Download or read book Homotopy Type Theory Univalent Foundations of Mathematics written by and published by Univalent Foundations. This book was released on with total page 484 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Homotopy Type Theory written by and published by . This book was released on 2013 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: The present work has its origins in our collective attempts to develop a new style of "informal type theory" that can be read and understood by a human being, as a complement to a formal proof that can be checked by a machine. Univalent foundations is closely tied to the idea of a foundation of mathematics that can be implemented in a computer proof assistant."--Page vi
Download or read book Homotopy Type Theory written by Univalent Foundations Program and published by . This book was released on 2013 with total page 589 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is the product of a yearlong collaboration at the Institute for Advanced Study. It describes (the beta version of) a new language for mathematics, which may some day replace set theory.
Download or read book Reflections on the Foundations of Mathematics written by Stefania Centrone and published by Springer Nature. This book was released on 2019-11-11 with total page 511 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book Modal Homotopy Type Theory written by David Corfield and published by Oxford University Press. This book was released on 2020-02-06 with total page 208 pages. Available in PDF, EPUB and Kindle. Book excerpt: "The old logic put thought in fetters, while the new logic gives it wings." For the past century, philosophers working in the tradition of Bertrand Russell - who promised to revolutionise philosophy by introducing the 'new logic' of Frege and Peano - have employed predicate logic as their formal language of choice. In this book, Dr David Corfield presents a comparable revolution with a newly emerging logic - modal homotopy type theory. Homotopy type theory has recently been developed as a new foundational language for mathematics, with a strong philosophical pedigree. Modal Homotopy Type Theory: The Prospect of a New Logic for Philosophy offers an introduction to this new language and its modal extension, illustrated through innovative applications of the calculus to language, metaphysics, and mathematics. The chapters build up to the full language in stages, right up to the application of modal homotopy type theory to current geometry. From a discussion of the distinction between objects and events, the intrinsic treatment of structure, the conception of modality as a form of general variation to the representation of constructions in modern geometry, we see how varied the applications of this powerful new language can be.
Download or read book Type Theory and Formal Proof written by Rob Nederpelt and published by Cambridge University Press. This book was released on 2014-11-06 with total page 465 pages. Available in PDF, EPUB and Kindle. Book excerpt: Type theory is a fast-evolving field at the crossroads of logic, computer science and mathematics. This gentle step-by-step introduction is ideal for graduate students and researchers who need to understand the ins and outs of the mathematical machinery, the role of logical rules therein, the essential contribution of definitions and the decisive nature of well-structured proofs. The authors begin with untyped lambda calculus and proceed to several fundamental type systems, including the well-known and powerful Calculus of Constructions. The book also covers the essence of proof checking and proof development, and the use of dependent type theory to formalise mathematics. The only prerequisite is a basic knowledge of undergraduate mathematics. Carefully chosen examples illustrate the theory throughout. Each chapter ends with a summary of the content, some historical context, suggestions for further reading and a selection of exercises to help readers familiarise themselves with the material.
Download or read book Categories for the Working Philosopher written by Elaine M. Landry and published by Oxford University Press. This book was released on 2017 with total page 486 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is the first book on category theory for a broad philosophical readership. There is no other discussion of category theory comparable in its scope. It is designed to show the interest and significant of category theory for philosophers working in a range of areas, including mathematics, proof theory, computer science, ontology, physics, biology, cognition, mathematical modelling, the structure of scientific theories, and the structure of the world. Moreover, it does this in a way that is accessible to non specialists. Each chapter is written by either a category-theorist or a philosopher working in one of the represented fields, in a way that builds on the concepts already familiar to philosophers working in these areas. The book is split into two halves. The 'pure' chapters focus on the use of category theory for mathematical, foundational, and logical purposes, while the 'applied' chapters consider the use of category theory for representational purposes, investigating category theory as a framework for theories of physics and biology, for mathematical modelling more generally, and for the structure of scientific theories. Book jacket.
Download or read book An Introduction to Proof Theory written by Paolo Mancosu and published by Oxford University Press. This book was released on 2021 with total page 431 pages. Available in PDF, EPUB and Kindle. Book excerpt: An Introduction to Proof Theory provides an accessible introduction to the theory of proofs, with details of proofs worked out and examples and exercises to aid the reader's understanding. It also serves as a companion to reading the original pathbreaking articles by Gerhard Gentzen. The first half covers topics in structural proof theory, including the Gödel-Gentzen translation of classical into intuitionistic logic (and arithmetic), natural deduction and the normalization theorems (for both NJ and NK), the sequent calculus, including cut-elimination and mid-sequent theorems, and various applications of these results. The second half examines ordinal proof theory, specifically Gentzen's consistency proof for first-order Peano Arithmetic. The theory of ordinal notations and other elements of ordinal theory are developed from scratch, and no knowledge of set theory is presumed. The proof methods needed to establish proof-theoretic results, especially proof by induction, are introduced in stages throughout the text. Mancosu, Galvan, and Zach's introduction will provide a solid foundation for those looking to understand this central area of mathematical logic and the philosophy of mathematics.
Download or read book Mathematics without Apologies written by Michael Harris and published by Princeton University Press. This book was released on 2017-05-30 with total page 468 pages. Available in PDF, EPUB and Kindle. Book excerpt: An insightful reflection on the mathematical soul What do pure mathematicians do, and why do they do it? Looking beyond the conventional answers—for the sake of truth, beauty, and practical applications—this book offers an eclectic panorama of the lives and values and hopes and fears of mathematicians in the twenty-first century, assembling material from a startlingly diverse assortment of scholarly, journalistic, and pop culture sources. Drawing on his personal experiences and obsessions as well as the thoughts and opinions of mathematicians from Archimedes and Omar Khayyám to such contemporary giants as Alexander Grothendieck and Robert Langlands, Michael Harris reveals the charisma and romance of mathematics as well as its darker side. In this portrait of mathematics as a community united around a set of common intellectual, ethical, and existential challenges, he touches on a wide variety of questions, such as: Are mathematicians to blame for the 2008 financial crisis? How can we talk about the ideas we were born too soon to understand? And how should you react if you are asked to explain number theory at a dinner party? Disarmingly candid, relentlessly intelligent, and richly entertaining, Mathematics without Apologies takes readers on an unapologetic guided tour of the mathematical life, from the philosophy and sociology of mathematics to its reflections in film and popular music, with detours through the mathematical and mystical traditions of Russia, India, medieval Islam, the Bronx, and beyond.
Download or read book Category Theory written by Steve Awodey and published by Oxford University Press. This book was released on 2010-06-17 with total page 328 pages. Available in PDF, EPUB and Kindle. Book excerpt: A comprehensive reference to category theory for students and researchers in mathematics, computer science, logic, cognitive science, linguistics, and philosophy. Useful for self-study and as a course text, the book includes all basic definitions and theorems (with full proofs), as well as numerous examples and exercises.
Download or read book Category Theory in Context written by Emily Riehl and published by Courier Dover Publications. This book was released on 2017-03-09 with total page 272 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book Categories for Types written by Roy L. Crole and published by Cambridge University Press. This book was released on 1993 with total page 362 pages. Available in PDF, EPUB and Kindle. Book excerpt: This textbook explains the basic principles of categorical type theory and the techniques used to derive categorical semantics for specific type theories. It introduces the reader to ordered set theory, lattices and domains, and this material provides plenty of examples for an introduction to category theory, which covers categories, functors, natural transformations, the Yoneda lemma, cartesian closed categories, limits, adjunctions and indexed categories. Four kinds of formal system are considered in detail, namely algebraic, functional, polymorphic functional, and higher order polymorphic functional type theory. For each of these the categorical semantics are derived and results about the type systems are proved categorically. Issues of soundness and completeness are also considered. Aimed at advanced undergraduates and beginning graduates, this book will be of interest to theoretical computer scientists, logicians and mathematicians specializing in category theory.
Download or read book Axiomatic Method and Category Theory written by Andrei Rodin and published by Springer Science & Business Media. This book was released on 2013-10-14 with total page 285 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book Twenty Five Years of Constructive Type Theory written by Giovanni Sambin and published by Clarendon Press. This book was released on 1998-10-15 with total page 292 pages. Available in PDF, EPUB and Kindle. Book excerpt: Per Martin-Löf's work on the development of constructive type theory has been of huge significance in the fields of logic and the foundations of mathematics. It is also of broader philosophical significance, and has important applications in areas such as computing science and linguistics. This volume draws together contributions from researchers whose work builds on the theory developed by Martin-Löf over the last twenty-five years. As well as celebrating the anniversary of the birth of the subject it covers many of the diverse fields which are now influenced by type theory. It is an invaluable record of areas of current activity, but also contains contributions from N. G. de Bruijn and William Tait, both important figures in the early development of the subject. Also published for the first time is one of Per Martin-Löf's earliest papers.
Download or read book Intuitionistic Type Theory written by Per Martin-Löf and published by . This book was released on 1984 with total page 116 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book More Concise Algebraic Topology written by J. P. May and published by University of Chicago Press. This book was released on 2012-02 with total page 544 pages. Available in PDF, EPUB and Kindle. Book excerpt: With firm foundations dating only from the 1950s, algebraic topology is a relatively young area of mathematics. There are very few textbooks that treat fundamental topics beyond a first course, and many topics now essential to the field are not treated in any textbook. J. Peter May’s A Concise Course in Algebraic Topology addresses the standard first course material, such as fundamental groups, covering spaces, the basics of homotopy theory, and homology and cohomology. In this sequel, May and his coauthor, Kathleen Ponto, cover topics that are essential for algebraic topologists and others interested in algebraic topology, but that are not treated in standard texts. They focus on the localization and completion of topological spaces, model categories, and Hopf algebras. The first half of the book sets out the basic theory of localization and completion of nilpotent spaces, using the most elementary treatment the authors know of. It makes no use of simplicial techniques or model categories, and it provides full details of other necessary preliminaries. With these topics as motivation, most of the second half of the book sets out the theory of model categories, which is the central organizing framework for homotopical algebra in general. Examples from topology and homological algebra are treated in parallel. A short last part develops the basic theory of bialgebras and Hopf algebras.
Download or read book Every Planar Map is Four Colorable written by Kenneth I. Appel and published by American Mathematical Soc.. This book was released on 1989 with total page 741 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this volume, the authors present their 1972 proof of the celebrated Four Color Theorem in a detailed but self-contained exposition accessible to a general mathematical audience. An emended version of the authors' proof of the theorem, the book contains the full text of the supplements and checklists, which originally appeared on microfiche. The thiry-page introduction, intended for nonspecialists, provides some historical background of the theorem and details of the authors' proof. In addition, the authors have added an appendix which treats in much greater detail the argument for situations in which reducible configurations are immersed rather than embedded in triangulations. This result leads to a proof that four coloring can be accomplished in polynomial time.
