A CRITICAL INTRODUCTION TO THE PHILOSOPHY OF GOTTLOB FREGE This page intentionally left blank A Critical Introduction to the Philosophy of Gottlob Frege GUILLERMO E. ROSADO HADDOCK University of Puerto Rico at Rio Piedras, Puerto Rico © Guillermo E. Rosado Haddock 2006 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise without the prior permission of the publisher. Guillermo E. Rosado Haddock has asserted his moral right under the Copyright, Designs and Patents Act, 1988, to be identifi ed as the author of this work. Published by Ashgate Publishing Limited Ashgate Publishing Company Gower House Suite 420 Croft Road 101 Cherry Street Aldershot Burlington, VT 05401-4405 Hampshire GU11 3HR USA England Ashgate website: http://www.ashgate.com British Library Cataloguing in Publication Data Rosado Haddock, Guillermo E., 1945- A critical introduction to the philosophy of Gottlob Frege 1.Frege, Gottlob, 1848-1925 2.Frege, Gottlob, 1848-1925. Begriffsschrift I.Title 193 Library of Congress Cataloging-in-Publication Data Rosado Haddock, Guillermo E., 1945- A critical introduction to the philosophy of Gottlob Frege / Guillermo E. Rosado Haddock. p. cm. Includes bibliographical references and index. ISBN 0-7546-5471-0 (hardback : alk. paper) 1. Frege, Gottlob, 1848-1925. I. Title. B3245.F24R67 2006 193—dc22 2005031884 ISBN-13: 978-0-7546-5471-1 ISBN-10: 0-7546-5471-0 Printed and bound in Great Britain by Antony Rowe Ltd, Chippenham, Wiltshire. Contents Preface vii Chapter 1 Philosophy in Begriffsschrift § 1 An Epistemological Distinction 1 § 2 On the Nature of Arithmetical Statements 2 § 3 Concept-Script versus Natural Language 3 § 4 Judgeable Content 4 § 5 Conceptual Content 5 § 6 The Universal Predicate of all Statements 6 § 7 Abandonment of some Traditional Distinctions 7 § 8 Identity Statements in Begriffsschrift 9 § 9 Functions in Begriffsschrift 11 § 10 Axiom Systems 15 Chapter 2 Die Grundlagen der Arithmetik: First Part (§§1-45) § 1 The Need for a Clarification of the Nature of Mathematics 17 § 2 The Methodological Principles 19 § 3 The Classification of Statements 22 § 4 The Naturalist Conception of Mathematics 23 § 5 The Nature of Geometry 25 § 6 Numbers and Properties of Objects 28 § 7 Psychologism in Mathematics 30 § 8 Numbers and the Process of Abstraction 33 Chapter 3 Die Grundlagen der Arithmetik: Second Part (§§46-109) § 1 Predications about Concepts 35 § 2 Concept and Object in Die Grundlagen der Arithmetik 37 § 3 First Attempt at a Definition of Number 39 § 4 Second Attempt at a Definition of Number 42 § 5 Third Attempt at a Definition of Number 49 § 6 A Brief Sketch of the System 57 § 7 Frege’s Summary and his Assessment of Kant’s Views 62 Chapter 4 The Basic Distinctions I: Sense and Referent § 1 Sense and Referent 67 § 2 The Context Principle and Frege’s Semantics of Sense and Referent 72 vi A Critical Introduction to the Philosophy of Gottlob Frege § 3 Critical Remarks I 75 § 4 Critical Remarks II 76 § 5 Critical Remarks III 81 § 6 Critical Remarks IV 83 § 7 Identity Statements 86 Chapter 5 The Basic Distinctions II: Function and Object § 1 The Nature of Functions 91 § 2 The Predicative Nature of Concepts 94 § 3 Value Ranges 99 § 4 Some Difficulties 102 § 5 Value Ranges and Extensions 105 Chapter 6 Fundamental Philosophical Issues in Grundgesetze § 1 Introduction 109 § 2 Frege’s Criticism of Psychologism in Logic 110 § 3 Logicism in Grundgesetze der Arithmetik 114 § 4 Frege’s Platonism 115 § 5 Frege’s Criticism of Formalism in Mathematics 117 § 6 On the Nature of Definitions 119 § 7 Some Critical Remarks 123 Chapter 7 Some Remaining Philosophical Issues § 1 The Zermelo-Russell Paradox and Frege’s Reaction to Russell’s Letter 129 § 2 The Controversy on the Foundations of Geometry 133 § 3 Frege’s Abandonment of Logicism 137 Bibliography 143 Name Index 151 Subject Index 155 Preface The publication of a new book in English on Frege could seem at first sight to be completely unnecessary. As Ivor Grattan-Guinness has put it on p. 177 of his recent book The Search for Mathematical Roots: 1870-1940, there is a massive Frege industry in English, though of a refurnished Frege, which Grattan-Guinness calls ‘Frege’ in order to distinguish him from the real Frege. In fact, a series of writings, mostly in English, have flourished in the last two decades, presenting Frege as a sort of epistemologist in the Kantian or, at least, neo-Kantian tradition. A similar massive bulk of writings, also mostly in English, has followed a programme of trying to derive as much arithmetic as possible from a weakening of Frege’s logical system. The last group of authors seems to conceive Frege primarily as a philosopher of mathematics – and they are right in doing so – but have adhered to a programme that not only has attained only modest goals, but that even if successful, would probably still be unsatisfactory – as argued at the very end of this book. A third trend in Fregean scholarship, and probably the oldest, is the view made famous by Michael Dummett’s first book on Frege, the very influential Frege: Philosophy of Language of 1973. According to that book, Frege was primarily a philosopher of language – though he published only one paper, namely ‘Über Sinn und Bedeutung’, that can be considered a writing on the philosophy of language. Moreover, in that book Dummett presents Frege as a sort of ‘philosophical Adam’, with almost no roots in the philosophical tradition. Dummett’s contention – which seems to have been already abandoned by its distinguished proponent – has been rightly criticized by many Fregean scholars and is nowadays upheld by almost nobody. However, it has produced an exaggerated reaction in those who try to see in Frege a sort of neo-Kantian epistemologist. The orthodoxy of the two most active schools of Fregeanism have originated another sort of extreme position among some important scholars, namely, that of not doing justice to the great value of Frege’s writings – I mean Frege, not ‘Frege’ – both for the development of contemporary logic and for philosophy. That perfectly comprehensible reaction can be seen in the above mentioned history of mathematical logic of Grattan-Guinness, as well as in William Tait’s paper ‘Frege versus Dedekind and Cantor’ and even in some writings of my friend – and coauthor of our Husserl or Frege?: Meaning, Objectivity and Mathematics – Claire Ortiz Hill, for example, in her book Rethinking Identity and Metaphysics. Although I am sympathetic to some of their criticisms of Frege, I think that the balance between criticism and recognition of the importance of Frege’s views gets lost. As a Fregean scholar who is not a Fregean, I hope that I have attained the proper balance in the present writing. This book has a long history, and even a predecessor. After teaching for the first time a graduate course on Frege some twenty-five years ago, and writing viii A Critical Introduction to the Philosophy of Gottlob Frege some reviews of books on Frege, I wrote a small book on Frege’s philosophy during a sabbatical in 1982-83. I committed two decisive errors with respect to that book: firstly, I wrote it in my mother tongue, Spanish, and published it privately in 1985. Due to my own carelessness, the book had many misprints and some imprecise references. Moreover, due to the fact that it was a private publication on such an exotic theme – at least for culturally underdeveloped Puerto Rico – the book had no distribution, and I was limited to sending copies as a gift to different scholars. Nonetheless, the book received three pleasant reviews by Mauricio Beuchot, by Lourdes Valdivia, and by my friend Matthias Schirn. Almost a decade ago, while participating in a congress in Brazil, my friend, the late Michael Wrigley, urged me to publish an English version of my old Frege book. I told him that I would have not only to translate the book, but also to revise and update it. Due to my commitment to other projects, the project remained unfulfilled for almost a decade. However, I now teach the graduate course on Frege’s philosophy more or less every four to five semesters, and my views on Frege’s philosophy have been maturing in the meantime. Thus, especially in my last two Frege courses I have had the opportunity to develop and polish my assessment of Frege’s philosophy. The success of my joint book with Claire Ortiz Hill mentioned above – published in 2000 and reprinted in a paperback edition in 2003 – has clearly also served as an encouragement. Although in comparison to my old Frege book in Spanish, the expository core of this book has not suffered great changes, my assessment of Frege’s philosophy having attained its maturation, the present book is not only somewhat longer – though still a short one – but also and foremost qualitatively superior to its ancestor, and not simply a sort of second edition. Moreover, though some of the Fregean issues discussed in the present book also have been considered in some of my papers included in my joint book with Claire Ortiz Hill, my assessment of Frege here is of his whole philosophy, not of particular issues. In fact, the present book is my most mature and complete assessment of Frege’s philosophy as a whole. The book is divided into seven chapters. The first chapter deals with Frege’s philosophical views in his 1879 Begriffsschrift. A good understanding of the philosophical views of the young Frege – not only his views on identity statements, but also and especially his distinction between the notions of conceptual content and judgeable content – is essential for an adequate assessment of some later issues. Chapters 2 and 3 are concerned with Frege’s philosophical masterpiece, Die Grundlagen der Arithmetik, and could very well serve as a commentary to that work. Nonetheless, they are not simply a commentary, but also an assessment of some Fregean views and even of a very influential recent criticism of Frege’s views by Paul Benacerraf. The remaining chapters are concerned with Frege’s post 1890 philosophical views. Chapters 4 and 5 are concerned mostly with Frege’s two fundamental distinctions between sense and referent – made independently by Husserl as early as 1890 and present in the latter’s review of Ernst Schröder’s Vorlesungen über die Algebra der Logik I of 1891 – and that between concept and object, which permeates the whole Fregean philosophy from Begriffsschrift onwards. Chapter 6 is concerned with some Preface ix important philosophical issues of Grundgesetze der Arithmetik, whereas the last chapter is concerned with three related important issues, namely, the Zermelo- Russell Paradox (as it should be called), Frege’s views on Hilbert’s axiomatization of geometry, and the old Frege’s attempt at a geometrical foundation of arithmetic. In view of the origin of the work in a graduate course, some repetitions, digressions and cross-references have not been completely avoided. That is the case, e.g., of the issue of the presence or disappearance of the Context Principle from Frege’s mature philosophy, as well as of his distinction between concept and object, which are considered already in the first part (Chapters 1-3), but their definitive treatment occurs in Chapters 4 and 5, respectively. In the present book, I have always referred to Frege’s works in their original language: German. Although in the references some English translations of Frege’s works are included, I find that in good scholarship there is no substitute for the works in their original language. Notwithstanding the fact that my mother tongue is neither German nor English, I have translated all the passages quoted from Frege’s writings, inserting the German original in the corresponding footnotes. Incidentally, to avoid interrupting the flow of the exposition, I have included quotations only when I am trying to make a point in the interpretation of Frege’s writings that could seem controversial to orthodox Fregean scholars. In the bulk of the book references in the footnotes to the pertinent Fregean writings are more than enough. To be a Fregean scholar and, in general, a scholar in very hard philosophy in a country so culturally deprived as Puerto Rico is not an easy matter. It means not only partial isolation from similar scholars around the world, but also total isolation from other academicians at your university and in your country. Thus, my graduate courses on Frege’s philosophy, on Husserl’s Logical Investigations, on the philosophy of mathematics or Logical Empiricism, on recent semantic theories of truth or classical model theory usually have had small audiences. Probably the greatest enduring joy that I have had in three decades of teaching is that there is always a very small but non-empty set of very capable students with philosophical acumen, interested in learning hard philosophy and its logical tools. Three of them – who wrote their MA Theses on Frege under my supervision – are mentioned in the references. Two of them, Dr. Wanda Torres Gregory and Mr. Pierre Baumann, have been extremely helpful in reading a preliminary version of the whole book and correcting my English. I have followed their advice most of the time, but not always, opting sometimes for a third option or, in a few cases, obstinately grasping to my original formulation. Thus, all the remaining errors in the book are my sole responsibility. My wife, Dr. Tinna Stoyanova and our friend, Dr. Joel Donato, Director of the LABCAD of the University of Puerto Rico at Río Piedras, have assisted me with the more technical aspects of the computer, by making the manuscript camera-ready. I am most grateful to all of them. Finally, I want to thank Prof. Jan Szrednicki for his encouragement and support during the final stages of the preparation of the book, as well as Paul Coulam, Sarah Charters and the editorial staff of Ashgate for their patience, and very especially Anthea Lockley