Frege. on. Frege's. Theorem. Dummett's long-awaited Frege: Philosophy of
Mathematics was published in the spring of 1991. I remember seeing it in a
bookstore in Amherst, Massachusetts, when I was there for a conference, and
buying it ...

Author: Richard G. Heck

Publisher: OUP Oxford

ISBN: 9780191619656

Category: Philosophy

Page: 336

View: 769

Frege's Theorem collects eleven essays by Richard G Heck, Jr, one of the world's leading authorities on Frege's philosophy. The Theorem is the central contribution of Gottlob Frege's formal work on arithmetic. It tells us that the axioms of arithmetic can be derived, purely logically, from a single principle: the number of these things is the same as the number of those things just in case these can be matched up one-to-one with those. But that principle seems so utterly fundamental to thought about number that it might almost count as a definition of number. If so, Frege's Theorem shows that arithmetic follows, purely logically, from a near definition. As Crispin Wright was the first to make clear, that means that Frege's logicism, long thought dead, might yet be viable. Heck probes the philosophical significance of the Theorem, using it to launch and then guide a wide-ranging exploration of historical, philosophical, and technical issues in the philosophy of mathematics and logic, and of their connections with metaphysics, epistemology, the philosophy of language and mind, and even developmental psychology. The book begins with an overview that introduces the Theorem and the issues surrounding it, and explores how the essays that follow contribute to our understanding of those issues. There are also new postscripts to five of the essays, which discuss changes of mind, respond to published criticisms, and advance the discussion yet further.

The proof of Theorem 154 is relatively easy. Its proof relies only upon the fact that
nothing ancestrally precedes zero in the Predseries, Frege's Theorem 126: –7(
Pred)(ar,0). This lemma follows immediately from Frege's Theorem 124:” JF(Q)(a
...

Author: Michael A. E. Dummett

Publisher: Harvard University Press

ISBN: 0674319427

Category: Mathematics

Page: 464

View: 403

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.

Again , the second conjunct follows from Frege's Theorem 126. What needs to be
established is therefore the following , which is Frege's Theorem 145 , a , from
which the third conjunct will follow by contraposition and generalization : F ( Pred
) ...

Author: Michael Beaney

Publisher: Taylor & Francis

ISBN: 0415306043

Category:

Page: 407

View: 149

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.

This follows from Frege's Theorem 243, mentioned earlier: (243) Func{Q)/\ Q*-aa.
... We have made use of none of the theorems Frege proves in the course of his
proof of Theorem 321, other than certain very general facts about the ancestral ...

Author: Richard G. Heck

Publisher: Oxford University Press on Demand

ISBN: 9780199233700

Category: Mathematics

Page: 296

View: 889

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.

This volume is the first collective study of a foundational text in modern philosophy and logic, Gottlob Frege's Basic Laws of Arithmetic which appeared in two volumes in 1893 and 1903.

Author: Philip A. Ebert

Publisher:

ISBN: 9780198712084

Category:

Page: 672

View: 525

This volume is the first collective study of a foundational text in modern philosophy and logic, Gottlob Frege's Basic Laws of Arithmetic which appeared in two volumes in 1893 and 1903. Twenty-two Frege scholars discuss a wide range of philosophical and logical topics arising from Basic Lawsof Arithmetic, and demonstrate the technical and philosophical richness of the work. Their original contributions make vivid the importance of this magnum opus not just for Frege scholars but for the study of the history of logic, mathematics, and philosophy.

8 On the Philosophical Significance of Frege's Theorem CRISPIN WRIGHT My
interest in Frege's philosophy of mathematics was kindled by studying
Grundlagen as a visiting graduate student in Oxford under Michael Dummett's
supervision in ...

Author: Michael A. E. Dummett

Publisher: Oxford University Press on Demand

ISBN: 0198239203

Category: Philosophy

Page: 308

View: 364

A distinguished international group of philosophers contribute new essays on central issues in philosophy of language and logic, in honour of the eminent Oxford philosopher Michael Dummett. They explore such topics as meaning, truth, content, time, and the foundations of mathematics; the dominant theme is the relation between language and thought.

The volume features much new material: introduction, postscript, bibliographies, and a new essay on a key problem.

Author: Bob Hale

Publisher: Oxford University Press

ISBN: 9780199266326

Category: Philosophy

Page: 455

View: 760

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.

... of Wright who had also independently rediscovered the Geach model, George
Boolos. He introduced the terminology still used in discussing these matters: "
Hume's principle" or "HP" (which Wright had called "N="), "Frege's theorem," "
Frege ...

Author: John P. Burgess

Publisher: Princeton University Press

ISBN: 0691122318

Category: Mathematics

Page: 257

View: 565

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.

But neither the explicit definition nor Basic Law V is used essentially in the proof
of any other arithmetical theorem; these other theorems are proven using only
second-order logic and Hume's Principle. Thus, Frege in fact proves that axioms
for ...

Author: Edward Craig

Publisher: Routledge

ISBN: 9781134344086

Category: Philosophy

Page: 1078

View: 582

The Shorter REP presents the very best of the acclaimed ten volume Routledge Encyclopedia of Philosophy in a single volume. It makes a selection of the most important entries available for the first time and covers all you need to know about philosophy, from Aristotle to Wittgenstein and animals and ethics to scientific method. Comprising over 900 entries and covering the major philosophers and philosophical topics, The Shorter REP includes the following special features: Unrivalled coverage of major philosophers, themes, movements and periods making the volume indispensable for any student or general reader Fully cross-referenced Revised versions of many of the most important entries, including fresh suggestions for further reading Over twenty brand new entries on important new topics such as Cloning and Sustainability entries by many leading philosophers such as Bernard Williams, Martha Nussbaum, Richard Rorty, Onora O'Neill, T.M. Scanlon and Anthony Appiah Striking new text design to help locate key entries quickly and easily An outstanding guide to all things philosophical, The Shorter Routledge Encyclopedia of Philosophy provides an unrivalled introduction to the subject for students and general readers alike.

To make this clearer, Frege replaces the familiar terms with terms that look like
variables. ... 391) This is a pseudo-theorem—in particular, it is Frege's '
pseudoproposition' restatement of: Two planes have either no point or a straight
line in ...

Author: Joan Weiner

Publisher: Open Court

ISBN: 9780812697520

Category: Philosophy

Page: 176

View: 353

What is the number one? How can we be sure that 2+2=4? These apparently ssimple questions have perplexed philosophers for thousands of years, but discussion of them was transformed by the German philosopher Gottlob Frege (1848-1925). Frege (pronounced Fray-guh)believed that arithmetic and all mathematics are derived from logic, and to prove this he developed a completely new approach to logic and numbers. Joan Weiner presents a very clear outline of Frege's life and ideas, showing how his thinking evolved through successive books and articles.

The derivability of Frege's Theorem is first explicitly asserted in Charles Parsons [
1964]; see remark at p. 194. My own 'rediscovery' of the theorem was
independent. I do not know what form of proof Parsons had in mind, but the
reconstruction ...

Author: Roy T. Cook

Publisher: Springer Science & Business Media

ISBN: 1402042655

Category: Mathematics

Page: 454

View: 830

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.

18 Frege's Theorem and the Peano Postulates Two thoughts about the concept
of number are incompatible: that any zero or more things have a (cardinal)
number, and that any zero or more things have a number (if and) only if they are
the ...

Author: George Boolos

Publisher: Harvard University Press

ISBN: 067453767X

Category: Philosophy

Page: 443

View: 578

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.

Introduction IM my article “ Remarks on Sense and Reference in Frege and
Husserl ” ] ( chapter 2 of the present book ) , I ... the best known disciple of Frege '
} ; and ( iv ) { " the Ultrafilter Theorem ' , ' Tarski's theorem on maximal dual ideals '
} .

Author: Claire Ortiz Hill

Publisher: Open Court Publishing

ISBN: 0812694171

Category: Philosophy

Page: 315

View: 223

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.

6.1 Urelements In Frege's Grundgesetze, urelements (objects which are not
Wertverläufe)e are allowed. More exactly, Frege does not set out axioms that
assure the provability of theorems such as the following: uv u = v z(z^u) = (z^v)
This was ...

Author: Gregory Landini

Publisher: Springer

ISBN: 9780230360150

Category: Philosophy

Page: 194

View: 496

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.

Frege spricht damit ein Kommutativtheorem aus , das wir folgendermaßen
formulieren können : ( K ) Für alle Sätze A , B gilt : der Gedanke , daß ( B und A )
ist identisch mit dem Gedanken , daß ( A und B ) . Dieses Theorem ist nach Freges ...

And Frege ' s claim here , that it must be possible to use a definition in a proof , is
very different from the fruitfulness ... a mere definition , but must conceal
something which would have either to be proved as a theorem or accepted as an
axiom .

Frege's theorem demonstrates that it is possible toderivethe Dedekind–Peano
axioms– including the axiomwhichproves thatevery natural numberhasa
successor –from N= and a suitablesystem of secondorder logic.5 George Boolos
(1998, pp ...

Author: G. Duke

Publisher: Springer

ISBN: 9780230378438

Category: Philosophy

Page: 212

View: 389

This historically-informed critical assessment of Dummett's account of abstract objects, examines in detail some of the Fregean presuppositions of Dummett's account whilst also engaging with phenomenological approaches and recent work on the problem of abstract entities.

There canbeno question that thisseries of developments, which might be
characterized as the rediscovery of Frege's theorem'— the theorem that the
second-order theory consisting of Hume's principle implies the infinity of the
natural numbers ...

... problem is solvable; rather, what he has in mind is. 41 s. feferman, Are There
Absolutely Unsolvable Problems? Gödel's Dichotomy, p. 149. 42 ibid., p. 148. 43
Kurt Gödel, Some basic theorems gödel's incompleteness theorem 211.

Author: Aleksy Molczanow

Publisher: BRILL

ISBN: 9789004222694

Category: Philosophy

Page: 231

View: 311

Drawing on the original conception of Kant’s synthetic a priori and the relevant related developments in philosophy, this book presents a reconstruction of the intellectual history of the conception of quantity and offers an entirely novel transcendental-metaphysical account of quantification.

The sentences belonging to the set are the theorems of the theory, and theorems
which do not belong to the set in virtue ... subject to limitations of computational
and memory capacities, concerning what any theorem says, that things are thus ...

Author: Mark Sainsbury

Publisher: Routledge

ISBN: 9781134483952

Category: Philosophy

Page: 256

View: 456

Frege is now regarded as one of the world's greatest philosophers, and the founder of modern logic. Mark Sainsbury argues that we must depart considerably from Frege's views if we are to work towards an adequate conception of natural language. This is an outstanding contribution to philosophy of language and logic and will be invaluable to all those interested in Frege and the philosophy of language.