Modal Homotopy Type Theory

Author : David Corfield
Publisher : Oxford University Press
Page : 208 pages
File Size : 46,8 Mb
Release : 2020-02-06
Category : Philosophy
ISBN : 9780192595034

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Modal Homotopy Type Theory by David Corfield Pdf

"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.

Author : David Corfield
Publisher : Oxford University Press, USA
Page : 0 pages
File Size : 50,8 Mb
Release : 2020
Category : Mathematics
ISBN : 0198853408

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Modal Homotopy Type Theory by David Corfield Pdf

Modal Homotopy Type Theory: The Prospect of a New Logic for Philosophy provides a reasonably gentle introduction to this new logic, thoroughly motivated by intuitive explanations of the need for all of its component parts, and illustrated through innovative applications of the calculus.

Author : Anonim
Publisher : Univalent Foundations
Page : 484 pages
File Size : 55,6 Mb
Release : 2024-06-30
Category : Electronic
ISBN : 8210379456XXX

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Homotopy Type Theory: Univalent Foundations of Mathematics by Anonim Pdf

Author : Elaine M. Landry
Publisher : Oxford University Press
Page : 486 pages
File Size : 40,5 Mb
Release : 2017
Category : Mathematics
ISBN : 9780198748991

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Categories for the Working Philosopher by Elaine M. Landry Pdf

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.

Author : Patrick Schultz,David I. Spivak
Publisher : Springer
Page : 235 pages
File Size : 52,5 Mb
Release : 2019-01-29
Category : Mathematics
ISBN : 9783030007041

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Temporal Type Theory by Patrick Schultz,David I. Spivak Pdf

This innovative monograph explores a new mathematical formalism in higher-order temporal logic for proving properties about the behavior of systems. Developed by the authors, the goal of this novel approach is to explain what occurs when multiple, distinct system components interact by using a category-theoretic description of behavior types based on sheaves. The authors demonstrate how to analyze the behaviors of elements in continuous and discrete dynamical systems so that each can be translated and compared to one another. Their temporal logic is also flexible enough that it can serve as a framework for other logics that work with similar models. The book begins with a discussion of behavior types, interval domains, and translation invariance, which serves as the groundwork for temporal type theory. From there, the authors lay out the logical preliminaries they need for their temporal modalities and explain the soundness of those logical semantics. These results are then applied to hybrid dynamical systems, differential equations, and labeled transition systems. A case study involving aircraft separation within the National Airspace System is provided to illustrate temporal type theory in action. Researchers in computer science, logic, and mathematics interested in topos-theoretic and category-theory-friendly approaches to system behavior will find this monograph to be an important resource. It can also serve as a supplemental text for a specialized graduate topics course.

Author : B. Jacobs
Publisher : Gulf Professional Publishing
Page : 784 pages
File Size : 54,6 Mb
Release : 2001-05-10
Category : Computers
ISBN : 0444508538

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Categorical Logic and Type Theory by B. Jacobs Pdf

This book is an attempt to give a systematic presentation of both logic and type theory from a categorical perspective, using the unifying concept of fibred category. Its intended audience consists of logicians, type theorists, category theorists and (theoretical) computer scientists.

Author : J. P. May,K. Ponto
Publisher : University of Chicago Press
Page : 544 pages
File Size : 53,6 Mb
Release : 2012-02
Category : Mathematics
ISBN : 9780226511788

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More Concise Algebraic Topology by J. P. May,K. Ponto Pdf

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.

Author : Rob Nederpelt,Herman Geuvers
Publisher : Cambridge University Press
Page : 465 pages
File Size : 54,5 Mb
Release : 2014-11-06
Category : Computers
ISBN : 9781107036505

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Type Theory and Formal Proof by Rob Nederpelt,Herman Geuvers Pdf

A gentle introduction for graduate students and researchers in the art of formalizing mathematics on the basis of type theory.

Author : Erich H. Reck,Georg Schiemer
Publisher : Oxford University Press
Page : 469 pages
File Size : 43,5 Mb
Release : 2020
Category : Mathematics
ISBN : 9780190641221

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The Prehistory of Mathematical Structuralism by Erich H. Reck,Georg Schiemer Pdf

This edited volume explores the previously underacknowledged 'pre-history' of mathematical structuralism, showing that structuralism has deep roots in the history of modern mathematics. The contributors explore this history along two distinct but interconnected dimensions. First, they reconsider the methodological contributions of major figures in the history of mathematics. Second, they re-examine a range of philosophical reflections from mathematically-inclinded philosophers like Russell, Carnap, and Quine, whose work led to profound conclusions about logical, epistemological, and metaphysic.

Author : Sergei Artemov,Anil Nerode
Publisher : Springer Nature
Page : 297 pages
File Size : 44,7 Mb
Release : 2019-12-13
Category : Mathematics
ISBN : 9783030367558

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Logical Foundations of Computer Science by Sergei Artemov,Anil Nerode Pdf

This book constitutes the refereed proceedings of the International Symposium on Logical Foundations of Computer Science, LFCS 2020, held in Deerfield Beach, FL, USA, in January 2020. The 17 revised full papers were carefully reviewed and selected from 30 submissions. The scope of the Symposium is broad and includes constructive mathematics and type theory; homotopy type theory; logic, automata, and automatic structures; computability and randomness; logical foundations of programming; logical aspects of computational complexity; parameterized complexity; logic programming and constraints; automated deduction and interactive theorem proving; logical methods in protocol and program verification; logical methods in program specification and extraction; domain theory logics; logical foundations of database theory; equational logic and term rewriting; lambda and combinatory calculi; categorical logic and topological semantics; linear logic; epistemic and temporal logics; intelligent and multiple-agent system logics; logics of proof and justification; non-monotonic reasoning; logic in game theory and social software; logic of hybrid systems; distributed system logics; mathematical fuzzy logic; system design logics; other logics in computer science.

Author : Emily Riehl
Publisher : Courier Dover Publications
Page : 272 pages
File Size : 44,7 Mb
Release : 2017-03-09
Category : Mathematics
ISBN : 9780486820804

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Category Theory in Context by Emily Riehl Pdf

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.

Author : Leon Horsten,Philip Welch
Publisher : Oxford University Press
Page : 289 pages
File Size : 44,7 Mb
Release : 2016
Category : Mathematics
ISBN : 9780198759591

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Gödel's Disjunction by Leon Horsten,Philip Welch Pdf

The logician Kurt Godel in 1951 established a disjunctive thesis about the scope and limits of mathematical knowledge: either the mathematical mind is not equivalent to a Turing machine (i.e., a computer), or there are absolutely undecidable mathematical problems. In the second half of the twentieth century, attempts have been made to arrive at a stronger conclusion. In particular, arguments have been produced by the philosopher J.R. Lucas and by the physicist and mathematician Roger Penrose that intend to show that the mathematical mind is more powerful than any computer. These arguments, and counterarguments to them, have not convinced the logical and philosophical community. The reason for this is an insufficiency if rigour in the debate. The contributions in this volume move the debate forward by formulating rigorous frameworks and formally spelling out and evaluating arguments that bear on Godel's disjunction in these frameworks. The contributions in this volume have been written by world leading experts in the field.

Author : Thomas Piecha,Peter Schroeder-Heister
Publisher : Springer
Page : 283 pages
File Size : 55,6 Mb
Release : 2015-10-24
Category : Philosophy
ISBN : 9783319226866

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Advances in Proof-Theoretic Semantics by Thomas Piecha,Peter Schroeder-Heister Pdf

This volume is the first ever collection devoted to the field of proof-theoretic semantics. Contributions address topics including the systematics of introduction and elimination rules and proofs of normalization, the categorial characterization of deductions, the relation between Heyting's and Gentzen's approaches to meaning, knowability paradoxes, proof-theoretic foundations of set theory, Dummett's justification of logical laws, Kreisel's theory of constructions, paradoxical reasoning, and the defence of model theory. The field of proof-theoretic semantics has existed for almost 50 years, but the term itself was proposed by Schroeder-Heister in the 1980s. Proof-theoretic semantics explains the meaning of linguistic expressions in general and of logical constants in particular in terms of the notion of proof. This volume emerges from presentations at the Second International Conference on Proof-Theoretic Semantics in Tübingen in 2013, where contributing authors were asked to provide a self-contained description and analysis of a significant research question in this area. The contributions are representative of the field and should be of interest to logicians, philosophers, and mathematicians alike.

Author : Paolo Mancosu,Sergio Galvan,Richard Zach
Publisher : Oxford University Press
Page : 431 pages
File Size : 42,5 Mb
Release : 2021
Category : Philosophy
ISBN : 9780192895936

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An Introduction to Proof Theory by Paolo Mancosu,Sergio Galvan,Richard Zach Pdf

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.

Author : Emily Riehl,Dominic Verity
Publisher : Cambridge University Press
Page : 781 pages
File Size : 48,8 Mb
Release : 2022-02-10
Category : Mathematics
ISBN : 9781108837989

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Elements of ?-Category Theory by Emily Riehl,Dominic Verity Pdf

This book develops the theory of infinite-dimensional categories by studying the universe, or ∞-cosmos, in which they live.