Categories for the Working Philosopher


Author: Elaine Landry

Publisher: Oxford University Press

ISBN: 019874899X

Category: Mathematics

Page: 528

View: 1812

This is the first volume on category theory for a broad philosophical readership. It is designed to show the interest and significance of category theory for a range of philosophical interests: mathematics, proof theory, computation, cognition, scientific modelling, physics, ontology, the structure of the world. Each chapter is written by either a category-theorist or a philosopher working in one of the represented areas, in an accessible waythat builds on the concepts that are already familiar to philosophers working in these areas.

From a Geometrical Point of View

A Study of the History and Philosophy of Category Theory


Author: Jean-Pierre Marquis

Publisher: Springer Science & Business Media

ISBN: 1402093845

Category: Science

Page: 310

View: 2628

From a Geometrical Point of View explores historical and philosophical aspects of category theory, trying therewith to expose its significance in the mathematical landscape. The main thesis is that Klein’s Erlangen program in geometry is in fact a particular instance of a general and broad phenomenon revealed by category theory. The volume starts with Eilenberg and Mac Lane’s work in the early 1940’s and follows the major developments of the theory from this perspective. Particular attention is paid to the philosophical elements involved in this development. The book ends with a presentation of categorical logic, some of its results and its significance in the foundations of mathematics. From a Geometrical Point of View aims to provide its readers with a conceptual perspective on category theory and categorical logic, in order to gain insight into their role and nature in contemporary mathematics. It should be of interest to mathematicians, logicians, philosophers of mathematics and science in general, historians of contemporary mathematics, physicists and computer scientists.

Relative category theory and geometric morphisms

a logical approach


Author: Jonathan Chapman,Frederick Rowbottom

Publisher: Oxford University Press, USA


Category: Mathematics

Page: 263

View: 4238

Topos theory provides an important setting and language for much of mathematical logic and set theory. It is well known that a typed language can be given for a topos to be regarded as a category of sets. This enables a fruitful interplay between category theory and set theory. However, one stumbling block to a logical approach to topos theory has been the treatment of geometric morphisms. This book presents a convenient and natural solution to this problem by developing the notion of a frame relative to an elementary topos. The authors show how this technique enables a logical approach to be taken to topics such as category theory relative to a topos and the relative Giraud theorem. The work is self-contained except that the authors presuppose a familiarity with basic category theory and topos theory. Logicians, set and category theorists, and computer scientist working in the field will find this work essential reading.

Theories, Sites, Toposes

Relating and Studying Mathematical Theories Through Topos-Theoretic 'bridges'


Author: Olivia Caramello

Publisher: Oxford University Press

ISBN: 019875891X

Category: Mathematics

Page: 336

View: 6631

According to Grothendieck, the notion of topos is "the bed or deep river where come to be married geometry and algebra, topology and arithmetic, mathematical logic and category theory, the world of the continuous and that of discontinuous or discrete structures." It is what he had "conceived of most broad to perceive with finesse, by the same language rich of geometric resonances, an "essence" which is common to situations most distant from each other, coming from one region or another of the vast universe of mathematical things." The aim of this book is to present a theory and a number of techniques which allow to give substance to Grothendieck's vision by building on the notion of classifying topos educed by categorical logicians. Mathematical theories (formalized within first-order logic) give rise to geometric objects called sites; the passage from sites to their associated toposes embodies the passage from the logical presentation of theories to their mathematical content, i.e. from syntax to semantics. The essential ambiguity given by the fact that any topos is associated in general with an infinite number of theories or different sites allows to study the relations between different theories, and hence the theories themselves, by using toposes as 'bridges' between these different presentations. The expression or calculation of invariants of toposes in terms of the theories associated with them or their sites of definition generates a great number of results and notions varying according to the different types of presentation, giving rise to a veritable mathematical morphogenesis.



Author: N.A

Publisher: N.A


Category: American literature

Page: N.A

View: 5235

From Riemann to Differential Geometry and Relativity


Author: Lizhen Ji,Athanase Papadopoulos,Sumio Yamada

Publisher: Springer

ISBN: 3319600397

Category: Mathematics

Page: 647

View: 7897

This book explores the work of Bernhard Riemann and its impact on mathematics, philosophy and physics. It features contributions from a range of fields, historical expositions, and selected research articles that were motivated by Riemann’s ideas and demonstrate their timelessness. The editors are convinced of the tremendous value of going into Riemann’s work in depth, investigating his original ideas, integrating them into a broader perspective, and establishing ties with modern science and philosophy. Accordingly, the contributors to this volume are mathematicians, physicists, philosophers and historians of science. The book offers a unique resource for students and researchers in the fields of mathematics, physics and philosophy, historians of science, and more generally to a wide range of readers interested in the history of ideas.

Real Non-Abelian Mixed Hodge Structures for Quasi-Projective Varieties: Formality and Splitting


Author: J. P. Pridham

Publisher: American Mathematical Soc.

ISBN: 1470419815

Category: Hodge theory

Page: 178

View: 4495

The author defines and constructs mixed Hodge structures on real schematic homotopy types of complex quasi-projective varieties, giving mixed Hodge structures on their homotopy groups and pro-algebraic fundamental groups. The author also shows that these split on tensoring with the ring R[x] equipped with the Hodge filtration given by powers of (x−i), giving new results even for simply connected varieties. The mixed Hodge structures can thus be recovered from the Gysin spectral sequence of cohomology groups of local systems, together with the monodromy action at the Archimedean place. As the basepoint varies, these structures all become real variations of mixed Hodge structure.

First Order Categorical Logic

Model-Theoretical Methods in the Theory of Topoi and Related Categories


Author: M. Makkai,G.E. Reyes

Publisher: Springer

ISBN: 3540371001

Category: Mathematics

Page: 318

View: 6573

Proper and Improper Forcing


Author: Saharon Shelah

Publisher: Cambridge University Press

ISBN: 1107168368

Category: Mathematics

Page: 1068

View: 2551

Since their inception, the Perspectives in Logic and Lecture Notes in Logic series have published seminal works by leading logicians. Many of the original books in the series have been unavailable for years, but they are now in print once again. This volume, the fifth publication in the Perspectives in Logic series, studies set-theoretic independence results (independence from the usual set-theoretic ZFC axioms), in particular for problems on the continuum. The author gives a complete presentation of the theory of proper forcing and its relatives, starting from the beginning and avoiding the metamathematical considerations. No prior knowledge of forcing is required. The book will enable a researcher interested in an independence result of the appropriate kind to have much of the work done for them, thereby allowing them to quote general results.

Understanding Understanding

Essays on Cybernetics and Cognition


Author: Heinz von Foerster

Publisher: Springer Science & Business Media

ISBN: 0387217223

Category: Computers

Page: 362

View: 7723

In these ground-breaking essays, Heinz von Foerster discusses some of the fundamental principles that govern how we know the world and how we process the information from which we derive that knowledge. The author was one of the founders of the science of cybernetics.

Sketches of an Elephant: A Topos Theory Compendium


Author: P. T. Johnstone

Publisher: Oxford University Press

ISBN: 9780198515982

Category: Mathematics

Page: 716

View: 6533

Topos Theory is an important branch of mathematical logic of interest to theoretical computer scientists, logicians and philosophers who study the foundations of mathematics, and to those working in differential geometry and continuum physics. This compendium contains material that was previously available only in specialist journals. This is likely to become the standard reference work for all those interested in the subject.