Frege's Theorem

Author: Richard G. Heck

Publisher: Oxford University Press

ISBN: 0199695644

Category: Mathematics

Page: 307

View: 5140

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: 7698

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.

Gottlob Frege: Frege's philosophy of mathematics

Author: Michael Beaney

Publisher: Taylor & Francis

ISBN: 9780415306041

Category:

Page: 416

View: 842

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.

Reading Frege's Grundgesetze

Author: Heck Jr.

Publisher: OUP Oxford

ISBN: 019165535X

Category: Philosophy

Page: 316

View: 3855

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 at the center of Frege's philosophy. Part I of Reading Frege's Grundgesetze develops an interpretation of the philosophy of logic that informs Grundgesetze, paying especially close attention to the difficult sections of Frege's book in which he discusses his notorious 'Basic Law V' and attempts to secure its status as a law of logic. Part II examines the mathematical basis of Frege's logicism, explaining and exploring Frege's formal arguments. Heck argues that Frege himself 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. He shows, by example, 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.

Routledge Philosophy GuideBook to Frege on Sense and Reference

Author: Mark Textor

Publisher: Routledge

ISBN: 1136930558

Category: Philosophy

Page: 304

View: 3339

Gottlob Frege (1848-1925) is considered the father of modern logic and one of the founding figures of analytic philosophy. He was first and foremost a mathematician, but his major works also made important contributions to the philosophy of language. Frege’s writings are difficult and deal with technical, abstract concepts. The Routledge Philosophy Guidebook to Frege On Sense and Reference helps the student to get to grips with Frege’s thought, and introduces and assesses: the background of Frege’s philosophical work Frege’s main papers and arguments, focussing on his distinction between sense and reference the continuing importance of Frege’s work to philosophy of logic and language. Ideal for those coming to Frege for the first time, and containing fresh insights for anyone interested in his philosophy, this Guidebook is essential reading for all students of philosophy of language, philosophical logic and the history of analytic philosophy.

Fixing Frege

Author: John P. Burgess

Publisher: Princeton University Press

ISBN: 0691187061

Category: Philosophy

Page: N.A

View: 3892

An Historical Introduction to the Philosophy of Mathematics: A Reader

Author: Russell Marcus,Mark McEvoy

Publisher: Bloomsbury Publishing

ISBN: 1472529480

Category: Philosophy

Page: 896

View: 586

A comprehensive collection of historical readings in the philosophy of mathematics and a selection of influential contemporary work, this much-needed introduction reveals the rich history of the subject. An Historical Introduction to the Philosophy of Mathematics: A Reader brings together an impressive collection of primary sources from ancient and modern philosophy. Arranged chronologically and featuring introductory overviews explaining technical terms, this accessible reader is easy-to-follow and unrivaled in its historical scope. With selections from key thinkers such as Plato, Aristotle, Descartes, Hume and Kant, it connects the major ideas of the ancients with contemporary thinkers. A selection of recent texts from philosophers including Quine, Putnam, Field and Maddy offering insights into the current state of the discipline clearly illustrates the development of the subject. Presenting historical background essential to understanding contemporary trends and a survey of recent work, An Historical Introduction to the Philosophy of Mathematics: A Reader is required reading for undergraduates and graduate students studying the philosophy of mathematics and an invaluable source book for working researchers.

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: 7212

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.

Logic, Logic, and Logic

Author: George Boolos,Richard C. Jeffrey

Publisher: Harvard University Press

ISBN: 9780674537675

Category: Philosophy

Page: 443

View: 8330

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.

Gottlob Frege: Basic Laws of Arithmetic

Author: Gottlob Frege

Publisher: Oxford University Press (UK)

ISBN: 0199281742

Category: Mathematics

Page: 33

View: 2781

This is the first complete English translation of Gottlob Frege's Grundgesetze der Arithmetik (originally published in two volumes, 1893 and 1903), with introduction and annotation. The importance of Frege's ideas within contemporary philosophy would be hard to exaggerate. He was, to allintents and purposes, the inventor of mathematical logic, and the influence exerted on modern philosophy of language and logic, and indeed on general epistemology, by the philosophical framework within which his technical contributions were conceived and developed has been so deep that he has astrong case to be regarded as the inventor of much of the agenda of modern analytical philosophy itself. Two of Frege's three principal books - the Begriffsschrift (1879) and Grundlagen der Arithmetik (1884) - have been available in English translation for many years, as have all the most important of his other, article-length writings. Grundgesetze was to have been the summit of Frege's life's work -a rigorous demonstration of how the fundamental laws of the classical pure mathematics of the natural and real numbers could be derived from principles which, in his view, were purely logical. A letter received from Bertrand Russell shortly before the publication of the second volume made Fregerealise that Axiom V of his system, governing identity for value-ranges, led to contradiction. But much of the main thrust of Frege's project can be salvaged. The continuing importance of the Grundgesetze lies not only in its bearing on issues in the foundations of mathematics but in its model of philosophical inquiry. Frege's ability to locate the essential questions, his integration of logical and philosophical analysis, and his rigorous approach tocriticism and argument in general are vividly in evidence in this, his most ambitious work.

Frege's Logic

Author: Danielle MACBETH,Danielle Macbeth

Publisher: Harvard University Press

ISBN: 0674040392

Category: Philosophy

Page: 218

View: 449

For many philosophers, modern philosophy begins in 1879 with the publication of Frege's Begriffsschrift, in which Frege presents the first truly modern logic in his symbolic language, Begriffsschrift, or concept-script. Macbeth's book, the first full-length study of this language, offers a highly original new reading of Frege's logic based directly on Frege's own two-dimensional notation and his various writings about logic.

Principia Mathematica

Author: Alfred North Whitehead,Bertrand Russell

Publisher: N.A

ISBN: N.A

Category: Logic, Symbolic and mathematical

Page: N.A

View: 8965

Philosophy of Arithmetic

Psychological and Logical Investigations with Supplementary Texts from 1887–1901

Author: Edmund Husserl

Publisher: Springer Science & Business Media

ISBN: 9401000603

Category: Mathematics

Page: 515

View: 5331

This volume is a window on a period of rich and illuminating philosophical activity that has been rendered generally inaccessible by the supposed "revolution" attributed to "Analytic Philosophy" so-called. Careful exposition and critique is given to every serious alternative account of number and number relations available at the time.

The Foundations of Arithmetic

A Logico-Mathematical Enquiry Into the Concept of Number

Author: Gottlob Frege,J. L. Austin

Publisher: Northwestern University Press

ISBN: 0810106051

Category: Mathematics

Page: 119

View: 6536

The Foundations of Arithmetic is undoubtedly the best introduction to Frege's thought; it is here that Frege expounds the central notions of his philosophy, subjecting the views of his predecessors and contemporaries to devastating analysis. The book represents the first philosophically sound discussion of the concept of number in Western civilization. It profoundly influenced developments in the philosophy of mathematics and in general ontology.

The Basic Laws of Arithmetic

Exposition of the System

Author: Gottlob Frege

Publisher: Univ of California Press

ISBN: 9780520047617

Category: Logic, Symbolic and mathematical

Page: 142

View: 4116

Logicism and Its Philosophical Legacy

Author: William Demopoulos

Publisher: Cambridge University Press

ISBN: 1107029805

Category: Mathematics

Page: 272

View: 8479

These essays apply the core conceptual innovation underlying Frege's theory of number to the general analysis of theoretical knowledge.

The Arché Papers on the Mathematics of Abstraction

Author: Roy T. Cook

Publisher: Springer Science & Business Media

ISBN: 9781402042652

Category: Mathematics

Page: 454

View: 7820

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.

Language, Thought, and Logic

Essays in Honour of Michael Dummett

Author: Richard G. Heck,Michael A. E. Dummett

Publisher: Oxford University Press on Demand

ISBN: 9780198239208

Category: Philosophy

Page: 308

View: 2835

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.

Logic, Logic, and Logic

Author: George Boolos,Richard C. Jeffrey

Publisher: Harvard University Press

ISBN: 9780674537675

Category: Philosophy

Page: 443

View: 8157

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.

Frege

Importance and Legacy

Author: Matthias Schirn

Publisher: De Gruyter

ISBN: 9783110150544

Category: Mathematics

Page: 466

View: 349