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Logic's Lost Genius

Author : Eckart Menzler-Trott
Publisher : American Mathematical Soc.
ISBN 13 : 1470428121
Total Pages : 466 pages
Book Rating : 4.4/5 (74 download)

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Book Synopsis Logic's Lost Genius by : Eckart Menzler-Trott

Download or read book Logic's Lost Genius written by Eckart Menzler-Trott and published by American Mathematical Soc.. This book was released on 2016-05-05 with total page 466 pages. Available in PDF, EPUB and Kindle. Book excerpt: Gerhard Gentzen (1909–1945) is the founder of modern structural proof theory. His lasting methods, rules, and structures resulted not only in the technical mathematical discipline called “proof theory” but also in verification programs that are essential in computer science. The appearance, clarity, and elegance of Gentzen's work on natural deduction, the sequent calculus, and ordinal proof theory continue to be impressive even today. The present book gives the first comprehensive, detailed, accurate scientific biography expounding the life and work of Gerhard Gentzen, one of our greatest logicians, until his arrest and death in Prague in 1945. Particular emphasis in the book is put on the conditions of scientific research, in this case mathematical logic, in National Socialist Germany, the ideological fight for “German logic”, and their mutual protagonists. Numerous hitherto unpublished sources, family documents, archival material, interviews, and letters, as well as Gentzen's lectures for the mathematical public, make this book an indispensable source of information on this important mathematician, his work, and his time. The volume is completed by two deep substantial essays by Jan von Plato and Craig Smoryński on Gentzen's proof theory; its relation to the ideas of Hilbert, Brouwer, Weyl, and Gödel; and its development up to the present day. Smoryński explains the Hilbert program in more than the usual slogan form and shows why consistency is important. Von Plato shows in detail the benefits of Gentzen's program. This important book is a self-contained starting point for any work on Gentzen and his logic. The book is accessible to a wide audience with different backgrounds and is suitable for general readers, researchers, students, and teachers.

Gentzen's Centenary

Author : Reinhard Kahle
Publisher : Springer
ISBN 13 : 331910103X
Total Pages : 563 pages
Book Rating : 4.3/5 (191 download)

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Book Synopsis Gentzen's Centenary by : Reinhard Kahle

Download or read book Gentzen's Centenary written by Reinhard Kahle and published by Springer. This book was released on 2015-11-02 with total page 563 pages. Available in PDF, EPUB and Kindle. Book excerpt: Gerhard Gentzen has been described as logic’s lost genius, whom Gödel called a better logician than himself. This work comprises articles by leading proof theorists, attesting to Gentzen’s enduring legacy to mathematical logic and beyond. The contributions range from philosophical reflections and re-evaluations of Gentzen’s original consistency proofs to the most recent developments in proof theory. Gentzen founded modern proof theory. His sequent calculus and natural deduction system beautifully explain the deep symmetries of logic. They underlie modern developments in computer science such as automated theorem proving and type theory.

Where is the Gödel-point hiding: Gentzen’s Consistency Proof of 1936 and His Representation of Constructive Ordinals

Author : Anna Horská
Publisher : Springer Science & Business Media
ISBN 13 : 3319021710
Total Pages : 81 pages
Book Rating : 4.3/5 (19 download)

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Book Synopsis Where is the Gödel-point hiding: Gentzen’s Consistency Proof of 1936 and His Representation of Constructive Ordinals by : Anna Horská

Download or read book Where is the Gödel-point hiding: Gentzen’s Consistency Proof of 1936 and His Representation of Constructive Ordinals written by Anna Horská and published by Springer Science & Business Media. This book was released on 2013-10-23 with total page 81 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book explains the first published consistency proof of PA. It contains the original Gentzen's proof, but it uses modern terminology and examples to illustrate the essential notions. The author comments on Gentzen's steps which are supplemented with exact calculations and parts of formal derivations. A notable aspect of the proof is the representation of ordinal numbers that was developed by Gentzen. This representation is analysed and connection to set-theoretical representation is found, namely an algorithm for translating Gentzen's notation into Cantor normal form. The topic should interest researchers and students who work on proof theory, history of proof theory or Hilbert's program and who do not mind reading mathematical texts.

Natural Deduction

Author : Dag Prawitz
Publisher : Courier Dover Publications
ISBN 13 : 0486446557
Total Pages : 132 pages
Book Rating : 4.4/5 (864 download)

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Book Synopsis Natural Deduction by : Dag Prawitz

Download or read book Natural Deduction written by Dag Prawitz and published by Courier Dover Publications. This book was released on 2006-02-24 with total page 132 pages. Available in PDF, EPUB and Kindle. Book excerpt: An innovative approach to the semantics of logic, proof-theoretic semantics seeks the meaning of propositions and logical connectives within a system of inference. Gerhard Gentzen invented proof-theoretic semantics in the early 1930s, and Dag Prawitz, the author of this study, extended its analytic proofs to systems of natural deduction. Prawitz's theories form the basis of intuitionistic type theory, and his inversion principle constitutes the foundation of most modern accounts of proof-theoretic semantics. The concept of natural deduction follows a truly natural progression, establishing the relationship between a noteworthy systematization and the interpretation of logical signs. As this survey explains, the deduction's principles allow it to proceed in a direct fashion — a manner that permits every natural deduction's transformation into the equivalent of normal form theorem. A basic result in proof theory, the normal form theorem was established by Gentzen for the calculi of sequents. The proof of this result for systems of natural deduction is in many ways simpler and more illuminating than alternative methods. This study offers clear illustrations of the proof and numerous examples of its advantages.

Logic from Russell to Church

Author : Dov M. Gabbay
Publisher : Elsevier
ISBN 13 : 0080885470
Total Pages : 1069 pages
Book Rating : 4.0/5 (88 download)

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Book Synopsis Logic from Russell to Church by : Dov M. Gabbay

Download or read book Logic from Russell to Church written by Dov M. Gabbay and published by Elsevier. This book was released on 2009-06-16 with total page 1069 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume is number five in the 11-volume Handbook of the History of Logic. It covers the first 50 years of the development of mathematical logic in the 20th century, and concentrates on the achievements of the great names of the period--Russell, Post, Gödel, Tarski, Church, and the like. This was the period in which mathematical logic gave mature expression to its four main parts: set theory, model theory, proof theory and recursion theory. Collectively, this work ranks as one of the greatest achievements of our intellectual history. Written by leading researchers in the field, both this volume and the Handbook as a whole are definitive reference tools for senior undergraduates, graduate students and researchers in the history of logic, the history of philosophy, and any discipline, such as mathematics, computer science, and artificial intelligence, for whom the historical background of his or her work is a salient consideration.• The entire range of modal logic is covered• Serves as a singular contribution to the intellectual history of the 20th century• Contains the latest scholarly discoveries and interpretative insights

Induction and Deduction in the Sciences

Author : F. Stadler
Publisher : Springer Science & Business Media
ISBN 13 : 1402021968
Total Pages : 355 pages
Book Rating : 4.4/5 (2 download)

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Book Synopsis Induction and Deduction in the Sciences by : F. Stadler

Download or read book Induction and Deduction in the Sciences written by F. Stadler and published by Springer Science & Business Media. This book was released on 2004-04-30 with total page 355 pages. Available in PDF, EPUB and Kindle. Book excerpt: The articles in this volume deal with the main inferential methods that can be applied to different kinds of experimental evidence. These contributions - accompanied with critical comments - by renowned scholars in the field of philosophy of science aim at removing the traditional opposition between inductivists and deductivists. They explore the different methods of explanation and justification in the sciences in different contexts and with different objectives. The volume contains contributions on methods of the sciences, especially on induction, deduction, abduction, laws, probability and explanation, ranging from logic, mathematics, natural to the social sciences. They present a highly topical pluralist re-evaluation of methodological and foundational procedures and reasoning, e.g. focusing in Bayesianism and Artificial Intelligence. They document the second international conference in Vienna on "Induction and Deduction in the Sciences" as part of the Scientific Network on "Historical and Contemporary Perspectives of Philosophy of Science in Europe", funded by the European Science Foundation (ESF).

Mathematical Problems

Author : Craig Smoryński
Publisher : Springer Nature
ISBN 13 : 3030509176
Total Pages : 406 pages
Book Rating : 4.0/5 (35 download)

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Book Synopsis Mathematical Problems by : Craig Smoryński

Download or read book Mathematical Problems written by Craig Smoryński and published by Springer Nature. This book was released on 2020-09-19 with total page 406 pages. Available in PDF, EPUB and Kindle. Book excerpt: The life and soul of any science are its problems. This is particularly true of mathematics, which, not referring to any physical reality, consists only of its problems, their solutions, and, most excitingly, the challenges they pose. Mathematical problems come in many flavours, from simple puzzles to major open problems. The problems stimulate, the stories of their successful solutions inspire, and their applications are wide. The literature abounds with books dedicated to mathematical problems — collections of problems, hints on how to solve them, and even histories of the paths to the solutions of some famous ones. The present book, aimed at the proverbial “bright high-school student”, takes a different, more philosophical approach, first dividing mathematical problems into three broad classes — puzzles, exercises, and open problems — and discussing their various roles in one’s mathematical education. Various chapters are devoted to discussing examples of each type of problem, along with their solutions and some of the developments arising from them. For the truly dedicated reader, more involved material is offered in an appendix. Mathematics does not exist in a vacuum, whence the author peppers the material with frequent extra-mathematical cultural references. The mathematics itself is elementary, for the most part pre-calculus. The few references to the calculus use the integral notation which the reader need not truly be familiar with, opting to read the integral sign as strange notation for area or as operationally defined by the appropriate buttons on his or her graphing calculator. Nothing further is required. Advance praise for Mathematical Problems "There are many books on mathematical problems, but Smoryński’s compelling book offers something unique. Firstly, it includes a fruitful classification and analysis of the nature of mathematical problems. Secondly, and perhaps most importantly, it leads the reader from clear and often amusing accounts of traditional problems to the serious mathematics that grew out of some of them." - John Baldwin, University of Illinois at Chicago "Smoryński manages to discuss the famous puzzles from the past and the new items in various modern theories with the same elegance and personality. He presents and solves puzzles and traditional topics with a laudable sense of humor. Readers of all ages and training will find the book a rich treasure chest." - Dirk van Dalen, Universiteit Utrecht

The Collected Papers of Gerhard Gentzen

Author : Lev D. Beklemishev
Publisher : Elsevier
ISBN 13 : 0080957749
Total Pages : 353 pages
Book Rating : 4.0/5 (89 download)

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Book Synopsis The Collected Papers of Gerhard Gentzen by : Lev D. Beklemishev

Download or read book The Collected Papers of Gerhard Gentzen written by Lev D. Beklemishev and published by Elsevier. This book was released on 2000-04-01 with total page 353 pages. Available in PDF, EPUB and Kindle. Book excerpt: The Collected Papers of Gerhard Gentzen

Rethinking Knowledge

Author : Carlo Cellucci
Publisher : Springer
ISBN 13 : 3319532375
Total Pages : 432 pages
Book Rating : 4.3/5 (195 download)

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Book Synopsis Rethinking Knowledge by : Carlo Cellucci

Download or read book Rethinking Knowledge written by Carlo Cellucci and published by Springer. This book was released on 2017-03-29 with total page 432 pages. Available in PDF, EPUB and Kindle. Book excerpt: This monograph addresses the question of the increasing irrelevance of philosophy, which has seen scientists as well as philosophers concluding that philosophy is dead and has dissolved into the sciences. It seeks to answer the question of whether or not philosophy can still be fruitful and what kind of philosophy can be such. The author argues that from its very beginning philosophy has focused on knowledge and methods for acquiring knowledge. This view, however, has generally been abandoned in the last century with the belief that, unlike the sciences, philosophy makes no observations or experiments and requires only thought. Thus, in order for philosophy to once again be relevant, it needs to return to its roots and focus on knowledge as well as methods for acquiring knowledge. Accordingly, this book deals with several questions about knowledge that are essential to this view of philosophy, including mathematical knowledge. Coverage examines such issues as the nature of knowledge; plausibility and common sense; knowledge as problem solving; modeling scientific knowledge; mathematical objects, definitions, diagrams; mathematics and reality; and more. This monograph presents a new approach to philosophy, epistemology, and the philosophy of mathematics. It will appeal to graduate students and researchers with interests in the role of knowledge, the analytic method, models of science, and mathematics and reality.

Saved from the Cellar

Author : Jan von Plato
Publisher : Springer
ISBN 13 : 3319421204
Total Pages : 321 pages
Book Rating : 4.3/5 (194 download)

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Book Synopsis Saved from the Cellar by : Jan von Plato

Download or read book Saved from the Cellar written by Jan von Plato and published by Springer. This book was released on 2017-03-31 with total page 321 pages. Available in PDF, EPUB and Kindle. Book excerpt: Gerhard Gentzen is best known for his development of the proof systems of natural deduction and sequent calculus, central in many areas of logic and computer science today. Another noteworthy achievement is his resolution of the embarrassing situation created by Gödel's incompleteness results, especially the second one about the unprovability of consistency of elementary arithmetic. After these successes, Gentzen dedicated the rest of his short life to the main problem of Hilbert's proof theory, the question of the consistency of analysis. He was arrested in the summer of 1945 with other professors of the German University of Prague and died soon afterward of starvation in a prison cell. Attempts at locating his lost manuscripts failed at the time, but several decades later, two slim folders of shorthand notes were found. In this volume, Jan von Plato gives an overview of Gentzen's life and scientific achievements, based on detailed archival and systematic studies, and essential for placing the translations of shorthand manuscripts that follow in the right setting. The materials in this book are singular in the way they show the birth and development of Gentzen's central ideas and results, sometimes in a well-developed form, and other times as flashes into the anatomy of the workings of a unique mind.

Robert Brandom's Normative Inferentialism

Author : Giacomo Turbanti
Publisher : John Benjamins Publishing Company
ISBN 13 : 9027265070
Total Pages : 245 pages
Book Rating : 4.0/5 (272 download)

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Book Synopsis Robert Brandom's Normative Inferentialism by : Giacomo Turbanti

Download or read book Robert Brandom's Normative Inferentialism written by Giacomo Turbanti and published by John Benjamins Publishing Company. This book was released on 2017-09-21 with total page 245 pages. Available in PDF, EPUB and Kindle. Book excerpt: The philosophy of language of Robert Brandom is based on a theoretical structure composed of three main elements: the normative analysis of linguistic practices, the inferential characterization of conceptual contents and the expressive articulation of the relations between the former two. Normative pragmatics aims to explain how linguistic practices are sufficient to confer contentful states in those who engage in them. Inferential semantics provides a theory of such pragmatic significances in terms of the inferential relations that articulate conceptual contents. Rational expressivism is the thesis that concept application is essentially a process of turning something that can only be done into something that can also be said. Such a threefold structure is the core of normative inferentialism. This book is a concise, self-contained and comprehensive presentation of this philosophical enterprise. It guides the reader through the analysis of Brandom's imposing theoretical apparatus, the discovery of the roots of his approach in American pragmatism and German idealism, till the exploration of some of its most interesting and recent outcomes in pragmatics and semantics. It is a valuable resource for both those who approach Brandom's work for the first time and those who are interested in the potential of normative inferentialism.

Dictionary of Logic as Applied in the Study of Language

Author : W. Marciszewski
Publisher : Springer Science & Business Media
ISBN 13 : 9401712530
Total Pages : 450 pages
Book Rating : 4.4/5 (17 download)

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Book Synopsis Dictionary of Logic as Applied in the Study of Language by : W. Marciszewski

Download or read book Dictionary of Logic as Applied in the Study of Language written by W. Marciszewski and published by Springer Science & Business Media. This book was released on 2013-06-29 with total page 450 pages. Available in PDF, EPUB and Kindle. Book excerpt: 1. STRUCTURE AND REFERENCES 1.1. The main part of the dictionary consists of alphabetically arranged articles concerned with basic logical theories and some other selected topics. Within each article a set of concepts is defined in their mutual relations. This way of defining concepts in the context of a theory provides better understand ing of ideas than that provided by isolated short defmitions. A disadvantage of this method is that it takes more time to look something up inside an extensive article. To reduce this disadvantage the following measures have been adopted. Each article is divided into numbered sections, the numbers, in boldface type, being addresses to which we refer. Those sections of larger articles which are divided at the first level, i.e. numbered with single numerals, have titles. Main sections are further subdivided, the subsections being numbered by numerals added to the main section number, e.g. I, 1.1, 1.2, ... , 1.1.1, 1.1.2, and so on. A comprehensive subject index is supplied together with a glossary. The aim of the latter is to provide, if possible, short defmitions which sometimes may prove sufficient. As to the use of the glossary, see the comment preceding it.

Paul Lorenzen -- Mathematician and Logician

Author : Gerhard Heinzmann
Publisher : Springer Nature
ISBN 13 : 3030658244
Total Pages : 268 pages
Book Rating : 4.0/5 (36 download)

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Book Synopsis Paul Lorenzen -- Mathematician and Logician by : Gerhard Heinzmann

Download or read book Paul Lorenzen -- Mathematician and Logician written by Gerhard Heinzmann and published by Springer Nature. This book was released on 2021-08-17 with total page 268 pages. Available in PDF, EPUB and Kindle. Book excerpt: This open access book examines the many contributions of Paul Lorenzen, an outstanding philosopher from the latter half of the 20th century. It features papers focused on integrating Lorenzen's original approach into the history of logic and mathematics. The papers also explore how practitioners can implement Lorenzen’s systematical ideas in today’s debates on proof-theoretic semantics, databank management, and stochastics. Coverage details key contributions of Lorenzen to constructive mathematics, Lorenzen’s work on lattice-groups and divisibility theory, and modern set theory and Lorenzen’s critique of actual infinity. The contributors also look at the main problem of Grundlagenforschung and Lorenzen’s consistency proof and Hilbert’s larger program. In addition, the papers offer a constructive examination of a Russell-style Ramified Type Theory and a way out of the circularity puzzle within the operative justification of logic and mathematics. Paul Lorenzen's name is associated with the Erlangen School of Methodical Constructivism, of which the approach in linguistic philosophy and philosophy of science determined philosophical discussions especially in Germany in the 1960s and 1970s. This volume features 10 papers from a meeting that took place at the University of Konstanz.

The Great Formal Machinery Works

Author : Jan von Plato
Publisher : Princeton University Press
ISBN 13 : 1400885035
Total Pages : 400 pages
Book Rating : 4.4/5 (8 download)

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Book Synopsis The Great Formal Machinery Works by : Jan von Plato

Download or read book The Great Formal Machinery Works written by Jan von Plato and published by Princeton University Press. This book was released on 2017-08-02 with total page 400 pages. Available in PDF, EPUB and Kindle. Book excerpt: The information age owes its existence to a little-known but crucial development, the theoretical study of logic and the foundations of mathematics. The Great Formal Machinery Works draws on original sources and rare archival materials to trace the history of the theories of deduction and computation that laid the logical foundations for the digital revolution. Jan von Plato examines the contributions of figures such as Aristotle; the nineteenth-century German polymath Hermann Grassmann; George Boole, whose Boolean logic would prove essential to programming languages and computing; Ernst Schröder, best known for his work on algebraic logic; and Giuseppe Peano, cofounder of mathematical logic. Von Plato shows how the idea of a formal proof in mathematics emerged gradually in the second half of the nineteenth century, hand in hand with the notion of a formal process of computation. A turning point was reached by 1930, when Kurt Gödel conceived his celebrated incompleteness theorems. They were an enormous boost to the study of formal languages and computability, which were brought to perfection by the end of the 1930s with precise theories of formal languages and formal deduction and parallel theories of algorithmic computability. Von Plato describes how the first theoretical ideas of a computer soon emerged in the work of Alan Turing in 1936 and John von Neumann some years later. Shedding new light on this crucial chapter in the history of science, The Great Formal Machinery Works is essential reading for students and researchers in logic, mathematics, and computer science.

Proof Theory

Author : Wolfram Pohlers
Publisher : Springer
ISBN 13 : 3540468250
Total Pages : 220 pages
Book Rating : 4.5/5 (44 download)

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Book Synopsis Proof Theory by : Wolfram Pohlers

Download or read book Proof Theory written by Wolfram Pohlers and published by Springer. This book was released on 2009-06-10 with total page 220 pages. Available in PDF, EPUB and Kindle. Book excerpt: Although this is an introductory text on proof theory, most of its contents is not found in a unified form elsewhere in the literature, except at a very advanced level. The heart of the book is the ordinal analysis of axiom systems, with particular emphasis on that of the impredicative theory of elementary inductive definitions on the natural numbers. The "constructive" consequences of ordinal analysis are sketched out in the epilogue. The book provides a self-contained treatment assuming no prior knowledge of proof theory and almost none of logic. The author has, moreover, endeavoured not to use the "cabal language" of proof theory, but only a language familiar to most readers.

The Problem of Plurality of Logics

Author : Pavel Arazim
Publisher : Bloomsbury Publishing
ISBN 13 : 1350146765
Total Pages : 208 pages
Book Rating : 4.3/5 (51 download)

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Book Synopsis The Problem of Plurality of Logics by : Pavel Arazim

Download or read book The Problem of Plurality of Logics written by Pavel Arazim and published by Bloomsbury Publishing. This book was released on 2021-08-12 with total page 208 pages. Available in PDF, EPUB and Kindle. Book excerpt: As the foundation of our rationality, logic has traditionally been considered fixed, stable and constant. This conception of the discipline has been challenged recently by the plurality of logics and in this book, Pavel Arazim extends the debate to offer a new view of logic as dynamic and without a definite, specific shape. The Problem of Plurality of Logics examines the origins of our standard view of logic alongside Kant's theories, the holistic view, the issue of logic's pragmatic significance and Robert Brandom's logical expressivism. Arazim then draws on proof-theoretical approaches to present a convincing argument for a dynamic version of logical inferentialism, which opens space for a new freedom to modify our own logic. He explores the scope, possibilities and limits of this freedom in order to highlight the future paths logic could take, as a motivation for further research. Marking a departure from logical monism and also from the recent doctrine of logical pluralism in its various forms, this book addresses current debates concerning the expressive role of logic and contributes to a lively area of discussion in analytic philosophy.

Interpreting Gödel

Author : Juliette Kennedy
Publisher : Cambridge University Press
ISBN 13 : 1139991752
Total Pages : 293 pages
Book Rating : 4.1/5 (399 download)

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Book Synopsis Interpreting Gödel by : Juliette Kennedy

Download or read book Interpreting Gödel written by Juliette Kennedy and published by Cambridge University Press. This book was released on 2014-08-21 with total page 293 pages. Available in PDF, EPUB and Kindle. Book excerpt: The logician Kurt Gödel (1906–1978) published a paper in 1931 formulating what have come to be known as his 'incompleteness theorems', which prove, among other things, that within any formal system with resources sufficient to code arithmetic, questions exist which are neither provable nor disprovable on the basis of the axioms which define the system. These are among the most celebrated results in logic today. In this volume, leading philosophers and mathematicians assess important aspects of Gödel's work on the foundations and philosophy of mathematics. Their essays explore almost every aspect of Godel's intellectual legacy including his concepts of intuition and analyticity, the Completeness Theorem, the set-theoretic multiverse, and the state of mathematical logic today. This groundbreaking volume will be invaluable to students, historians, logicians and philosophers of mathematics who wish to understand the current thinking on these issues.