Frege's Theorem

Author: Richard G. Heck

Publisher: Oxford University Press

ISBN: 0199695644

Category: Mathematics

Page: 307

View: 8252

Richard Heck explores a key idea in the work of the great philosopher/logician Gottlob Frege: that the axioms of arithmetic can be logically derived from a single principle. Heck uses the theorem to explore historical, philosophical, and technical issues in philosophy of mathematics and logic, relating them to key areas of contemporary philosophy.

Frege's Philosophy of Mathematics

Author: William Demopoulos

Publisher: Harvard University Press

ISBN: 9780674319424

Category: Mathematics

Page: 464

View: 6728

Widespread interest in Frege's general philosophical writings is, relatively speaking, a fairly recent phenomenon. But it is only very recently that his philosophy of mathematics has begun to attract the attention it now enjoys. This interest has been elicited by the discovery of the remarkable mathematical properties of Frege's contextual definition of number and of the unique character of his proposals for a theory of the real numbers. This collection of essays addresses three main developments in recent work on Frege's philosophy of mathematics: the emerging interest in the intellectual background to his logicism; the rediscovery of Frege's theorem; and the reevaluation of the mathematical content of The Basic Laws of Arithmetic. Each essay attempts a sympathetic, if not uncritical, reconstruction, evaluation, or extension of a facet of Frege's theory of arithmetic. Together they form an accessible and authoritative introduction to aspects of Frege's thought that have, until now, been largely missed by the philosophical community.

The Reason's Proper Study

Essays Towards a Neo-Fregean Philosophy of Mathematics

Author: Bob Hale,Crispin Wright

Publisher: Oxford University Press

ISBN: 9780199266326

Category: Philosophy

Page: 455

View: 9146

Bob Hale and Crispin Wright draw together here the key writings in which they have worked out their distinctive neo-Fregean approach to the philosophy of mathematics. The two main components in Frege's mathematical philosophy were his platonism and his logicism - the claims, respectively, that mathematics is a body of knowledge about independently existing objects, and that this knowledge may be acquired on the basis of general logical laws and suitable definitions. The central thesis ofthis collection is that Frege was - his own eventual recantation notwithstanding - substantially right in both claims. Where neo-Fregeanism principally differs from Frege is in taking a more optimistic view of the kind of contextual explanation (proceeding via what are now commonly called abstraction principles) of the fundamental concepts of arithmetic and analysis which Frege considered and rejected. On this basis, neo-Fregeanism promises defensible and attractive answers to some of the most important ontological and epistemological questions in the philosophy of mathematics. In addition to fourteen previously published papers, the volume features a new paper on the Julius Caesar problem; a substantial new introduction mapping out the programme and the contributions made to it by the various papers; a postscript explaining which issues most require further attention; and bibliographies both of references and of further useful sources. The Reason's Proper Study will be recognized as the most powerful presentation yet of the neo-Fregean programme; it will prove indispensable reading not just to philosophers of mathematics but to all who are interested in the fundamental metaphysical and epistemological issues on which the programme impinges.

Language, Thought, and Logic

Essays in Honour of Michael Dummett

Author: Michael A. E. Dummett

Publisher: Oxford University Press on Demand

ISBN: 9780198239208

Category: Language Arts & Disciplines

Page: 308

View: 1474

In this exciting new collection, a distinguished international group of philosophers contribute new essays on central issues in philosophy of language and logic, in honor of Michael Dummett, one of the most influential philosophers of the late twentieth century. The essays are focused on areas particularly associated with Professor Dummett. Five are contributions to the philosophy of language, addressing in particular the nature of truth and meaning and the relation between language and thought. Two contributors discuss time, in particular the reality of the past. The last four essays focus on Frege and the philosophy of mathematics. The volume represents some of the best work in contemporary analytical philosophy.

Gottlob Frege: Frege's philosophy in context

Author: Michael Beaney

Publisher: Taylor & Francis

ISBN: 9780415306027


Page: 424

View: 7930

This collection brings together recent scholarship on Frege, including new translations of German material which is made available to Anglophone scholars for the first time.

Gottlob Frege: Frege's philosophy of mathematics

Author: Michael Beaney

Publisher: Taylor & Francis

ISBN: 9780415306041


Page: 416

View: 3548

This collection brings together recent scholarship on Frege, including new translations of German material which is made available to Anglophone scholars for the first time.

Logik und Mathematik

Frege-Kolloquium Jena 1993

Author: Ingolf Max,Werner Stelzner

Publisher: Walter de Gruyter

ISBN: 3110887797

Category: Philosophy

Page: 564

View: 462

Reading Frege's Grundgesetze

Author: Richard G. Heck

Publisher: Oxford University Press on Demand

ISBN: 0199233705

Category: Mathematics

Page: 296

View: 7323

Gottlob Frege's Grundgesetze der Arithmetik, or Basic Laws of Arithmetic, was intended to be his magnum opus, the book in which he would finally establish his logicist philosophy of arithmetic. But because of the disaster of Russell's Paradox, which undermined Frege's proofs, the more mathematical parts of the book have rarely been read. Richard G. Heck, Jr., aims to change that, and establish it as a neglected masterpiece that must be placed atthe center of Frege's philosophy. He argues that Frege knew that his proofs could be reconstructed so as to avoid Russell's Paradox, and presents Frege's arguments in a way that makes them available to a wide audience.Heck demonstrates that careful attention to the structure of Frege's arguments, to what he proved, to how he proved it, and even to what he tried to prove but could not, has much to teach us about Frege's philosophy.

All We Need Is a Paradigm

Essays on Science, Economics, and Logic from the Harvard Review of Philosophy

Author: S. Phineas Upham

Publisher: Open Court Publishing Company


Category: Philosophy

Page: 226

View: 1265

The Harvard Review of Philosophy has long been a forum for new thoughts in the field — this collection includes some of the most important essays from that publication. Exploring the unexpected ways that philosophy impacts our world, this book considers the discipline as an essential element in our understanding of science, economics, and logic. This fascinating read for both laypeople and those familiar with philosophical concepts delves deep into questions of human nature, intellectual thought, and the manner in which our world operates. Using several different approaches, including puzzles, essays, and songs, this book challenges our basic assumptions about how things work.

Fixing Frege

Author: John P. Burgess

Publisher: Princeton University Press

ISBN: 9780691122311

Category: Mathematics

Page: 257

View: 459

Gottlob Frege's attempt to found mathematics on a grand logical system came to grief when Bertrand Russell discovered a contradiction in it. This book surveys consistent restrictions in both the old and new versions of Frege's system, determining just how much of mathematics can be reconstructed in each.

Frege's Conception of Logic

Author: Patricia Blanchette

Publisher: OUP USA

ISBN: 0199891613

Category: Mathematics

Page: 190

View: 5089

In Frege's Conception of Logic Patricia A. Blanchette explores the relationship between Gottlob Frege's understanding of conceptual analysis and his understanding of logic. She argues that the fruitfulness of Frege's conception of logic, and the illuminating differences between that conception and those more modern views that have largely supplanted it, are best understood against the backdrop of a clear account of the role of conceptual analysis in logical investigation. The first part of the book locates the role of conceptual analysis in Frege's logicist project. Blanchette argues that despite a number of difficulties, Frege's use of analysis in the service of logicism is a powerful and coherent tool. As a result of coming to grips with his use of that tool, we can see that there is, despite appearances, no conflict between Frege's intention to demonstrate the grounds of ordinary arithmetic and the fact that the numerals of his derived sentences fail to co-refer with ordinary numerals. In the second part of the book, Blanchette explores the resulting conception of logic itself, and some of the straightforward ways in which Frege's conception differs from its now-familiar descendants. In particular, Blanchette argues that consistency, as Frege understands it, differs significantly from the kind of consistency demonstrable via the construction of models. To appreciate this difference is to appreciate the extent to which Frege was right in his debate with Hilbert over consistency- and independence-proofs in geometry. For similar reasons, modern results such as the completeness of formal systems and the categoricity of theories do not have for Frege the same importance they are commonly taken to have by his post-Tarskian descendants. These differences, together with the coherence of Frege's position, provide reason for caution with respect to the appeal to formal systems and their properties in the treatment of fundamental logical properties and relations.

Logic, Logic, and Logic

Author: George Boolos,Richard C. Jeffrey

Publisher: Harvard University Press

ISBN: 9780674537675

Category: Philosophy

Page: 443

View: 8266

George Boolos was one of the most prominent and influential logician-philosophers of recent times. This collection, nearly all chosen by Boolos himself shortly before his death, includes thirty papers on set theory, second-order logic, and plural quantifiers; on Frege, Dedekind, Cantor, and Russell; and on miscellaneous topics in logic and proof theory, including three papers on various aspects of the Gödel theorems. Boolos is universally recognized as the leader in the renewed interest in studies of Frege's work on logic and the philosophy of mathematics. John Burgess has provided introductions to each of the three parts of the volume, and also an afterword on Boolos's technical work in provability logic, which is beyond the scope of this volume.

The Arché Papers on the Mathematics of Abstraction

Author: Roy T. Cook

Publisher: Springer Science & Business Media

ISBN: 9781402042652

Category: Mathematics

Page: 454

View: 5572

This volume collects together a number of important papers concerning both the method of abstraction generally and the use of particular abstraction principles to reconstruct central areas of mathematics along logicist lines. Attention is focused on extending the Neo-Fregean treatment to all of mathematics, with the reconstruction of real analysis from various cut- or cauchy-sequence-related abstraction principles and the reconstruction of set theory from various restricted versions of Basic Law V as case studies.

Husserl Or Frege?

Meaning, Objectivity, and Mathematics

Author: Claire Ortiz Hill,Guillermo E. Rosado Haddock

Publisher: Open Court Publishing

ISBN: 9780812694178

Category: Philosophy

Page: 315

View: 7165

Most areas of philosopher Edmund Husserl’s thought have been explored, but his views on logic, mathematics, and semantics have been largely ignored. These essays offer an alternative to discussions of the philosophy of contemporary mathematics. The book covers areas of disagreement between Husserl and Gottlob Frege, the father of analytical philosophy, and explores new perspectives seen in their work.

Frege’s Notations

What They Are and How They Mean

Author: Gregory Landini,Michael Beaney

Publisher: Springer

ISBN: 0230360157

Category: Philosophy

Page: 194

View: 6585

A new approach to reading Frege's notations that adheres to the modern view that terms and well-formed formulas are any disjoint syntactic categories. On this new approach, we can at last read Frege's notations in their original form revealing striking new solutions to many of the outstanding problems of interpreting his philosophy.

Die Grundlagen der Arithmetik

Eine logische mathematische Untersuchung über den Begriff der Zahl

Author: Gottlob Frege

Publisher: Hansebooks

ISBN: 9783337357986


Page: 160

View: 7935

Die Grundlagen der Arithmetik - Eine logische mathematische Untersuchung über den Begriff der Zahl ist ein unveränderter, hochwertiger Nachdruck der Originalausgabe. Hansebooks ist Herausgeber von Literatur zu unterschiedlichen Themengebieten wie Forschung und Wissenschaft, Reisen und Expeditionen, Kochen und Ernährung, Medizin und weiteren Genres. Der Schwerpunkt des Verlages liegt auf dem Erhalt historischer Literatur. Viele Werke historischer Schriftsteller und Wissenschaftler sind heute nur noch als Antiquitäten erhältlich. Hansebooks verlegt diese Bücher neu und trägt damit zum Erhalt selten gewordener Literatur und historischem Wissen auch für die Zukunft bei.

Two Soviet Studies on Frege

Translated from the Russian and edited by Ignacio Angelelli

Author: B.V. Birjukov

Publisher: Springer Science & Business Media

ISBN: 940103611X

Category: Social Science

Page: 102

View: 4218

1 The significance of the two papers by B. V. Birjukov on Frege within Soviet studies on logic and its history is indicated by G. 1. Ruzavin and P. V. Tavanec in their article 'Fundamental Periods in the Evolution of Formal Logic' in the collective volume Philo 2 sophical Questions of Contemporary Formal Logic. There (page 18) while the organization of "systematic studies on history of logic" is proposed as "one of the fundamental tasks for Marxist logicians", reference is made to a series of recent publications which suggest that such a task is already being accomplished. These are A. S. 3 Axmanov's The Logical Doctrine of Aristotle , v. F. Asmus' 'Criticism of the Bourgeois Idealist Logical Doctrine in the Era of Imperia lism'4, in Voprosy Logiki (Logical Questions), P. S. Popov's A 5 History of Modern Logic and B. V. Birjukov's 'G. Frege's Theory of Sense' in the collective work Applications of Logic in Science and 6 Technology. In this book, published by the Academy of Sciences of the USSR, Moscow, in a printing of 10 000 copies, Birjukov's article fills 56 pages. Before this one, however, Birjukov published another study on Frege: 'On Frege's Works on Philosophical Problems of Mathe matics' in the collective volume Philosophical Questions of Natural Sciences 7, published in a printing of 8000 copies by the Moscow University Press. This article fills 45 pages.