Geometry Intuition And Concepts

Author : Jost-Hinrich Eschenburg
Publisher : Springer Nature
Page : 168 pages
File Size : 43,9 Mb
Release : 2022-10-31
Category : Mathematics
ISBN : 9783658386405

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Geometry - Intuition and Concepts by Jost-Hinrich Eschenburg Pdf

This book deals with the geometry of visual space in all its aspects. As in any branch of mathematics, the aim is to trace the hidden to the obvious; the peculiarity of geometry is that the obvious is sometimes literally before one's eyes.Starting from intuition, spatial concepts are embedded in the pre-existing mathematical framework of linear algebra and calculus. The path from visualization to mathematically exact language is itself the learning content of this book. This is intended to close an often lamented gap in understanding between descriptive preschool and school geometry and the abstract concepts of linear algebra and calculus. At the same time, descriptive geometric modes of argumentation are justified because their embedding in the strict mathematical language has been clarified. The concepts of geometry are of a very different nature; they denote, so to speak, different layers of geometric thinking: some arguments use only concepts such as point, straight line, and incidence, others require angles and distances, still others symmetry considerations. Each of these conceptual fields determines a separate subfield of geometry and a separate chapter of this book, with the exception of the last-mentioned conceptual field "symmetry", which runs through all the others: - Incidence: Projective geometry - Parallelism: Affine geometry - Angle: Conformal Geometry - Distance: Metric Geometry - Curvature: Differential Geometry - Angle as distance measure: Spherical and Hyperbolic Geometry - Symmetry: Mapping Geometry. The mathematical experience acquired in the visual space can be easily transferred to much more abstract situations with the help of the vector space notion. The generalizations beyond the visual dimension point in two directions: Extension of the number concept and transcending the three illustrative dimensions. This book is a translation of the original German 1st edition Geometrie – Anschauung und Begriffe by Jost-Hinrich Eschenburg, published by Springer Fachmedien Wiesbaden GmbH, part of Springer Nature in 2020. The translation was done with the help of artificial intelligence (machine translation by the service DeepL.com). A subsequent human revision was done primarily in terms of content, so that the book will read stylistically differently from a conventional translation. Springer Nature works continuously to further the development of tools for the production of books and on the related technologies to support the authors.

Author : David Eisenbud,Joe Harris
Publisher : Springer Science & Business Media
Page : 340 pages
File Size : 41,5 Mb
Release : 2006-04-06
Category : Mathematics
ISBN : 9780387226392

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The Geometry of Schemes by David Eisenbud,Joe Harris Pdf

Grothendieck’s beautiful theory of schemes permeates modern algebraic geometry and underlies its applications to number theory, physics, and applied mathematics. This simple account of that theory emphasizes and explains the universal geometric concepts behind the definitions. In the book, concepts are illustrated with fundamental examples, and explicit calculations show how the constructions of scheme theory are carried out in practice.

Author : Emily Carson,Renate Huber
Publisher : Springer Science & Business Media
Page : 356 pages
File Size : 51,6 Mb
Release : 2006-01-24
Category : Mathematics
ISBN : 1402040393

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Intuition and the Axiomatic Method by Emily Carson,Renate Huber Pdf

Following developments in modern geometry, logic and physics, many scientists and philosophers in the modern era considered Kant’s theory of intuition to be obsolete. But this only represents one side of the story concerning Kant, intuition and twentieth century science. Several prominent mathematicians and physicists were convinced that the formal tools of modern logic, set theory and the axiomatic method are not sufficient for providing mathematics and physics with satisfactory foundations. All of Hilbert, Gödel, Poincaré, Weyl and Bohr thought that intuition was an indispensable element in describing the foundations of science. They had very different reasons for thinking this, and they had very different accounts of what they called intuition. But they had in common that their views of mathematics and physics were significantly influenced by their readings of Kant. In the present volume, various views of intuition and the axiomatic method are explored, beginning with Kant’s own approach. By way of these investigations, we hope to understand better the rationale behind Kant’s theory of intuition, as well as to grasp many facets of the relations between theories of intuition and the axiomatic method, dealing with both their strengths and limitations; in short, the volume covers logical and non-logical, historical and systematic issues in both mathematics and physics.

Author : Thomas C. Vinci
Publisher : Oxford University Press, USA
Page : 265 pages
File Size : 42,5 Mb
Release : 2015
Category : Philosophy
ISBN : 9780199381166

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Space, Geometry, and Kant's Transcendental Deduction of the Categories by Thomas C. Vinci Pdf

In section 20 in the B edition 'Deduction', Kant states that his purpose is achieved: to show that all intuitions in general are subject to the categories. The standard reading understands this to mean that all our representational ideas, including those originating in sense experience, are structured by categories: there are 'no judgments of perception' in the doctrine of the 'First Critique', only judgments of experience. Against this reading the book argues that while all intuitions for Kant are unified intuitions, not all are unified by the categories, thus allowing for judgments of perception.

Author : Nancy Smythe
Publisher : Springer Science & Business Media
Page : 233 pages
File Size : 44,6 Mb
Release : 2012-12-06
Category : Philosophy
ISBN : 9789400996687

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Forms of Intuition by Nancy Smythe Pdf

Author : L. Magnani
Publisher : Springer Science & Business Media
Page : 256 pages
File Size : 44,9 Mb
Release : 2012-12-06
Category : Science
ISBN : 9789401096225

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Philosophy and Geometry by L. Magnani Pdf

Philosophers have studied geometry since ancient times. Geometrical knowledge has often played the role of a laboratory for the philosopher's conceptual experiments dedicated to the ideation of powerful theories of knowledge. Lorenzo Magnani's new book Philosophy and Geometry illustrates the rich intrigue of this fascinating story of human knowledge, providing a new analysis of the ideas of many scholars (including Plato, Proclus, Kant, and Poincaré), and discussing conventionalist and neopositivist perspectives and the problem of the origins of geometry. The book also ties together the concerns of philosophers of science and cognitive scientists, showing, for example, the connections between geometrical reasoning and cognition as well as the results of recent logical and computational models of geometrical reasoning. All the topics are dealt with using a novel combination of both historical and contemporary perspectives. Philosophy and Geometry is a valuable contribution to the renaissance of research in the field.

Author : B.H. Arnold
Publisher : Courier Corporation
Page : 192 pages
File Size : 40,9 Mb
Release : 2015-02-23
Category : Mathematics
ISBN : 9780486275765

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Intuitive Concepts in Elementary Topology by B.H. Arnold Pdf

Classroom-tested and much-cited, this concise text is designed for undergraduates. It offers a valuable and instructive introduction to the basic concepts of topology, taking an intuitive rather than an axiomatic viewpoint. 1962 edition.

Author : Meridon Vestal Garner,B. G. Nunley
Publisher : Unknown
Page : 186 pages
File Size : 47,8 Mb
Release : 1971
Category : Geometria
ISBN : 0876203500

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Geometry by Meridon Vestal Garner,B. G. Nunley Pdf

Here, explored in an intuitive manner, are the basic concepts of geometry required for effective teaching of elementary school mathematics. The text investigates the metric and non-metric properties of both plane and three-dimensional geometric figures. Also included is a study of plane coordinate geometry. Very precise mathematically, this text will serve as the basis for a more rigorous study of geometry. Answers to selected problems are also provided.

Author : Imre Bárány,K. Böröczky
Publisher : Unknown
Page : 456 pages
File Size : 53,7 Mb
Release : 1997
Category : Geometry
ISBN : UOM:39015050320566

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Intuitive Geometry by Imre Bárány,K. Böröczky Pdf

Author : Henri Cartan
Publisher : Courier Corporation
Page : 178 pages
File Size : 41,8 Mb
Release : 2012-07-06
Category : Mathematics
ISBN : 9780486139111

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Differential Forms by Henri Cartan Pdf

"Cartan's work provides a superb text for an undergraduate course in advanced calculus, but at the same time it furnishes the reader with an excellent foundation for global and nonlinear algebra."—Mathematical Review "Brilliantly successful."—Bulletin de l'Association des Professeurs de Mathematiques "The presentation is precise and detailed, the style lucid and almost conversational . . . clearly an outstanding text and work of reference."—Annales Cartan's Formes Differentielles was first published in France in 1967. It was based on the world-famous teacher's experience at the Faculty of Sciences in Paris, where his reputation as an outstanding exponent of the Bourbaki school of mathematics was first established. Addressed to second- and third-year students of mathematics, the material skillfully spans the pure and applied branches in the familiar French manner, so that the applied aspects gain in rigor while the pure mathematics loses none of its dignity. This book is equally essential as a course text, as a work of reference, or simply as a brilliant mathematical exercise.

Author : L. Magnani
Publisher : Springer Science & Business Media
Page : 280 pages
File Size : 41,5 Mb
Release : 2001-11-30
Category : Gardening
ISBN : 1402002416

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Philosophy and Geometry by L. Magnani Pdf

Philosophers have studied geometry since ancient times. Geometrical knowledge has often played the role of a laboratory for the philosopher's conceptual experiments dedicated to the ideation of powerful theories of knowledge. Lorenzo Magnani's new book Philosophy and Geometry illustrates the rich intrigue of this fascinating story of human knowledge, providing a new analysis of the ideas of many scholars (including Plato, Proclus, Kant, and Poincaré), and discussing conventionalist and neopositivist perspectives and the problem of the origins of geometry. The book also ties together the concerns of philosophers of science and cognitive scientists, showing, for example, the connections between geometrical reasoning and cognition as well as the results of recent logical and computational models of geometrical reasoning. All the topics are dealt with using a novel combination of both historical and contemporary perspectives. Philosophy and Geometry is a valuable contribution to the renaissance of research in the field.

Author : Daniel Perrin
Publisher : Springer Science & Business Media
Page : 263 pages
File Size : 49,7 Mb
Release : 2007-12-16
Category : Mathematics
ISBN : 9781848000568

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Algebraic Geometry by Daniel Perrin Pdf

Aimed primarily at graduate students and beginning researchers, this book provides an introduction to algebraic geometry that is particularly suitable for those with no previous contact with the subject; it assumes only the standard background of undergraduate algebra. The book starts with easily-formulated problems with non-trivial solutions and uses these problems to introduce the fundamental tools of modern algebraic geometry: dimension; singularities; sheaves; varieties; and cohomology. A range of exercises is provided for each topic discussed, and a selection of problems and exam papers are collected in an appendix to provide material for further study.

Author : Francesca Biagioli
Publisher : Springer
Page : 239 pages
File Size : 46,9 Mb
Release : 2016-08-22
Category : Philosophy
ISBN : 9783319317793

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Space, Number, and Geometry from Helmholtz to Cassirer by Francesca Biagioli Pdf

This book offers a reconstruction of the debate on non-Euclidean geometry in neo-Kantianism between the second half of the nineteenth century and the first decades of the twentieth century. Kant famously characterized space and time as a priori forms of intuitions, which lie at the foundation of mathematical knowledge. The success of his philosophical account of space was due not least to the fact that Euclidean geometry was widely considered to be a model of certainty at his time. However, such later scientific developments as non-Euclidean geometries and Einstein’s general theory of relativity called into question the certainty of Euclidean geometry and posed the problem of reconsidering space as an open question for empirical research. The transformation of the concept of space from a source of knowledge to an object of research can be traced back to a tradition, which includes such mathematicians as Carl Friedrich Gauss, Bernhard Riemann, Richard Dedekind, Felix Klein, and Henri Poincaré, and which finds one of its clearest expressions in Hermann von Helmholtz’s epistemological works. Although Helmholtz formulated compelling objections to Kant, the author reconsiders different strategies for a philosophical account of the same transformation from a neo-Kantian perspective, and especially Hermann Cohen’s account of the aprioricity of mathematics in terms of applicability and Ernst Cassirer’s reformulation of the a priori of space in terms of a system of hypotheses. This book is ideal for students, scholars and researchers who wish to broaden their knowledge of non-Euclidean geometry or neo-Kantianism.

Author : Mateusz Hohol
Publisher : Routledge
Page : 188 pages
File Size : 46,7 Mb
Release : 2019-09-12
Category : Psychology
ISBN : 9780429508592

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Foundations of Geometric Cognition by Mateusz Hohol Pdf

The cognitive foundations of geometry have puzzled academics for a long time, and even today are mostly unknown to many scholars, including mathematical cognition researchers. Foundations of Geometric Cognition shows that basic geometric skills are deeply hardwired in the visuospatial cognitive capacities of our brains, namely spatial navigation and object recognition. These capacities, shared with non-human animals and appearing in early stages of the human ontogeny, cannot, however, fully explain a uniquely human form of geometric cognition. In the book, Hohol argues that Euclidean geometry would not be possible without the human capacity to create and use abstract concepts, demonstrating how language and diagrams provide cognitive scaffolding for abstract geometric thinking, within a context of a Euclidean system of thought. Taking an interdisciplinary approach and drawing on research from diverse fields including psychology, cognitive science, and mathematics, this book is a must-read for cognitive psychologists and cognitive scientists of mathematics, alongside anyone interested in mathematical education or the philosophical and historical aspects of geometry.

Author : Eugene M. Izhikevich
Publisher : MIT Press
Page : 459 pages
File Size : 46,7 Mb
Release : 2010-01-22
Category : Medical
ISBN : 9780262514200

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Dynamical Systems in Neuroscience by Eugene M. Izhikevich Pdf

Explains the relationship of electrophysiology, nonlinear dynamics, and the computational properties of neurons, with each concept presented in terms of both neuroscience and mathematics and illustrated using geometrical intuition. In order to model neuronal behavior or to interpret the results of modeling studies, neuroscientists must call upon methods of nonlinear dynamics. This book offers an introduction to nonlinear dynamical systems theory for researchers and graduate students in neuroscience. It also provides an overview of neuroscience for mathematicians who want to learn the basic facts of electrophysiology. Dynamical Systems in Neuroscience presents a systematic study of the relationship of electrophysiology, nonlinear dynamics, and computational properties of neurons. It emphasizes that information processing in the brain depends not only on the electrophysiological properties of neurons but also on their dynamical properties. The book introduces dynamical systems, starting with one- and two-dimensional Hodgkin-Huxley-type models and continuing to a description of bursting systems. Each chapter proceeds from the simple to the complex, and provides sample problems at the end. The book explains all necessary mathematical concepts using geometrical intuition; it includes many figures and few equations, making it especially suitable for non-mathematicians. Each concept is presented in terms of both neuroscience and mathematics, providing a link between the two disciplines. Nonlinear dynamical systems theory is at the core of computational neuroscience research, but it is not a standard part of the graduate neuroscience curriculum—or taught by math or physics department in a way that is suitable for students of biology. This book offers neuroscience students and researchers a comprehensive account of concepts and methods increasingly used in computational neuroscience. An additional chapter on synchronization, with more advanced material, can be found at the author's website, www.izhikevich.com.