Foundational Theories Of Classical And Constructive Mathematics

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Foundational Theories of Classical and Constructive Mathematics

Giovanni Sommaruga
Foundational Theories of Classical and Constructive Mathematics Book PDF

Author : Giovanni Sommaruga
Publisher : Springer Science & Business Media
Page : 312 pages
File Size : 42,6 Mb
Release : 2011-03-24
Category : Mathematics
ISBN : 9789400704312

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Foundational Theories of Classical and Constructive Mathematics by Giovanni Sommaruga Pdf

The book "Foundational Theories of Classical and Constructive Mathematics" is a book on the classical topic of foundations of mathematics. Its originality resides mainly in its treating at the same time foundations of classical and foundations of constructive mathematics. This confrontation of two kinds of foundations contributes to answering questions such as: Are foundations/foundational theories of classical mathematics of a different nature compared to those of constructive mathematics? Do they play the same role for the resp. mathematics? Are there connections between the two kinds of foundational theories? etc. The confrontation and comparison is often implicit and sometimes explicit. Its great advantage is to extend the traditional discussion of the foundations of mathematics and to render it at the same time more subtle and more differentiated. Another important aspect of the book is that some of its contributions are of a more philosophical, others of a more technical nature. This double face is emphasized, since foundations of mathematics is an eminent topic in the philosophy of mathematics: hence both sides of this discipline ought to be and are being paid due to.

Foundations of Constructive Mathematics

M.J. Beeson
Foundations of Constructive Mathematics Book PDF

Author : M.J. Beeson
Publisher : Springer Science & Business Media
Page : 484 pages
File Size : 46,6 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783642689529

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Foundations of Constructive Mathematics by M.J. Beeson Pdf

This book is about some recent work in a subject usually considered part of "logic" and the" foundations of mathematics", but also having close connec tions with philosophy and computer science. Namely, the creation and study of "formal systems for constructive mathematics". The general organization of the book is described in the" User's Manual" which follows this introduction, and the contents of the book are described in more detail in the introductions to Part One, Part Two, Part Three, and Part Four. This introduction has a different purpose; it is intended to provide the reader with a general view of the subject. This requires, to begin with, an elucidation of both the concepts mentioned in the phrase, "formal systems for constructive mathematics". "Con structive mathematics" refers to mathematics in which, when you prove that l a thing exists (having certain desired properties) you show how to find it. Proof by contradiction is the most common way of proving something exists without showing how to find it - one assumes that nothing exists with the desired properties, and derives a contradiction. It was only in the last two decades of the nineteenth century that mathematicians began to exploit this method of proof in ways that nobody had previously done; that was partly made possible by the creation and development of set theory by Georg Cantor and Richard Dedekind.

Revolutions and Revelations in Computability

Ulrich Berger,Johanna N. Y. Franklin,Florin Manea,Arno Pauly
Revolutions and Revelations in Computability Book PDF

Author : Ulrich Berger,Johanna N. Y. Franklin,Florin Manea,Arno Pauly
Publisher : Springer Nature
Page : 374 pages
File Size : 42,7 Mb
Release : 2022-06-25
Category : Computers
ISBN : 9783031087400

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Revolutions and Revelations in Computability by Ulrich Berger,Johanna N. Y. Franklin,Florin Manea,Arno Pauly Pdf

This book constitutes the proceedings of the 18th Conference on Computability in Europe, CiE 2022, in Swansea, UK, in July 2022. The 19 full papers together with 7 invited papers presented in this volume were carefully reviewed and selected from 41 submissions. The motto of CiE 2022 was “Revolutions and revelations in computability”. This alludes to the revolutionary developments we have seen in computability theory, starting with Turing's and Gödel's discoveries of the uncomputable and the unprovable and continuing to the present day with the advent of new computational paradigms such as quantum computing and bio-computing, which have dramatically changed our view of computability and revealed new insights into the multifarious nature of computation.

Finite Mathematics as the Foundation of Classical Mathematics and Quantum Theory

Felix Lev
Finite Mathematics as the Foundation of Classical Mathematics and Quantum Theory Book PDF

Author : Felix Lev
Publisher : Springer Nature
Page : 291 pages
File Size : 49,7 Mb
Release : 2020-11-03
Category : Science
ISBN : 9783030611019

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Finite Mathematics as the Foundation of Classical Mathematics and Quantum Theory by Felix Lev Pdf

This book delves into finite mathematics and its application in physics, particularly quantum theory. It is shown that quantum theory based on finite mathematics is more general than standard quantum theory, whilst finite mathematics is itself more general than standard mathematics.As a consequence, the mathematics describing nature at the most fundamental level involves only a finite number of numbers while the notions of limit, infinite/infinitesimal and continuity are needed only in calculations that describe nature approximately. It is also shown that the concepts of particle and antiparticle are likewise approximate notions, valid only in special situations, and that the electric charge and baryon- and lepton quantum numbers can be only approximately conserved.

Categories for the Working Philosopher

Elaine Landry
Categories for the Working Philosopher Book PDF

Author : Elaine Landry
Publisher : Oxford University Press
Page : 432 pages
File Size : 52,6 Mb
Release : 2017-11-17
Category : Philosophy
ISBN : 9780191065828

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Categories for the Working Philosopher by Elaine Landry Pdf

Often people have wondered why there is no introductory text on category theory aimed at philosophers working in related areas. The answer is simple: what makes categories interesting and significant is their specific use for specific purposes. These uses and purposes, however, vary over many areas, both "pure", e.g., mathematical, foundational and logical, and "applied", e.g., applied to physics, biology and the nature and structure of mathematical models. Borrowing from the title of Saunders Mac Lane's seminal work "Categories for the Working Mathematician", this book aims to bring the concepts of category theory to philosophers working in areas ranging from mathematics to proof theory to computer science to ontology, from to physics to biology to cognition, from mathematical modeling to the structure of scientific theories to the structure of the world. Moreover, it aims to do 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 areas, and in a way that builds on the concepts that are already familiar to philosophers working in these areas.

Research in History and Philosophy of Mathematics

Maria Zack,Dirk Schlimm
Research in History and Philosophy of Mathematics Book PDF

Author : Maria Zack,Dirk Schlimm
Publisher : Springer
Page : 203 pages
File Size : 51,9 Mb
Release : 2018-09-14
Category : Mathematics
ISBN : 9783319909837

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Research in History and Philosophy of Mathematics by Maria Zack,Dirk Schlimm Pdf

This volume contains thirteen papers that were presented at the 2017 Annual Meeting of the Canadian Society for History and Philosophy of Mathematics/Société canadienne d’histoire et de philosophie des mathématiques, which was held at Ryerson University in Toronto. It showcases rigorously reviewed modern scholarship on an interesting variety of topics in the history and philosophy of mathematics from Ancient Greece to the twentieth century. A series of chapters all set in the eighteenth century consider topics such as John Marsh’s techniques for the computation of decimal fractions, Euler’s efforts to compute the surface area of scalene cones, a little-known work by John Playfair on the practical aspects of mathematics, and Monge’s use of descriptive geometry. After a brief stop in the nineteenth century to consider the culture of research mathematics in 1860s Prussia, the book moves into the twentieth century with an examination of the historical context within which the Axiom of Choice was developed and a paper discussing Anatoly Vlasov’s adaptation of the Boltzmann equation to ionized gases. The remaining chapters deal with the philosophy of twentieth-century mathematics through topics such as an historically informed discussion of finitism and its limits; a reexamination of Mary Leng’s defenses of mathematical fictionalism through an alternative, anti-realist approach to mathematics; and a look at the reasons that mathematicians select specific problems to pursue. Written by leading scholars in the field, these papers are accessible to not only mathematicians and students of the history and philosophy of mathematics, but also anyone with a general interest in mathematics.

The Map and the Territory

Shyam Wuppuluri,Francisco Antonio Doria
The Map and the Territory Book PDF

Author : Shyam Wuppuluri,Francisco Antonio Doria
Publisher : Springer
Page : 641 pages
File Size : 43,7 Mb
Release : 2018-02-13
Category : Science
ISBN : 9783319724782

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The Map and the Territory by Shyam Wuppuluri,Francisco Antonio Doria Pdf

This volume presents essays by pioneering thinkers including Tyler Burge, Gregory Chaitin, Daniel Dennett, Barry Mazur, Nicholas Humphrey, John Searle and Ian Stewart. Together they illuminate the Map/Territory Distinction that underlies at the foundation of the scientific method, thought and the very reality itself. It is imperative to distinguish Map from the Territory while analyzing any subject but we often mistake map for the territory. Meaning for the Reference. Computational tool for what it computes. Representations are handy and tempting that we often end up committing the category error of over-marrying the representation with what is represented, so much so that the distinction between the former and the latter is lost. This error that has its roots in the pedagogy often generates a plethora of paradoxes/confusions which hinder the proper understanding of the subject. What are wave functions? Fields? Forces? Numbers? Sets? Classes? Operators? Functions? Alphabets and Sentences? Are they a part of our map (theory/representation)? Or do they actually belong to the territory (Reality)? Researcher, like a cartographer, clothes (or creates?) the reality by stitching multitudes of maps that simultaneously co-exist. A simple apple, for example, can be analyzed from several viewpoints beginning with evolution and biology, all the way down its microscopic quantum mechanical components. Is there a reality (or a real apple) out there apart from these maps? How do these various maps interact/intermingle with each other to produce a coherent reality that we interact with? Or do they not? Does our brain uses its own internal maps to facilitate “physicist/mathematician” in us to construct the maps about the external territories in turn? If so, what is the nature of these internal maps? Are there meta-maps? Evolution definitely fences our perception and thereby our ability to construct maps, revealing to us only those aspects beneficial for our survival. But the question is, to what extent? Is there a way out of the metaphorical Platonic cave erected around us by the nature? While “Map is not the territory” as Alfred Korzybski remarked, join us in this journey to know more, while we inquire on the nature and the reality of the maps which try to map the reality out there. The book also includes a foreword by Sir Roger Penrose and an afterword by Dagfinn Follesdal.

The Best Writing on Mathematics 2012

Mircea Pitici,David Mumford
The Best Writing on Mathematics 2012 Book PDF

Author : Mircea Pitici,David Mumford
Publisher : Princeton University Press
Page : 321 pages
File Size : 47,8 Mb
Release : 2013
Category : Mathematics
ISBN : 9780691156552

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The Best Writing on Mathematics 2012 by Mircea Pitici,David Mumford Pdf

Collects essays on mathematics, from the mathematical aspects of origami and the mathematics of dating to the frequency and distribution of prime numbers and a ball in five dimensions.

Types for Proofs and Programs

Thorsten Altenkirch,Conor McBride
Types for Proofs and Programs Book PDF

Author : Thorsten Altenkirch,Conor McBride
Publisher : Springer
Page : 277 pages
File Size : 40,9 Mb
Release : 2007-09-13
Category : Computers
ISBN : 9783540744641

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Types for Proofs and Programs by Thorsten Altenkirch,Conor McBride Pdf

The refereed post-proceedings of the International Workshop of the Types Working Group are presented in this volume. The 17 papers address all current issues in formal reasoning and computer programming based on type theory, including languages and computerized tools for reasoning; applications in several domains, such as analysis of programming languages; certified software; formalization of mathematics; and mathematics education.

Axiomatics

Alma Steingart
Axiomatics Book PDF

Author : Alma Steingart
Publisher : University of Chicago Press
Page : 300 pages
File Size : 54,9 Mb
Release : 2023-01-10
Category : Mathematics
ISBN : 9780226824208

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Axiomatics by Alma Steingart Pdf

The first history of postwar mathematics, offering a new interpretation of the rise of abstraction and axiomatics in the twentieth century. Why did abstraction dominate American art, social science, and natural science in the mid-twentieth century? Why, despite opposition, did abstraction and theoretical knowledge flourish across a diverse set of intellectual pursuits during the Cold War? In recovering the centrality of abstraction across a range of modernist projects in the United States, Alma Steingart brings mathematics back into the conversation about midcentury American intellectual thought. The expansion of mathematics in the aftermath of World War II, she demonstrates, was characterized by two opposing tendencies: research in pure mathematics became increasingly abstract and rarified, while research in applied mathematics and mathematical applications grew in prominence as new fields like operations research and game theory brought mathematical knowledge to bear on more domains of knowledge. Both were predicated on the same abstractionist conception of mathematics and were rooted in the same approach: modern axiomatics. For American mathematicians, the humanities and the sciences did not compete with one another, but instead were two complementary sides of the same epistemological commitment. Steingart further reveals how this mathematical epistemology influenced the sciences and humanities, particularly the postwar social sciences. As mathematics changed, so did the meaning of mathematization. Axiomatics focuses on American mathematicians during a transformative time, following a series of controversies among mathematicians about the nature of mathematics as a field of study and as a body of knowledge. The ensuing debates offer a window onto the postwar development of mathematics band Cold War epistemology writ large. As Steingart’s history ably demonstrates, mathematics is the social activity in which styles of truth—here, abstraction—become synonymous with ways of knowing.

Reflections on the Foundations of Mathematics

Stefania Centrone,Deborah Kant,Deniz Sarikaya
Reflections on the Foundations of Mathematics Book PDF

Author : Stefania Centrone,Deborah Kant,Deniz Sarikaya
Publisher : Springer Nature
Page : 511 pages
File Size : 49,5 Mb
Release : 2019-11-11
Category : Mathematics
ISBN : 9783030156558

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Reflections on the Foundations of Mathematics by Stefania Centrone,Deborah Kant,Deniz Sarikaya Pdf

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.

Handbook of Computability and Complexity in Analysis

Vasco Brattka,Peter Hertling
Handbook of Computability and Complexity in Analysis Book PDF

Author : Vasco Brattka,Peter Hertling
Publisher : Springer Nature
Page : 427 pages
File Size : 44,5 Mb
Release : 2021-06-04
Category : Computers
ISBN : 9783030592349

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Handbook of Computability and Complexity in Analysis by Vasco Brattka,Peter Hertling Pdf

Computable analysis is the modern theory of computability and complexity in analysis that arose out of Turing's seminal work in the 1930s. This was motivated by questions such as: which real numbers and real number functions are computable, and which mathematical tasks in analysis can be solved by algorithmic means? Nowadays this theory has many different facets that embrace topics from computability theory, algorithmic randomness, computational complexity, dynamical systems, fractals, and analog computers, up to logic, descriptive set theory, constructivism, and reverse mathematics. In recent decades computable analysis has invaded many branches of analysis, and researchers have studied computability and complexity questions arising from real and complex analysis, functional analysis, and the theory of differential equations, up to (geometric) measure theory and topology. This handbook represents the first coherent cross-section through most active research topics on the more theoretical side of the field. It contains 11 chapters grouped into parts on computability in analysis; complexity, dynamics, and randomness; and constructivity, logic, and descriptive complexity. All chapters are written by leading experts working at the cutting edge of the respective topic. Researchers and graduate students in the areas of theoretical computer science and mathematical logic will find systematic introductions into many branches of computable analysis, and a wealth of information and references that will help them to navigate the modern research literature in this field.

Formal Semantics in Modern Type Theories

Stergios Chatzikyriakidis,Zhaohui Luo
Formal Semantics in Modern Type Theories Book PDF

Author : Stergios Chatzikyriakidis,Zhaohui Luo
Publisher : John Wiley & Sons
Page : 256 pages
File Size : 41,5 Mb
Release : 2021-02-17
Category : Language Arts & Disciplines
ISBN : 9781786301284

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Formal Semantics in Modern Type Theories by Stergios Chatzikyriakidis,Zhaohui Luo Pdf

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.

Feferman on Foundations

Gerhard Jäger,Wilfried Sieg
Feferman on Foundations Book PDF

Author : Gerhard Jäger,Wilfried Sieg
Publisher : Springer
Page : 551 pages
File Size : 50,6 Mb
Release : 2018-04-04
Category : Mathematics
ISBN : 9783319633343

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Feferman on Foundations by Gerhard Jäger,Wilfried Sieg Pdf

This volume honours the life and work of Solomon Feferman, one of the most prominent mathematical logicians of the latter half of the 20th century. In the collection of essays presented here, researchers examine Feferman’s work on mathematical as well as specific methodological and philosophical issues that tie into mathematics. Feferman’s work was largely based in mathematical logic (namely model theory, set theory, proof theory and computability theory), but also branched out into methodological and philosophical issues, making it well known beyond the borders of the mathematics community. With regard to methodological issues, Feferman supported concrete projects. On the one hand, these projects calibrate the proof theoretic strength of subsystems of analysis and set theory and provide ways of overcoming the limitations imposed by Gödel’s incompleteness theorems through appropriate conceptual expansions. On the other, they seek to identify novel axiomatic foundations for mathematical practice, truth theories, and category theory. In his philosophical research, Feferman explored questions such as “What is logic?” and proposed particular positions regarding the foundations of mathematics including, for example, his “conceptual structuralism.” The contributing authors of the volume examine all of the above issues. Their papers are accompanied by an autobiography presented by Feferman that reflects on the evolution and intellectual contexts of his work. The contributing authors critically examine Feferman’s work and, in part, actively expand on his concrete mathematical projects. The volume illuminates Feferman’s distinctive work and, in the process, provides an enlightening perspective on the foundations of mathematics and logic.

Epistemology versus Ontology

P. Dybjer,Sten Lindström,Erik Palmgren,B.G. Sundholm
Epistemology versus Ontology Book PDF

Author : P. Dybjer,Sten Lindström,Erik Palmgren,B.G. Sundholm
Publisher : Springer Science & Business Media
Page : 399 pages
File Size : 43,9 Mb
Release : 2012-07-10
Category : Philosophy
ISBN : 9789400744356

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Epistemology versus Ontology by P. Dybjer,Sten Lindström,Erik Palmgren,B.G. Sundholm Pdf

This book brings together philosophers, mathematicians and logicians to penetrate important problems in the philosophy and foundations of mathematics. In philosophy, one has been concerned with the opposition between constructivism and classical mathematics and the different ontological and epistemological views that are reflected in this opposition. The dominant foundational framework for current mathematics is classical logic and set theory with the axiom of choice (ZFC). This framework is, however, laden with philosophical difficulties. One important alternative foundational programme that is actively pursued today is predicativistic constructivism based on Martin-Löf type theory. Associated philosophical foundations are meaning theories in the tradition of Wittgenstein, Dummett, Prawitz and Martin-Löf. What is the relation between proof-theoretical semantics in the tradition of Gentzen, Prawitz, and Martin-Löf and Wittgensteinian or other accounts of meaning-as-use? What can proof-theoretical analyses tell us about the scope and limits of constructive and predicative mathematics?