Search Results: sketches-of-an-elephant-a-topos-theory-compendium-volume-1-vol-1-oxford-logic-guides

Sketches of an Elephant: A Topos Theory Compendium

Author: P. T. Johnstone

Publisher: Oxford University Press

ISBN: 9780198515982

Category: Mathematics

Page: 716

View: 1887

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.

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: 6700

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.

Category Theory in Context

Author: Emily Riehl

Publisher: Courier Dover Publications

ISBN: 0486820807

Category: Mathematics

Page: 272

View: 9822

Introduction to concepts of category theory — categories, functors, natural transformations, the Yoneda lemma, limits and colimits, adjunctions, monads — revisits a broad range of mathematical examples from the categorical perspective. 2016 edition.

From Sets and Types to Topology and Analysis

Towards Practicable Foundations for Constructive Mathematics

Author: Laura Crosilla,Peter Schuster

Publisher: Oxford University Press on Demand

ISBN: 0198566514

Category: Mathematics

Page: 350

View: 1686

Bridging the foundations and practice of constructive mathematics, this text focusses on the contrast between the theoretical developments - which have been most useful for computer science - and more specific efforts on constructive analysis, algebra and topology.

Reality and Measurement in Algebraic Quantum Theory

NWW 2015, Nagoya, Japan, March 9-13

Author: Masanao Ozawa,Jeremy Butterfield,Hans Halvorson,Miklós Rédei,Yuichiro Kitajima,Francesco Buscemi

Publisher: Springer

ISBN: 9811324875

Category: Mathematics

Page: 396

View: 4877

This volume contains papers based on presentations at the “Nagoya Winter Workshop 2015: Reality and Measurement in Algebraic Quantum Theory (NWW 2015)”, held in Nagoya, Japan, in March 2015. The foundations of quantum theory have been a source of mysteries, puzzles, and confusions, and have encouraged innovations in mathematical languages to describe, analyze, and delineate this wonderland. Both ontological and epistemological questions about quantum reality and measurement have been placed in the center of the mysteries explored originally by Bohr, Heisenberg, Einstein, and Schrödinger. This volume describes how those traditional problems are nowadays explored from the most advanced perspectives. It includes new research results in quantum information theory, quantum measurement theory, information thermodynamics, operator algebraic and category theoretical foundations of quantum theory, and the interplay between experimental and theoretical investigations on the uncertainty principle. This book is suitable for a broad audience of mathematicians, theoretical and experimental physicists, and philosophers of science.

Mathematical Reviews

Author: N.A

Publisher: N.A

ISBN: N.A

Category: Mathematics

Page: N.A

View: 5575

Algebraic Set Theory

Author: Andri Joyal,Ieke Moerdijk

Publisher: Cambridge University Press

ISBN: 9780521558303

Category: Mathematics

Page: 123

View: 3583

This book offers a new algebraic approach to set theory. The authors introduce a particular kind of algebra, the Zermelo-Fraenkel algebras, which arise from the familiar axioms of Zermelo-Fraenkel set theory. Furthermore, the authors explicitly construct these algebras using the theory of bisimulations. Their approach is completely constructive, and contains both intuitionistic set theory and topos theory. In particular it provides a uniform description of various constructions of the cumulative hierarchy of sets in forcing models, sheaf models and realizability models. Graduate students and researchers in mathematical logic, category theory and computer science should find this book of great interest, and it should be accessible to anyone with a background in categorical logic.

Mathematical Logic and Theoretical Computer Science

Author: Kueker

Publisher: CRC Press

ISBN: 9780824777463

Category: Mathematics

Page: 408

View: 1526

Algebraic Logic

Author: Semen G. Gindikin

Publisher: Springer Science & Business Media

ISBN: 9780387961798

Category: Mathematics

Page: 356

View: 8991

The popular literature on mathematical logic is rather extensive and written for the most varied categories of readers. College students or adults who read it in their free time may find here a vast number of thought-provoking logical problems. The reader who wishes to enrich his mathematical background in the hope that this will help him in his everyday life can discover detailed descriptions of practical (and quite often -- not so practical!) applications of logic. The large number of popular books on logic has given rise to the hope that by applying mathematical logic, students will finally learn how to distinguish between necessary and sufficient conditions and other points of logic in the college course in mathematics. But the habit of teachers of mathematical analysis, for example, to stick to problems dealing with sequences without limit, uniformly continuous functions, etc. has, unfortunately, led to the writing of textbooks that present prescriptions for the mechanical construction of definitions of negative concepts which seem to obviate the need for any thinking on the reader's part. We are most certainly not able to enumerate everything the reader may draw out of existing books on mathematical logic, however.

Categorical Quantum Models and Logics

Author: Chris Heunen

Publisher: Amsterdam University Press

ISBN: 9085550246

Category: Electronic books

Page: 214

View: 9766

This dissertation studies the logic behind quantum physics, using category theory as the principal tool and conceptual guide. To do so, principles of quantum mechanics are modeled categorically. These categorical quantum models are justified by an embedding into the category of Hilbert spaces, the traditional formalism of quantum physics. In particular, complex numbers emerge without having been prescribed explicitly. Interpreting logic in such categories results in orthomodular property lattices, and furthermore provides a natural setting to consider quantifiers. Finally, topos theory, incorporating categorical logic in a refined way, lets one study a quantum system as if it were classical, in particular leading to a novel mathematical notion of quantum-

Reductive Logic and Proof-search

Proof Theory, Semantics, and Control

Author: David J. Pym,Eike Ritter

Publisher: Oxford University Press on Demand

ISBN: 0198526334

Category: Mathematics

Page: 208

View: 2449

This book is a specialized monograph on the development of the mathematical and computational metatheory of reductive logic and proof-search, areas of logic that are becoming important in computer science. A systematic foundational text on these emerging topics, it includes proof-theoretic, semantic/model-theoretic and algorithmic aspects. The scope ranges from the conceptual background to reductive logic, through its mathematical metatheory, to its modern applications in the computational sciences. Suitable for researchers and graduate students in mathematical, computational and philosophical logic, and in theoretical computer science and artificial intelligence, this is the latest in the prestigous world-renowned Oxford Logic Guides, which contains Michael Dummet's Elements of intuitionism (2nd Edition), Dov M. Gabbay, Mark A. Reynolds, and Marcelo Finger's Temporal Logic Mathematical Foundations and Computational Aspects , J. M. Dunn and G. Hardegree's Algebraic Methods in Philosophical Logic, H. Rott's Change, Choice and Inference: A Study of Belief Revision and Nonmonotonic Reasoning , and P. T. Johnstone's Sketches of an Elephant: A Topos Theory Compendium: Volumes 1 and 2 .

Classifying Spaces and Classifying Topoi

Author: Izak Moerdijk

Publisher: Springer

ISBN: 3540449124

Category: Mathematics

Page: 98

View: 1034

This monograph presents a new, systematic treatment of the relation between classifying topoi and classifying spaces of topological categories. Using a new generalized geometric realization which applies to topoi, a weak homotopy equival- ence is constructed between the classifying space and the classifying topos of any small (topological) category. Topos theory is then applied to give an answer to the question of what structures are classified by "classifying" spaces. The monograph should be accessible to anyone with basic knowledge of algebraic topology, sheaf theory, and a little topos theory.

Nominal Sets

Names and Symmetry in Computer Science

Author: Andrew M. Pitts

Publisher: Cambridge University Press

ISBN: 1107244684

Category: Computers

Page: N.A

View: 2892

Nominal sets provide a promising new mathematical analysis of names in formal languages based upon symmetry, with many applications to the syntax and semantics of programming language constructs that involve binding, or localising names. Part I provides an introduction to the basic theory of nominal sets. In Part II, the author surveys some of the applications that have developed in programming language semantics (both operational and denotational), functional programming and logic programming. As the first book to give a detailed account of the theory of nominal sets, it will be welcomed by researchers and graduate students in theoretical computer science.

Logic and Its Applications

Author: Andreas Blass,Yi Zhang

Publisher: American Mathematical Soc.

ISBN: 0821834746

Category: Mathematics

Page: 306

View: 6090

Two conferences, Logic and Its Applications in Algebra and Geometry and Combinatorial Set Theory, Excellent Classes, and Schanuel Conjecture, were held at the University of Michigan (Ann Arbor). These events brought together model theorists and set theorists working in these areas. This volume is the result of those meetings. It is suitable for graduate students and researchers working in mathematical logic.

Duality and Definability in First Order Logic

Author: Michael Makkai

Publisher: American Mathematical Soc.

ISBN: 0821825658

Category: Mathematics

Page: 106

View: 5400

Using the theory of categories as a framework, this book develops a duality theory for theories in first order logic in which the dual of a theory is the category of its models with suitable additional structure. This duality theory resembles and generalizes M. H. Stone's famous duality theory for Boolean algebras. As an application, the author derives a result akin to the well-known definability theorem of E. W. Beth. This new definability theorem is related to theorems of descent in category theory and algebra and can also be stated as a result in pure logic without reference to category theory. Containing novel techniques as well as applications of classical methods, this carefuly written book shows an attention to both organization and detail and will appeal to mathematicians and philosophers interested in category theory.

Relational Methods in Computer Science

Author: Chris Brink,Wolfram Kahl,Gunther Schmidt

Publisher: Springer Science & Business Media

ISBN: 9783211829714

Category: Computers

Page: 272

View: 1341

The calculus of relations has been an important component of the development of logic and algebra since the middle of the nineteenth century, when Augustus De Morgan observed that since a horse is an animal we should be able to infer that the head of a horse is the head of an animal. For this, Aristotelian syllogistic does not suffice: We require relational reasoning. George Boole, in his Mathematical Analysis of Logic of 1847, initiated the treatment of logic as part of mathematics, specifically as part of algebra. Quite the opposite conviction was put forward early this century by Bertrand Russell and Alfred North Whitehead in their Principia Mathematica (1910 - 1913): that mathematics was essentially grounded in logic. Logic thus developed in two streams. On the one hand algebraic logic, in which the calculus of relations played a particularly prominent part, was taken up from Boole by Charles Sanders Peirce, who wished to do for the "calculus of relatives" what Boole had done for the calculus of sets. Peirce's work was in turn taken up by Schroder in his Algebra und Logik der Relative of 1895 (the third part of a massive work on the algebra of logic). Schroder's work, however, lay dormant for more than 40 years, until revived by Alfred Tarski in his seminal paper "On the calculus of binary relations" of 1941 (actually his presidential address to the Association for Symbolic Logic).

Interpolation and Definability

Modal and Intuitionistic Logics

Author: Dov M. Gabbay,Larisa Maksimova

Publisher: Oxford University Press on Demand

ISBN: 0198511744

Category: Computers

Page: 508

View: 7001

This book is a specialized monograph on interpolation and definability, a notion central in pure logic and with significant meaning and applicability in all areas where logic is applied, especially computer science, artificial intelligence, logic programming, philosophy of science and natural language.Suitable for researchers and graduate students in mathematics, computer science and philosophy, this is the latest in the prestigous world-renowned Oxford Logic Guides, which contains Michael Dummet's Elements of intuitionism (second edition), J. M. Dunn and G. Hardegree's Algebraic Methods in Philosophical Logic, H. Rott's Change, Choice and Inference: A Study of Belief Revision and NonmonotonicReasoning, P. T. Johnstone's Sketches of an Elephant: A Topos Theory Compendium: Volumes 1 and 2, and David J. Pym and Eike Ritter's Reductive Logic and Proof Search: Proof theory, semantics and control.

Axiomatic Method and Category Theory

Author: Andrei Rodin

Publisher: Springer Science & Business Media

ISBN: 3319004042

Category: Philosophy

Page: 285

View: 2790

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.

Perspectives on Universal Logic

Author: Jean-Yves Béziau,Alexandre Costa-Leite

Publisher: Polimetrica s.a.s.

ISBN: 8876990771

Category: Mathematics

Page: 434

View: 9420

Folk Devils and Moral Panics

Author: Stanley Cohen

Publisher: Taylor & Francis

ISBN: 1136807047

Category: Social Science

Page: 328

View: 9965

'Richly documented and convincingly presented' -- New Society Mods and Rockers, skinheads, video nasties, designer drugs, bogus asylum seeks and hoodies. Every era has its own moral panics. It was Stanley Cohen’s classic account, first published in the early 1970s and regularly revised, that brought the term ‘moral panic’ into widespread discussion. It is an outstanding investigation of the way in which the media and often those in a position of political power define a condition, or group, as a threat to societal values and interests. Fanned by screaming media headlines, Cohen brilliantly demonstrates how this leads to such groups being marginalised and vilified in the popular imagination, inhibiting rational debate about solutions to the social problems such groups represent. Furthermore, he argues that moral panics go even further by identifying the very fault lines of power in society. Full of sharp insight and analysis, Folk Devils and Moral Panics is essential reading for anyone wanting to understand this powerful and enduring phenomenon. Professor Stanley Cohen is Emeritus Professor of Sociology at the London School of Economics. He received the Sellin-Glueck Award of the American Society of Criminology (1985) and is on the Board of the International Council on Human Rights. He is a member of the British Academy.

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