Axiomatic Method And Category Theory
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Axiomatic Method And Category Theory
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Author : Andrei Rodin
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-10-14
Axiomatic Method And Category Theory written by Andrei Rodin and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-10-14 with Philosophy categories.
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.
Entropy And Diversity
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Author : Tom Leinster
language : en
Publisher: Cambridge University Press
Release Date : 2021-04-22
Entropy And Diversity written by Tom Leinster and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2021-04-22 with Language Arts & Disciplines categories.
Discover the mathematical riches of 'what is diversity?' in a book that adds mathematical rigour to a vital ecological debate.
Basic Category Theory
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Author : Tom Leinster
language : en
Publisher: Cambridge University Press
Release Date : 2014-07-24
Basic Category Theory written by Tom Leinster and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-07-24 with Mathematics categories.
A short introduction ideal for students learning category theory for the first time.
Categories Allegories
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Author : P.J. Freyd
language : en
Publisher: Elsevier
Release Date : 1990-11-08
Categories Allegories written by P.J. Freyd and has been published by Elsevier this book supported file pdf, txt, epub, kindle and other format this book has been release on 1990-11-08 with Mathematics categories.
General concepts and methods that occur throughout mathematics – and now also in theoretical computer science – are the subject of this book. It is a thorough introduction to Categories, emphasizing the geometric nature of the subject and explaining its connections to mathematical logic. The book should appeal to the inquisitive reader who has seen some basic topology and algebra and would like to learn and explore further.The first part contains a detailed treatment of the fundamentals of Geometric Logic, which combines four central ideas: natural transformations, sheaves, adjoint functors, and topoi. A special feature of the work is a general calculus of relations presented in the second part. This calculus offers another, often more amenable framework for concepts and methods discussed in part one. Some aspects of this approach find their origin in the relational calculi of Peirce and Schroeder from the last century, and in the 1940's in the work of Tarski and others on relational algebras. The representation theorems discussed are an original feature of this approach.
Intuition And The Axiomatic Method
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Author : Emily Carson
language : en
Publisher: Springer Science & Business Media
Release Date : 2006-07-02
Intuition And The Axiomatic Method written by Emily Carson and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006-07-02 with Philosophy categories.
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.
Elements Of Category Theory
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Author : Emily Riehl
language : en
Publisher: Cambridge University Press
Release Date : 2022-02-10
Elements Of Category Theory written by Emily Riehl and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2022-02-10 with Mathematics categories.
This book develops the theory of infinite-dimensional categories by studying the universe, or ∞-cosmos, in which they live.
Axiomatic Set Theory
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Author : G. Takeuti
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-12-01
Axiomatic Set Theory written by G. Takeuti and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-12-01 with Mathematics categories.
This text deals with three basic techniques for constructing models of Zermelo-Fraenkel set theory: relative constructibility, Cohen's forcing, and Scott-Solovay's method of Boolean valued models. Our main concern will be the development of a unified theory that encompasses these techniques in one comprehensive framework. Consequently we will focus on certain funda mental and intrinsic relations between these methods of model construction. Extensive applications will not be treated here. This text is a continuation of our book, "I ntroduction to Axiomatic Set Theory," Springer-Verlag, 1971; indeed the two texts were originally planned as a single volume. The content of this volume is essentially that of a course taught by the first author at the University of Illinois in the spring of 1969. From the first author's lectures, a first draft was prepared by Klaus Gloede with the assistance of Donald Pelletier and the second author. This draft was then rcvised by the first author assisted by Hisao Tanaka. The introductory material was prepared by the second author who was also responsible for the general style of exposition throughout the text. We have inc1uded in the introductory material al1 the results from Boolean algebra and topology that we need. When notation from our first volume is introduced, it is accompanied with a deflnition, usually in a footnote. Consequently a reader who is familiar with elementary set theory will find this text quite self-contained.
Reflections On The Foundations Of Mathematics
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Author : Stefania Centrone
language : en
Publisher: Springer Nature
Release Date : 2019-11-11
Reflections On The Foundations Of Mathematics written by Stefania Centrone and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019-11-11 with Mathematics categories.
This edited work presents contemporary mathematical practice in the foundational mathematical theories, in particular set theory and the univalent foundations. It shares the work of significant scholars across the disciplines of mathematics, philosophy and computer science. Readers will discover systematic thought on criteria for a suitable foundation in mathematics and philosophical reflections around the mathematical perspectives. The volume is divided into three sections, the first two of which focus on the two most prominent candidate theories for a foundation of mathematics. Readers may trace current research in set theory, which has widely been assumed to serve as a framework for foundational issues, as well as new material elaborating on the univalent foundations, considering an approach based on homotopy type theory (HoTT). The third section then builds on this and is centred on philosophical questions connected to the foundations of mathematics. Here, the authors contribute to discussions on foundational criteria with more general thoughts on the foundations of mathematics which are not connected to particular theories. This book shares the work of some of the most important scholars in the fields of set theory (S. Friedman), non-classical logic (G. Priest) and the philosophy of mathematics (P. Maddy). The reader will become aware of the advantages of each theory and objections to it as a foundation, following the latest and best work across the disciplines and it is therefore a valuable read for anyone working on the foundations of mathematics or in the philosophy of mathematics.
Sets For Mathematics
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Author : F. William Lawvere
language : en
Publisher: Cambridge University Press
Release Date : 2003-01-27
Sets For Mathematics written by F. William Lawvere and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2003-01-27 with Mathematics categories.
In this book, first published in 2003, categorical algebra is used to build a foundation for the study of geometry, analysis, and algebra.
Category Theory In Context
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Author : Emily Riehl
language : en
Publisher: Courier Dover Publications
Release Date : 2017-03-09
Category Theory In Context written by Emily Riehl and has been published by Courier Dover Publications this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-03-09 with Mathematics categories.
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.