David Hilbert's Lectures on the Foundations of Geometry, 1891-1902

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Author: David Hilbert,Michael Hallett,Ulrich Majer

Publisher: Springer Science & Business Media

ISBN: 9783540643739

Category: Literary Criticism

Page: 661

View: 3210

This volume contains notes for lectures on the foundations of geometry held by Hilbert from 1891-1902. These contain material which never found its way into print. The volume also reprints the first edition of Hilbert’s celebrated Grundlagen der Geometrie.

The Philosophy of Science: A-M

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Author: Sahotra Sarkar,Jessica Pfeifer

Publisher: Taylor & Francis

ISBN: 9780415977098

Category: Science

Page: 965

View: 7345

The first in-depth reference to the field that combines scientific knowledge with philosophical inquiry, this encyclopedia brings together a team of leading scholars to provide nearly 150 entries on the essential concepts in the philosophy of science. The areas covered include biology, chemistry, epistemology and metaphysics, physics, psychology and mind, the social sciences, and key figures in the combined studies of science and philosophy. (Midwest).

What is a Mathematical Concept?

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Author: Elizabeth de Freitas,Nathalie Sinclair,Alf Coles

Publisher: Cambridge University Press

ISBN: 1107134633

Category: Education

Page: 316

View: 1340

Leading thinkers in mathematics, philosophy and education offer new insights into the fundamental question: what is a mathematical concept?

Hilbert's Programs and Beyond

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Author: Wilfried Sieg

Publisher: Oxford University Press

ISBN: 0195372220

Category: Computers

Page: 439

View: 9028

David Hilbert was one of the great mathematicians who expounded the centrality of their subject in human thought. In this collection of essays, Wilfried Sieg frames Hilbert's foundational work, from 1890 to 1939, in a comprehensive way and integrates it with modern proof theoretic investigations.

Abstraction and Infinity

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Author: Paolo Mancosu

Publisher: Oxford University Press

ISBN: 0191063800

Category: Philosophy

Page: 240

View: 4851

Paolo Mancosu provides an original investigation of historical and systematic aspects of the notions of abstraction and infinity and their interaction. A familiar way of introducing concepts in mathematics rests on so-called definitions by abstraction. An example of this is Hume's Principle, which introduces the concept of number by stating that two concepts have the same number if and only if the objects falling under each one of them can be put in one-one correspondence. This principle is at the core of neo-logicism. In the first two chapters of the book, Mancosu provides a historical analysis of the mathematical uses and foundational discussion of definitions by abstraction up to Frege, Peano, and Russell. Chapter one shows that abstraction principles were quite widespread in the mathematical practice that preceded Frege's discussion of them and the second chapter provides the first contextual analysis of Frege's discussion of abstraction principles in section 64 of the Grundlagen. In the second part of the book, Mancosu discusses a novel approach to measuring the size of infinite sets known as the theory of numerosities and shows how this new development leads to deep mathematical, historical, and philosophical problems. The final chapter of the book explore how this theory of numerosities can be exploited to provide surprisingly novel perspectives on neo-logicism.

Mathematical Methods in Computer Vision

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Author: Peter J. Olver,Allen Tannenbaum

Publisher: Springer Science & Business Media

ISBN: 9780387004976

Category: Business & Economics

Page: 153

View: 9314

"Comprises some of the key work presented at two IMA Wokshops on Computer Vision during fall of 2000."--Pref.

The Implicit Function Theorem

History, Theory, and Applications

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Author: Steven G. Krantz,Harold R. Parks

Publisher: Springer Science & Business Media

ISBN: 1461459818

Category: Mathematics

Page: 163

View: 9087

The implicit function theorem is part of the bedrock of mathematical analysis and geometry. Finding its genesis in eighteenth century studies of real analytic functions and mechanics, the implicit and inverse function theorems have now blossomed into powerful tools in the theories of partial differential equations, differential geometry, and geometric analysis. There are many different forms of the implicit function theorem, including (i) the classical formulation for Ck functions, (ii) formulations in other function spaces, (iii) formulations for non-smooth function, and (iv) formulations for functions with degenerate Jacobian. Particularly powerful implicit function theorems, such as the Nash–Moser theorem, have been developed for specific applications (e.g., the imbedding of Riemannian manifolds). All of these topics, and many more, are treated in the present uncorrected reprint of this classic monograph. ​ Originally published in 2002, The Implicit Function Theorem is an accessible and thorough treatment of implicit and inverse function theorems and their applications. It will be of interest to mathematicians, graduate/advanced undergraduate students, and to those who apply mathematics. The book unifies disparate ideas that have played an important role in modern mathematics. It serves to document and place in context a substantial body of mathematical ideas.​

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

From Grundlagen der Geometrie to Grundlagen der Physik

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Author: L. Corry

Publisher: Springer Science & Business Media

ISBN: 1402027788

Category: Science

Page: 513

View: 9748

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 Cambridge Companion to Frege

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Author: Tom Ricketts,Michael Potter

Publisher: Cambridge University Press

ISBN: 0521624282

Category: Literary Criticism

Page: 639

View: 7914

Offers a comprehensive and accessible exploration of the scope and importance of Gottlob Frege's work.

Convex Variational Problems

Linear, nearly Linear and Anisotropic Growth Conditions

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Author: Michael Bildhauer

Publisher: Springer

ISBN: 3540448853

Category: Mathematics

Page: 220

View: 7423

The author emphasizes a non-uniform ellipticity condition as the main approach to regularity theory for solutions of convex variational problems with different types of non-standard growth conditions. This volume first focuses on elliptic variational problems with linear growth conditions. Here the notion of a "solution" is not obvious and the point of view has to be changed several times in order to get some deeper insight. Then the smoothness properties of solutions to convex anisotropic variational problems with superlinear growth are studied. In spite of the fundamental differences, a non-uniform ellipticity condition serves as the main tool towards a unified view of the regularity theory for both kinds of problems.

Completeness and Reduction in Algebraic Complexity Theory

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Author: Peter Bürgisser

Publisher: Springer Science & Business Media

ISBN: 3662041790

Category: Mathematics

Page: 168

View: 5729

This is a thorough and comprehensive treatment of the theory of NP-completeness in the framework of algebraic complexity theory. Coverage includes Valiant's algebraic theory of NP-completeness; interrelations with the classical theory as well as the Blum-Shub-Smale model of computation, questions of structural complexity; fast evaluation of representations of general linear groups; and complexity of immanants.

Geometry and the Imagination

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Author: David Hilbert,Stephan Cohn-Vossen

Publisher: American Mathematical Soc.

ISBN: 0821819984

Category: Mathematics

Page: 357

View: 2411

This remarkable book endures as a true masterpiece of mathematical exposition. The book is overflowing with mathematical ideas, which are always explained clearly and elegantly, and above all, with penetrating insight. It is a joy to read, both for beginners and experienced mathematicians. Geometry and the Imagination is full of interesting facts, many of which you wish you had known before. The book begins with examples of the simplest curves and surfaces, including thread constructions of certain quadrics and other surfaces. The chapter on regular systems of points leads to the crystallographic groups and the regular polyhedra in $\mathbb{R}^3$. In this chapter, they also discuss plane lattices. By considering unit lattices, and throwing in a small amount of number theory when necessary, they effortlessly derive Leibniz's series: $\pi/4 = 1 - 1/3 + 1/5 - 1/7 + - \ldots$. In the section on lattices in three and more dimensions, the authors consider sphere-packing problems, including the famous Kepler problem. One of the most remarkable chapters is ``Projective Configurations''. In a short introductory section, Hilbert and Cohn-Vossen give perhaps the most concise and lucid description of why a general geometer would care about projective geometry and why such an ostensibly plain setup is truly rich in structure and ideas. The chapter on kinematics includes a nice discussion of linkages and the geometry of configurations of points and rods that are connected and, perhaps, constrained in some way. This topic in geometry has become increasingly important in recent times, especially in applications to robotics. This is another example of a simple situation that leads to a rich geometry. It would be hard to overestimate the continuing influence Hilbert-Cohn-Vossen's book has had on mathematicians of this century. It surely belongs in the "pantheon" of great mathematics books.

The Theory of Algebraic Number Fields

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Author: David Hilbert

Publisher: Springer Science & Business Media

ISBN: 3662035456

Category: Mathematics

Page: 351

View: 2829

A translation of Hilberts "Theorie der algebraischen Zahlkörper" best known as the "Zahlbericht", first published in 1897, in which he provides an elegantly integrated overview of the development of algebraic number theory up to the end of the nineteenth century. The Zahlbericht also provided a firm foundation for further research in the theory, and can be seen as the starting point for all twentieth century investigations into the subject, as well as reciprocity laws and class field theory. This English edition further contains an introduction by F. Lemmermeyer and N. Schappacher.

Mathematical Problems from Applied Logic II

Logics for the XXIst Century

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Author: Dov Gabbay,Sergei Goncharov,Michael Zakharyaschev

Publisher: Springer Science & Business Media

ISBN: 0387692452

Category: Mathematics

Page: 354

View: 6553

This book presents contributions from world-renowned logicians, discussing important topics of logic from the point of view of their further development in light of requirements arising from successful application in Computer Science and AI language. Coverage includes: the logic of provability, computability theory applied to biology, psychology, physics, chemistry, economics, and other basic sciences; computability theory and computable models; logic and space-time geometry; hybrid systems; logic and region-based theory of space.

Geometry

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Author: Roger Fenn

Publisher: Springer Science & Business Media

ISBN: 1447103254

Category: Mathematics

Page: 313

View: 5603

Intended to introduce readers to the major geometrical topics taught at undergraduate level in a manner that is both accessible and rigorous, the author uses world measurement as a synonym for geometry - hence the importance of numbers, coordinates and their manipulation - and has included over 300 exercises, with answers to most of them.

Mathematical Thought From Ancient to Modern Times

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Author: Morris Kline

Publisher: Oxford University Press

ISBN: 0199770468

Category: Mathematics

Page: 432

View: 2086

The major creations and developments in mathematics from the beginnings in Babylonia and Egypt through the first few decades of the twentieth century are presented with clarity and precision in this comprehensive historical study.

Axiomatic Method and Category Theory

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Author: Andrei Rodin

Publisher: Springer Science & Business Media

ISBN: 3319004042

Category: Philosophy

Page: 285

View: 7217

This volume explores the many different meanings of the notion of the axiomatic method, offering an insightful historical and philosophical discussion about how these notions changed over the millennia. The author, a well-known philosopher and historian of mathematics, first examines Euclid, who is considered the father of the axiomatic method, before moving onto Hilbert and Lawvere. He then presents a deep textual analysis of each writer and describes how their ideas are different and even how their ideas progressed over time. Next, the book explores category theory and details how it has revolutionized the notion of the axiomatic method. It considers the question of identity/equality in mathematics as well as examines the received theories of mathematical structuralism. In the end, Rodin presents a hypothetical New Axiomatic Method, which establishes closer relationships between mathematics and physics. Lawvere's axiomatization of topos theory and Voevodsky's axiomatization of higher homotopy theory exemplify a new way of axiomatic theory building, which goes beyond the classical Hilbert-style Axiomatic Method. The new notion of Axiomatic Method that emerges in categorical logic opens new possibilities for using this method in physics and other natural sciences. This volume offers readers a coherent look at the past, present and anticipated future of the Axiomatic Method.