X = {x : x is a foundation of arithmetic}. Instead it has come to stand for a philosophico-logical domain of knowledge, concerned with axiom systems, structures, and analyses of arithmetic concepts. It is a bit as if "rock" had come to mean "geology." The conflation of subject matter and its study (...) is a serious one, because in the end, one can lose sight of what one should be doing in the first place. Perhaps it is taking matters too literally, but it seems that there is something to be said for taking the term to represent X. Doing so and accepting the term to have some kind of significance, it is then natural to focus on the question of what a foundation of arithmetic should be; and, if one exists, what one is. Whatever the case, that is what shall be done in this paper. (shrink)