199 research outputs found

    Gottlob Frege and Gongsun Long in Dialogue: An Exploration of Two Classical Paradoxes from the East and West

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    This work addresses the critical discussion featured in the contemporary literature about two well-known paradoxes belonging to different philosophical traditions, namely Frege’s puzzling claim that “the concept horse is not a concept” and Gongsun Long’s “white horse is not horse”. We first present the source of Frege’s paradox and its different interpretations, which span from plain rejection to critical analysis, to conclude with a more general view of the role of philosophy as a fight against the misunderstandings that come from the different uses of language (a point later developed by the “second” Wittgenstein). We then provide an overview of the ongoing discussions related to the Bai Ma Lun paradox, and we show that its major interpretations include—as in the case of Frege’s paradox—dismissive accounts that regard it as either useless or wrong, as well as attempts to interpret and repair the argument. Resting on our reading of Frege’s paradox as an example of the inescapability of language misunderstandings, we advance a similar line of interpretation for the paradox in the Bai Ma Lun: both the paradoxes, we suggest, can be regarded as different manifestations of similar concerns about language, and specifically about the difficulty of referring to concepts via language

    Gottlob Frege and Gongsun Long in Dialogue

    Get PDF
    This work addresses the critical discussion featured in the contemporary literature about two well-known paradoxes belonging to different philosophical traditions, namely Frege’s puzzling claim that “the concept horse is not a concept” and Gongsun Long’s “white horse is not horse”. We first present the source of Frege’s paradox and its different interpretations, which span from plain rejection to critical analysis, to conclude with a more general view of the role of philosophy as a fight against the misunderstandings that come from the different uses of language (a point later developed by the “second” Wittgenstein). We then provide an overview of the ongoing discussions related to the Bai Ma Lun paradox, and we show that its major interpretations include—as in the case of Frege’s paradox—dismissive accounts that regard it as either useless or wrong, as well as attempts to interpret and repair the argument. Resting on our reading of Frege’s paradox as an example of the inescapability of language misunderstandings, we advance a similar line of interpretation for the paradox in the Bai Ma Lun: both the paradoxes, we suggest, can be regarded as different manifestations of similar concerns about language, and specifically about the difficulty of referring to concepts via language

    Digital writing technologies in higher education : theory, research, and practice

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    This open access book serves as a comprehensive guide to digital writing technology, featuring contributions from over 20 renowned researchers from various disciplines around the world. The book is designed to provide a state-of-the-art synthesis of the developments in digital writing in higher education, making it an essential resource for anyone interested in this rapidly evolving field. In the first part of the book, the authors offer an overview of the impact that digitalization has had on writing, covering more than 25 key technological innovations and their implications for writing practices and pedagogical uses. Drawing on these chapters, the second part of the book explores the theoretical underpinnings of digital writing technology such as writing and learning, writing quality, formulation support, writing and thinking, and writing processes. The authors provide insightful analysis on the impact of these developments and offer valuable insights into the future of writing. Overall, this book provides a cohesive and consistent theoretical view of the new realities of digital writing, complementing existing literature on the digitalization of writing. It is an essential resource for scholars, educators, and practitioners interested in the intersection of technology and writing

    Adelic Geometry via Topos Theory

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    Our starting point has to do with a key tension running through number theory: although all completions of the rationals Q should be treated symmetrically, this is complicated by fundamental disanalogies between the p-adics vs. the reals. Whereas prior work has typically been guided by classical point-set reasoning, this thesis explores various ways of pulling this problem away from the underlying set theory, revealing various surprises that are obscured by the classical perspective. Framing these investigations is the following test problem: construct and describe the topos of completions of Q (up to equivalence). Chapter 2 begins with the preliminaries: we set up the topos-theoretic framework of point-free topology, with a view towards highlighting the distinction between classical vs. geometric mathematics, before introducing the number-theoretic context. A key theme is that geometric mathematics possesses an intrinsic continuity, which forces us to think more carefully about the topological character of classical algebraic constructions. Chapter 3 represents the first step towards constructing the topos of completions. Here, we provide a pointfree account of real exponentiation and logarithms, which will allow us to define the equivalence of completions geometrically. Chapter 4 provides a geometric proof of Ostrowski's Theorem for both upper-valued abosolute values on Z as well as Dedekind-valued absolute values on Q, along with some key insights about the relationship between the multiplicative seminorms and upper reals. In a slightly more classical interlude, Chapter 5 extends these insights to obtain a surprising generalisation of a foundational result in Berkovich geometry. Namely, by replacing the use of classical rigid discs with formal balls, we obtain a classification of the points of Berkovich Spectra M(K{R^{-1}T}) via the language of filters [more precisely, what we call: R-good filters] even when the base field K is trivially-valued. Returning to geometricity, Chapter 6 builds upon Chapters 3 and 4 to investigate the space of places of Q via descent arguments. Here, we uncover an even deeper surprise. Although the non-Archimedean places correspond to singletons (as is classically expected), the Archimedean place corresponds to the subspace of upper reals in [0, 1], a sort of blurred unit interval. The chapter then analyses the topological differences between the non-Archimedean vs. Archimedean places. In particular, we discover that while the topos corresponding to Archimedean place witnesses non-trivial forking in the connected components of its sheaves, the topos corresponding to the non-Archimedean place eliminates all kinds of forking phenomena. We then conclude with some insights and observations, framed by the question: "How should the connected and the disconnected interact?

    Quality education

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    This book investigates the intersections between education, social justice, gendered violence and human rights in South African schools and universities. The rich and multifarious tapestry of scholarship and literature emanating from South African classrooms provides a fascinating lens through which we can understand the complex consequences of the economies of education, social justice imperatives, gendered violence on the lives of women and children, and marginalised communities. The scholarship in the book challenges readers to imagine alternative futures predicated on the transformational capacity of a democratic South Africa. Contributors to this volume examine the many ways in which social justice and gendered violence mirrors, expresses, projects and articulates the larger phenomenon of human rights violations in Africa and how, in turn, the discourse of human rights informs the ways in which we articulate, interrogate, conceptualise, enact and interpret quality education. The book also wrestles with the linguistic contradictions and ambiguities in the articulation of quality education in public and private spaces. This book is essential reading for scholars seeking solid grounding in exploring quality education, the instances of epistemic disobedience, the political implications of place and power, and human rights in theory and practice

    Proceedings of the 15th International Conference on Technology in Mathematics Teaching: Making and Strengthening "Connections and Connectivity" for Teaching Mathematics with Technology

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    The 15th International Conference on Technology in Mathematics Teaching (ICTMT 15) took place on September 13–16, 2022, in the Danish School of Education, Aarhus University, located on campus Emdrup, in the Northwestern district of Copenhagen, Denmark. There were a total of 66 participants from 15 different countries. ICTMT 15 certainly focused on the impacts that the coronavirus pandemic has had on global mathematics education. However, it looked at the impacts of digital technology from a much wider perspective. In particular, the conference aimed to highlight how technology facilitates the multiple “Connections and Connectivity” between us all to achieve the goals of purposeful mathematics education in the early 21st century. By “Connections” we mean the interrelationships between researchers, teachers, students, parents, policymakers, and industry (big and small). “Connectivity” includes oral, aural, textual and gestural communications as mediated by the internet, learning environments and classroom activities. Together, “Connections and Connectivity” describes the relationships between people, between different ideas and strategies to teach, and between people and environments. It offers a frame through which to interpret assessment in mathematics education as a more formative process from the point of view of both teachers and students

    Označivanje uzročnosti u dječjem usmenom i pismenom prepričavanju

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    One of the main prerequisites for understanding and producing coherent oral discourse or written text is successful understanding and production of causal relations. During both production, children have at their disposal a wide range of linguistic modes to mark it, some of which are more explicit and others more implicit. In this study, retelling was used as a method to elicit narratives that served as a tool for analysing causal relations. Retelling enables exploring the linguistic reformulation of the syntactic structures of a previously stored story and the analysis of the overlap between the language content which child is exposed to (language input) and the language that child produces (language output). Two groups of children, aged 10 (N = 23) and 12 (N = 30), were exposed to the story at two time points; in the first they had to retell it orally and in the second they had to write it. The conducted analyses showed that 12-years old children produced in total more causal relations than 10-year-old in written modality only. This difference is explained by the greater writing competence of 12-year-old children in the production of more complex syntactic structures. Furthermore, both groups of children in both modalities dominantly used the same causal markers that are primarily grammatical. All these findings point to the children’s ability to reformulate causal relations regardless of the language content to which they were previously exposed.Jedan od glavnih preduvjeta za razumijevanje i oblikovanje koherentnoga usmenoga ili pisanoga diskursa jest uspješno razumijevanje i stvaranje uzročno-posljedičnih odnosa. Tijekom obiju proizvodnja djeca imaju na raspolaganju širok raspon jezičnih sredstava za obilježavanje uzročnosti, od eksplicitnijih do implicitnijih. U ovom je istraživanju upotrijebljeno prepričavanje kao metoda za poticanje stvaranja priča na temelju kojih je raspodjelom u pet markerskih tipova analizirano izražavanje uzročno-posljedičnih odnosa. Prepričavanje omogućuje istraživanje reformulacije sintaktičkih struktura prethodno pohranjene priče i analizu preklapanja između jezičnoga sadržaja kojemu je dijete izloženo (jezični unos) i jezika koji dijete proizvodi (jezični izlaz). Dvije skupine djece, u dobi od 10 (N=23) i 12 (N=30), bile su izložene priči u dvjema vremenskim točkama; u prvoj su je morali prepričati usmeno, a u drugoj pismeno. Provedene analize pokazale su da su 12-godišnjaci proizveli ukupno više uzročno-posljedičnih veza od 10-godišnjaka, ali samo u pisanom obliku. Ta se razlika objašnjava većom pismenom kompetencijom 12-godišnje djece u proizvodnji složenih sintaktičkih struktura. Nadalje, obje skupine djece služile su se u obama modalitetima dominantno istim uzročnim markerima koji su u prvom redu gramatički. Rezultati upućuju na dječju sposobnost reformuliranja uzročnih odnosa neovisno o jezičnom sadržaju kojemu su prethodno bila izložena
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