An Axiomatic Approach To Geometry

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An Axiomatic Approach to Geometry

Author : Francis Borceux
Publisher : Springer Science & Business Media
ISBN 13 : 3319017306
Total Pages : 403 pages
Book Rating : 4.3/5 (19 download)

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Book Synopsis An Axiomatic Approach to Geometry by : Francis Borceux

Download or read book An Axiomatic Approach to Geometry written by Francis Borceux and published by Springer Science & Business Media. This book was released on 2013-10-31 with total page 403 pages. Available in PDF, EPUB and Kindle. Book excerpt: Focusing methodologically on those historical aspects that are relevant to supporting intuition in axiomatic approaches to geometry, the book develops systematic and modern approaches to the three core aspects of axiomatic geometry: Euclidean, non-Euclidean and projective. Historically, axiomatic geometry marks the origin of formalized mathematical activity. It is in this discipline that most historically famous problems can be found, the solutions of which have led to various presently very active domains of research, especially in algebra. The recognition of the coherence of two-by-two contradictory axiomatic systems for geometry (like one single parallel, no parallel at all, several parallels) has led to the emergence of mathematical theories based on an arbitrary system of axioms, an essential feature of contemporary mathematics. This is a fascinating book for all those who teach or study axiomatic geometry, and who are interested in the history of geometry or who want to see a complete proof of one of the famous problems encountered, but not solved, during their studies: circle squaring, duplication of the cube, trisection of the angle, construction of regular polygons, construction of models of non-Euclidean geometries, etc. It also provides hundreds of figures that support intuition. Through 35 centuries of the history of geometry, discover the birth and follow the evolution of those innovative ideas that allowed humankind to develop so many aspects of contemporary mathematics. Understand the various levels of rigor which successively established themselves through the centuries. Be amazed, as mathematicians of the 19th century were, when observing that both an axiom and its contradiction can be chosen as a valid basis for developing a mathematical theory. Pass through the door of this incredible world of axiomatic mathematical theories!

From Affine to Euclidean Geometry

Author : W. Szmielew
Publisher : Springer
ISBN 13 : 9789027712431
Total Pages : 212 pages
Book Rating : 4.7/5 (124 download)

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Book Synopsis From Affine to Euclidean Geometry by : W. Szmielew

Download or read book From Affine to Euclidean Geometry written by W. Szmielew and published by Springer. This book was released on 1983-08-31 with total page 212 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Axiomatic Geometry

Author : John M. Lee
Publisher : American Mathematical Soc.
ISBN 13 : 0821884786
Total Pages : 490 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis Axiomatic Geometry by : John M. Lee

Download or read book Axiomatic Geometry written by John M. Lee and published by American Mathematical Soc.. This book was released on 2013-04-10 with total page 490 pages. Available in PDF, EPUB and Kindle. Book excerpt: The story of geometry is the story of mathematics itself: Euclidean geometry was the first branch of mathematics to be systematically studied and placed on a firm logical foundation, and it is the prototype for the axiomatic method that lies at the foundation of modern mathematics. It has been taught to students for more than two millennia as a mode of logical thought. This book tells the story of how the axiomatic method has progressed from Euclid's time to ours, as a way of understanding what mathematics is, how we read and evaluate mathematical arguments, and why mathematics has achieved the level of certainty it has. It is designed primarily for advanced undergraduates who plan to teach secondary school geometry, but it should also provide something of interest to anyone who wishes to understand geometry and the axiomatic method better. It introduces a modern, rigorous, axiomatic treatment of Euclidean and (to a lesser extent) non-Euclidean geometries, offering students ample opportunities to practice reading and writing proofs while at the same time developing most of the concrete geometric relationships that secondary teachers will need to know in the classroom. -- P. [4] of cover.

The Axiomatic Method with Special Reference to Geometry and Physics

Author : Leon Henkin
Publisher :
ISBN 13 :
Total Pages : 562 pages
Book Rating : 4.3/5 (91 download)

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Book Synopsis The Axiomatic Method with Special Reference to Geometry and Physics by : Leon Henkin

Download or read book The Axiomatic Method with Special Reference to Geometry and Physics written by Leon Henkin and published by . This book was released on 1959 with total page 562 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Geometry of Vector Sheaves

Author : Anastasios Mallios
Publisher : Springer Science & Business Media
ISBN 13 : 9780792350040
Total Pages : 468 pages
Book Rating : 4.3/5 (5 download)

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Book Synopsis Geometry of Vector Sheaves by : Anastasios Mallios

Download or read book Geometry of Vector Sheaves written by Anastasios Mallios and published by Springer Science & Business Media. This book was released on 1998 with total page 468 pages. Available in PDF, EPUB and Kindle. Book excerpt: This text is part of a two-volume monograph which obtains fundamental notions and results of the standard differential geometry of smooth manifolds, without using differential calculus. Here, the sheaf-theoretic character is emphasized. This has theoretical advantages such as greater perspective, clarity and unification, but also practical benefits ranging from elementary particle physics, via gauge theories and theoretical cosmology (differential spaces), to non-linear PDEs (generalized functions). Thus, more general applications, which are no longer smooth in the classical sense, can be coped with. The treatise might also be construed as a new systematic endeavour to confront the ever-increasing notion that the world around us is far from being smooth enough.

Geometry of Vector Sheaves

Author : Anastasios Mallios
Publisher : Springer Science & Business Media
ISBN 13 : 9401150060
Total Pages : 457 pages
Book Rating : 4.4/5 (11 download)

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Book Synopsis Geometry of Vector Sheaves by : Anastasios Mallios

Download or read book Geometry of Vector Sheaves written by Anastasios Mallios and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 457 pages. Available in PDF, EPUB and Kindle. Book excerpt: This two-volume monograph obtains fundamental notions and results of the standard differential geometry of smooth (CINFINITY) manifolds, without using differential calculus. Here, the sheaf-theoretic character is emphasised. This has theoretical advantages such as greater perspective, clarity and unification, but also practical benefits ranging from elementary particle physics, via gauge theories and theoretical cosmology (`differential spaces'), to non-linear PDEs (generalised functions). Thus, more general applications, which are no longer `smooth' in the classical sense, can be coped with. The treatise might also be construed as a new systematic endeavour to confront the ever-increasing notion that the `world around us is far from being smooth enough'. Audience: This work is intended for postgraduate students and researchers whose work involves differential geometry, global analysis, analysis on manifolds, algebraic topology, sheaf theory, cohomology, functional analysis or abstract harmonic analysis.

Axiomatic Projective Geometry

Author : A. Heyting
Publisher : Elsevier
ISBN 13 : 1483259315
Total Pages : 160 pages
Book Rating : 4.4/5 (832 download)

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Book Synopsis Axiomatic Projective Geometry by : A. Heyting

Download or read book Axiomatic Projective Geometry written by A. Heyting and published by Elsevier. This book was released on 2014-05-12 with total page 160 pages. Available in PDF, EPUB and Kindle. Book excerpt: Bibliotheca Mathematica: A Series of Monographs on Pure and Applied Mathematics, Volume V: Axiomatic Projective Geometry, Second Edition focuses on the principles, operations, and theorems in axiomatic projective geometry, including set theory, incidence propositions, collineations, axioms, and coordinates. The publication first elaborates on the axiomatic method, notions from set theory and algebra, analytic projective geometry, and incidence propositions and coordinates in the plane. Discussions focus on ternary fields attached to a given projective plane, homogeneous coordinates, ternary field and axiom system, projectivities between lines, Desargues' proposition, and collineations. The book takes a look at incidence propositions and coordinates in space. Topics include coordinates of a point, equation of a plane, geometry over a given division ring, trivial axioms and propositions, sixteen points proposition, and homogeneous coordinates. The text examines the fundamental proposition of projective geometry and order, including cyclic order of the projective line, order and coordinates, geometry over an ordered ternary field, cyclically ordered sets, and fundamental proposition. The manuscript is a valuable source of data for mathematicians and researchers interested in axiomatic projective geometry.

Intuition and the Axiomatic Method

Author : Emily Carson
Publisher : Springer Science & Business Media
ISBN 13 : 9781402040399
Total Pages : 356 pages
Book Rating : 4.0/5 (43 download)

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Book Synopsis Intuition and the Axiomatic Method by : Emily Carson

Download or read book Intuition and the Axiomatic Method written by Emily Carson and published by Springer Science & Business Media. This book was released on 2006-01-24 with total page 356 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Studies in Logic and the Foundations of Mathematics - The Axiomatic Method with Special Reference to Geometry and Physics

Author : L. Brouwer
Publisher : Brouwer Press
ISBN 13 : 1443728128
Total Pages : 504 pages
Book Rating : 4.4/5 (437 download)

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Book Synopsis Studies in Logic and the Foundations of Mathematics - The Axiomatic Method with Special Reference to Geometry and Physics by : L. Brouwer

Download or read book Studies in Logic and the Foundations of Mathematics - The Axiomatic Method with Special Reference to Geometry and Physics written by L. Brouwer and published by Brouwer Press. This book was released on 2008-11 with total page 504 pages. Available in PDF, EPUB and Kindle. Book excerpt: Many of the earliest books, particularly those dating back to the 1900s and before, are now extremely scarce and increasingly expensive. We are republishing these classic works in affordable, high quality, modern editions, using the original text and artwork.

Discovering Geometry

Author : Wladimir-Georges Boskoff
Publisher :
ISBN 13 : 9789737556684
Total Pages : 134 pages
Book Rating : 4.5/5 (566 download)

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Book Synopsis Discovering Geometry by : Wladimir-Georges Boskoff

Download or read book Discovering Geometry written by Wladimir-Georges Boskoff and published by . This book was released on 2011 with total page 134 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Axiomatic Thinking II

Author : Fernando Ferreira
Publisher : Springer Nature
ISBN 13 : 3030777995
Total Pages : 293 pages
Book Rating : 4.0/5 (37 download)

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Book Synopsis Axiomatic Thinking II by : Fernando Ferreira

Download or read book Axiomatic Thinking II written by Fernando Ferreira and published by Springer Nature. This book was released on 2022-09-17 with total page 293 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this two-volume compilation of articles, leading researchers reevaluate the success of Hilbert's axiomatic method, which not only laid the foundations for our understanding of modern mathematics, but also found applications in physics, computer science and elsewhere. The title takes its name from David Hilbert's seminal talk Axiomatisches Denken, given at a meeting of the Swiss Mathematical Society in Zurich in 1917. This marked the beginning of Hilbert's return to his foundational studies, which ultimately resulted in the establishment of proof theory as a new branch in the emerging field of mathematical logic. Hilbert also used the opportunity to bring Paul Bernays back to Göttingen as his main collaborator in foundational studies in the years to come. The contributions are addressed to mathematical and philosophical logicians, but also to philosophers of science as well as physicists and computer scientists with an interest in foundations.

Geometry and Discrete Mathematics

Author : Benjamin Fine
Publisher : Walter de Gruyter GmbH & Co KG
ISBN 13 : 3110740931
Total Pages : 412 pages
Book Rating : 4.1/5 (17 download)

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Book Synopsis Geometry and Discrete Mathematics by : Benjamin Fine

Download or read book Geometry and Discrete Mathematics written by Benjamin Fine and published by Walter de Gruyter GmbH & Co KG. This book was released on 2022-08-22 with total page 412 pages. Available in PDF, EPUB and Kindle. Book excerpt: In the two-volume set ‘A Selection of Highlights’ we present basics of mathematics in an exciting and pedagogically sound way. This volume examines many fundamental results in Geometry and Discrete Mathematics along with their proofs and their history. In the second edition we include a new chapter on Topological Data Analysis and enhanced the chapter on Graph Theory for solving further classical problems such as the Traveling Salesman Problem.

Geometry

Author : Israel M. Gelfand
Publisher : Springer Nature
ISBN 13 : 1071602993
Total Pages : 420 pages
Book Rating : 4.0/5 (716 download)

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Book Synopsis Geometry by : Israel M. Gelfand

Download or read book Geometry written by Israel M. Gelfand and published by Springer Nature. This book was released on 2020-02-22 with total page 420 pages. Available in PDF, EPUB and Kindle. Book excerpt: This text is the fifth and final in the series of educational books written by Israel Gelfand with his colleagues for high school students. These books cover the basics of mathematics in a clear and simple format – the style Gelfand was known for internationally. Gelfand prepared these materials so as to be suitable for independent studies, thus allowing students to learn and practice the material at their own pace without a class. Geometry takes a different approach to presenting basic geometry for high-school students and others new to the subject. Rather than following the traditional axiomatic method that emphasizes formulae and logical deduction, it focuses on geometric constructions. Illustrations and problems are abundant throughout, and readers are encouraged to draw figures and “move” them in the plane, allowing them to develop and enhance their geometrical vision, imagination, and creativity. Chapters are structured so that only certain operations and the instruments to perform these operations are available for drawing objects and figures on the plane. This structure corresponds to presenting, sequentially, projective, affine, symplectic, and Euclidean geometries, all the while ensuring students have the necessary tools to follow along. Geometry is suitable for a large audience, which includes not only high school geometry students, but also teachers and anyone else interested in improving their geometrical vision and intuition, skills useful in many professions. Similarly, experienced mathematicians can appreciate the book’s unique way of presenting plane geometry in a simple form while adhering to its depth and rigor. “Gelfand was a great mathematician and also a great teacher. The book provides an atypical view of geometry. Gelfand gets to the intuitive core of geometry, to the phenomena of shapes and how they move in the plane, leading us to a better understanding of what coordinate geometry and axiomatic geometry seek to describe.” - Mark Saul, PhD, Executive Director, Julia Robinson Mathematics Festival “The subject matter is presented as intuitive, interesting and fun. No previous knowledge of the subject is required. Starting from the simplest concepts and by inculcating in the reader the use of visualization skills, [and] after reading the explanations and working through the examples, you will be able to confidently tackle the interesting problems posed. I highly recommend the book to any person interested in this fascinating branch of mathematics.” - Ricardo Gorrin, a student of the Extended Gelfand Correspondence Program in Mathematics (EGCPM)

David Hilbert and the Axiomatization of Physics (1898–1918)

Author : L. Corry
Publisher : Springer Science & Business Media
ISBN 13 : 1402027788
Total Pages : 513 pages
Book Rating : 4.4/5 (2 download)

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Book Synopsis David Hilbert and the Axiomatization of Physics (1898–1918) by : L. Corry

Download or read book David Hilbert and the Axiomatization of Physics (1898–1918) written by L. Corry and published by Springer Science & Business Media. This book was released on 2013-06-29 with total page 513 pages. Available in PDF, EPUB and Kindle. Book excerpt: David Hilbert (1862-1943) was the most influential mathematician of the early twentieth century and, together with Henri Poincaré, the last mathematical universalist. His main known areas of research and influence were in pure mathematics (algebra, number theory, geometry, integral equations and analysis, logic and foundations), but he was also known to have some interest in physical topics. The latter, however, was traditionally conceived as comprising only sporadic incursions into a scientific domain which was essentially foreign to his mainstream of activity and in which he only made scattered, if important, contributions. Based on an extensive use of mainly unpublished archival sources, the present book presents a totally fresh and comprehensive picture of Hilbert’s intense, original, well-informed, and highly influential involvement with physics, that spanned his entire career and that constituted a truly main focus of interest in his scientific horizon. His program for axiomatizing physical theories provides the connecting link with his research in more purely mathematical fields, especially geometry, and a unifying point of view from which to understand his physical activities in general. In particular, the now famous dialogue and interaction between Hilbert and Einstein, leading to the formulation in 1915 of the generally covariant field-equations of gravitation, is adequately explored here within the natural context of Hilbert’s overall scientific world-view. This book will be of interest to historians of physics and of mathematics, to historically-minded physicists and mathematicians, and to philosophers of science.

The Foundations of Geometry

Author : David Hilbert
Publisher : Prabhat Prakashan
ISBN 13 :
Total Pages : 94 pages
Book Rating : 4./5 ( download)

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Book Synopsis The Foundations of Geometry by : David Hilbert

Download or read book The Foundations of Geometry written by David Hilbert and published by Prabhat Prakashan. This book was released on 1950-01-01 with total page 94 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Geometry and Its Applications

Author : Walter A. Meyer
Publisher : Elsevier
ISBN 13 : 0080478034
Total Pages : 560 pages
Book Rating : 4.0/5 (84 download)

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Book Synopsis Geometry and Its Applications by : Walter A. Meyer

Download or read book Geometry and Its Applications written by Walter A. Meyer and published by Elsevier. This book was released on 2006-02-21 with total page 560 pages. Available in PDF, EPUB and Kindle. Book excerpt: Meyer's Geometry and Its Applications, Second Edition, combines traditional geometry with current ideas to present a modern approach that is grounded in real-world applications. It balances the deductive approach with discovery learning, and introduces axiomatic, Euclidean geometry, non-Euclidean geometry, and transformational geometry. The text integrates applications and examples throughout and includes historical notes in many chapters. The Second Edition of Geometry and Its Applications is a significant text for any college or university that focuses on geometry's usefulness in other disciplines. It is especially appropriate for engineering and science majors, as well as future mathematics teachers. Realistic applications integrated throughout the text, including (but not limited to): Symmetries of artistic patterns Physics Robotics Computer vision Computer graphics Stability of architectural structures Molecular biology Medicine Pattern recognition Historical notes included in many chapters

Hilbert, Göttingen and the Development of Modern Mathematics

Author : Joan Roselló
Publisher : Cambridge Scholars Publishing
ISBN 13 : 152752762X
Total Pages : 295 pages
Book Rating : 4.5/5 (275 download)

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Book Synopsis Hilbert, Göttingen and the Development of Modern Mathematics by : Joan Roselló

Download or read book Hilbert, Göttingen and the Development of Modern Mathematics written by Joan Roselló and published by Cambridge Scholars Publishing. This book was released on 2019-02-01 with total page 295 pages. Available in PDF, EPUB and Kindle. Book excerpt: David Hilbert is one of the outstanding mathematicians of the twentieth century and probably the most influential. This book highlights Hilbert’s contributions to mathematics, putting them in their historical, social and cultural context. In doing so, particular attention is paid to Hilbert’s axiomatic method and his proposal for the foundations of mathematics, the so-called Hilbert’s program. The book also discusses the development of algebraic number theory, the theory of integral equations, modern algebra and the structural image of mathematics. In addition, it considers the famous list of Mathematical Problems presented in Paris in 1900, the mathematical tradition of the University of Göttingen, the great debate on the foundations of mathematics in the twenties between formalists and intuitionists, and, finally, Hilbert’s work on the theory of relativity and the foundations of quantum mechanics. The book will primarily appeal to an academic audience, although it will also be of interest to general-interest science readers.