Mathematics

Frege's Theorem

Author: Richard G. Heck

Publisher: Oxford University Press

ISBN: 0199695644

Category: Mathematics

Page: 307

View: 2826

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.

Gottlob Frege: Frege's philosophy of mathematics

Author: Michael Beaney

Publisher: Taylor & Francis

ISBN: 9780415306041

Category:

Page: 416

View: 6237

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 in context

Author: Michael Beaney

Publisher: Taylor & Francis

ISBN: 9780415306027

Category:

Page: 424

View: 7562

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 logic

Author: Michael Beaney

Publisher: Taylor & Francis

ISBN: 9780415306034

Category:

Page: 417

View: 3053

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.
Mathematics

Frege's Philosophy of Mathematics

Author: William Demopoulos

Publisher: Harvard University Press

ISBN: 9780674319424

Category: Mathematics

Page: 464

View: 4213

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.

Die Grundlagen der Arithmetik

eine logisch-mathematische untersuchung über den begriff der zahl

Author: Gottlob Frege

Publisher: Georg Olms Verlag

ISBN: 9783487400280

Category:

Page: 119

View: 5534

Philosophy

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: 8733

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 Arts & Disciplines

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: 7558

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.
Philosophy

Logik und Mathematik

Frege-Kolloquium Jena 1993

Author: Ingolf Max,Werner Stelzner

Publisher: Walter de Gruyter

ISBN: 3110887797

Category: Philosophy

Page: 564

View: 8433

Philosophy

Frege's Theory of Sense and Reference

Its Origin and Scope

Author: Wolfgang Carl

Publisher: Cambridge University Press

ISBN: 9780521398169

Category: Philosophy

Page: 220

View: 3266

This book provides a completely new and systematic account of Frege's philosophy by focusing on its cornerstone: the theory of sense and reference.
Mathematics

Reading Frege's Grundgesetze

Author: Richard G. Heck

Publisher: Oxford University Press on Demand

ISBN: 0199233705

Category: Mathematics

Page: 296

View: 2975

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.
Mathematics

Fixing Frege

Author: John P. Burgess

Publisher: Princeton University Press

ISBN: 9780691122311

Category: Mathematics

Page: 257

View: 9438

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.
Philosophy

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: 5066

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.
Social Science

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: 3587

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.
Philosophy

Frege’s Notations

What They Are and How They Mean

Author: Gregory Landini,Michael Beaney

Publisher: Springer

ISBN: 0230360157

Category: Philosophy

Page: 194

View: 5000

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.
Mathematics

Frege's Conception of Logic

Author: Patricia Blanchette

Publisher: OUP USA

ISBN: 0199891613

Category: Mathematics

Page: 190

View: 714

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.
Philosophy

Logic, Logic, and Logic

Author: George Boolos,Richard C. Jeffrey

Publisher: Harvard University Press

ISBN: 9780674537675

Category: Philosophy

Page: 443

View: 1378

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.
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

ISBN: N.A

Category: Philosophy

Page: 226

View: 9608

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.
Philosophy

Frege

Philosophy of Mathematics

Author: Michael A. E. Dummett

Publisher: Harvard University Press

ISBN: 9780674319356

Category: Philosophy

Page: 331

View: 1094

No one has figured more prominently in the study of the German philosopher Gottlob Frege than Michael Dummett. His magisterial Frege: Philosophy of Language is a sustained, systematic analysis of Frege's thought, omitting only the issues in philosophy of mathematics. In this work Dummett discusses, section by section, Frege's masterpiece The Foundations of Arithmetic and Frege's treatment of real numbers in the second volume of Basic Laws of Arithmetic, establishing what parts of the philosopher's views can be salvaged and employed in new theorizing, and what must be abandoned, either as incorrectly argued or as untenable in the light of technical developments. Gottlob Frege (1848-1925) was a logician, mathematician, and philosopher whose work had enormous impact on Bertrand Russell and later on the young Ludwig Wittgenstein, making Frege one of the central influences on twentieth-century Anglo-American philosophy; he is considered the founder of analytic philosophy. His philosophy of mathematics contains deep insights and remains a useful and necessary point of departure for anyone seriously studying or working in the field.