An Introduction to Proof Theory: Normalization, Cut-Elimination, and Consistency Proofs

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.

Politics & Social Sciences

Mancosu, Paolo

Galvan, Sergio

Zach, Richard

Philosophy

Oxford University Press

Product Description 
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.\nAbout the Author 
Paolo Mancosu, 
UC Berkeley,Sergio Galvan, 
Catholic University of Milan,Richard Zach, 
University of Calgary\nPaolo Mancosu is Willis S. and Marion Slusser Professor of Philosophy at the University of California at Berkeley. He is the author of numerous articles and books in logic and philosophy of mathematics. During his career he has taught at Stanford, Oxford, and Yale. He was awarded a fellowship at the 
Wissenschaftskolleg zu Berlin in 1997-1998, a stipendiary position as 
Directeur de recherche invité; au CNRS in Paris in 2004-2005, a Guggenheim Fellowship in 2008-2009, a position as member at the Institute of Advanced Study in Princeton in 2009, a visiting professorship as LMU-UCB Research in the 
Humanities at LMU in 2014, and a Humboldt Research Award in 2017-2018.\nSergio Galvan is emeritus Professor of Logic at the Catholic University of Milan, Italy. His main areas of research are proof-theory and philosophical logic. In the first area he focuses on sequent calculi and natural deduction, the metamathematics of arithmetic systems (from Q to PA), Gödel's 
incompleteness theorems, and Gentzen's cut-elimination theorem. In the second area, his major interest is in the philosophical interpretations (deontic, epistemic and metaphysical) of modal logic. Recently, he has been working on the relationships between formal proof and intuition in mathematics, 
and on the metaphysics of essence and the ontology of possibilia.\nRichard Zach is Professor of Philosophy at the University of Calgary, Canada. He works in logic, history of analytic philosophy, and the philosophy of mathematics. In logic, his main interests are non-classical logics and proof theory. He has also written on the development of formal logic and 
historical figures associated with this development such as Hilbert, Gödel, and Carnap. He has held visiting appointments at the University of California, Irvine, McGill University, and the University of Technology, Vienna.

The seventeenth century saw dramatic advances in mathematical theory and practice. With the recovery of many of the classical Greek mathematical texts, new techniques were introduced, and within 100 years, the rules of analytic geometry, geometry of indivisibles, arithmetic of infinites, and calculus were developed. Although many technical studies have been devoted to these innovations, Mancosu provides the first comprehensive account of the relationship between mathematical advances of the seventeenth century and the philosophy of mathematics of the period. Starting with the Renaissance debates on the certainty of mathematics, Mancosu leads the reader through the foundational issues raised by the emergence of these new mathematical techniques, including the influence of the Aristotelian conception of science in Cavalieri and Guldin, the foundational relevance of Descartes' Geometrie, the relation between geometrical and epistemological theories of the infinite, and the Leibnizian calculus and the opposition to infinitesimalist procedures. In the process Mancosu draws a sophisticated picture of the subtle dependencies between technical development and philosophical reflection in seventeenth century mathematics.

Philosophy of Mathematics and Mathematical Practice in the Seventeenth Century

History

Asia

From Brouwer To Hilbert: The Debate on the Foundations of Mathematics in the 1920s offers the first comprehensive introduction to the most exciting period in the foundation of mathematics in the twentieth century. The 1920s witnessed the seminal foundational work of Hilbert and Bernays in proof theory, Brouwer's refinement of intuitionistic mathematics, and Weyl's predicativist approach to the foundations of analysis. This impressive collection makes available the first English translations of twenty-five central articles by these important contributors and many others. The articles have been translated for the first time from Dutch, French, and German, and the volume is divided into four sections devoted to (1) Brouwer, (2) Weyl, (3) Bernays and Hilbert, and (4) the emergence of intuitionistic logic. Each section opens with an introduction which provides the necessary historical and technical context for understanding the articles. Although most contemporary work in this field takes its start from the groundbreaking contributions of these major figures, a good, scholarly introduction to the area was not available until now. Unique and accessible, From Brouwer To Hilbert will serve as an ideal text for undergraduate and graduate courses in the philosophy of mathematics, and will also be an invaluable resource for philosophers, mathematicians, and interested non-specialists.

From Brouwer to Hilbert: The Debate on the Foundations of Mathematics in the 1920s

Oxford University Press, USA

The seventeenth century saw dramatic advances in mathematical theory and practice than any era before or since. With the recovery of many of the classical Greek mathematical texts, new techniques were introduced, and within 100 years, analytic geometry, the geometry of indivisibles, the arithmetic of infinites, and the calculus had been developed. Although many technical studies have been devoted to these innovations, Paolo Mancosu provides the first comprehensive account of the relationship between mathematical advances of the seventeenth century and the philosophy of mathematics of the period. Beginning with the Renaissance debates on the certainty of mathematics, Mancosu leads the reader through the foundational issues raised by the emergence of these new mathematical techniques, including the influence of the Aristotelian conception of science in Cavalieri and Guldin, the foundational relevance of Descartes' Geometrie, the relationship between empiricist epistemology and infinitistic theorems in geometry, and the debates concerning the foundations of the Leibnizian calculus In the process Mancosu draws a sophisticated picture of the subtle dependencies between technical development and philosophical reflection in seventeenth century mathematics.

Literature & Fiction

Dramas & Plays

Paolo Mancosu presents a series of innovative studies in the history and the philosophy of logic and mathematics in the first half of the twentieth century. The Adventure of Reason is divided into five main sections: history of logic (from Russell to Tarski); foundational issues (Hilbert's program, constructivity, Wittgenstein, Godel); mathematics and phenomenology (Weyl, Becker, Mahnke); nominalism (Quine, Tarski); semantics (Tarski, Carnap, Neurath). Mancosu exploits extensive untapped archival sources to make available a wealth of new material that deepens in significant ways our understanding of these fascinating areas of modern intellectual history. At the same time, the book is a contribution to recent philosophical debates, in particular on the prospects for a successful nominalist reconstruction of mathematics, the nature of finitist intuition, the viability of alternative definitions of logical consequence, and the extent to which phenomenology can hope to account for the exact sciences.

The Adventure of Reason: Interplay Between Philosophy of Mathematics and Mathematical Logic, 1900-1940

Paolo Mancosu provides an original investigation of historical and systematic aspects of the notions of abstraction and infinity and their interaction. A familiar way of introducing concepts in mathematics rests on so-called definitions by abstraction. An example of this is Hume's Principle, which introduces the concept of number by stating that two concepts have the same number if and only if the objects falling under each one of them can be put in one-one correspondence. This principle is at the core of neo-logicism. In the first two chapters of the book, Mancosu provides a historical analysis of the mathematical uses and foundational discussion of definitions by abstraction up to Frege, Peano, and Russell. Chapter one shows that abstraction principles were quite widespread in the mathematical practice that preceded Frege's discussion of them and the second chapter provides the first contextual analysis of Frege's discussion of abstraction principles in section 64 of the Grundlagen. In the second part of the book, Mancosu discusses a novel approach to measuring the size of infinite sets known as the theory of numerosities and shows how this new development leads to deep mathematical, historical, and philosophical problems. The final chapter of the book explore how this theory of numerosities can be exploited to provide surprisingly novel perspectives on neo-logicism.

Abstraction and Infinity

Paolo Mancosu provides an original investigation of historical and systematic aspects of the notions of abstraction and infinity and their interaction. A familiar way of introducing concepts in mathematics rests on so-called definitions by abstraction. An example of this is Hume's Principle, which introduces the concept of number by stating that two concepts have the same number if and only if the objects falling under each one of them can be put in one-one correspondence. This principle is at the core of neo-logicism. In the first two chapters of the book, Mancosu provides a historical analysis of the mathematical uses and foundational discussion of definitions by abstraction up to Frege, Peano, and Russell. Chapter one shows that abstraction principles were quite widespread in the mathematical practice that preceded Frege's discussion of them and the second chapter provides the first contextual analysis of Frege's discussion of abstraction principles in section 64 of the Grundlagen. In the second part of the book, Mancosu discusses a novel approach to measuring the size of infinite sets known as the theory of numerosities and shows how this new development leads to deep mathematical, historical, and philosophical problems. The final chapter of the book explore how this theory of numerosities can be exploited to provide surprisingly novel perspectives on neo-logicism.

Paolo Mancosu presents a series of innovative studies in the history and the philosophy of logic and mathematics in the first half of the twentieth century. The Adventure of Reason is divided into five main sections: history of logic (from Russell to Tarski); foundational issues (Hilbert's program, constructivity, Wittgenstein, Godel); mathematics and phenomenology (Weyl, Becker, Mahnke); nominalism (Quine, Tarski); semantics (Tarski, Carnap, Neurath). Mancosu exploits extensive untapped archival sources to make available a wealth of new material that deepens in significant ways our understanding of these fascinating areas of modern intellectual history. At the same time, the book is a contribution to recent philosophical debates, in particular on the prospects for a successful nominalist reconstruction of mathematics, the nature of finitist intuition, the viability of alternative definitions of logical consequence, and the extent to which phenomenology can hope to account for the exact sciences.

The conflict between Soviet Communists and Boris Pasternak over the publication of Doctor Zhivago did not end when he won the Nobel Prize, or even when the author died. Paolo Mancosu tells how Pasternak's expulsion from the Soviet Writers' Union left him in financial difficulty. After Pasternak's death, Olga Ivinskaya, his companion, literary assistant, and the inspiration for Zhivago's Lara, also received some of the Zhivago royalties. After the KGB intercepted Pasternak's will on her behalf, the Soviets arrested and sentenced her to eight years of labor camp. The ensuing international outrage inspired a secret campaign in the West to win her freedom. Mancosu's new book--the first to explore the post-Nobel history of Pasternak and Ivinskaya--provides extraordinary detail on these events, in a thrilling account that involves KGB interceptions, fabricated documents, smugglers, and much more. While a general reader will respond to the dramatic human story, specialists will be rewarded with a rich assemblage of new archival material, especially letters of Pasternak and Ivinskaya from the Feltrinelli Archives and the Hoover Institution Library and Archives.

Moscow Has Ears Everywhere: New Investigations on Pasternak and Ivinskaya (698)

Europe

Hoover Institution Press

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Books by Mancosu, Paolo

See the best price to sell, buy, or rent books by Mancosu, Paolo. BookScouter helps to compare book prices from 25+ online bookstores and 30+ buyback vendors with a single search.

An Introduction to Proof Theory: Normalization, Cut-Elimination, and Consistency Proofs

An Introduction to Proof Theory: Normalization, Cut-Elimination, and Consistency Proofs

Philosophy of Mathematics and Mathematical Practice in the Seventeenth Century

From Brouwer to Hilbert: The Debate on the Foundations of Mathematics in the 1920s

Philosophy of Mathematics and Mathematical Practice in the Seventeenth Century

The Adventure of Reason: Interplay Between Philosophy of Mathematics and Mathematical Logic, 1900-1940

Abstraction and Infinity

Abstraction and Infinity

The Adventure of Reason: Interplay Between Philosophy of Mathematics and Mathematical Logic, 1900-1940

Moscow Has Ears Everywhere: New Investigations on Pasternak and Ivinskaya (698)