Homotopy Type Theory
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Homotopy Type Theory Univalent Foundations Of Mathematics
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Author :
language : en
Publisher: Univalent Foundations
Release Date :
Homotopy Type Theory Univalent Foundations Of Mathematics written by and has been published by Univalent Foundations this book supported file pdf, txt, epub, kindle and other format this book has been release on with categories.
Modal Homotopy Type Theory
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Author : David Corfield
language : en
Publisher: Oxford University Press
Release Date : 2020-02-06
Modal Homotopy Type Theory written by David Corfield and has been published by Oxford University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020-02-06 with Philosophy categories.
"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.
Homotopy Type Theory
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Author : Univalent Foundations Program
language : en
Publisher:
Release Date : 2013
Homotopy Type Theory written by Univalent Foundations Program and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013 with Homotopy theory categories.
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.
Homotopy Type Theory
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Author :
language : en
Publisher:
Release Date : 2013
Homotopy Type Theory written by and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013 with Homotopy theory categories.
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
Category Theory
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Author : Zoran Majkic
language : en
Publisher: Walter de Gruyter GmbH & Co KG
Release Date : 2023-03-06
Category Theory written by Zoran Majkic and has been published by Walter de Gruyter GmbH & Co KG this book supported file pdf, txt, epub, kindle and other format this book has been release on 2023-03-06 with Computers categories.
This book analyzes the generation of the arrow-categories of a given category, which is a foundational and distinguishable Category Theory phenomena, in analogy to the foundational role of sets in the traditional set-based Mathematics, for defi nition of natural numbers as well. This inductive transformation of a category into the infinite hierarchy of the arrowcategories is extended to the functors and natural transformations. The author considers invariant categorial properties (the symmetries) under such inductive transformations. The book focuses in particular on Global symmetry (invariance of adjunctions) and Internal symmetries between arrows and objects in a category (in analogy to Field Theories like Quantum Mechanics and General Relativity). The second part of the book is dedicated to more advanced applications of Internal symmetry to Computer Science: for Intuitionistic Logic, Untyped Lambda Calculus with Fixpoint Operators, Labeled Transition Systems in Process Algebras and Modal logics as well as Data Integration Theory.
The Liminal Codex
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Author : Pasquale De Marco
language : en
Publisher: Pasquale De Marco
Release Date : 2025-03-17
The Liminal Codex written by Pasquale De Marco and has been published by Pasquale De Marco this book supported file pdf, txt, epub, kindle and other format this book has been release on 2025-03-17 with Mathematics categories.
Embark on an intellectual odyssey into the realm of type theory, a language that captures the essence of structure and change, shaping our understanding of computation, logic, and mathematics. In this comprehensive and engaging book, you will delve into the intricate world of categories, where structure takes center stage. You will explore the dynamic realm of functional programming, where change unfolds in elegant patterns. You will witness the unification of structure and change in polymorphic type theory, a framework that captures the essence of both stability and transformation. Along your journey, you will encounter the rigorous foundations of category theory, a mathematical tapestry that provides a unifying framework for understanding structure in its myriad forms. You will delve into the concepts of soundness and completeness, the cornerstones of trust in formal systems, ensuring that your reasoning is both correct and comprehensive. Venture into the realm of applications, where type theory unveils its transformative power. Witness how type theory has revolutionized programming languages, enabling the construction of reliable and efficient software. Explore its impact on artificial intelligence, providing a solid foundation for reasoning and learning. Uncover its role in logic and mathematics, formalizing intricate concepts and unlocking new avenues of exploration. Peer into the frontiers of research, where type theory continues to push the boundaries of knowledge. Encounter dependent type theory, a framework that interweaves structure and propositions, and homotopy type theory, a bridge between topology and type theory. Discover the burgeoning field of category theory in computer science, which is reimagining the foundations of computation itself. Join us on this intellectual odyssey as we unravel the mysteries of type theory, uncovering its profound implications for computation, logic, and mathematics. Prepare to be captivated by the elegance and power of a language that captures the essence of structure and change, shaping our understanding of the world around us. If you like this book, write a review!
Categories For The Working Philosopher
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Author : Elaine M. Landry
language : en
Publisher: Oxford University Press
Release Date : 2017
Categories For The Working Philosopher written by Elaine M. Landry and has been published by Oxford University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017 with Mathematics categories.
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.
Modern Perspectives In Type Theoretical Semantics
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Author : Stergios Chatzikyriakidis
language : en
Publisher: Springer
Release Date : 2017-02-07
Modern Perspectives In Type Theoretical Semantics written by Stergios Chatzikyriakidis and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-02-07 with Language Arts & Disciplines categories.
This book is a collective volume that reports the state of the art in the applications of type theory to linguistic semantics. The volume fills a 20 year gap from the last published book on the issue and aspires to bring researchers closer to cutting edge alternatives in formal semantics research. It consists of unpublished work by some key researchers on various issues related to the type theoretical study of formal semantics and further exemplifies the advantages of using modern type theoretical approaches to linguistic semantics. Themes that are covered include modern developments of type theories in formal semantics, foundational issues in linguistic semantics like anaphora, modality and plurals, innovational interdisciplinary research like the introduction of probability theory to type theories as well as computational implementations of type theoretical approaches. This volume will be of great interest to formal semanticists that are looking for alternative ways to study linguistic semantics, but will also be of interest to theoretical computer scientists and mathematicians that are interested in the applications of type theory.
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.
Formal Semantics In Modern Type Theories
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Author : Stergios Chatzikyriakidis
language : en
Publisher: John Wiley & Sons
Release Date : 2021-02-17
Formal Semantics In Modern Type Theories written by Stergios Chatzikyriakidis and has been published by John Wiley & Sons this book supported file pdf, txt, epub, kindle and other format this book has been release on 2021-02-17 with Language Arts & Disciplines categories.
This book studies formal semantics in modern type theories (MTTsemantics). Compared with simple type theory, MTTs have much richer type structures and provide powerful means for adequate semantic constructions. This offers a serious alternative to the traditional settheoretical foundation for linguistic semantics and opens up a new avenue for developing formal semantics that is both model-theoretic and proof-theoretic, which was not available before the development of MTTsemantics. This book provides a reader-friendly and precise description of MTTs and offers a comprehensive introduction to MTT-semantics. It develops several case studies, such as adjectival modification and copredication, to exemplify the attractiveness of using MTTs for the study of linguistic meaning. It also examines existing proof assistant technology based on MTT-semantics for the verification of semantic constructions and reasoning in natural language. Several advanced topics are also briefly studied, including dependent event types, an application of dependent typing to event semantics.