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One of our best-known cultural critics examines the concept of representation and its many and varied forms in literature and art.
In an age when more and more items. are made to be quickly disposable or soon become obsolete due to either progress or other man caused reasons it seems almost anachronistic to write a book in the classical sense. A mathematics book becomes an indespensible companion, if it is worthy of such a relation, not by being rapidly read from cover to cover but by frequent browsing, consultation and other occasional use. While trying to create such a work I tried not to be encyclopedic but rather select only those parts of each chosen topic which I could present clearly and accurately in a formulation which is likely to last. The material I chose is all mathematics which is interesting and important both for the mathematician and to a large extent also for the mathematical physicist. I regret that at present I could not give a similar account on direct integrals and the representation theory of certain classes of Lie groups. I carefully kept the level of presentation throughout the whole book as uniform as possible. Certain introductory sections are kept shorter and are perhaps slightly more detailed in order to help the newcomer prog ress with it at the same rate as the more experienced person is going to proceed with his study of the details.
In this book, Raymond Duval shows how his theory of registers of semiotic representation can be used as a tool to analyze the cognitive processes through which students develop mathematical thinking. To Duval, the analysis of mathematical knowledge is in its essence the analysis of the cognitive synergy between different kinds of semiotic representation registers, because the mathematical way of thinking and working is based on transformations of semiotic representations into others. Based on this assumption, he proposes the use of semiotics to identify and develop the specific cognitive processes required to the acquisition of mathematical knowledge. In this volume he presents a method to do so, addressing the following questions: • How to situate the registers of representation regarding the other semiotic “theories” • Why use a semio-cognitive analysis of the mathematical activity to teach mathematics • How to distinguish the different types of registers • How to organize learning tasks and activities which take into account the registers of representation • How to make an analysis of the students’ production in terms of registers Building upon the contributions he first presented in his classic book Sémiosis et pensée humaine, in this volume Duval focuses less on theoretical issues and more on how his theory can be used both as a tool for analysis and a working method to help mathematics teachers apply semiotics to their everyday work. He also dedicates a complete chapter to show how his theory can be applied as a new strategy to teach geometry. “Understanding the Mathematical Way of Thinking – The Registers of Semiotic Representations is an essential work for mathematics educators and mathematics teachers who look for an introduction to Raymond Duval’s cognitive theory of semiotic registers of representation, making it possible for them to see and teach mathematics with fresh eyes.” Professor Tânia M. M. Campos, PHD.
This book is open access and available on www.bloomsburycollections.com. It is funded by Knowledge Unlatched. The major principles and systems of C. S. Peirce's ground-breaking theory of signs and signification are now generally well known. Less well known, however, is the fact that Peirce initially conceived these systems within a 'Philosophy of Representation', his latter-day version of the traditional grammar, logic and rhetoric trivium. In this book, Tony Jappy traces the evolution of Peirce's Philosophy of Representation project and examines the sign systems which came to supersede it. Surveying the stages in Peirce's break with this Philosophy of Representation from its beginnings in the mid-1860s to his final statements on signs between 1908 and 1911, this book draws out the essential theoretical differences between the earlier and later sign systems. Although the 1903 ten-class system has been extensively researched by scholars, this book is the first to exploit the untapped potential of the later six-element systems. Showing how these systems differ from the 1903 version, Peirce's Twenty-Eight Classes of Signs and the Philosophy of Representation offers an innovative and valuable reinterpretation of Peirce's thinking on signs and representation. Exploring the potential of the later sign-systems that Peirce scholars have hitherto been reluctant to engage with and extending Peirce's semiotic theory beyond the much canvassed systems of his Philosophy of Representation, this book will be essential reading for everyone working in the field of semiotics.
The Images of Time presents a philosophical investigation of the nature of time and the mind's ways of representing it. Robin Le Poidevin examines how we perceive time and change, the means by which memory links us with the past, the attempt to represent change and movement in art, and the nature of fictional time. These apparently disparate questions all concern the ways in which we represent aspects of time, in thought, experience, art and fiction. They also raisefundamental problems for our philosophical understanding, both of mental representation, and of the nature of time itself.Le Poidevin brings together issues in philosophy, psychology, aesthetics, and literary theory in examining the mechanisms underlying our representation of time in various media, and brings these to bear on metaphysical debates over the real nature of time. These debates concern which aspects of time are genuinely part of time's intrinsic nature, and which, in some sense, are mind-dependent.Arguably, the most important debate concerns time's passage: does time pass in reality, or is the division of events into past, present, and future simply a reflection of our temporal perspective - a result of the interaction between a 'static' world and minds capable of representing it? Le Poidevin argues that, contrary to what perception and memory lead us to suppose, time does not really pass, and this surprising conclusion can be reconciled with the characteristic features of temporalexperience.
Six Stories is a radically new look at the intersection of science and art through “failed” images.
This book is a sequel to the book by the same authors entitled Theory of Groups and Symmetries: Finite Groups, Lie Groups, and Lie Algebras.The presentation begins with the Dirac notation, which is illustrated by boson and fermion oscillator algebras and also Grassmann algebra. Then detailed account of finite-dimensional representations of groups SL(2, C) and SU(2) and their Lie algebras is presented. The general theory of finite-dimensional irreducible representations of simple Lie algebras based on the construction of highest weight representations is given. The classification of all finite-dimensional irreducible representations of the Lie algebras of the classical series sℓ(n, C), so(n, C) and sp(2r, C) is exposed.Finite-dimensional irreducible representations of linear groups SL(N, C) and their compact forms SU(N) are constructed on the basis of the Schur-Weyl duality. A special role here is played by the theory of representations of the symmetric group algebra C[Sr] (Schur-Frobenius theory, Okounkov-Vershik approach), based on combinatorics of Young diagrams and Young tableaux. Similar construction is given for pseudo-orthogonal groups O(p, q) and SO(p, q), including Lorentz groups O(1, N-1) and SO(1, N-1), and their Lie algebras, as well as symplectic groups Sp(p, q). The representation theory of Brauer algebra (centralizer algebra of SO(p, q) and Sp(p, q) groups in tensor representations) is discussed.Finally, the covering groups Spin(p, q) for pseudo-orthogonal groups SO↑(p, q) are studied. For this purpose, Clifford algebras in spaces Rp, q are introduced and representations of these algebras are discussed.
This eBook contains ten articles on the topic of representation of abstract concepts, both simple and complex, at the neural level in the brain. Seven of the articles directly address the main competing theories of mental representation – localist and distributed. Four of these articles argue – either on a theoretical basis or with neurophysiological evidence – that abstract concepts, simple or complex, exist (have to exist) at either the single cell level or in an exclusive neural cell assembly. There are three other papers that argue for sparse distributed representation (population coding) of abstract concepts. There are two other papers that discuss neural implementation of symbolic models. The remaining paper deals with learning of motor skills from imagery versus actual execution. A summary of these papers is provided in the Editorial.