Table Of Content

Translated and edited by Philip A. Ebert & Marcus Rossberg ISBN 978-0-19-877730-4 OXFORD with · Wright 1111111 11 UNIVERSITY PRESS 9 780198 777304 www.oup.com BASIC LAWS OF ARITHMETIC The German philosopher and mathematician Gottlob Frege (1848-1925) was the father of analytic philosophy and to all intents and purposes the inventor of modern logic. Basic Laws of Arithmetic, originally published in German in two volumes (1893, 1903), is Frege's magnum opus. It was to be the pinnacle of Frege's life work. It represents the final stage of his logicist project-idea that arithmetic and analysis are reducible to logic-and contains his mature philosophy of mathematics and logic. The aim of Basic Laws of Arithmetic is to demonstrate the logical nature of mathematical the orems by providing gapless proofs in Frege's formal system using only basic laws of logic, logical inference, and explicit definitions. The work contains a philosophical foreword, an introduction to Frege's logic, a derivation of arithmetic from this logic, a critique of contemporary approaches to the real numbers, and the beginnings of a logicist treatment of real analysis. A letter received from Bertrand Russell shortly before the publication of the second volume made Frege realise that his Basic Law V leads into inconsistency. Frege discusses a revision to Basic Law V in an afterword to the second volume. The continuing importance of Basic Laws of Arithmetic lies not only in its bearing on issues in the foundations of mathematics and logic 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 to criticism and argument in general are vividly in evidence in this, his most ambitious work. Philip A. Ebert and Marcus Rossberg present the first full English translation of both volumes of Frege's major work preserving the original formalism and pagina tion. The edition contains a foreword by Crispin Wright and an extensive appendix providing an introduction to Frege's formal system by Roy T. Cook. Basic Laws of Arithmetic. Derived using concept-script BY Gottlob Frege VOLUMES I & II TRANSLATED AND EDITED BY Philip A. Ebert & Marcus Rossberg WITH Crispin Wright AND THE ADVICE OF Michael Beaney Robert May Roy T. Cook Eva Picardi Gottfried Gabriel William Stirton Michael Hallett Christian Thiel Richard G. Heck, Jr. Kai F. Wehmeier APPENDIX BY Roy T. Cook OXFORD UNIVERSITY PRESS OXFORD UNIVERSITY PRESS Great Clarendon Street, Oxford, OX2 6DP, United Kingdom Oxford University Press is a department of the University of Oxford. It furthers the University's objective of excellence in research, scholarship, Prefa ce to the paperback edition and education by publishing worldwide. Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries © Philip A. Ebert, Marcus Rossberg, Crispin Wright, and Roy T. Cook 2013 We corrected minor typographical errors in this paperback edition. We would like to The moral rights of the authors have been asserted thank Fabio Lampert, Shay Logan, and William Stirton for drawing our attention to First published 2013 some of them, and, in particular, Christian Thiel, who kindly subjected large parts First published in paperback 2016 of the volume to his scrutiny. Regarding Frege's Afterword, volume II, page 256, we All rights reserved. No part of this publication may be reproduced, stored in followed Peter Geach and Max Black's suggestion and corrected what appears to be a a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted slip by Frege, as noted in our translators' note b on page 273-both in the hardcover by law, by licence or under terms agreed with the appropriate reprographics edition and here. An addition to this footnote now refers to a publication by Christian rights organization. Enquiries concerning reproduction outside the scope of the Thiel, who argues for an alternative reading of the relevant passage according to which above should be sent to the Rights Department, Oxford University Press, at the address above Frege's gloss would not be incorrect. You must not circulate this work in any other form and you must impose this same condition on any acquirer Philip Ebert, Stirling, Scotland Published in the United States of America by Oxford University Press Marcus R.ossberg, Storrs, Conn., USA March 2016 198 Madison Avenue, New York, NY 10016, United States of America British Library Cataloguing in Publication Data Data available Library of Congress Cataloging in Publication Data Data available ISBN 978-0-19-928174-9 (Hbk.) ISBN 978-0-19-877730-4 (Pbk.) Links to third party websites are provided by Oxford in good faith and for information only. Oxford disclaims any responsibility for the materials contained in any third party website referenced in this work. Contents Foreword Crispin Wright vii Translators' Introduction Philip A. Ebert and Marcus Rossberg xiii Basic Laws of Arithmetic Gottlob Frege First volume Foreword v Table of contents XX VII Introduction 1 Part I: Exposition of the concept-script 5 Part II: Proofs of the basic laws of cardinal number 70 Appendices 239 8econd volume Table of contents v Part II: Proofs of the basic laws of cardinal number (continued) 1 Part III: The real numb,ers 69 Appendices 244 Afterword 253 Translators' Notes 269 Corrections 275 Bibliography 281 Appendix: How to Read Grundgesetze Roy T. Cook A-1 Index I-1 Foreword Crispin Wright The importance of Frege's ideas within contemporary English-speaking analytical philosophy would be hard to exaggerate. Yet while two of Frege's three principal books-Begriffsschrift of 1879 and Die Grundlagen der Arithmetik of 1884-have been available in good quality English translation for many years, his most major work, Grundgesetze der Arithmetik, published in two volumes in 1893 and 1903, has never previously been completely translated into English. Part of the explanation of the omission has no doubt been the long-standing impression that Frege's project in that book had completely failed. Grundgesetze was to have been the climax of his life's researches-a rigorous demonstration within the system of concept-script how the fundamental laws of the classical pure mathematics of the natural and real numbers could be derived from principles that were purely logical. Bertrand Russell's famous letter, received shortly before the publication of the second volume, disclosed that Basic Law V of Frege's system, governing identity for value-ranges, led to contradiction. A hastily proposed revision was later independently shown by a number of writers to be satisfiable only within singleton domains but Frege would himself have rapidly realised that it would not underwrite the proofs his project required, including crucially, of course, the derivation of the infinity of the series of natural numbers. Within a short space of time, Frege had apparently convinced himself that the problem was not one of infelicitous formulation but undermined his whole approach. Volume III of Grundgesetze, which was to contain the formal derivations of the theories of the real and (presumably) the complex numbers, never appeared. Frege's own despairing assessment of his achievement prevailed for the first four fifths of the twentieth century. Writing in 1973, no less an authority than Sir Michael Dummett remarked that Frege's philosophy of mathematics remained of "merely historical interest". Yet quite soon afterwards that assessment began to seem com pletely wrong. A sea change began with developments effected in work by the late George Boolos and, independently, by the present writer in the 1980s. The broad upshot of these researches was that the replacement of Basic Law V in the system of Grundgesetze by what came to be known as Hume's Principle, associating bijectable concepts with the same cardinal number, provided for, first, a consistent theory in which classical number theory could be developed and, second, a theory of consid erable philosophical interest. Principles of the form shared by Basic Law V and Hume's Principle-principles licensing the transition from an equivalence relation on entities of a certain kind to identity statements configuring terms standing for objects of viii Crispin Wright Foreword ix a new kind (the abstracts )-came to be known as abstraction principles. These have However it occurred to me, about mid-way through the five years of the Logical and provided the central focus of study in a neo--Fregean (sometimes called neo-logicist, or Metaphysical Foundations of Classical Mathematics project, that a perhaps unique "abstractionist") programme for the foundations for classical mathematics aiming at a opportunity had now ripened to accomplish the task by teamwork. In Philip Ebert broad subservience to Frege's own original philosophical objective: a demonstration and Marcus Rossberg, both then approaching completion of doctoral theses written in that arithmetic and analysis flow from logic and definitions-with abstraction principles, association with the AHRC project, the project team included two native German so it is claimed, playing the definitional, or quasi-definitional role. speakers who were fluent in English, who had trained intensively in Frege's philosophy The last 30 years have seen considerable philosophical and technical advances in over a period of years and were completely familiar with his philosophy of mathematics this programme, with much of the work being done by the core group and wider in particular. My own experience, before I took up philosophy, in the translation of network of scholars associated with the AHRC-funded project on the Logical and Latin and Greek encouraged me to think that I could help with the accurate rendition Metaphysical Foundations of Classical Mathematics, which I had the good fortune to of Frege's prose into decent philosophical English and with the development and lead at the St Andrews philosophical research centre, Arche, from 2000 to 2005. On maintenance of a suitable authorial "voice". Furthermore I knew we could count on the technical side, it emerged that (presumptively) consistent abstractionist systems the support of a group who believed absolutely in the importance of making Frege's can be developed for theories which are at least interpretable as real and complex legacy more fully available to the mostly non-German-speaking community of analytical analysis, but that the development of a reasonably strong pure abstractionist set philosophers worldwide with interests in the field. A small foreseeable surplus in the theory-strong enough, for instance, to recover the sets in the standard iterative AHRC funding suggested that it might be possible to dedicate a couple of half-time hierarchy up to but excluding the first inaccessible cardinal-seems to present serious research fellowships to the translation over the final year of the award---surely ample obstacles of principle. Philosophically, much interesting work has been done on a time to accomplish at least a complete basic translation ... wide range of central questions, including the claim of at least some abstraction It remained to persuade Philip and Marcus to take the project on. In fact, they principles to constitute good implicit definitions of the concepts they introduce; the needed no persuading. In the context, then as now, of an extrerp.ely bearish job nature of their distinction from "bad" cases where no such claim can be upheld; the market in philosophy and the pressing need to complete their doctorates and get connection in general between implicit definition and basic a priori knowledge; the some publications out if they were to compete effectively in it, their decision to do so ramifications of the issues here for the inferentialist conception of the meanings of the impressed and continues to impress me as a remarkably courageous one. logical constants developed in work of Gentzen and Prawitz; and, most generally, the So we launched the project in April 2003 using the tried and tested Arche seminar metaphysics of abstract entities that belongs with abstractionism, and the resources it method, a small group receiving weekly small translation bulletins-no more than two offers to address traditional empiricist (nominalist) scruples. All this has provided to three pages of text-from Marcus and Philip and then working over them together, a vibrant research context, provoking major contributions from some of the world's struggling with alternative interpretations, debating suggestions and agreeing revisions. most distinguished contemporary philosophers of mathematics, and with many of the The aim at this stage was merely to produce a passably accurate text in passably best contributions-as so often in philosophy-being made, of course, from a critical intelligible English. Initially our practice in the seminars was simply to come up with a standpoint. list of suggestions, leaving the translators to collate them later and work them into the By the early years of the millennium, a point had thus been reached when Frege's translation text. This had the drawbacks both that progress was excruciatingly slow approach to foundations, and modern descendants of his ideas, had become a central and that we had little opportunity to get a sense of the collective effect of the changes preoccupation of much of the best work on the philosophy of mathematics and on being proposed or check for consistency of renditions of particular phrases. But things associated metaphysical and epistemological issues. There was now, for the first improved exponentially when Walter Pedriali assumed the role of court stenographer time, a compelling argument for an authoritative complete English translation of and armed with personal laser pointers, whose use somehow occasioned much hilarity, Grundgesetze based purely on grounds of contemporary scholarly interests and need. we took to editing the text of the translation online on the Arche seminar room Moreover with the advent of the lb.TEX typesetting system, the formidable technical smartboard. Even so, it rapidly became clear that there was no hope whatever of barriers confronting any attempt to republish a book containing large tracts of Frege's completing anything but a fragment of the task within the nine months or so for which idiosyncratic logical notation had become surmountable. we had AHRC funding. In February 2005 we therefore decided to submit an outline Still, there remained another, potentially insuperable barrier: the sheer amount of research grant application for a full two-years of further sponsorship, including fully labour, and sheer range of linguistic, philosophical and technical expertise, necessary to funded 24-month postdoctoral fellowships for the translators, to the Leverhulme Trust. do justice to more than 500 pages of intricate formal proofs and conceptually challenging We subsequently received the go-ahead to submit a full application and learned of its nineteenth century German prose. There are vanishingly few single scholars in the success in December 2005. At that time, as now, the Trust's policy was to reward world with the range of competences demanded by such a project and, among that applications that prioritised the research vision of a principal investigator. Since I small number, still fewer who would be willing and able to take it on, in a scholarly was not offering to lead any research, as usually conceived, and the research vision milieu on both sides of the Atlantic that places a large and ever-increasing premium involved was entirely that of Frege, we had cause to be extremely grateful to the Trust on personal research publication on top of the normal pressures exerted by teaching for the flexibility it showed in responding positively to our application. It was the and administration on a typical professional scholar's working time. resulting crucial two-year period of Leverhulme support, providing the opportunity to x Crispin Wright Foreword xi devote several days a week to the translation, that gave us the opportunity to get real the millennium and which has provided a model for a whole new wave of similarly momentum into the project and hone the skills it required. inspired research centres across the UK and Europe. The Grundgesetze translation The backing of the Leverhulme Trust was one of two crucial factors that have project was conceived and entered upon in the spirit of that paradigm, and conducted enabled us to bring this work to completion. The other was the unstinting support of and completed with the energy, dedication, and good humour that it characteristically our group of international expert Grundgesetze enthusiasts who have given us continual fosters. I think the result shines like a beacon as evidence of what can be accomplished aid, comfort, and advice throughout the process and to whom a great proportion by collaborative processes in philosophical research and scholarship, and will be a last of the credit is due if what we now publish measures up to the scholarly standards ing resource to generations of students of Frege, and of the foundations of mathematics. which Frege's flawed masterpiece deserves. Many of this group attended at three delightful Grundgesetze workshops, at St. Andrews in 2006, when we worked over the entire first draft translation of volume I, and again in 2008, where the entire New York University and draft translation of volume II was discussed, and then with the support of the NYU The Northern Institute of Philosophy, Aberdeen December 2012 Philosophy Department, in New York in the spring of 2010, where selected passages from both volumes were subjected to yet another collaborative overhaul. I would like to express my thanks here to all who attended these workshops or sent us written comments at various stages of the translation: to Michael Beaney, Roy Cook, Gottfried Gabriel, Michael Hallett, Richard Heck, Michael Kremer, Wolfgang Kiinne, Robert May, Eva Picardi, Peter Simons, William Stirton, Peter Sullivan, Jamie Tappenden, Christian Thiel, and Kai Wehmeier. A collaboration on this scale takes a lot of logistical management and I would also like to record my gratitude to Sharon Coull for administrating the liaison with these colleagues and working so hard and ably to ensure that the workshops ran smoothly and enjoyably. Three members of the wider team played especially important roles. William Stirton regularly travelled from Edinburgh to attend project seminars in the early days and subsequently provided an immense amount of help with the setting of the formulas and proof-reading the final files. Roy Cook offered expert advice on Frege's logic and proofs throughout the process, and has contributed the invaluable guide to Frege's logic and symbolism included here as an Appendix. Richard Heck enlisted his unsurpassed scholarship and philosophical understanding of Frege's writings to the aid of the project from the start, providing both intellectual and moral support in equal munificent measure. At the end of the Leverhulme funding, the core group broke up, Marcus and Philip leaving Arche for posts at the Universities of Connecticut and Stirling, and I to work at New York University and to the launch of the Northern Institute of Philosophy at Aberdeen. The translation continued by Skype, with only occasional face-to-face meetings when the three of us were on the same continent. This was the period of "rants and frustrations" and "ruined weekends" that Marcus and Philip refer to below. Now, however, at last, it is done, in the best Odyssean tradition almost ten years after we started. Two things are worth emphasising about how it has been done. First, the work has been, despite the uncertainties and delays, extraordinarily enjoyable. I have never previously, and don't expect again, to experience such a warmth of camaraderie around an academic project, and it has been a delight to work closely, over an extended period, with so many talented scholars whom I now number among my friends. Second, I believe that the whole process stands as a tribute to the mode of interactive, collaborative philosophical working which the long-term members of the Logical and Metaphysical Foundations of Classical Mathematics project-Marcus, Philip, Roy Cook, Nikolaj Jang Pedersen, Graham Priest, Agustin Rayo, Stewart Shapiro, Robbie Williams, and Elia Zardini-pioneered at Arche in the early years of Translators' Introduction The importance of Grundgesetze der Arithmetik Gottlob Frege's Grundgesetze der Arithmetik was originally published in two volumes: the first in 1893, the second in 1903. It was to be the pinnacle of Frege's life's work. The aim was to demonstrate that arithmetic and analysis are reducible to logic-a position later called "Logicism". Frege's project began with the publication of Begriffsschrift in 1879, which contains the first version of his logical system, also named 'Begriffsschrift' (concept-script), the mature formulation of which Frege would prese'nt in Grundgesetze. His work was groundbreaking in many ways: it contained the first occurrence of the quantifier in formal logic, including a treatment of second-order quantification; the first formal treatment of multiple and embedded generality; and it also offered the first formulation of a logical system containing relations. Begriffsschrift is widely acknowledged to constitute one of the greatest advances in logic since Aristotle. As W.V. Quine put it: "Logic is an old subject, and since 1879 it has been a great one" (Quine 1950, p. vii). Begriffsschrift was followed by Die Grundlagen der Arithmetik in 1884. Having previously completed a manuscript of a more formal treatment of Logicism around 1882-a lost ancestor of Grundgesetze-Frege developed a philosophical foundation for his position in Grundlagen. It was likely on the advice of Carl Stumpf1 that Frege decided to write a less technical and more accessible introduction to Logicism. Grundlagen is regarded by many as a philosophical masterpiece and by some, most notably Sir Michael Dummett, as "the most brilliant piece of philosophical writing of its length ever penned" (Auxier and Hahn, 2007, p. 9). The first part contains devastating criticisms of well-known approaches to the philosophy of arithmetic, including those of John Stuart Mill and Immanuel Kant, as well as those of Ernst Schroder, Hermann Hankel, and others. In the second part, Frege develops his Logicism. In Grundlagen, Frege eschews the use of a formal system, merely offering non-formal sketches of how a version of the Peano-Dedekind axioms for arithmetic can be derived from pure logic. He closes Grundlagen by suggesting that the Logicist approach is not restricted to arithmetic: it also has the potential to account for higher mathematics, in particular, 1 Stumpf was a German philosopher and psychologist, a student of Franz Brentano and Hermann Lotze, who at the time was professor of philosophy in Prague. Ill a letter by Stumpf (September 9, 1882)-in response to a letter that Frege wrote to him (but, it seems, wrongly filed as a letter to Anton Marty who was Stumpf's colleague in Prague)-Frege is encouraged to spell out his Logicist programme in a more accessible manner, without recourse to his formal language. See Gabriel et al. (1976), p. 257, and the editor's note on p. 162. xiv Basic Laws of Arithmetic: Translators' Introduction The importance of Grundgesetze xv real and complex analysis. At the end of Grundlagen, Frege was thus left with the However, it was not to be. As is well known, in 1902 Frege received a letter monumental task properly to establish Logicism: he needed to identify a small number from Bertrand Russell, when the second volume was already in press. In this letter, of basic laws of logic; offer a small number of indisputably sound rules of inference; Russell proposes his famous antinomy, which made Frege realise that his Basic Law and, finally, provide gapless proofs in his formal system of the basic laws of arithmetic, V, governing the identity of value-ranges, leads into inconsistency.3 Frege discusses using only the identified basic laws and rules of logic together with suitable explicit a revision to Basic Law V-which he labels: V'- in the afterword to volume II of definitions. This was to be the task of his magnum opus: Grundgesetze der Arithmetik. Grundgesetze. However, it can be surmised that Frege himself realised (probably The first volume of Grundgesetze is structured to reflect these three main tasks sometime after 1906) that V' was unsuitable for his project-it is inconsistent with and includes a substantial philosophical foreword. In this foreword, Frege provides the assumption that there are at least two distinct objects.4 Frege did not publish a extensive criticism of psychological approaches to logic-in particular, against Benno third volume. Given the inconsistency of his Basic Law V and Frege's inability to find Erdmann's Logik (1892)-and also offers an explanation of how his logical system a suitable substitute that could be regarded as a basic law of logic, Frege gave up on has changed since 1879 and why the publi\:'.ation of Grundgesetze was delayed. Frege Logicism late in his life.5 notes that a nearly completed manuscript was discarded in light of "a deep-reaching A recent resurgence of Logicism, and so-called Neo-Fregeanism, has again sparked development" (vol. I, p. X) in his logical views, in particular his adoption of the now interest in Frege's original writings and, in particular, in Grundgesetze. Despite infamous Basic Law V. The remainder of the volume is divided into two parts. In part the inconsistency of the formal system of Grundgesetze, Frege's proof strategy and I, Frege introduces the language of concept-script, his modes of inference, his basic technical accomplishment have come under renewed investigation. There is, for laws, and a number of important explicit definitions. Part II of Grundgesetze contains example, the recent "discovery" of so-called Frege's Theorem:6 the proof that the the proofs of the basic laws of cardinal number-i.e., the most important theorems axioms of arithmetic can be derived in second-order logic using Hume's Principle~a of arithmetic proven within Frege's logical system-as well as Frege's treatment of principle governing the identity of cardinal numbers: the number of Fs equals the recursion and countable infinity (see Heck (2012) for details). The proofs are contained number of Gs if, and only if, the Fs and the Gs are in one-to-one correspondence. in sections labelled "Construction". Each such section is preceded by a section called Frege first introduced Hume's Principle in Grundlagen (see §§63 and 72) but rejected "Analysis" in which Frege outlines in non-technical prose the general proof strategy in it as a foundation for arithmetic because of the infamous Julius Caesar Problem order to facilitate understanding of the subsequent formal proofs. Volume I finishes ( Grundlagen, §§56 and 66): it cannot be decided on the basis of Hume's Principle rather abruptly. One may speculate that Frege was given a page limit by the publisher. alone, whether Julius Caesar (or any other object that is not given as a number) is It was to be another ten years until the publication of volume II. It seamlessly identical to the number Two, say. In Grundgesetze, Frege proves both directions of continues where volume I left off: with no introduction, but merely a brief reminder Hume's Principle (vol.I, propositions (32), §65, and ( 49), §69; see also §38 where Frege of two theorems of volume I that had not previously been indexed for further use mentions the principle). Once the two directions of Hume's Principle are proven Frege but which are employed in the subsequent proofs. Frege thus finishes part II of makes no further essential use of Basic Law V in the development of arithmetic. 7 Grundgesetze, providing sixty-eight further pages of proofs of arithmetical theorems. There have also been attempts to offer revisions to Basic Law V, e.g., in Boolos (1998); Part III occupies the rest of the second volume. Following a strategy analogous to lastly, there has been recent work on identifying consistent fragments of the logic of the one he uses in Grundlagen, Frege first provides strong, and in parts polemical, Grundgesetze.8 Yet it is not just these formal aspects of Grundgesetze that called for a criticisms of alternative approaches to real analysis of many of his contemporaries. One new engagement with Frege's magnum opus. The philosophical arguments in part III, particular focus is his criticism of (game) formalism, as advocated by Hermann Hankel in particular, Frege's account of definitions, his conception of Basic Law V, his critical and Frege's colleague in Jena, Johannes Thomae. The critical sections of part III are assessment of contemporary accounts of real analysis, and Frege's own approach to real followed by a brief non-formal description of the approximate shape of Frege's approach analysis have again become relevant to current debates in the philosophy of logic and to a logicist foundation of real analysis as based on the notion of "magnitude". (Frege mathematics. Given that only parts of Grundgesetze had previously been translated, points out that he will not "follow this path in every detail" (vol. II, p.162).) Part the need for a complete English translation became more and more pressing. III.2 contains the beginnings of a formal treatment of real analysis, and finishes with an outline of what remains to be accomplished at the close of volume II. Evidently (published as Frege (1983)), is presumed lost due to an airstrike towards the end of the second world war-however, see Wehmeier and Schmidt am Busch (2005). Frege had plans for, or may even had already written parts of, a third volume in order 3 The antinomy Russell suggests is not well formed in Frege's system-in his response to Russell, to finish part III. Perhaps it was also to contain a part IV dedicated-as already noted Frege provides the correct formulation. See the Frege--Russell correspondence from 1902, in Gabriel in Grundlagen-to a logicist treatment of the complex numbers.2 et al. (1976), and, in particular, Russell (1902) and Frege (1902). 4 See Quine (1955) and Cook (2013). 2 Whether and how much of the continuation of part III was written is unfortunately unclear. 5 See his posthumously published note written around 1924/25 in Frege (1924/25). We may again speculate that Frege was, in turn, pushed to the publication of volume II by having 6 It was published in Wright (1983). More recent presentations of the proof can be found in Boolos exceeded a certain page limit. It seems natural to assume that he had some of the proofs that (1987) (discursive), Boolos (1990) (rigourous), Boolos (1995), Boolos (1996), and Zalta (2012). A volume III was meant to contain. Indeed there is evidence that Frege worked on further aspects of very insightful discussion of Frege's Theorem, including its origins and recent (re-)discovery, can be his theory of irrational numbers-see Dummett (1991), p. 242, fn. 3. According to the inventory of found in Heck (2011). Frege's Nachlaj], published in Veraart (1976), there were manuscripts that might have contained 7 For an excellent discussion see Heck (2012). such material. Frege's Nachlaj], except for a small part that was transcribed by Heinrich Scholz 8 See e.g. Wehmeier (1999), Heck (2000), Fine (2002), and Burgess (2005).