Concepts of proof in mathematics, philosophy, and computer science
In the last decades, mathematical logic has developed into a technically quite sophisticated area of mathematics. Nevertheless, inspirations from philosophy and computer science continue to be important and noticeable. The series publishes conference proceedings as well as monographs written by lead...
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| Hauptverfasser: | , |
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| Format: | E-Book Buch |
| Sprache: | Englisch |
| Veröffentlicht: |
Berlin
De Gruyter
2016
Walter de Gruyter GmbH |
| Ausgabe: | 1 |
| Schriftenreihe: | Ontos Mathematical Logic |
| Schlagworte: | |
| ISBN: | 1501510800, 9781501510809 |
| Online-Zugang: | Volltext |
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| Abstract | In the last decades, mathematical logic has developed into a technically quite sophisticated area of mathematics. Nevertheless, inspirations from philosophy and computer science continue to be important and noticeable. The series publishes conference proceedings as well as monographs written by leading researchers in mathematical logic. |
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| AbstractList | In the last decades, mathematical logic has developed into a technically quite sophisticated area of mathematics. Nevertheless, inspirations from philosophy and computer science continue to be important and noticeable. The series publishes conference proceedings as well as monographs written by leading researchers in mathematical logic. A proof is a successful demonstration that a conclusion necessarily follows by logical reasoning from axioms which are considered evident for the given context and agreed upon by the community. It is this concept that sets mathematics apart from other disciplines and distinguishes it as the prototype of a deductive science. Proofs thus are utterly relevant for research, teaching and communication in mathematics and of particular interest for the philosophy of mathematics. In computer science, moreover, proofs have proved to be a rich source for already certified algorithms. This book provides the reader with a collection of articles covering relevant current research topics circled around the concept 'proof'. It tries to give due consideration to the depth and breadth of the subject by discussing its philosophical and methodological aspects, addressing foundational issues induced by Hilbert's Programme and the benefits of the arising formal notions of proof, without neglecting reasoning in natural language proofs and applications in computer science such as program extraction. This book provides the reader with research arising from the Humboldt-Kolleg 'Proof' held in Bern in fall 2013, which gathered more than sixty leading experts actively involved with the concept 'proof' in philosophy, mathematics and computer science. This volume aims to do justice to the breadth and depth of the subject and presents relevant current conceptions and technical advances featuring 'proof' in the fields mentioned above. |
| Author | Probst, Dieter Schuster, Peter |
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| Discipline | Mathematics |
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| Editor | Probst, Dieter Schuster, Peter |
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| ISBN | 1501510800 9781501510809 |
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| Keywords | Theoretische Informatik Mathematical Logic Philosophy of Mathematics Mathematische Logik Theoretical Computer Science Philosophie der Mathematik |
| LCCN | 2016030276 |
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| Language | English |
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| Notes | Includes bibliographical references |
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| PublicationPlace | Berlin |
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| PublicationSeriesTitle | Ontos Mathematical Logic |
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| Snippet | In the last decades, mathematical logic has developed into a technically quite sophisticated area of mathematics. Nevertheless, inspirations from philosophy... A proof is a successful demonstration that a conclusion necessarily follows by logical reasoning from axioms which are considered evident for the given context... This book provides the reader with research arising from the Humboldt-Kolleg 'Proof' held in Bern in fall 2013, which gathered more than sixty leading experts... |
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| SubjectTerms | Logic, Symbolic and mathematical Mathematical Logic Mathematics MATHEMATICS / History & Philosophy Mathematische Logik Philosophie der Mathematik PHILOSOPHY / Epistemology PHILOSOPHY / Logic PHILOSOPHY / Methodology Philosophy of Mathematics Proof theory Theoretical Computer Science Theoretische Informatik |
| TableOfContents | Intro -- Contents -- Introduction -- Herbrand Confluence for First-Order Proofs with π 2-Cuts -- Proof-Oriented Categorical Semantics -- Logic for Gray-code Computation -- The Continuum Hypothesis Implies Excluded Middle -- Theories of Proof-Theoretic Strength -- Some Remarks about Normal Rings -- On Sets of Premises -- Non-Deterministic Inductive Definitions and Fullness -- Cyclic Proofs for Linear Temporal Logic -- Craig Interpolation via Hypersequents -- A General View on Normal Form Theorems for Lukasiewicz Logic with Product -- Relating Quotient Completions via Categorical Logic -- Some Historical, Philosophical and Methodological Remarks on Proof in Mathematics -- Cut Elimination in Sequent Calculi with Implicit Contraction, with a Conjecture on the Origin of Gentzen's Altitude Line Construction -- Hilbert's Programme and Ordinal Analysis -- Aristotle's Deductive Logic: a Proof-Theoretical Study -- Remarks on Barr's Theorem: Proofs in Geometric Theories Contents -- Preface -- Cut Elimination in Sequent Calculi with Implicit Contraction, with a Conjecture on the Origin of Gentzen’s Altitude Line Construction Hajime Ishihara, Takako Nemoto -- Aristotle’s Deductive Logic: a Proof-Theoretical Study On Sets of Premises Jan von Plato -- Ulrik Buchholtz, Gerhard Jäger, Thomas Strahm -- Cyclic Proofs for Linear Temporal Logic Ulrich Berger, Kenji Miyamoto, Helmut Schwichtenberg, Hideki Tsuiki -- Dieter Probst, Peter Schuster -- Maria Emilia Maietti, Giuseppe Rosolini -- Wolfram Pohlers -- Thierry Coquand, Henri Lombardi -- Michael Rathjen Logic for Gray-code Computation Some Remarks about Normal Rings Hilbert’s Programme and Ordinal Analysis Sara Negri, Jan von Plato -- Some Historical, Philosophical and Methodological Remarks on Proof in Mathematics Roman Kuznets -- Non-Deterministic Inductive Definitions and Fullness Theories of Proof-Theoretic Strength Ψ (ΓΩ +1) Marco Benini -- Ioannis Kokkinis, Thomas Studer -- Roman Murawski -- Douglas S. Bridges -- Proof-Oriented Categorical Semantics Introduction Serafina Lapenta, Ioana Leuştean -- Relating Quotient Completions via Categorical Logic Bahareh Afshari, Stefan Hetzl, Graham E. Leigh -- Frontmatter -- Craig Interpolation via Hypersequents Kosta Došen -- The Continuum Hypothesis Implies Excluded Middle Herbrand Confluence for First-Order Proofs with Π2-Cuts A General View on Normal Form Theorems for Łukasiewicz Logic with Product Remarks on Barr’s Theorem: Proofs in Geometric Theories |
| Title | Concepts of proof in mathematics, philosophy, and computer science |
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