Unexplored Dimensions: Karl Menger on Economics and Philosophy (1923-1938) (Advances in Austrian Economics, Vol 12)

ADVANCES IN AUSTRIAN ECONOMICS Series Editors: R. Koppl and S. Horwitz Associate Editors: J. Birner and P. Kurrild-Klitg...

Author: Giandomenica Becchio (ed.)

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ADVANCES IN AUSTRIAN ECONOMICS Series Editors: R. Koppl and S. Horwitz Associate Editors: J. Birner and P. Kurrild-Klitgaard Recent Volumes: Volume 6: Volume 7: Volume 8:

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ADVANCES IN AUSTRIAN ECONOMICS VOLUME 12

UNEXPLORED DIMENSIONS: KARL MENGER ON ECONOMICS AND PHILOSOPHY (1923–1938) EDITED BY

GIANDOMENICA BECCHIO University of Torino, Torino, Italy

United Kingdom – North America – Japan India – Malaysia – China

Emerald Group Publishing Limited Howard House, Wagon Lane, Bingley BD16 1WA, UK First edition 2009 Copyright r 2009 Emerald Group Publishing Limited Reprints and permission service Contact: [email protected] No part of this book may be reproduced, stored in a retrieval system, transmitted in any form or by any means electronic, mechanical, photocopying, recording or otherwise without either the prior written permission of the publisher or a licence permitting restricted copying issued in the UK by The Copyright Licensing Agency and in the USA by The Copyright Clearance Center. No responsibility is accepted for the accuracy of information contained in the text, illustrations or advertisements. The opinions expressed in these chapters are not necessarily those of the Editor or the publisher. British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN: 978-1-84855-998-1 ISSN: 1529-2134 (Series)

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ADVISORY BOARD Don Bellante University of South Florida, USA

Uskali Ma¨ki University of Helsinki, Finland

Stephan Boehm University of Graz, Austria

Ferdinando Meacci Universita´ degli Studi di Padova, Italy

Peter J. Boettke George Mason University, USA

Mark Perlman University of Pittsburgh, USA

James Buchanan George Mason University, USA

John Pheby University of Luton, England, UK

Bruce Caldwell University of North Carolina, USA

Warren Samuels Michigan State University, USA

Jacques Garello Universitu e d’Aix-Marseille, France Roger Garrison Auburn University, USA

Barry Smith State University of New York, USA

Jack High George Mason University, USA

Erich Streissler University of Vienna, Austria

Masazuni Ikemoto Senshu University, Japan

Martti Vihanto Turku University, Finland

Richard N. Langlois The University of Connecticut, USA

Richard Wagner George Mason University, USA

Brian Loasby University of Stirling, Scotland, UK

Lawrence H. White University of Missouri, USA

Ejan Mackaay University of Montreal, Canada

Ulrich Witt Max Planck Institute, Germany

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PREFACE Karl Menger (1902–1985) was the mathematician son of the famous economist Carl Menger. He was professor of geometry at the University of Vienna from 1927 to 1938. During that period, which was crucial from a historical and philosophical point of view, he joined the Vienna Circle and founded his Mathematical Colloquium. Menger’s memoirs of those Viennese years are recollected in his Reminiscences of the Vienna Circle, appeared in 1994 as the twentieth volume of the Vienna Circle Collection. The drafts of this volume are parts of the Karl Menger papers, held in the Rare Book, Manuscript and Special Collection Library, Duke University. In this resource, there is his whole archive, which includes the notes Menger composed for his reminiscences. Checking those notes and comparing them with the published book, it is clear that many parts of his recollections were unpublished. In particular, his more autobiographical sketches had not been published. The present volume of Advances in Austrian Economics offers the transcription of those unpublished parts of Menger’s notes, which could be seen as a sort of autobiography for Menger. The foreword by Professor Brian McGuinness, who was one of the editors of Menger’s reminiscences (published in 1994), provides a charming portrait of Menger as a quirky, but admirable exemplar of ‘the world that might have existed if the liberal ideas that the Crown Prince Rudolf imbibed from Carl Menger the father had carried the day.’ I have written two introductory chapters: an historical reconstruction of the half-published Viennese autobiography of Menger and a paper on Menger’s contributions on ethics and economics, which he made during the period described in his autobiography. I thank Prof. Eve Menger, the daughter of Karl Menger, for permission to use her father’s notes. I thank Prof. Brian McGuinness for his kindness in providing several services. He read the text; he kindly engaged me in discussions on Menger, the history of the book (including the published and unpublished parts) and the Viennese philosophical milieu during Menger’s time, and, last but not least, he wrote his engaging foreword. I am inﬁnitely indebted to Prof. Roger Koppl who suggested that I prepare the present ix

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volume and offered his support and encouragement as I worked on it. I also thank Prof. Roy Weintraub and Prof. Robert Leonard very much for having patiently given me some hints on Menger’s philosophical and theoretical outlook. All that I have written remains my responsibility, however. Giandomenica Becchio Editor

FOREWORD From passages quoted in Giandomenica Becchio’s excellent introduction, it will be clear that I found Karl Menger an ideal subject when I was a series editor (of the Vienna Circle Collection) and he a contributor. He produced splendid copy, did his own translations, found his own references and was altogether easy to deal with – when he had his own way, of course, but that was usually the best course in any case. Alas he died before he and we could assemble a full set of reminiscences. The volume we brought out is still in demand and was greeted with enthusiasm by those interested in the intellectual history of its period. All the more it is a joy to learn that further papers of this nature have now been made available for publication. Karl Menger represented the ﬁne ﬂower of the educated civil society – the Bildungsbu¨rgertum – of the Austro-Hungarian cultural area. His family came from what is now Poland and were gentry, though they seemed to have dropped the predicate von to which they were entitled. His father was the founder of the Austrian School of Economics and had at one time been tutor to the Crown Prince. His mother (much younger) was a writer and musician. Both died just when he reached (very early) his intellectual maturity and so did not live to see what they launched upon the world. In fact, he himself went into a Magic Mountain period of illness and convalescence, perhaps a period of mourning, perhaps of preparation for the full efﬂorescence. For his gifts were many and varied. He tried his hand as a writer. He became a mathematician. He preserved and read his father’s philosophical library. Nearly every occasion would produce some surprise – we had a lot of material in Gabelsberger shorthand – it turned out that he used to win prizes for writing it in his schooldays. He came to London and we had to hunt through secondhand bookshops for a 19th-century Grammar of Design. I quoted a Schu¨ttelreim of Paul Engelmann’s, he capped it with any number of double – Schu¨ttelreime, enough to give one a taste for the genre. In Chicago, he gave me a Cook’s tour of the architecture, which he loved to explain. When we got to the University of Chicago (where I was to talk), he tried to sit anonymously at the back as if he were my driver. In fact, it was him and his friends that I was to talk about. In Oxford, I was able to xi

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bring him into the company of George Temple and other local greats for diverting conversation as fast as speed chess. We persuaded him (at least I played my part) to come back to Austria at last. He had been reluctant to set foot there since the war, yet few knew and appreciated its beauties more than he. No meadow in the world could match those of spring in the Austrian hills. It may be so, but all of that and a lot more – the unspoiled farmhouses and country estates where the Wittgenstein, Bru¨cke or Oser families, his friends managed still to live their 19th-century life, the opera, all the charm, had been lost to him by a decision of his own, a determination to be self-sufﬁcient in his new world. It is the cue perhaps to many turns in his career. Quirks of course he had – they were the badge of an e´migre´ scholar. His seatbelt must be put on in a particular way, his health and his diet required constant attention (Frau Dr Kaufmann, Felix’s widow, thought him a hypochondriac), travel required Napoleonic planning, but such things were part of his wish and ability to ﬁnd a theory for everything, from taxis to abstract groups. But Austrianness or Austro-Hungarianness was always there. The soft though accent-free voice, the absence of chauvinism, the assumption that his interlocutors paid tribute to the same set of values as himself, even though each had his own psychological oddity, all characters, as it were, from Stefan Zweig. There was an optimism there, which must have been dashed by experience. Contrast his brilliant account of how groups with conﬂicting values might best (from the point of each!) live together, with the world of Canetti’s Crowds and Power, or indeed with the fact that the groups in Menger’s Austria chieﬂy wanted the other groups not to exist. The reasonableness of it all, the fact that anything higher is banished to a region of pure aesthesis, as when Menger says that the chant of the Archangels in the ‘Prologue in Heaven’ from Goethe’s Faust is the most beautiful thing in the German language, ‘but of course it doesn’t mean anything’, mark Menger as belonging to the world that might have existed if the liberal ideas that the Crown Prince Rudolf imbibed from Carl Menger the father had carried the day. For his own life, given his acute appreciation of the aesthetic as well as the intellectually satisfying, it presumably sufﬁced and it certainly made him a fascinating companion or guide. Brian McGuinness

THE GENESIS OF THE HALF-PUBLISHED VIENNESE AUTOBIOGRAPHY OF KARL MENGER (1923–1938): NEW LIGHT ON THE VIENNA CIRCLE AND THE MATHEMATICAL COLLOQUIUM The role of the Viennese philosophical milieu in the 1930s was fundamental for the history of many disciplines, from logic and mathematics to social sciences, including economic theory. The mathematician Karl Menger (1902–1985), the son of Carl Menger, the founder of the Austrian school of economics, was undoubtedly one of the most representative scholars of that time, not only because of his researches in topology, logic, ethics and economics (Leonard, 1998; Sigmund, 2006), but also because of his Mathematical Colloquium, an informal circle in which new inquiries into geometry and logic were expounded, as well as ‘new applications of the exact sciences to problems of a sociological character’, as Menger himself wrote in an Italian paper (Menger, 1935) recently translated (Menger, 1998). In most recent years, several papers on Menger’s contribution have appeared. His works and his personal history attract not only mathematicians (Sigmund, 1998; Kass, 1996) and logicians (Dawson, 1998) but also philosophers (Gilles, 1981; Golland, 1996) and economists (Leonard, 1998; Punzo, 1989, 1991; Weintraub, 1983, 2002). The main reason for this is probably the fact that Menger’s interests covered all these ﬁelds of research, but another reason is his crucial work in Vienna during the 1930s whereby, during the following decades, he kept alive the history of the so-called golden autumn of that ‘Viennese Enlightenment’, which tragically ended in 1938 (Golland & Sigmund, 2000). 1

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In 1994, Menger’s reminiscences appeared posthumously as the twentieth volume of the Vienna Circle Collection (Menger, 1994). The editors, Louise Golland, Abe Sklar and Brian McGuinness, pointed out in a note on the text that, although Menger died before ﬁnishing the book and he was often worried about it: ‘it was therefore a matter of agreeable surprise to his friends and heirs both that it was so near completion and that he had clearly indicated a wish that it be published’. According to Schweizer (one of the editors of the most recent recollection of Menger’s papers (Menger, 2002, 2003), ‘these reminiscences may be viewed as the beginnings of an intellectual autobiography’ (Schweizer, 2002, p. 2). Menger’s drafts of this book, together with his complete private archive, are kept in the Special and Rare Books Collection at the Perkins Library of Duke University. On reading them, it is quite evident that many parts of Menger’s drafts were not published and some others appeared in other publications (Menger, 1974, 1980, 1982). This booklet is the transcription of still unpublished parts of Karl Menger’s memoirs with an introduction that reconstructs their history. There are two main reasons for publishing these unedited parts in an economic journal mainly dealing with Austrian themes. First, Karl Menger’s role in the history of economic theory is well known: his early contributions [his father’s second edition of Grundsa¨tze in 1923; his papers on the St. Petersburg paradox (Menger, [1934] 1979) and on the law of diminishing returns (Menger, [1936] 1979)], as well as his crucial role in the development of the theory of general economic equilibrium by his Mathematical Colloquium, through the contributions of Schlesinger, Wald and von Neumann, gained him an important place in the history of mathematical economics after the Second World War. The role of Menger’s Colloquium in building the neoclassical paradigm of general economic equilibrium has long been recognized. In a letter to Menger (5 February 1964),1 Arrow, who was writing ‘a systematic treatment of the theory of general competitive equilibrium’ and ‘would like to indicate at least brieﬂy the intellectual history of the subject’ asked Menger for some explanations on ‘the intellectual relationship between Wald’s ﬁrst paper on the existence of equilibrium and von Neumann’s article on an expanding economy’.2 Most recently, on the occasion of the reprinting of the Proceedings of the Colloquium (Menger, 1998), Debreu has stressed the extent to which Menger’s Colloquium ‘had a long lasting inﬂuence on economic theory’ (Debreu, 1998, p. 1). Even if, during the Viennese years of the Colloquium (1928–1936), its proceedings were not common reading among economists, ‘the ideas of Schlesinger, Wald and von Neumann, published in the volume

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6,7 and 8 of the Proceedings in 1935, 1936 and 1937, deserved immediate notice’ (ibid. p. 2). The second reason is that Menger’s contacts with the exiled Vienna Circle (and in general with other former Viennese members, or simply friends of his Colloquium, such as Go¨del or Morgenstern) during the decades after the Anschluss, when they were all living in the United States, yield better understanding of the history of subsequent developments in the economic theory (or economic theories) developed in Vienna (not only in Menger’s Colloquium but also in the Vienna Circle) and thereafter in the United States. As Debreu recognizes, Menger’s Colloquium generated subsequent developments in the general economic equilibrium theory: from Nash’s introduction of Kakutani’s theorem into economics in 1950 to ArrowDebreu and McKenzie’s model in 1954. Debreu adds that Slater’s application of ‘the von Neumann-Kakutani result to his study of a class of constrained maximum problems that had been considered by Kuhn and Tucker’ appeared in ‘a little known Cowles Commission Discussion Paper’ in 1950 (Debreu, 1998, p. 3). Another reason can be added: Menger’s unpublished drafts are mainly autobiographical, but they also deal with the philosophical climate in Vienna before and during the interwar period, so that the reader can reconstruct that particular and almost unique time. As is well known, philosophical inquiry in Vienna during the interwar period was proliﬁc. Historical reconstruction of the Viennese philosophical milieu around Menger cannot but start with the founding of the Vienna Circle3 and the Mathematical Colloquium.

1. A BRIEF HISTORICAL OVERVIEW ON THE TWO VIENNESE CIRCLES BEFORE THEIR EXILE For better understanding the connections between the Viennese circles in which Menger was involved, it is necessary to make some remarks on the Viennese context where they developed. Since the end of the 19th century up to the interwar period, Vienna was a very lively city from a cultural point of view, the birthplace of modernism (Janik & Toulmin, 1973). In the age of the late Habsburg monarchy as well as in the post-First War ‘Red Vienna’, the intellectual, scientiﬁc and artistic life of the Austrian capital was so fervent that those years are recalled by historians as the Viennese Enlightenment, the gay apocalypse and the golden autumn: ‘two generations

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were enough to cover the whole period. The economist Carl Menger (1841– 1920) shaped the beginning, and his son, the mathematician Karl Menger (1902–1985), witnessed the end’ (Golland & Sigmund, 2000, p. 34). After the First World War, from an economic point of view, a high inﬂation overwhelmed the country; while from a political point of view, ‘the new Austria was fragmented and labyrinthine’ (Leonard, 1998, p. 6): the Christian socialists were the conservative part of the society, but one third of citizens supported the new social-democratic party, which had the majority in Vienna. During that period many progressive intellectual circles developed. They were groups of discussion, some of them were connected with the university, such as Hans Kelsen’s and the Miseskreis, led by Ludwig von Mises (Kurrild-Kiltgaard, 2003),4 many others were private meetings such as the Geistkreis, dealing with researches in social sciences, founded in 1921 by Hayek and Furth ‘who were dissatisﬁed with the character of Othmar Spann’s doctoral seminar and wanted to create an independent and broader alternative’ (Kurrild-Kiltgaard, 2003, p. 48).5 Among those mostly far-away the academic world, there were the socialistic discussion groups and the psychoanalytic ones; the groups devoted to educational reforms, some literary circles (mainly inﬂuenced by Russian writers, Tolstoy in particular) and many philosophical groups, some oriented to historicism, some to kantianism, many others to Kierkegaard and phenomenology (Menger, 1994). The inﬂuence of these private meetings in the Viennese cultural life was so great that it was rightly said that: ‘seen from the perspective of a sociology of science, Viennese modernism developed via ‘circles’, characterized by both academic and private dynamics’ (Jabloner, 1998, p. 376). One of the most impressive characteristics of these circles was the active interactions between them. The foundation and rise of the Vienna Circle (founded in 1922 by Moritz Schlick) and of the Mathematical Colloquium (founded in 1928 by Karl Menger) belonged to this history. They were the most important informal institutions in which the scholars of the time could share their reﬂections to develop and improve science (Dawson & Sigmund, 2006; Stadler, 2001; Weintraub, 1983; Punzo, 1989, 1991). Contrary to what is commonly believed, however, there were differences of view within the Vienna Circle and between the Circle and the Colloquium (Sigmund, 2002; Stadler, 2006, Becchio, 2008). The main issues on which different positions were taken up were the validity of empirical observation as the foundation for all knowledge, the controversial ideal of physicalism as the sole language of science, the dispute between logicism and formalism in

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mathematics and logic, the nature of ethical propositions and the proper meaning of ‘values’ and the political struggle between liberalism and socialism. At the same time, there was a certain consensus in the Vienna Circle on the merits of the logical analysis of language (following Wittgenstein’s Tractatus), on the need to develop a rigorous epistemology for a scientiﬁc vision of the world and on the unity of scientiﬁc explanation and knowledge in general. These matters formed the core of the international movement in the 1930s, when the philosophical position of the Vienna Circle was most prominently represented by Carnap’s analysis of language and by Neurath’s physicalism and its program for a uniﬁed rational reconstruction of science (including human sciences) (Stadler, 2006). There was general consensus in the Mathematical Colloquium on a logico-deductive treatment of exact science (from geometry to arithmetic) and on a new foundation for ethics and economics based on logical coherence (Menger, 1935). In the mid-1930s the distance between the Circle and the Colloquium became evident. In 1934, with the advent of the right-wing Austrian government, difﬁcult times began for those scholars who were Jewish in particular, and for those not linked with the regime in general (Schlick was killed in 1936, Schlesinger committed suicide in the same year). In 1938 the Anschluss deﬁnitively destroyed the Viennese milieu: most of its members moved to the United States or to England, where the Circle’s ideas became increasingly widespread. Just as it is possible to speak of the ‘exiled Vienna Circle’ (Stadler, 2001; Richardson & Uebel, 2007), so it is possible to regard the close relationship between Menger and other members of the Colloquium (Abraham Wald, Kurt Go¨del, Franz Alt, Georg No¨beling) and with John von Neumann and Oskar Morgenstern,6 as ‘the exiled Mathematical Colloquium’. Like the other Viennese scholars, these had all moved to the United States in the second half of the 1930s, with the exception of No¨beling (who became professor in Erlangen, Germany, in 1940). Wald accepted an invitation from the Cowles Commission and then became professor at Columbia University (New York); Alt emigrated to Aberdeen, then moved to New York and took up a post at the American Institute of Physics; Go¨del went to Princeton, and likewise von Neumann and Morgenstern, where they published what is perhaps the most famous book in the history of mathematical economics (von Neumann Morgenstern, 1944). Menger had spent the 1930/1931 academic year in the United States visiting Harvard and the Rice Institute in Houston, meeting numerous prominent American mathematicians and philosophers of the time. He then returned in 1937, when he accepted a

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position at the University of Notre Dame, Indiana, and never thereafter lived in Vienna (Menger, 1994). In 1946 Karl Menger moved to Chicago to join the Illinois Institute of Technology on invitation by the chairman of the mathematics department, Lester R. Ford, who had been at the Rice Institute at the time of Menger’s visit there. At the Illinois Institute of Technology, Menger founded his new Mathematical Colloquium.

2. FROM VIENNA TO THE UNITED STATES: THE PROPOSED (BUT NOT ACCEPTED) ROLE OF KARL MENGER IN THE PROJECT TO UNIFY SCIENCE In United States the exiled Vienna Circle maintained contacts and organized six conferences, ‘International Congresses for the Unity of Science’: Paris (1935 and 1937), Copenhagen (1936), Cambridge, UK (1938), Cambridge, Massachusetts (1939) and Chicago (1941). The decision to launch this project was taken in August 1934 in Prague where numerous Viennese, German and American philosophers met for a philosophical congress (Stadler, 2001). As a result of these international meetings, the so-called Unity of Science Movement was formed7 and the project for the International Encyclopaedia of Uniﬁed Science began (Morris 1960). The two prime movers of this project were Otto Neurath in Europe (he moved from Vienna to The Hague in 1935, and after the Nazi occupation of Holland in 1941 he escaped to England) and Charles Morris in Chicago. As Menger’s correspondence reveals, in July 1934, before Morris attended the congress in Prague, he spent some weeks in Vienna ‘to make connection with members of the Wiener Kreis’.8 The Special Collections Research Center, Regenstein Library, University of Chicago, conserves the archive of the Unity of Science Movement. It forms a small part of Charles Morris’ personal archive. The collection contains items from 1934 (the Prague congress) to 1968 (the year of publication of the last volume of the Encyclopaedia), and it also includes the correspondence of the movement’s members,9 manuscripts, organizational materials and abstracts from the International Congresses and documents related to the International Encyclopaedia of Uniﬁed Science. As the political situation in Vienna grew increasingly precarious and difﬁcult in the mid1930s, it was Morris who ﬁnanced the organization of the entire project. In a letter of 28 January 1935, he requested W. Weaver (Rockefeller Foundation)

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to furnish ﬁnancial support for the uniﬁed science movement. In this letter, Morris described the movement: I represent an international group of scientists and logicians interested in the uniﬁcation of science meaning by that term not a synthesis of factual results but the analysis and systematization of the language of science. The enterprise is in no sense philosophical, if by philosophy one means more than work on the foundations of science and completion of the structure of science. . . . The three purposes of this group are 1) to unite scientists and logicians interested in the uniﬁcation of the language, and secondarily the factual results of science, 2) to publish ultimately an encyclopaedia of the sciences, written for the ﬁrst time in terms of a uniﬁed conceptual apparatus, 3) to act to some extent as a uniﬁed center against the anti-scientiﬁc tendencies which threaten to limit or destroy the slowly acquired scientiﬁc habit of mind and technique.10

The correspondence between Morris and Neurath illustrates the genesis of the Encyclopaedia. This was intended to be a new research project based on the general aims described in the above-quoted letter and not simply an extension of the Vienna Circle’s philosophical positions. As Morris pointed out in a letter of 11 November 1936 to Neurath: ‘I am interested in the Unity of Science movement as such, and not in any special group such as the Wiener Kreis, the pragmatists, or any other’.11 Their correspondence also reveals the efforts made to involve European and American scholars in the project. In a letter of 30 March 1936, Morris announced to Neurath that the University of Chicago Press had agreed to publish the Encyclopaedia. Hence the next step was to set up ‘special committees’ for the various disciplines: Morris suggested Enriques for the history of science, Łukasiewicz for the history of logic, Carnap for logic and Reichenbach for the methodology of logic.12 In a letter of 11 May 1936, as contributors Neurath gave Morris the names of Mannoury, Feigl, Dubislaw, Karl Menger, Radakovic, Waismann, Zilsel and Popper and suggested to Morris that ‘it would be useful to use Ernest Nagel as collaborator and other young Americans’. Hence Menger was included by Neurath in the initial project of the Encyclopaedia, even if Neurath himself wrote to Morris that ‘maybe for mathematics it is better Go¨del than Menger’ (2 November 1936). With regard to Menger and the vain attempt to include him in the Unity of Science Movement, Menger’s drafts reveal that the ﬁrst indirect contact between the Viennese philosophers of the Circle and Morris was mediated by Menger, who met Morris at the Rice Institute in Texas (spring semester 1931), where Menger lectured and Morris worked at the Department of Philosophy, and with whom he talked ‘a great deal about the Vienna Circle’.

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Menger’s correspondence shows that in a letter of 19 February 1937, Neurath invited Menger to write ‘Mathematics’ for the Encyclopaedia and to attend the congress for the Unity of Science in Paris in the same year. However, Menger declined his invitation, as well as refusing to become a member of the Advisory Commission for the Encyclopaedia.13 Neurath communicated Menger’s decision in a letter of 12 March 1937 to Morris and Carnap: Menger is concentrated in special problems of mathematics and not in the position to write about Mathematics. He suggested Waismann. I agree with him to invite Waismann, but as collaborator for the other volumes not for this ﬁrst group of pamphlets. I think we shall invite Tarski. Please tell me you both what you suggest. It was I think our common opinion to invite Tarski if Menger would be not in the position to make this pamphlet . . . [he] could use the title ‘Mathematics and logic’. I understand your hesitations in the case that Menger or Go¨del would make this pamphlet.

In the fall of 1936 Neurath had gone to Chicago to organize every detail for the Encyclopaedia with Morris and Carnap. In a letter of 10 October 1936 about Neurath’s itinerary and the people who he was expected to meet, Morris gave a list of persons on the faculty interested in the Encyclopaedia project, and he invited Neurath to regard Henry Schutz, the economist, as someone that they ‘should consider in making assignments for the Encyclopaedia’. By the late 1930s the entire Vienna Circle had resettled outside Austria. Its journal, Erkenntnis was replaced by a new one, which Morris suggested to Neurath should be called Journal of Uniﬁed Science.14 In 1939 Morris and Carnap started organizing the ‘Foundation of the Unity of Science’ and in the same year the ﬁrst volume of the Encyclopaedia appeared. Although there was strong initial interest in the movement among philosophers, the activities of the Unity of Science Movement were severely restricted by the outbreak of the Second World War. Only the ﬁrst two of the planned 200 volumes of the Encyclopaedia were published. The last congress was held in 1941 at the University of Chicago, and the Journal, published in Holland, was suspended in 1940.15 Neurath died suddenly at the end of 1945, and organization of the movement as well as of the Encyclopaedia was interrupted for a long period. After some years, the project to unify science was resumed by Charles Morris, Rudolph Carnap (both in Chicago) and Philipp Frank, whose role grew increasingly important. In 1947, the Institute for the Unity of Science was founded in Ithaca (New York) by Carnap, Morris, Reichenbach, Frank and Milton Konvitz, with Frank as President. When Frank moved to Boston, the legal head ofﬁce of

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the Institute followed him. Its purpose was ‘to encourage the integration of knowledge by scientiﬁc methods, to conduct research in the psychological and sociological backgrounds of science, to compile and publish bibliographies and other forms of literature with respect of the integration of scientiﬁc knowledge, to support the international movement for the Unity of Science and to serve as a centre for the continuation of the publications of the Unity of Science movement’ (Frank, 1947).16 As Frank stated in a letter to Morris (8 November 1949), the main aim of the new Institute was ‘to prevent logical empiricism from becoming a narrow sector but rather a signiﬁcant part of a wide movement in science and philosophy’. According to Frank, ‘to start the international cooperation, we have to keep the European scientist informed about the work which is done in the USA’. In this regard, Frank informed Morris about the agreement between himself and the editors of the international philosophical journal Synthese, published in Amsterdam, for the publication of ‘a special series of articles running in our communications to the history of logical empiricism and to its relations to other similar trends’. Philipp Frank, like Neurath before him, tried to involve Menger in the activities of the new Institute. In 1952 Frank sent Menger a letter in which he asked him to write ‘a small book on the philosophical interpretations of mathematics’ within a series of books on the philosophical foundations and interpretations of the different sciences; Frank explained him that ‘the books should be on such a level that they are understandable for all people with an average college education and a certain interest in science and philosophy’.17 Menger again declined. To be noted is that the letter was written on a card of the ‘Institute for the Unity of Science’, and inspection of the list of board members shows that it included all the members of the old Vienna Circle who had moved to the United States: Frank, the president (Harvard), Morris (Chicago), Nagel (Columbia), Carnap (Chicago), Feigl (Minnesota), Hempel (Yale), Reichenbach (UCLA). This testiﬁes to the evident re-establishment of relationships among the exiled Viennese scientists in the United States after the Second World War. In 1965 Morris succeeded Frank as President of the Institute for the Unity of Science. In the same year, Synthese was discontinued, and it was replaced by a new journal, Methodology and Science, brought out by another publisher. In 1968, G. Tintner published Methodology of Mathematical Economics and Econometrics, which was chronologically the last published volume of the International Encyclopedia (Tintner, 1968). While waiting for this ﬁnal

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volume to appear, Morris wrote to Frank (2 June 1965): ‘it has been a ﬁne enterprise, though only a fragment of what Neurath had planned’.

3. RENEWED INTEREST IN THE HISTORY OF THE VIENNA CIRCLE: THE REQUESTED (AND WELL ACCEPTED) ROLE OF KARL MENGER IN RECOLLECTING THE VIENNESE HERITAGE As the Karl Menger archive testiﬁes, relations between Menger and the other members of the exiled Mathematical Colloquium were very close throughout the decades after their move from Vienna to the United States: Menger, indeed, maintained contacts with all his former colleagues.18 His friendship with Oskar Morgenstern was particularly long lasting, thanks to their mutual interest in the relationship between mathematics, logic and economics (Leonard, 1998). Their strong bond of friendship was shared with Go¨del, who was constantly in contact with both of them. As we have seen, Menger was never interested in Neurath and Morris’ project for the International Encyclopaedia; nor was he involved in the later movement for the unity of science promoted by Frank, Morris and Carnap. Like Menger, no other member of the exiled Mathematical Colloquium was involved in those projects, or even attended the international congress.19 But the situation changed some years later. Once the publication of Encyclopaedia and the ‘unity of science movement’ had ceased, Menger became a highly active and proliﬁc protagonist of what can be termed a ‘revival’ of the Viennese heritage in the history of philosophy and science. At the very beginning of the 1970s, the Dutch historian of philosophy, Henk L. Mulder (1921–1998), started to work on a project to reconstruct the history of the Vienna Circle. In 1973 Mulder, McGuinness and Bob Cohen began publication of the Vienna Circle Collection, a series of books containing translations of important studies by members of the Vienna Circle and related authors. Between 1973–2004, twenty-three volumes were published, initially by D. Reidel Publishing Company, later by Kluwer Academic Publishers and Springer-Verlag. The editorial committee consisted of Mulder himself, Robert Cohen (Boston University) and Brian McGuinness (Queen’s College, Oxford); some members of the editorial advisory board were exiled Viennese scholars such as Herbert Feigl and Karl Menger (Viktor Kraft was the sole who remained in Austria). The ﬁrst volume of the Vienna Circle Collection was an anthology of Neurath’s

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papers on the relationship between neo-empiricism and sociology (Neurath, 1973); and the last was a collection of Neurath’s economic writings (Neurath, 2004). In 1978 Mulder founded the Vienna Circle Foundation (Wiener Kreis Stichting) to promote studies on the work of the members of the Vienna Circle and to create a sort of Vienna Circle Archive.20 Menger was soon involved in the project for the Vienna Circle Collection. A year after its foundation, his book on ethics Moral, Wille und Weltgestaltung, Grundlegung zur Logic der Sitten (1934) was translated into English and printed as the Collection’s sixth volume (Menger, 1974). In the postscript to the English edition, Menger himself recalls how on several occasions during the past decades, ‘even before the current revival of interest in the Vienna circle, reﬂected in the publication of the Collection’, he had been asked to translate his book into English. Morgenstern in particular repeatedly exhorted him to do so; ‘suggestions’ – Menger adds – ‘also came from H. Mulder, who has done so very much to keep the interest in the Vienna Circle alive, to broaden and deepen it’. Finally Menger edited the translation and thus joined the project to revive the Vienna Circle. This was not simply a personal matter: the private correspondence between Brian McGuinness and Karl Menger testiﬁes to Menger’s deep interest in the project as a whole. Menger played an active role in choosing the subjects of the various volumes of the Collection, and he became – almost unconsciously – an ‘editor or consulting editor of the collection’ (letter of 2 July 1976). Moreover, and especially, Menger became a sort of ‘historical memory’, an expert witness for the reconstruction of the Vienna Circle’s history. McGuinness and Menger’s correspondence contains numerous examples of Menger’s role as a valuable eye witness to events within the Vienna Circle. In a letter of 2 August 1974, McGuinness asked Menger to conﬁrm Feigl’s memory on the role of Boltzmann in the Circle, which had been soon dismissed under the inﬂuence of Carnap and Wittgenstein: ‘I asked Feigl and he said that the Vienna Circle members, originally admirers of Boltzmann came to regard his ‘‘realism’’ as meaningless, under the inﬂuence of Carnap’s Aufbau and Wittgenstein’s Tractatus’. Naturally, Menger’s memoirs were particularly interesting because of his friendship with Wittgenstein’s family in the Viennese years.21 McGuinness often thanked Menger for explaining the role of several scholars in the Circle and the relationships among them. In a letter of 2 July 1976, McGuinness thanked Menger for sending him useful ‘remarks about Kaufmann’ and added: ‘Now I am bound to say that

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I ﬁnd Kaufmann one of the most intelligible writers of the phenomenological school and therefore of particular interest. But of course he is a marginal ﬁgure to the V[ienna] C[ircle] C[ollection]’ (Kaufmann, 1978). In the same letter, McGuinness referred to Wittgenstein: ‘I had another question which someone suggested I might ask you, but do not feel obliged to write anything about it if you have in fact nothing to say on the matter. The question really concerned whether Wittgenstein ever got in touch with Brouwer. We know, of course that he heard Brouwer’s lectures in Vienna in 1928 or at any rate one of them, but there have been rumors of a later visit to Brouwer in Holland. If you happened to know anything of that it would be very interesting to hear of it’. On reading this letter, it is evident that, in the mid-1970s, Menger had in mind two projects for the Vienna Circle Collection: he was planning a collection of his selected papers and an autobiography, at least of his Viennese period – a sort of ‘Viennese autobiography’. McGuinness expressed his enthusiasm for both projects, although he had some doubts about the feasibility of the second one, at least as part of the Collection. He wrote: ‘Your plan for the collected papers seems to be an excellent idea. When you come onto the possible intellectual autobiography that you spoke of, please do get in touch with me if you think that Reidel and the V[ienna] C[ircle] C[ollection] is not the suitable home for it. It should be easily possible to ﬁnd another one and I at any rate should be glad to help in any way’.22 Menger’s ﬁrst project was realized in 1979 when a selection of his papers on logic, mathematics and economics, written between 1928 and 1978, were assembled and printed as the tenth volume of the Collection (Menger, 1979). On that occasion, Menger wrote a letter (16 March 1979) to the publishing company in which he expressed his enthusiasm over the project of the Vienna Circle Collection and its results. Menger wrote: ‘It has been a pleasure for me to work with you in bringing out my book, Selected Papers . . . I wish to express my sincere thanks for the beautiful result . . . I wish to congratulate your publishing house on this expansion of the impressive series, which is a true monument to the Vienna Circle. Please communicate my congratulations to everyone connected with the series’. In the same letter, Menger complained of a lack of interest in the person and works of Hans Hahn, who was not included in the Vienna Circle Collection: ‘I only regret that Hans Hahn, who with Schlick, Neurath, Carnap and Waismann constituted the very core of the group is not represented by a small volume of his philosophical writings, as planned, I believe, by McGuinness. In fact, on the back of the dust jacket of my book Hahn’s name is not even mentioned’. Menger was referring to a previous agreement between him and

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McGuinness, who had sought to involve Menger in preparing a volume on Hahn’s philosophical papers and had asked him to write its introduction: Dear Professor Menger, Could I return to the project mentioned at the end of 1976 of including in our Vienna Circle collection a volume devoted to Hahn? The time is now ripe to do something about it. I am to be editor and I have written to Frau Minor to get family permission for the publication. In due course I will write to the various publishers for their permission too. Meanwhile we have translated the three ﬁrst items on the enclosed list (which is Mulder’s) and I suppose should include the rest (if we can get hold of them all). But are there others known to you which it would be suitable to include in an essentially philosophical volume? . . . An even more presumptuous question is the following: would you be prepared to write an introduction (anything from 10 pages up) for the volume? Who would be better than you? And (if I don’t ﬂatter myself with false hopes) you will soon have ﬁnished your volume to which we much look forward. Please, if you can’t agree straightway, bear this possibility in mind and let us recur to it when we have ﬁnalized a list of contents for the volume.23

The genesis of Hahn’s volume was rather complicated. It was decided in 1978: McGuinness’ letters to Menger testify to their shared involvement in obtaining permission for publication from Hahn’s daughter and contain suggestions on the papers to select (letters of 14 February 1978 and 8 May 1978). Finally, Hahn’s Philosophical Papers volume came out in the fall of 1979, and a year later, it was printed with an introduction by Karl Menger (Menger, 1980).

4. KARL MENGER’S FINAL YEARS: RECOLLECTION OF HIS MEMORIES FOR THE VIENNA CIRCLE COLLECTION AND FOR HIMSELF What happened to ‘the possible intellectual autobiography’ of which Menger spoke in 1976? Menger worked on the autobiography for a long time. Its drafts are kept in the Perkins Library, Duke University, together with Menger’s entire personal archive (notes, correspondence and miscellanea). Some parts of the drafts were published in Menger’s books concerning the Vienna Circle [Postscript to the English edition of Morality, Decision and Social Organization (Menger, 1974), Introduction to Hahn’s Philosophical Papers (Menger, 1980), Memories of Moritz Schlick (Menger, 1982) and ﬁnally Reminiscences of the Vienna Circle and the Mathematical Colloquium (Menger, 1994)]. But many parts of Menger’s drafts remained unpublished.

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The following two letters from Kurt Go¨del and Herbert Feigl to Menger testify that Menger had been working on his autobiography since the early 1970s: the letters were in fact answers to Menger’s requests to help him in recalling those crucial years when the Vienna Circle and the Mathematical Colloquium were founded and developed. Go¨del clearly explained his personal distance from the logical positivism of the Vienna Circle and the absence of relations between him and Wittgenstein:24 Princeton, April 20, 1972 Dear Professor Menger, I am sorry my reply to your letter of Jan 15 comes so late. I hope it will still be of some use to you. In consequence of frequent tiredness I hardly even answered letters before a few weeks time. But in this case there was moreover a special reason, namely that I have always inhibitions to write about my relationship to the Vienna circle. I never was a logical positivist in the sense in which this term is commonly understood.25 On the other hand, by some publications, the impression is created that I was.26 As far as your speciﬁc questions are concerned I can see now that they have nothing to do with this problem. The answers are as follows: 1. I was never introduced to Wittgenstein and I have never spoken a word with him. I only saw him once in my life when he attended a lecture in Vienna. I think it was Brouwer’s. 2. As far as my theorem about undecidable propositions is concerned it is indeed clear from the passages you cite, that Wittgenstein did not understand it (or pretended not to understand it). He interprets it as a kind of logical paradox, while in fact it is just the opposite, namely a mathematical theorem within an absolutely uncontroversial part of mathematics, namely ﬁnitary number theory or combinatorics. Incidentally the whole passage you cite seems non-sense to me. See e.g. ‘superstitious fear of mathematicians, of a contradiction’. 3. I don’t remember any talk given by Schechter in the Vienna Circle. However this means very little27 because I am sure I did not enter the Circle before the academic year 1925/26 and very probably not before the calendar year 1926. Moreover I practically never attended after the spring term of 1933. I attended pretty regularly in 1926, 1927. I don’t remember exactly how frequently I did 1928–1933. Yours sincerely Kurt Go¨del

Feigl replied to Menger in regard to Wittgenstein as well, adding that he had also written a historical reconstruction of the Vienna Circle (he was referring to Feigl, 1969a, 1969b)28: Minneapolis, May 3, 1972 Dear Menger, Sorry this response is delayed –because of my illness I get only once a week to my ofﬁceand the secretary neglected to forward your kind letter . . . I wish I could answer your

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questions with some assurance, but I have no diaries, protocols and my recollections are dim. I think that K. Go¨del joined the Wr Kreis in 1927 (or at the latest in 1928). Wouldn’t he (Princeton Instit.) respond to your questions? [Oh, I just see from your letter that you probably asked him, but are dubious about his reply]. As to Wittgenstein – I met with him, sometime together with Schlick &/or Carnap &/or Waismann (sometime just the two of us, sometime at my ﬁance´e’s Maria Kasper – now – since 1931 – my wife) – 1927–1929 – and I guess he became unavailable to me after he stopped seeing Carnap. (I think he later corrupted the philosophers very regrettably – no doubt a result effect of his ‘hypnotic’ personality and manners. I’m almost entirely on B. Russell’s side of the current controversy. You have ‘Konkurrenz’ in re, to writing about the Wr Kreis. I trust I sent you the encl. reprint of my (not very deep) memoir 2 or 3 years ago. Rose Rand (intending to emigrate to Israel) has steadily refused to part with her protocols (my copy is lost, there may be one in Carnap’s NachlaX). R. Rand apparently intends to write the history of the Wr Kreis in Israel – and says she has a publisher for it. H. Mulder, an eager archivist in Amsterdam couldn’t get the protocols from her despite the efforts of Marie Neurath, myself and others. I will not write anymore on this subject (I have enough other projects that’ll keep me occupied for the rest of my ‘natural life’ . . . With all good wishes, most cordially yours, Herbert Feigl

The editors of Menger’s Reminiscences were Brian McGuinness, Abe Sklar and Louise Golland. In a private email recently sent to me by Brian McGuinness, he explained that the aim of the publisher was to put together a clear text book for older readers, but it never had the popular success that might have been expected (there were plans for a paperback edition, but they never came to fruition). McGuinness received the manuscript in English and he attended to the copy-editing for annotation; the role of Golland was to act as the channel to the family for McGuinness; and, on this speciﬁc occasion, Sklar had a very marginal role in production of the book. According to McGuinness, it would have been better to see the raw material from which Menger was assembling the memoir until his death. However, Menger was very keen to do his own editing in the earlier collections, and because he had various linguistic skills such as translation into English verse, McGuinness did not have much more to do than proof-reading. Since the letter of 2 July 1976, quoted earlier, McGuinness had suspected that Menger’s project to write his autobiographical sketches would not have been of interest to Reidel. Hence he suggested that Menger should ‘get in touch with [him] if you think that Reidel and the V[ienna] C[ircle] C[ollection] is not the suitable home for it’. McGuinness was aware that the publisher preferred to promote a book written by Menger on the history and development of the Vienna Circle from its foundation until Schlick’s assassination, rather than his Viennese autobiography.

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The result was a sort of compromise, perhaps engineered by Menger himself: in 1994 Reminiscences of the Vienna Circle and the Mathematical Colloquium was published posthumously. This was a collection of memories that enabled the reader to reconstruct the intellectual milieu in Vienna during the interwar period, with particular insights into the philosophical context, as well put by review: ‘this volume, along with others in the Vienna Circle Collection, furthers our appreciation of the cultural and historical context within which logical positivism itself developed’ (Friedmanm, 1995). But Menger’s most autobiographical chapters were not published, nor were his reﬂections on the history of philosophy before the founding of the Vienna Circle. Furthermore, his criticisms of the philosophical position taken up by the Vienna Circle during the 1930s [as set out in Menger, 1974, 1980 (Becchio, 2008)] were not reprinted, even though they belonged to the corpus of the drafted text. There are various reasons for this. The main and simplest one is that twenty years elapsed from the beginning of the project (early 1970s) to its ﬁnal realization, and in the meantime Menger died in 1985, nine years before the book appeared. A possible reconstruction of the facts could be as follows. As we have seen, the two letters from Feigl and Go¨del (1972) testiﬁed to Menger’s endeavour to reconstruct the historical relationship between the Vienna Circle and his Mathematical Colloquium. As he gathered his memories on this topic, it may be that Menger decided to write his autobiography (as he proposed to McGuinness in 1976). Later, although his autobiographical memoirs could not be wholly included in the volume of the Vienna Circle Collection, he continued to write his Viennese autobiography.29 The fact that Menger’s drafts are divided into ‘Book A’ (historical remarks) and ‘Book B’ (autobiographical reminiscences) suggests that, at a certain time, he had two publications in mind: his autobiography and a book on the history and the general philosophical framework of the Vienna Circle (to be included in the Vienna Circle Collection). His death interrupted publication of his autobiography, while his Reminiscences of the Vienna Circle and the Mathematical Colloquium was edited posthumously nine years later.

5. THE CONTENTS OF MENGER’S UNPUBLISHED VIENNESE AUTOBIOGRAPHY AND SOME CONCLUDING REMARKS Part I of this book deals with the philosophical background and framework of the Vienna Circle – from Menger’s particular point of view, of course; part

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II forms what can be called his Viennese autobiography. It consists not only of mere personal recollections from 1923 to 1938, but also of a historical and cultural reconstruction that sheds new light on the philosophical context of the Vienna Circle and the Mathematical Colloquium. What Menger was concerned to explain was the difference in attitude between German and Austrian philosophy. He repeatedly emphasised that, in the Austrian tradition, philosophers and scientists (such as Brentano and Bolzano, whom he considered to be ‘remote predecessors’ of the Vienna Circle and its ‘immediate predecessors’, namely Mach, Boltzmann, Mauthner) were averse to any kind of metaphysics. Menger also stressed the Viennese opposition to Husserl’s phenomenology, although introducing Husserl’s thought was essential to yield better understanding of Viennese developments in logic and philosophy: for instance, Husserl’s conception of intuitive knowledge was thoroughly discussed in Vienna, but it was almost immediately rejected, primarily by Schlick. Menger also identiﬁed this antimetaphysical attitude in Kant’s philosophy (rather paradoxically if we adopt Kant’s point of view), which had permeated the whole of Europe during the 19th century and whose inﬂuence was still unavoidable even in Vienna. Menger’s speciﬁc intention was to demonstrate the great interest of Viennese scholars, above all philosophers and mathematicians, in the most recent developments in logic and ‘fundamental ideas’ (i.e., what today is called metatheory) and in the debate surrounding them. These subjects were discussed at the University and at the Philosophical Society, as well as at private meetings in Viennese cafe´s, by scholars and students, most of them interested in scientiﬁc issues. As is well known, the idea of founding what was to become the Vienna Circle was ﬁrst put forward at these private meetings. References to the Vienna Circle in Menger’s drafts show that Menger wanted to emphasize that the group was not as cohesive as Neurath claimed [from creation of the Verein Ernst Mach, in 1928, as the Circle’s ofﬁcial institute, to publication of the manifesto Wissenschaftliche Weltauffassung (Hahn, Neurath, & Carnap, 1929)]. According to Menger, in fact, there were ‘very few subjects on which there was a consensus in the group’ because its members were ‘highly individualistic’. Nevertheless, Menger admired the fact that the Circle consisted of scientists mainly interested in logic and with ‘a common background of scientiﬁc and mathematical knowledge . . . while many a traditional philosopher in Central Europe at that time was struggling with basic scientiﬁc ideas or [was] ignorant of mathematics’. This was the main reason that Menger himself joined the group, upon

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invitation by Hahn, between 1927 and 1928, and he continued to attend its meetings in the following years when he headed his Mathematical Colloquium open to mathematicians and logicians interested in investigating the logical paradigms of exact and sociological science. Clearly apparent in Menger’s drafts is the distance between the two circles after the introduction of ‘the battle cry Einheitswissenschaft (literally, UnityScience) for the entire movement’, an aversion to which Menger shared with Go¨del and Wald. On the contrary, Menger reveals that the Vienna Circle showed great interest in Wald’s mathematical works, but not in the impressive results in logic achieved in 1931 by Go¨del, who in January of that year presented them to Menger’s Colloquium (chaired by No¨beling, because Menger was visiting the United States) and in September to the Ko¨nigsberg meeting. An explanation for Go¨del’s results is set out in Menger’s draft. Collaboration between the two circles, or better between Menger and Hahn, resumed in 1933, when they organized three series of annual conferences under the title Crisis and Reconstruction in the Exact Sciences proposed by Menger himself. The main purpose of these conferences was to raise money, which was also used to publish Friedrich Waismann’s Introduction to Mathematical Thinking (Waismann, 1951), the genesis of which was rather difﬁcult. Hahn’s unexpected death in the summer of 1934 and the simultaneous advent of the Unity of Science Movement through the endeavours of Neurath, Carnap and Morris again led Menger and his Colloquium far from the Vienna Circle. This booklet may contribute to better understanding of the historical genesis and subsequent development of that complex relation between those circles that belongs to the history of philosophy, logic and economic theory.

NOTES 1. Menger’s correspondence is kept in the Rare Book and Special Collection, Perkins Library, Duke University, henceforth Karl Menger papers, Perkins Library, Duke University. 2. Karl Menger papers, Perkins Library, Duke University. 3. Menger usually called it the ‘Schlick Circle’ because there were many ‘kreise’ in Vienna in those time. 4. ‘Kelsen and Mises were – despite philosophical differences – close friends; they had known each other since school, when they were about 10 years old, and they had worked together at the Department of Economic Warfare during the war. They later spent several years together in Geneva after both having escaped Austria’ (KurrildKiltgaard, 2003, p. 35).

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5. Othmar Spann was professor of economics in the University of Vienna and later became one of the most inﬂuent theorist of the Austrian fascism and of the corporative state. 6. Morgenstern and von Neumann were not ofﬁcially members of the Colloquium, but their role in it, their inﬂuence on the subsequent development of its inquiries, and their friendships with its members, were so deeply important and widely recognized that they may be included in the Colloquium (Weintraub, 1983; Punzo, 1989; Golland & Sigmund, 2000). 7. The special issue of Philosophy and Phenomenological Research Vol. 6, No. 4, June, 1946, is entirely devoted to the Unity of Science Movement. 8. Letter of 5 July 1934 from Charles Morris to Karl Menger, Karl Menger papers, Perkins Library, Duke University. 9. Most notably Percy W. Bridgman, Rudolph Carnap, John Dewey, Herbert Feigl, Philipp Frank, Joergen Joergensen, Victor Lenzen, Charles W. Morris, Otto Neurath, Hans Reichenbach, Louis Rougier, and Bertrand Russell. 10. Charles Morris Correspondence, Guide to the Unity of Science Movement Records, 1934–1968, Regenstein Library, University of Chicago: henceforth Charles Morris correspondence, Regenstein Library, University of Chicago. 11. Charles Morris correspondence, Regenstein Library, University of Chicago. 12. Furthermore, Morris promised Neurath that he would organize a congress in the United States in 1938 or 1939 (although he pointed out that 1939 was better than 1938 because in the United States the movement was ‘just beginning to grow’): Charles Morris correspondence, Regenstein Library, University of Chicago. In fact, the congress was held in 1939 at Cambridge (Mass). 13. Otto Neurath to Karl Menger (19 February 1937 and 14 March 1937), Karl Menger Papers, Perkins Library, Duke University. 14. Letter from Morris to Neurath (31 May 1938) and letter from Morris to Donald Bean, head of the Chicago University Press (20 November 1938). See also Neurath to Morris (5 April 1940): ‘the transformation of Erkenntnis into the Journal of Uniﬁed Science is completed’, Charles Morris correspondence, Regenstein Library, University of Chicago. 15. In a letter from Morris to Carnap, Reichenbach, Frank, and Neurath (29 September 1941). Morris stated in regard to the Journal that he was carrying on a personal struggle to ﬁnd funds so that it could be published in Chicago, although Neurath would have liked it to be published in the United Kingdom (by Blackwell): Charles Morris correspondence, Regenstein Library, University of Chicago. 16. The manifesto of this new movement was published in Institute for the Unity of Science. From Minutes of Board of Regents, Synthese, 6(3/4) (1947–1948), p. 158. 17. Philipp Frank to Karl Menger, 4 June 1952, in Karl Menger Papers, Perkins Library, Duke University. 18. As testiﬁed by several letters in his correspondence, he was particularly active in organizing tuition for Wald’s children after the death of their parents in a plane crash in 1951. 19. In a letter of 23 June 1941, Morris urged Go¨del to attend the Congress, but he refused.

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20. This archive comprises the scientiﬁc papers of Moritz Schlick and Otto Neurath; Mulder came into contact ﬁrst with the daughter of Schlick, who happened to live in the Netherlands, and then with Neurath’s widow, who was living in London (Mulder, 1985). 21. McGuinness wrote: ‘How interesting that you knew his aunt Clara. She (and his father) were the only family he kept photographs of’ (letter of 2 August 1974). Karl Menger Papers, Perkins Library, Duke University. 22. Letter of 2 July 1976, Karl Menger Papers, Perkins Library, Duke University. 23. Letter from Brian McGuinness to Karl Menger (6 January 1978), Karl Menger Papers, Perkins Library, Duke University. 24. Letter from Kurt Go¨del to Karl Menger, Karl Menger Papers, Perkins Library, Duke University. 25. Explained in Wissenschaftliche Weltaufassung, der Wiener Kreis, edited by Verein Ernst Mach, 1929. 26. Probably in part, due to my fault. 27. Even if I was present I may have forgotten it. I remember only very few of the talks given. There were not many that really interested me. 28. Letter from Herbert Feigl to Karl Menger, Karl Menger Papers, Perkins Library, Duke University. 29. Professor McGuinness recently explained me that they were never offered to the editors for their publication and McGuinness could not get access to further Menger material beyond that supplied by Sklar and Golland.

ETHICS AND ECONOMICS IN KARL MENGER This section provides a summary of Menger’s main contributions to ethics and economics. They were composed during the period described by Menger in his autobiography. Many references of these works are presented in Menger’s drafts published in this volume.

1. THE ST. PETERSBURG PARADOX Menger studied mathematics and logic in the 1920s in Vienna under Hahn and thereafter in Amsterdam under Brouwer, one of the leading exponents of intuitionism. Between 1921 and 1927, Menger worked on topology and geometry (above all set theory and curve theory, Golland & Sigmund, 2000). But he also concerned himself with economic theory: an interest possibly explained by the fact that he was the son of one of the foremost economists of the time. After the death of his father, in 1921, he started to revise the second edition of the Grundsa¨tze, which was published two years later. In the same year, he wrote his ﬁrst paper on economic theory. This dealt with ‘the role of uncertainty in economics’ (as appeared in the title of the paper itself), and it urged economists and psychologists to ﬁnd a general function able to describe the constant behaviour of agents and deviations from it. The paper was a reﬂection on the St. Petersburg paradox formulated in the 18th century by Nicholas Bernoulli and which highlighted the conﬂict between rational expectations and common sense. Nicholas’s cousin, Daniel Bernoulli, claimed that mathematical expectations should be replaced with ‘moral’ expectations and was thus the ﬁrst to investigate the meaning of socalled expected utility for a gambler who persists in playing the same game. He deﬁned the notion of expected utility by decomposing the valuation of a risky venture as the sum of utilities from outcomes weighted by the probabilities of outcomes. He illustrated his famous paradox with the 21

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following situation: a coin will be tossed until a head appears; if the ﬁrst head appears on the nth toss, then the payoff is 2n and the game is over. The paradox, of course, is that the mathematical expectation in this game is inﬁnite.1 Yet, although the expected payoff is inﬁnite, one would not suppose, at least intuitively, that a real-world person would be willing to pay an inﬁnite amount of money to play the game. In fact, there is a quite high probability that he would collect only a few units of money. Daniel Bernoulli’s solution involved two ideas: (i) that people’s utility from wealth, u(W), is not linearly related to wealth (W) but rather increases at a decreasing rate – the famous idea of diminishing marginal utility; (ii) that a person’s valuation of a risky venture is not based on the expected return from that venture, but rather the expected utility from that venture.2 Consequently, people would only be willing to pay a ﬁnite amount of money to play the game, even though its expected return is inﬁnite. On replacing the ﬁrst factor with the subjective (or, as he put it, ‘moral’) value of the gain – which, Bernoulli emphasized, depends not only on the amount A of the gain but also on the wealth W of the evaluating person – he had assumed the subjective value to be proportional to A and inversely proportional to W. Hence, the subjective value became A c log 1 þ W where c is the number independent that may differ among persons but is constant for each person. By considering the subjective value, we can obtain a ﬁnite expectation and then a solution of the paradox. Menger disagreed with this view for various reasons. Also, the subjective expectation is inﬁnite. There are many cases where man’s behaviour fails to conform to mathematical expectations: games in which a player can win only one very large amount with a very small probability or games offering a single moderate amount with a very high probability. Furthermore, we can always ﬁnd a sequence of payoffs x1, x2, x3, . . . , which yield inﬁnite expected value, and then propose, say, that u(xn) ¼ 2n, so that expected utility is also inﬁnite. Menger therefore proposed that utility must also be bounded above for paradoxes of this type to be resolved. Menger’s explanation was that most people systematically underrate very small and very high probabilities in their economic actions, while they overrate medium probabilities. Moreover, Menger pointed out, besides a person’s wealth W, one must also consider the amount U that he needs to continue his present standard of living as well as the amount Uu that he

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absolutely needs to go on living. And the difference WU plays a crucial role in a rational person’s attitude towards risk. Finally, other ‘personal’ parameters may modify the way a person gambles. Hence, Menger stressed that deviations of behaviour are very complex: even in a game with a ﬁnite number of solutions, individual choices may disregard mathematical expectations because individual choices are conditioned by psychological motivations or by taste (Leonard, 1995). It is perhaps no coincidence that Menger’s interest in uncertainty in economic theory arose in the same period when he was editing his father’s treatise. It is well known that in Carl Menger’s theory (as well as in the subsequent Austrian tradition), human knowledge is incomplete, rationality is not full and uncertainty performs a major role in human actions. Moreover, Chapter II of his father’s Grundsa¨tze contains a section on uncertainty in economics. Unlike his father (as well as the following Austrian tradition), Menger sought to formalize the economic issue in mathematical terms. The ‘uncertainty’ to which Menger referred in the title of the paper is the randomness of choices when they are made in a context of knowable probabilities. This laid the basis for the subsequent development of inquiry by mathematical economics into the theory of expected utility. Menger’s paper was not published when it was written in 1923. When Menger wrote it, Hans Mayer, the then editor of Zeitschrift fu¨r Nationalo¨konomie, refused to publish the paper because it made excessive use of mathematical formulas. The paper was discussed in 1927 at a meeting of the Viennese Economic Society and provoked differing reactions. It was only published in 1934, when Oskar Morgenstern – according to Menger ‘one of the very few Austrian economists who were free from prejudices against mathematical methods in economics’ (Menger, 1967, p. 211) – was appointed editor of the Zeitschrift (Becchio, 2008). In the same year, Menger published his book on ethics. Before describing its content, it is essential to recall the philosophical and methodological framework on which Menger’s inquiry on social science was developing.

2. MENGER AND THE VIENNESE DEBATE ON LOGICS During the 1920s, the philosophical climate in Vienna was inﬂuenced by the fact – strongly stressed by Karl Menger in his Reminiscences – that ‘Austrians never contributed to the German type of metaphysics that

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culminated in Fichte, Schelling and Hegel’ (Menger, 1994, p. 18). Austrian philosophy was mainly oriented to discussions on epistemology and logics: ‘the great thinkers born in the Austrian empire, Bolzano and Mach, used to philosophize along scientiﬁc lines’ (p. 18). The rejection of both idealism and neo-Kantianism culminated in the advent of the new empiricism developed by the Vienna Circle (founded in 1922 by Moritz Schlick) where there was a certain consensus on the merits of the logical analysis of language (following Wittgenstein’s Tractatus), on the need to develop a rigorous epistemology for a scientiﬁc vision of the world and on the unity of scientiﬁc explanation and knowledge in general (Stadler, 2001).3 New inquiries into epistemology and logic were expounded in a different way also in the Mathematical Colloquium (founded by Menger in 1928), as well as ‘new applications of the exact sciences to problems of a sociological character’, as Menger himself put it in an Italian paper (Menger, 1935; recently translated as Menger, 1998). Ofﬁcial members of the Colloquium were Abraham Wald, Kurt Go¨del, Franz Alt and Georg No¨beling; many other guests, such as John von Neumann and Oskar Morgenstern, gave lectures.4 There was general agreement within the Mathematical Colloquium that logical deductive treatment should be given exact science (from geometry to arithmetic) and that ethics and economics should be given a new foundation based on logical coherence (Menger, 1935). In 1932, Menger delivered a lecture in Vienna on ‘the new logic’, which was published a year later and then translated into English and published in 1937 (Menger, 1937, 1979). The paper was a historical reconstruction of the crisis of logic in the 19th and the 20th centuries. Menger’s paper described the evolution of certain exact sciences (like geometry and physics) over the past decades as a development towards more scientiﬁc results. Menger rhetorically claimed that ‘[just] one subject, however, is generally supposed to be unchanging and unshakable. That subject is logic’ (Menger, 1937, p. 300).5 According to Menger, the refoundation of logic had consisted of ﬁve developments: a calculus of classes (Boole’s, Pierce’s and Schro¨der’s algebras of logic); a calculus of propositions (Frege, Whitehead-Russell, Hilbert-Ackermann, Carnap), which led to Wittgenstein’s deﬁnition of tautologies as ‘propositions always true’; a calculus of functions (Frege, Pierce); a calculus of relations (ﬁrst formulated by Peano); and ﬁnally, an expanded calculus of functions developed by Russell. According to Menger, the ﬁrst three developments ‘may be interpreted as mere reﬁnements on the old logic’; the fourth and ﬁfth ones were revolutionary and had provoked an outright crisis in the old logic.

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Menger maintained that there were three possible solutions for that still ongoing crisis: Russell’s logicism, Hilbert’s formalism and Brouwer’s intuitionism. Russell thought that language could be analyzed into a perfect logical structure and also that mathematics could become a part of logic; his position was shared by the scientists and philosophers of the Wiener Kreis, who were also deeply inﬂuenced by Wittgenstein, for whom logic was something that both the world and the language must have in common (it is only because language has something in common with the world that it can be used to picture the world; so it is only because of logic that our sentences have meaning at all). Hilbert’s formalism was based on the axiomatization of logical and mathematical theory: every branch of mathematics starts with a number of axioms or statements that are assumed to be true and with which all other statements in that branch can be proven. And the system is consistent. Unlike logicism and Hilbertism, intuitionism (Kronecker, Brouwer, Heyting) was a non-classical logic that refused to reduce either mathematics to logic or logic to mathematics. This was because mathematical and logical proofs simply work differently: in particular, Brouwer, Menger’s mentor in the mid-1920s, showed that in some cases, the law of excluded middle does not hold in mathematics (it is impossible in inﬁnite sets). In spite of the fact that since the late 1920s, Menger had clearly rejected Brouwer’s intuitionism, Brouwer’s inﬂuence on Menger was kept alive, so to speak, by his interest in non-classical (or non-standard) logics. Many-valued logic as a separate subject was created by the Polish logician and philosopher Łukasiewicz in 1920 and developed ﬁrst in Poland.6 Essentially, in parallel to the Łukasiewicz approach, the American mathematician Post in 1921 introduced the idea of additional truth degrees and applied it to problems of the representability of functions. Menger severely criticized the idea that logic was unique. Contrary to standard logic (ordinary two-valued logic), he strongly supported the n-valued logics of Post and Łukasiewicz in which neither the law of excluded middle nor the law of non-contradiction operated.7 Menger considered the so-called uniqueness of logic to be a dogma, and he was highly sceptical of the notion; he shared with Post and Łukasiewicz the idea that logic could take various forms.8 Menger’s acceptance of many-valued logics ‘drew him gently but inexorably away from Hanh on logic and Neurath on the unity of science’ (Menger, 1979, p. 16). As Menger himself stated later, his tolerant attitude towards the logical foundations of mathematics was very close to Popper’s criticism of

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essentialism, that is, ‘the futility of deﬁning a science [and] the arbitrariness of precisely circumscribing its object’ (ibidem).

3. TOWARD A LOGIC OF ETHICS Menger’s book Moral, Wille und Weltgestaltung, Grundlegung zur Logic der Sitten (Menger, 1974)9 represented a formal approach to moral problems, based on an unambiguous deﬁnition of ethical language; a form of positive, or not normative, ethics. When Menger’s book appeared in Vienna, in 1934, it was welcomed by Oskar Morgenstern as ‘the only examination of a strictly formal nature about social groups’ (Leonard, 1995, p. 746) and by Karl Popper as ‘one of a few books in which the author attempts to depart from the stupid talk in ethics’ (Popper, 1944–1945). Both Popper and Menger, in fact, totally disagreed with the neopositivists of the Vienna Circle, who – inﬂuenced by Wittgenstein – considered norms to be ‘meaningless’ because they cannot be deduced from facts. According to Popper, Menger did not fall in this fallacy and tried to develop a positive logic of norms. Also, Hayek paid attention to Menger’s work on ethics. When discussing compatibility among the plans of different individuals (Hayek, 1937), he explicitly cited Menger’s investigations in social theory and he hoped that he would be able develop his ﬁrst insights into an ‘exact sociological theory’ (Hayek, 1937) as he had promised. In a footnote, Hayek wrote, ‘It has long been a subject of wonder to me why there should have been no systematic attempts in sociology to analyse social relations in terms of correspondence and non-correspondence, or compatibility and non-compatibility, of individual aims and desires. It seems that the mathematical technique of analysis situs (topology) and particularly such concepts developed by it as that of homeomorphism might prove very useful in this connection, although it may appear doubtful whether even this technique, at any rate in the present state of its development, is adequate to the complexity of the structures with which we have to deal. A ﬁrst attempt made recently in this direction by an eminent mathematician (Karl Menger, 1934) has so far not yet led to very illuminating results, but we may look forward with interest to the treatise on exact sociological theory Menger has promised’ (Hayek, 1937, p. 38).10 Menger was convinced that ethics as normative science was impossible because experience reveals only what ‘is’, not what ‘ought to be’. His book on ethics should – he declared – be considered an application of ‘exact thinking in the ﬁeld of ethics’ free from any inﬂuence ‘by subjective feelings’

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(Menger, 1974, p. 1). He would ‘steer clear of the search for ultimate meanings or essences such as ‘‘the concept of morality’’ or ‘‘the principle of virtue’’’ (Leonard, 1998, p. 20). Menger’s aim was to identify ‘speciﬁc rules of conduct’ in a strict logical sense. He maintained that it is necessary to free ethics from metaphysical inquiries into ‘the concept of morality’ or ‘virtue’ or ‘the essence of good’, and he sought ‘to conﬁne [these] cognitive studies to facts’. Menger’s investigations were concerned ‘with the application of mathematical modes of thinking and deal only with the formal side of questions’. His ethics was a formalist model: not a scale of values but rather a set of rules, a sort of Hilbertian programme extended to the moral ﬁeld. The main reference in formalist ethics is of course to Kant and his ‘practical reason’ founded on three categorical imperatives (based on the conformity of any action with universality, autonomy and to notion of humans as ends in themselves) in opposition to hypothetical ones (religion, laws, hedonistic pleasure and personal ideals), which do not confer morality on an action. Menger stated that his morality was very far from Kantianism. He considered ethics to understand individual decisions, and subsequently social organizations, and he argued that the Kantian categorical imperative was neither a necessary nor a sufﬁcient condition for constituting cohesive (or peaceful) groups. Hence, ethical imperatives are always hypothetical, never categorical. The unresolved question in Kantianism was ‘what concrete precepts result from the categorical imperative in speciﬁc situations’ (Menger, 1974, p. 9). To answer this question (how to apply the categorical imperative to a decision), it is necessary to consider a decision that implies a cognitive activity. This is the point where, according to Menger, ethics and logic are strictly connected. The ethical problem in Menger’s thought was understanding how social coexistence comes about in concrete situations where ‘there are several mutually incompatible decisions to consider’ (Menger, 1974, p. 10), and the categorical imperative needs supplementary stipulations and additional norms to generate the well-being of a group. An epistemology of morality could be developed by founding it on a logical basis expressed by mathematical tools able to construct an ‘ethics without morality and immorality’. Menger consequently focused his attention on a set of decisions by individuals whether or not to adhere to certain norms: the main point was understanding why, when and how a person accepts a deﬁnite system of norms, basing his decisions on that system. According to Menger, rational foundations for decisions are possible. By way of example, he cited the idea of maximization in economic theory and stated that if economists wished to claim that ‘the optimal

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distribution of commodities and the greatest welfare of mankind could be achieved under certain system of organization which they describe’, they ‘indeed must ﬁrst take the trouble to study logic’ (Menger, 1974, p. 31), and he added ironically that such an exercise would ‘without doubt increase their self-criticism’. Cognitive action is the starting point for the explanation of how individual decisions and social changes arise, although volition has a central role as well. But the problem is that, if we can study only cognition (the logical side of decision-making), volition remains a matter of individual taste. According to Menger, it is possible to develop a logic of ethics, which considers every single norm or system of norms within a social group in which people (in connection with any norm) can approve the norm, disapprove of it or be indifferent to it. Thus, for every norm (n), people are always divided into three groups: those who always follow the rule; those who sometimes do and sometimes do not follow it and those who never follow it. It may happen that one (or two) of these sets is empty; if everybody always (or never) adheres to the norm the other two sets are empty. If someone is indifferent, he may or may not follow the rule: he hence forms another group comprising indifferent people who sometimes follow the rule and sometimes does not, so that the groups become three in number. Then, in a society that comprises a system of n norms, there are 3n possible groups. These groups are 1. jointly exhaustive ¼ each member of the class belongs to one of them; 2. mutually disjoint ¼ there are no members in common: ‘each member of the class belongs to one and only one of the total groups of consentience’; 3. within a group, agreement relations are transitive: two individuals (P) who completely agree with a third also completely agree with each another (Menger, 1974, pp. 43–44). In 1938, Menger published a paper in which he explained more clearly the subject of his previous book, that is, how it can be possible to form a cohesive group (Menger, 1938). If we consider a group of men G as a total group, it can be divided into two fundamental groups with no overlaps: G1 and G2 (e.g., men and women). Each member of G has four possible attitudes towards the association (G1 or G2); with everybody (G1,2) or with nobody (G0). Groups are thus represented as Gba , where a is membership of a group and b is compatibility or mutual acceptance. Thus, we have eight main classes: 1;2 0 0 G11 ; G21 ; G12 ; G22 ; G1;2 1 ; G2 ; G1 and G2

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1;2 Groups G11 ; G22 ; G1;2 1 ; G2 are consistent ¼ any member of G is willing to associate with any member of his own group (the last two also with members of the other group). Groups G21 ; G12 ; G01 G02 are inconsistent ¼ no member of G is willing to associate with a member of his own group (the last two with nobody). Groups G11 G22 are antipathetic each other ¼ any member of G is willing to associate only with a member of his own group. 1 1;2 Groups G21 ; G1;2 2 and G2 G1 are mixed. Menger’s aim is to understand when is it possible to unify all the members of G into a consistent group. The answer lies in the tolerance of the mixed 1;2 groups G1;2 1 and G2 that makes possible the overlapping between G1 and G2. Kant was the one who gave a great importance to the formal aspects of ethics, but his categorical imperative is unable to guarantee general harmony because it operates only in groups G11 and G22 , that is, only in those groups in which there is no need to ﬁnd a tool of cohesiveness or peace, because they are formed by member who shared the same set of rules. In the postscript to the English edition of his book (Menger, 1974), Menger summed up its purpose as an attempt to apply logico-mathematical thought to ethical matters. He wrote that it was possible to construct an exact system of thought in regard not to dealing with personal dilemmas but to treating social problems in what today is called group decision theory. On 27 December, 1961, Oskar Morgenstern wrote to Menger, ‘I do not know whether you have followed the literature on the problem of ‘‘fair division’’. You will ﬁnd an interesting discussion in the book by Luce and Raiffa, Games and Decisions. This whole area has a close connection with your work on the logic of ethics, which is unfortunately quite unknown to these authors because of the time of its publication and its being available only in German. ( . . . ) The relation of the explorations in your book to game theory and fair division certainly bears further study, and I would be happy if I could stimulate you to resume when they were begun’ (Karl Menger papers). Morgenstern was referring to the well-known book by Luce and Raiffa (1957) and dedicated to the memory of John von Neumann. That book was about game theory and placed particular emphasis on the social science point of view. Chapter thirteen dealt with individual decision-making under uncertainty, and Chapter fourteen with group decision-making, devoting a section to ‘games of fair division’.11 Menger returned to his paper many years later, in 1983, when he proposed a general criterion for explaining how cohesive social groups come into being. The model was the same as in 1938, but the paper comprised

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some additions on the ethics of the Vienna Circle and on game theory that are worth recalling. Menger criticized both Kant’s formalistic morality and the ethics proposed by the Vienna Circle and founded on Wittgenstein’s Tractatus. According to Menger, Kant’s categorical imperative was unable to form a cohesive group because ‘in most speciﬁc situations it is impossible to deduce speciﬁc precepts for behaviour unless the imperative is supplemented by the value judgments’, and it is very difﬁcult to ﬁnd a ‘maxim that can become a general law’. Menger was also dissatisﬁed with the Vienna Circle’s notion that, after the complete elimination of value judgments from ethics, only historical and ethnographical descriptions of moral beliefs and conditions were possible. The multiplicity of beliefs and evaluations seemed to recommend the formal study of inner judgments and attitudes among human groups with incompatible wishes and of conﬂicting decisions between individuals and individuals, individuals and groups, groups and groups (Menger, 1983). In this paper, Menger claimed that experiments should ‘test the presence and demonstrate the evolution of ethical norms’ (Perlman & McCann, 1998, p. 441).

4. MENGER AND HIS IDEA OF META-ECONOMICS In his paper on the ‘Law of Diminishing Returns. A study in metaeconomics’, published in 1936,12 Menger was the ﬁrst of using this term ‘meta-economics’ to build a meta-theory of economics (Buracas, 2004, p. 1). As Menger himself explained, ‘following a suggestion of Hilbert, modern logicians refer to the study of the logical relations between the statements of a theory as the corresponding meta-theory. In this terminology, the contents of the present paper can be described as a chapter in meta-economics’ (Menger, 1979, p. 280). When Menger used the expression ‘metaeconomics’, he was referring to the ‘logical relations between economic propositions as distinct from the study of economic material’ (Conrides, 1983, p. 4). Hence, in this paper, Menger focused on those aspects of economics to which mathematics could be applied according to a meta-mathematical model a` la Hilbert and he deﬁned it as ‘the ﬁrst instance in economics of a clear separation between the question of logical interrelations among various propositions and the question of empirical validity’ (Menger, 1979, p. 300). The occasion was given by a talk between Menger and Ludwig von Mises who had claimed that ‘certain propositions of economics can be

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proved [and] as an example he mentioned the law of diminishing returns [referring him] to the literature for the proofs’ (Menger, 1979, p. 279). Menger wrote a very articulate paper in which he introduced the fact that a sequence of inferences becomes a proof when propositions are assumed to have the same meaning and when logical quantiﬁers (‘many’, ‘all’, ‘some’ and so on) are properly applied, but this methodological and logical operation is often made without any attention and rigor and often what is regarded as a proof is not ‘precise enough to be subject to deductive treatment’ (ibid.). Hence, Menger analyzed the proof of the law of diminishing return on land according to the formulation given by Wicksell in 1909, Bo¨hm-Bawerk in 1912 and von Mises in 1933: ‘additional applications of capital and labor on a piece of land increase the total product, but after a certain point this output increases relatively less than further costs’ (Menger, 1979, p. 281). The law of diminishing imply law of diminishing product increment and of diminishing average product. These two laws are not the same, they are different. The law of diminishing product increment claims an increase of cost outlay yields a smaller increase of the product when added to a larger outlay than when added to a smaller outlay, provided that both exceed a certain outlay level (which may depend on the amount of the land used. (Menger, 1979, p. 281)

The law of diminishing average product claims as the cost outlay rises, the average product of every piece of and falls after a certain point has been reached. (Menger, 1979, p. 281)

Menger showed how economists often claim that the law of diminishing product increment and of diminishing average product are the same law expressed in different words: this is a logical mistake. Moreover, the law of diminishing returns could be divided into many other sub-propositions that differ from the law and each other, hence ‘the question arise, which of these two propositions do economists mean when they speak of the law of diminishing return’ (Menger, 1979, p. 283). Menger’s conclusion is that there are many forms of the law of return; some formulation is ‘empirical valid’, but the meta-problem is ‘how the various propositions are related, which ones are consequences of others – these and similar questions are purely logical and have nothing to do with experience’ (Menger, 1979, p. 300).

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5. SOME CONCLUDING REMARKS Now that some features of Menger’s thought on economics and ethics have been explained, some considerations can be ventured. The main characteristic of his reﬂections was the endeavour to ﬁnd a logical pattern able to explain the mental mechanisms involved in decision-making by individuals and groups. Menger used an individual approach in his economic analyses (expected utility in his commentary on the St. Petersburg paradox) and a holistic approach when discussing the cohesiveness of groups. Like the early Vienna Circle, he was ﬁrmly convinced that the social sciences could be treated from a scientiﬁc point of view, but with some differences. Firstly, he rejected the idea that social sciences such as ethics and economics were, following Wittgenstein’s contention, to be considered ‘meaningless’ because they were too deeply embedded in human actions. He shared this critique (albeit in very different ways that led to very different results) against the neopositivism of the Vienna Circle with Popper and Hayek. Inﬂuenced by the debate on the relationship between mathematics and logics of those years, he considered it urgent to re-build social science and he chose an axiomatic and deductive model, applied to social contexts (to give coherence to the system). Nevertheless, Menger’s demystiﬁed ethics and Menger’s meta-economics present some problems. Menger tried to develop an ‘exact theory of social groups and relations’ based on the idea that there exists a logic able to drive social behaviour in a different way from the logic of an individual to determine when a group is consistent. As we have seen, Menger deﬁned groups by starting from a division into two non-overlapping groups and then building subgroups until cohesive groups are found. This procedure has no faults from a formal point of view, but in realistic terms, it seems rather problematic. Firstly, uncertainty emerges when deﬁning the behaviour of groups; even if we accept the deﬁnition of cohesiveness, the model remains static and it does not envisage the possible presence of groups with a separate identity within cohesive sets (this is not the case of women and men, but it could be the case of smokers and non-smokers: there could be a class of people who normally do not smoke but sometimes smoke: in this case what is their behaviour?).13 Moreover even if the formal structure of ethics in Menger’s thought seems to work, it is not clear what he meant for ‘norm’; Menger did not gave an exact deﬁnition of norm: his formal ethics is anti-Kantian (Menger rejected the idea according to which ‘ought’ implies ‘can’) and it is a sort of revalidation of Hume’s law (you

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cannot deduce an ‘ought’ from an ‘is’), but, above all in his second paper on social groups (1938), even if ethics is logically structured on an axiomatic systems of groups, it seemed to be an empty box able to explain the characteristics of peaceful groups but unable to describe in which way social dynamics are formed. About Menger’s meta-economics, there are some problems as well. Metaeconomics checked the validity of the economic discourse in a logical perspective, but in which sense a valid proposition can be considered true? Menger considered valid every proposition that is deduced in a logical proper way from another one, but his focus on the logical coherence of the economic discourse led apart the question of the urgency of empirical validity of a preposition. It was not by chance that Hayek was very disappointed by Menger’s following inquiries into social group decision theory after the publication of his book on morality: when he reprinted Economics and Knowledge (1937) as a chapter in Individualism and Economic Order (1948), the footnote in which Hayek claimed that he was waiting for the following research of Menger on social groups had been deleted.14

NOTES 1. 1st toss: payoff ¼ 1; 2nd toss: payoff ¼ 4 and so on and EðwÞ ¼ ð1=2n Þ 2n ¼ ð1=2Þ 2 þ ð1=4Þ22 þ ð1=8Þ23 þ ¼ 1 þ 1 þ 1 þ ¼ 1. 2. In the St. Petersburg case, the value of the game to an agent (assuming initial wealth is zero) is EðuÞ ¼ ð1=2n Þ uð2n Þ ¼ ð1=2Þ ð2Þ þ ð1=4Þ uð22 Þ þ ð1=8Þ uð23 Þ þ o1, which Bernoulli conjectured is ﬁnite because of the principle of diminishing marginal utility. 3. These matters formed the core of the international movement in the 1930s, when the philosophical position of the Vienna Circle was most prominently represented by Carnap’s analysis of language and by Neurath’s physicalism and its program for a uniﬁed rational reconstruction of science (including the human sciences) (Stadler, 2006). 4. Morgenstern and von Neumann were not ofﬁcially members of the Colloquium, but their role in it, their inﬂuence on the subsequent development of its inquiries and their friendships with its members were so deeply important and widely recognized that they may be included in the Colloquium (Weintraub, 1983; Punzo, 1989; Golland & Sigmund, 2000). As the Karl Menger archive testiﬁes, relations between Menger and the other members of the Mathematical Colloquium were very close during the 1930s and in the following decades, when most of them moved to the United States. Menger’s friendship with Oskar Morgenstern was particularly longlasting because of their shared interest in the relationship among mathematics, logic

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and economics (Leonard, 1995, 1998). They also had amicable relations with Go¨del, who was constantly in contact with both of them. 5. Aristotelian logic founded on the three principles of identity, non-contradiction and excluded middle had ﬁrst been criticized by Leibniz, who realized that the logic based on ‘subject-predicate propositions’ should be replaced by a logic of relations. Leibnitz consequently proposed his lingua universalis, ‘which should permit all scientiﬁc propositions to be stated in precise form . . . and treat mathematically all methods of inference’ (Menger, 1937, p. 301). Leibniz’s endeavour was obscured by Kantian logic, which took his rightful place in the history of science at precisely the time when mathematicians, whose efforts were directed at enumerating all the axioms of geometry and arithmetic, were asserting the inadequacy of the old logic for that purpose. 6. Łukasiewicz’s prime intention was to use a third, additional truth value for ‘possible’, and thereby model the modalities ‘it is necessary that’ and ‘it is possible that’, although its intended application to modal logic did not materialize. The outcome of these investigations, however, was the Łukasiewicz systems and a series of theoretical results concerning these systems. 7. In particular, Menger regarded Łukasiewicz’s three-valued logic as being able to include uncertainty: ‘the third value being the excluded middle of the traditional twovalued system’ (Leonard, 1998, p. 16). 8. Menger gave the name of ‘logical tolerance’ to this theoretical stance that there is not one but many logics (any expression in language is acceptable as long as there are sufﬁcient rules governing its logical application). On several occasions, he defended what Carnap later called the principle of ‘logical tolerance’ (Carnap, 1934) and objected to employment of the word ‘meaningless’ to signify that, for example – in Wittgenstein’s terms – all mathematical propositions are tautologies. 9. It was translated into English as Morality, Decision and Social Organization. Toward a Logic of Ethics, only in 1974, and printed as the Vienna Circle Collection’s sixth volume (Menger, 1974). 10. Hayek was referring to a paper on social relations and groups that Menger had presented in the same year at the Third Annual Conference of the Cowles Commission in Colorado: ‘An Exact Theory of Social Relations and Groups’, in Report of Third Annual Research Conference on Economics and Statistics, Cowles Commission for Research in Economics, Colorado Springs, 1937, pp. 71–73), later published (Menger, 1938). 11. Luce and Raiffa maintained that a fair rule is a mode of conduct considered social desirable. The rules are so mixed that when players act rationally following their own selﬁsh interest, the outcome is shared by all the participants, and the outcome of a fair procedure is Pareto optimal. A group decision welfare function is built by passing from individual values to social preferences. The main difﬁculty is devising a system that is sufﬁciently egalitarian and ﬂexible to cope with the dynamic vicissitudes of individual tastes. In Arrow’s social welfare problem, there are preference rankings of alternatives by members of a society and a fair method for aggregating this set of individual rankings into a single ranking for the society (from a n-tuple of individual rankings to a single ranking for the society to construct a social welfare function). As well known, Arrow’s model states ﬁve requirements of fairness, and its result is the famous impossibility theorem.

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12. In 1936, Menger published ‘Bemerkungen zu den Ertragsgesetzen’, which was translated into English in 1954 as ‘Remarks on the Law of Diminishing Returns. A Study in Meta-Economics’, and was, according to Schumpeter, ‘a shining example of the general tendency towards increased rigor that is an important characteristic of the economics of our own period’ (Schumpeter, 1954, p. 1037). 13. In truth, Menger was partly aware of this criticism, and in his last paper (Menger, 1983), he proposed introducing social experiments under minimal assumptions, as well as arrangements made feasible by computers. 14. I am indebted to Professor Roger Koppl for pointing this out to me.

REFERENCES Becchio, G. (2008). The complex role of Karl Menger in the Viennese economic theory. Review of Austrian Economics, 21, 61–79. Buracas, A. (2004). On paradigm of metaeconomics: Essence and sense. Management of Organizations: Systematic Research, 54, 1–12. Carnap, R. (1937 [1934]). The logical syntax of language. London: Kegan Paul. Conrides, T. (1983). Karl Menger’s contributions to social thought. Mathematical Social Sciences, 6, 1–11. Dawson, J. W., Jr. (1998). Logical contributions to the Menger colloquium. In: E. Dierker & K. Sigmund (Eds), Ergebnisse eines Mathematischen Kolloquiums (pp. 33–42). Wien, New York: Springer. Dawson, J. W. Jr., & Sigmund, K. (2006). Go¨del’s Vienna. The Mathematical Intelligencer, 28, 44–53. Debreu, G. (1998). Economics in a mathematics colloquium, Foreword to Menger (1998) Ergebnisse eines Mathematischen Kolloquiums. Dore, M., Chakravarty, S., & Goodwin, R. (Eds). (1989). John von Neumann and modern economics (pp. 129–165). Oxford: Clarendon Press. Feigl, H. (1969a). The Wiener Kreis in America. In: D. Fleming & B. Baylin (Eds), The intellectual migration. Cambridge, MA: Harvard University Press, reprinted in Feigl 1981, pp. 57–93. Feigl, H. (1969b). The origin and spirit of logical positivism. In: Aichenstein & Barker (Eds), The legacy of logical positivism (pp. 3–24). Baltimore: Johns Hopkins Press. Frank, P., (1947:mei-1948:mei). The institute for the unity of science, its background and its purpose. Synthese, 6 (3/4), 160–167. Friedmanm, M. (1995). Karl Menger reminiscences of the Vienna Circle and the mathematical colloquium. ISIS, 86(4), 675–676. Gilles, D. (1981). Karl Menger as a philosopher. The British Journal for the Philosophy of Science, 32(2), 183–196. Golland, L. A. (1996). Formalism in economics. Journal of the History of Economic Thought, 18, 1–12. Golland, L. A., & Sigmund, K. (2000). Exact thought in a demented time: Karl Menger and his mathematical colloquium. The Mathematical Intelligencer, 34–45. Hahn, H., Neurath, O., & Carnap, R. (1929). Wissenschaftliche Weltauffassung. Der Wiener Kreis. Wien: Arthur Wolf Verlag. Hayek, F. A. (1937). Economics and knowledge. Economica, 4, 33–54. In Hayek, 1948, 33–56. Chicago: University of Chicago Press.. Hayek, F. A. (1948). Individualism and economic order. Chicago: University of Chicago Press. Janik, A., & Toulmin, S. (1996). Wittgenstein’s Vienna. Ivan R. Dee Publisher: Chicago.

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Jabloner, C. (1998). Kelsen and his circle: The Viennese years. European Journal of International Law, 9, 368–385. Kass, S. (1996). Karl Menger. Notice of the American Mathematical Society, 43(5), 558–561. Kaufmann, F. (1978). The inﬁnite in Mathematics. Logico-mathematical writings. In: B. McGuinness (Ed.), Vienna Circle collection (Vol. 9). Dordrecht and Boston: Reidel Publishing Co. Kurrild-Kiltgaard, P. (2003). The Viennese connection: Alfred Schutz and the Austrian School. The Quarterly Journal of Austrian Economics, 6(2), 35–67. Leonard, R. (1995). From Parlor Games to social sciences: von Neumann, Morgenstern, and the creation of game theory, 1928–1944. Journal of Economic Literature, 33, 730–761. Leonard, R. (1998). Ethic and the excluded middle. Karl Menger and the social science in Interwar Vienna. ISIS, 89, 1–26. Luce, D. R., & Raiffa, H. (1957). Games and decisions. New York: Wiley. Menger, K. (1935). Sull’ Indirizzo di Idee e sulle Tendenze Principali del Colloquio Matematico di Vienna. Annali della Scuola Normale Superiore di Pisa Scienze ﬁsiche e matematiche, II, IV, 327–339. Menger, K. (1937). The new logic. Philosophy of Science, 3, 299–336. Menger, K. (1938). An exact theory of social groups and relations. The American Journal of Sociology, 5, 790–798. Menger, K. (1967). The role of uncertainty in economics. In: M. Shubik (Ed.), Essays in mathematical economics in honor of O. Morgenstern (pp. 211–223). Princeton: Princeton University Press. Menger, K. (1974). Morality, decision and social organization toward a logic of ethics. Vienna Circle collection, 6. Reidel Publishing Co: Dordrecht and Boston. Menger, K. (1979). Selected papers in logic and foundations, didactics, economics. Vienna Circle collection, 10. Reidel Publishing Co: Dordrecht and Boston. Menger, K. (1980). ‘‘Preface’’ to Hans Hahn, empiricism, logic and mathematics. Philosophical papers, pp. 9–18. In: McGuinness B. (Ed), Vienna Circle collection, 13, Dordrecht and Boston: Reidel Publishing Co. Menger, K. (1982). Memories of Moritz Schlick. In: G. Eugene (Ed.), Rationality and science: a memorial volume for Moritz Schlick in celebration of the centennial of his birth. Wien: Springer. Menger, K. (1983). On social groups and relations. Mathematical Social Sciences, 6, 13–26. Menger, K. (1994). Reminiscences of the Vienna Circle and the mathematical colloquium. Dordrech: Kluver Academic Publishers. Menger, K. (1998). Ergebnisse eines mathematischen Kolloquiums. In: E. Dierker & K. Sigmund (Eds.), Wien, New York: Springer. Menger, K. (2002). Selecta Mathematica vol. 1. In: B. Schweizer, A. Sklar, K. Sigmund, P. Gruber, E. Hlawka, L. Reich & L. Schmetterer (Eds), Wien: Springer. Menger, K. (2003) Selecta Mathematica vol. 2. In: B. Schweizer, A. Sklar, K. Sigmund, P. Gruber, E. Hlawka, L. Reich & L. Schmetterer (Eds), Wien: Springer. Menger, K. Karl Menger Papers, Perkins Library, Duke University: Durham, NC. Morris, C. (1960). On the history of the international Encyclopedia of uniﬁed science. Synthese, 12(4), 517–521. Morris Charles Correspondence, Regenstein Library, University of Chicago. Mulder, H. (1985). The Vienna Circle Archive and the literary remains of Moritz Schlick and Otto Neurath. Synthese, 64(3), 375–387.

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Neumann von, J., & Morgenstern, O. (1944). Theory of games and economic behavior. Princeton: Princeton University Press. Neurath, O. (1973). Empiricism and sociology. Dordrecht: Reidel. Neurath, O. (2004). In: T. E. Uebel & R. Cohen (Eds), Economic writings. Selection 1904–1945. Dordrecht: Kluwer Academic Publishers. Perlman, M., & McCann, R. (1998). The pillars of economic understanding. ideas and traditions. Ann Arbor, Michigan: The University of Michigan Press. Popper, K. (1944-1945). The open society and its enemies. London: Routledge. Punzo, L. (1989). Karl Menger’s mathematical colloquium. In: M. Dore, S. Chakravarty & R. Goodwin (Eds), John von Neumann and modern economics (pp. 129–165). Oxford: Clarendon Press. Punzo, L. (1991). The school of mathematical formalism and the Viennese Circle of mathematical economists. Journal of the History of Economic Thought, 13, 1–18. Richardson, A., & Uebel, T. (2007). The Cambridge companion to logical empiricism. Cambridge: Cambridge University Press. Sarkar, S., & Pfeifer, J. (2006). The philosophy of science. An encyclopedia. New York-London: Routledge. Schumpeter, J. A. (1954). History of economic analysis. New York: Oxford University Press. Schweizer, B. (2002). Introduction. In: B. Schweizer, A. Sklar, K. Sigmund, P. Gruber, E. Hlawka, L. Reich & L. Schmetterer (Eds), K. Menger Selecta Mathematica (Vol. 1, pp. 1–5). Wien, New York: Springer. Sigmund, K. (1998) Menger’s Ergebnisse – a biographical introduction in K. Menger (1998) Ergebnisse eines Mathematischen Kolloquiums, pp. 5–31. Sigmund, K. (2002) Karl Menger and Vienna’s golden autumn. In: K. Menger Selecta Mathematica Vol. 1. B. Schweizer, A. Sklar, K. Sigmund, P. Gruber, E. Hlawka, L. Reich, L. Schmetterer (Eds), pp. 7–21. Stadler, F. (2001). The Vienna Circle. Studies in the origins, development, and inﬂuence of logical empiricism. Wien, New York: Springer. Stadler, F. (2006). The Vienna Circle. In: S. Sarkar & J. Pfeifer (Eds), A philosophy of science, an encyclopedia (pp. 858–863). London: Routledge. Tintner, G. (1968) Methodology of mathematical economics and econometrics. In: The International Encyclopedia of Uniﬁed Science, Vol. 2, n. 6, Chicago: University of Chicago Press. Waismann, F. (1951). Introduction to mathematical thinking. New York: Frederick Ungar Publishing Co. Weintraub, E. R. (1983). On the existence of a competitive equilibrium: 1930–1954. Journal of Economic Literature, XXI, 1–39. Weintraub, E. R. (2002). How economics became a mathematical science. Durham: Duke University Press.

KARL MENGER’S DRAFTS – EDITOR’S NOTE ON THE TEXT Before Menger’s drafts are presented, an explanation of their main similarities and differences is required.1 The ﬁrst three chapters of Menger’s Reminiscences deal with the historical, cultural and philosophical background in Vienna. Chapters 1 and 2 are quite similar to those of Menger’s drafts. There is only a deleted annotation on ‘the Mises Circle’: after a description of the economists belonging to that circle (Haberler, Hayek and Machlup) as those ‘who were later to continue the tradition of Austrian marginalism in the English-speaking world’ (Menger, 1994, p. 17), Menger’s drafts contain the following words omitted from Reminiscences: ‘though coloured, especially in Hayek and Machlup, with the extreme advocacy of laissez-faire preached by Mises himself and with his idiosyncrasies against the use of any mathematics in economics’. Menger’s drafts of Chapter 3 are organized differently, and they comprise numerous unpublished parts, although there are some similarities as well. In the drafts there is a long description of the philosophical climate in Vienna before and during the interwar period, which includes two sections entitled by Menger ‘Pre-Circle Austrian Philosophy’. There are two sub-sections on the remote predecessors of the Circle (Bolzano, Brentano, Meinong and Husserl) and on its immediate predecessors (Mach, Boltzmann and Mauthner) – and obviously a ﬁnal section on the Vienna Circle. These unpublished parts of what is now Chapter 3 are printed as Part I of this book. Thereafter, Chapter 4 of Menger’s Reminiscences deals with Menger’s curves and dimension theory (the reason why Menger was invited to attend a meeting of the Circle after his return from Amsterdam to Vienna in 1927). This part is quite similar in Menger’s drafts, but it is included as the ﬁnal part of a chapter on Brouwer. Chapter 5 of Menger’s Reminiscences contains some descriptions (‘vignettes’) of the members of the Vienna Circle. In Menger’s drafts these ‘vignettes’ were included in Part I as its last section (The Vienna Circle). Chapters 6–10 of Menger’s Reminiscences deal with Wittgenstein, his philosophical thought, the inﬂuence of his Tractatus on the Circle, and ﬁnally 41

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reminiscences about his family. These are quite similar to those in Menger’s drafts. We know from the correspondence between Menger and McGuinness that Menger was a friend of Clara, Ludwig’s elderly aunt, and that this fact had impressed McGuinness. Curiously enough, one ﬁnds the following deleted remark in Menger’s drafts: ‘Many years later I learned, I believe from Dr B. McGuinness, that a photo of Clara was, besides one of his father, the only family portrait that Ludwig took with him when leaving Austria’.2 The Wittgenstein chapters in Menger’s Reminiscences are followed by Chapter 11, ‘Discussions in the Circle 1927–30’. This chapter is only a couple of pages in length, and it is forwarded by a note from the editors informing the reader that Menger had planned a complete chapter on this subject, but they had decided to ‘print just the scant material concerning the period before Menger’s visit to Poland’ (Menger, 1994, p, 140), because the following chapter dealt with ‘Poland and the Vienna Circle’(Chapter 12). The editors also called the reader’s attention to the fact that some parts of the deleted material could be found in other publications by Menger (Menger, 1980, 1982). In Menger’s drafts, these unpublished parts are denoted ‘Book B’, and so are all the pages dealing with Menger’s more autobiographical memoirs. Book B covers Menger’s journey to the United States in 1930/1931 (Chapter 13 of Menger’s Reminiscences) and discussions within the Circle during the mid-1930s until Schlick’s assassination in 1936 (Chapters 14 and 15). The unpublished pages of Book B are now printed in Part II of this book. They constitute Menger’s Viennese autobiography from 1923 to 1938. The footnotes are taken from Menger; editorial notes are speciﬁed, written in italics and put in square brackets.

NOTES 1. For more speciﬁc details, see the editorial notes in the text. 2. Menger was referring to the above-quoted letter of 2 August 1974 from B. McGuinness.

PART I THE PHILOSOPHICAL ATMOSPHERE IN VIENNA BEFORE AND DURING THE INTERWAR PERIOD [The following pages are made up of those unprinted parts of chapter three of Menger’s Reminiscences which deals with the philosophical background of the Vienna Circle. Some overlaps and reproductions are inevitable for the coherence of the discourse.]

Like many continental cities, Vienna had a large number of coffee houses where people sat reading newspapers or meeting acquaintances and carrying on conversations. The productive coffee house discussions in Vienna were directed more towards belles lettres than in other cities and less towards logic and mathematics. For my part I untypically disliked the atmosphere of those places. But this is not the main reason why I cannot report a great deal that has come out of them. For if there had been much I should have heard about it. It was in Vienna that several important Western writings on empiricist philosophy were translated and made more widely known to the readers of the German language. Theodor Gomperz, the author of a widely read book, Greek Thinkers, brought out J. S. Mill’s works in German in the 1870s. Later, W. Jerusalem translated James. Mach’s student, Fritz Adler, published a German translation of P. Duhem’s important book La theorie physique, son objet et sa structure twenty years before an English translation appeared under the title Physics: Its object and its Structure. Another book by Duhem, L’e´volution de la me´canique, was translated in 1912 by Philipp Frank, then Dozent at the University of Vienna, who was to play a great role in the Vienna Circle. At the University of Vienna after Boltzmann’s death, the chair for the philosophy of inductive sciences remained vacant for a few years and then went to the Austrian philosopher, A. Sto¨hr (1855–1921), who was strongly inﬂuenced by Mach and also interested in a critique of language. In his

Psychology he wrote: ‘If there were no words, there would be no nonsense, at worst, errors . . . . Nonsense cannot be thought, it can only be talked’. Sto¨hr coined the term glossurey responsible for a type of metaphysical problem that he repudiated. In 1898 he wrote a book, Algebra of Grammatics [sic], which may be worth examining from a modern point of view. Ten years later, he published a logic treated from the point of view of psychologism and in connection with linguistics. When I entered the University of Vienna in the fall of 1920, several older students highly recommended Sto¨hr’s course and I planned to attend it. But before Sto¨hr could start, he became very ill and died.

CHAPTER 1 PRE-CIRCLE AUSTRIAN PHILOSOPHY Before talking about the Vienna Circle, I wish to sketch the earlier history of philosophy in Austria insofar as it is connected with ideas of the circle – by contrast or by similarity – but without any claim as to completeness of the list of authors considered or the work described.1 There is one general fact about Austrian philosophy that clearly emerges from history, and that is, its immunity from the extreme form of German metaphysics, which in the early 19th century culminated in Fichte, Schelling and Hegel. Austrian philosophers professed a variety of views, and the differences between some of them were rather profound, but they were united in rejecting speculative metaphysics. This type of thinking never captivated the Viennese. Another observation, while exempliﬁed in Austrian philosophy, has a much wider scope, indeed almost universal validity. Philosophers as well as scientists – even some scientists of the highest rank whose scientiﬁc works are models of clarity – in the ﬁeld of epistemology often fail to express themselves in a perfectly organized and coherent way and display inconsistencies in some of their writings. Moreover, they of course sometimes change their views from one writing to the next but not always in a linear development of their ideas. Moreover, they of course sometimes change their views from one writing to the next but not always in a linear development of their ideas: they venture a suggestion, drop the idea in subsequent publications only to resume it at a still later time. All these facts, which naturally apply to undogmatic and critical philosophers even more than to builders of metaphysical systems, often puzzle the expositor; inevitably, they also diminish the value of his presentation to the reader. This is especially true for expositors and readers who are logicians or mathematicians. y 1. While the German metaphysicians were totally engrossed in contemplation of ‘the absolute’ and revelled in speculations about continuity and inﬁnity, the

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Austrian mathematician-philosopher Bernhard Bolzano (1781–1848) in Prague did serious work in logic and methodology, soberly studied continuous functions and inﬁnite sets and criticized the logically confused speculations of the metaphysicians of his time severely. Bolzano’s Wissenschaftslehre (Theory of Science) is an encyclopaedia (in four volumes) of methodology. The book assigns a fundamental role to ‘ideas in themselves’ and ‘propositions in themselves’ (Vorstellungen an sich and Sa¨tze an sich) – the latter being true or false regardless of whether they have ever been verbalized or even thought. Proposition in themselves, Bolzano emphasized, must not be confused with judgments actually made or with the symbols expressing them, just as ideas in themselves must be distinguished from thoughts actually possessed by someone and from words denoting them. Though this position is of course more Platonic than positivistic, some of the members of the Vienna Circle were even more opposed to another basic thesis of Bolzano’s: that each science has not only its particular assumptions but that the various sciences require fundamentally unlike methods. From the point of view of a philosopher of science, Bolzano’s principal methodological achievement was the insight that mathematics could be developed by purely logical reasoning. At his time, many books on mathematical analysis (i.e. on functions, their differentiations and integration, and related topics) abounded in appeal to intuition; some proofs referred to accompanying drawings with the remark ‘as is evident from the ﬁgure . . . ’ – a procedure which in textbooks on geometry was not uncommon until recently. Bolzano strongly felt that analysis ought to be based exclusively on logic-arithmetical considerations; and he began to carry out this program, with special success in studies of continu[ity]. The speculations of philosophers about continuity were altogether nebulous. Bolzano’s contemporary, J. F. Herbart,2 called continuity ‘union in separation and separation in unity’. But even the descriptions of continuous functions by the mathematician[s] of the early 1800s were hazy. Consider, for example, the square function, which assumes the value 9 for 3, and, more generally, the value x2 for any argument x. That this square function is continuous at 3 used to be expressed by saying that the value of the function for an argument inﬁnitely close to 3 is inﬁnitely close to 9 (or that its value for the immediate neighbour of 3 is the immediate neighbour of 9). Bolzano called the square function continuous at 3 because its value for 3þd differs from its value for 3 by as little as we please, provided d is any number sufﬁciently3 close to 0. Bolzano described properties of continuous functions in theorems for which he supplied what is called rigorous proofs, that is, proofs by pure logical

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reasoning.4 One of the most important of these theorems states what in plane geometry is expressed by saying that a continuous curve joining a point above, and a point below, a horizontal line crosses that line in at least one point. The analytic theorem deals with functions that are continuous in an interval (i.e. for each number of an interval) and asserts that if such a function assumes a positive value for one number, and a negative value for another, then it assumes the value 0 for at least one number between these two arguments. In the 1920s, it became known – mainly through the work of Czech mathematicians – that Bolzano had also anticipated Weierstrass’ famous discovery of a continuous function that is nowhere differentiable or, geometrically speaking, of a continuous curve without a tangent at any point. Not only had, in 1806, Ampere published a paper purporting to prove that all continuous functions are differentiable but even in the 1870s most of Weierstrass’ contemporaries refused to believe in the possibility of continuous curves without tangents. Through his study of inﬁnite sets, Bolzano became one of the most important forerunners of George Cantor, the founder of set theory. Bolzano compared two sets such as the following: S 1 ; consisting of all numbers between 0 and 1 and S 2 ; consisting of all numbers between 0 and 2 Yet, when he paired to each number, x belonging to S2, its half (the number ½x) belonging to S1, he had thereby paired each number y belonging to S1 to one and only one number belonging to S2, namely to the number 2y. He thus had established what today is called a one-to-one correspondence between a whole and one of its parts – something that cannot possibly be established, between a ﬁnite set and one of its parts. Yet, S1 is a part of S2. Similarly, on a scaled line, the points between 0 and 1 can be brought into a one-to-one correspondence with the points between 0 and 2, even though the latter segment includes the former as a part. And although Bolzano did not take Cantor’s decisive ﬁnal step of calling two sets whose elements are (or can be brought into) a one-to-one correspondence equinumeral, he felt that he had obtained, for inﬁnite sets, a result contradicting Euclid’s famous assumption that the whole is greater than the part – a result which for this reason appeared to him to be paradoxical. Thus, the discovery5 is conﬁned to inﬁnite sets, whence Bolzano described it in a booklet, The Paradoxes of Inﬁnity, published in 1848. To appreciate the

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precision and the fertility of Bolzano’s reasoning, one should compare it with the spurious arguments in the Critique of Pure Reason, where Kant attempts to establish pairs of contradictory sentences; for example, so-called proofs for the thesis that the world ‘has a beginning in time and is bounded with regard to space’ and the antithesis that the world has neither a beginning in time nor bounds in space, but is inﬁnite with regard to both time and space. Bolzano’s Wissenschaftslehere (Theory of Science) is an encyclopaedia (in four volumes) of methodology. The book assigns a fundamental role to ‘ideas in themselves’ and ‘proposition in themselves’ (Vorstellungen an sich and Sa¨tze an sich), latter being true or false regardless of whether they have ever been verbalized or even thought. Propositions in themselves, Bolzano emphasized, must not be confused with judgments actually made or with the symbols expressing them, just as ideas in themselves must be distinguished from thoughts actually possessed by someone and from words denoting them. Though this position is of course more Platonic than positivistic, some of the members of the Vienna Circle were even more opposed to another basic thesis of Bolzano’s: that each science has not only its particular assumptions but that various sciences require fundamentally unlike methods. On the philosophers of his time, Bolzano, who was a priest, had no inﬂuence whatever. Only ﬁfty years after his death, did his work begin to receive recognition. It was then especially acclaimed by Austrian philosophers. Being the son of an Italian father and a German mother in the heartland of the Czech, Bolzano was a typical product of the Central European melting pot in the Austrian empire. But he belonged to the small ethical elite that preached friendship between the two nationalities in Bohemia, and at least up to the outbreak of World War II, Bolzano was revered by Czech and German alike. y 2. In contrast to the lonely, uninﬂuential man in Prague, Franz Brentano (1838–1917), who came to the University of Vienna from Germany in 1874, founded a large and widely ramiﬁed school. It dominated the academic philosophy throughout the Austrian empire for decades: A. Meinong and his student E. Mally taught in Gratz; A. Ho¨ﬂer, another one of Meinong’s students in Vienna; Ehrenfels, the Swiss Marty and his student O. Kraus, in Prague, K. Twardowski was one of the ﬁrst Polish philosophers; T. Masaryk became the ﬁrst president of the Czechoslovakian republic after World War I; E. Husserl went to the universities of Go¨ttingen and Freiburg in Germany. Like Bolzano, Brentano articulately opposed the metaphysical speculations of Kant’s successors. Instead of contemplating the absolute, he stressed – at least programmatically – the concrete and the particular.

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In his youth, as a priest, Brentano studied scholastic philosophy and published a book on Aristotle. He later also wrote about logic, but without any apparent appreciation for the logical achievements of some of the schoolmen or for the spectacular development of formal logic during the second half of the 19th century. One of Brentano’s chief ideas was expressed in the thesis that every mental act has an object – an ‘intentional object’, as he said (using a scholastic term), towards which the act is directed. The main topics of what he called ‘empirical psychology’ are the relations between the mental phenomena and their objects. The object may either be simply before the mind, without the mind taking any attitude towards the object, as in the case of an idea.6 Or the mind may take an intellectual or emotional stand towards the object: accepting or rejecting it; loving or hating it. An intellectual stand towards an object is expressed in a judgment, and, conversely, according to Brentano, every judgment accepts or rejects an intentional object, for example, the judgment ‘all men are mortal’ (which he restated in the form ‘an immortal man does not exist’) rejects the object, immortal man. An emotional stand leads to statements concerning love or hate. Brentano claimed that each attitude, be it intellectual or emotional, is either correct or incorrect. Brentano’s view on emotional attitudes is developed in his work on ethics, which was acclaimed by his followers, especially by O. Kraus, and in England, G. E. Moore wrote in the Preface to his Principia Ethica (1903) that Brentano’s opinion far more closely resembled his own that do those of any other ethical writer. Brentano’s theory culminated in what purported to be a deﬁnition of the good as ‘that which is worthy of love’ and ‘that which can be loved with a love that is correct’. This deﬁnition was implemented by maxims such as the rule that, of various possible aims, one should always choose the best. For my part, I must confess that I have never been able to see in Brentano’s approach to ethics more than verbiage. In his later years, Brentano began a rudimentary critique of language directed mainly towards eliminating nouns that do not designate anything and nouns that correspond to ﬁctional entities. y 3. Alexius Meinong (1853–1920), after publishing a book on Hume, developed Brentano’s idea of the objects of mental phenomena. In his Gegenstandslehre (Theory of Objects), he extended the realm of those objects so as to include non-concrete and non-particular objects. For example, he considered existence and non-existence of objects to be objects. Moreover, he studied admittedly non-existent objects such as mermaids and even impossible objects such as a round square or a square circle insisting that all these objects

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lend themselves to the predication of qualities such as ‘the golden mountain is golden’. As Meinong put it, the ‘Sosein’ (being as it is) of an object is not affected by its ‘Nichtsein’ (not-being). Meinong moreover distinguished between subsistence and existence (bestehen and sein). The law of contradiction, he claimed, holds for all that exists, but not necessarily for what merely subsists; impossible objects violate that law. A favourite example that philosophers have mentioned for ages whatever they speak of impossible objects is a square circle (or round square). A round square, according to Meinong, is round and is not round. Meinong was in correspondence with G. E. Moore and Bertrand Russell in Cambridge. The latter in his youth studied with great interest Meinong’s writings, especially the theory of objects. This study stimulated, if only by somewhat tenuous connection, the development of Russell’s own theory of description. Most philosophers considered statements about Meinong’s non-existing objects such as ‘the present King of France is bald’, to be neither true nor false but meaningless. Russell pointed out that that sentence (because of the use of the deﬁnite article in the phrase ‘the present King of France’) implies that there exists one, and only one, individual who at present is King of France, which assertion is false rather than meaningless. Under Meinong’s patronage in the 1890s, A. Ho¨ﬂer wrote textbooks on logic and psychology that were widely used in Austrian secondary schools for at least three decades (I think they were obligatory). According to these books, logic is the theory of correct thinking. Its method is ‘the same that psychology applies to all psychological phenomena; logic describes (observes, collects, classiﬁes) the various forms of correct thinking as employed in every day life and in science . . . and then reduces them to laws that are as simple as possible’. Meinong later abandoned this ‘psychologism’ in logic partly under the inﬂuence of Husserl. y 4. Husserl’s ﬁrst book, Philosophy of Arithmetic (1891), was inﬂuenced by John Stuart Mill’s psychologism. But impressed by a critique from the pen of the logician G. Frege, Husserl relinquished this view. His Logical Investigations (1900) were devoted to the refutation of psychologism in preparation of what Husserl called ‘pure logic’. According to him, logical laws have a priori validity and are established by inner evidence rather than on inductive or psychological foundation. They neither presuppose nor imply matters of fact or even the existence of mental phenomena. Bolzano’s ‘proposition in themselves’ come to mind. So does his view that each branch of science has its own axioms and methods when one reads in Husserl’s book that the domain of truth is objectively divided into sections, the domain of each science being an objectively closed unit. Husserl indeed highly praised Bolzano’s

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Wissenschaftslehre and made that practically forgotten work widely known to philosophers. (Ho¨ﬂer brought out a new edition). In 1912, Husserl published a paper about philosophy as a rigorous science. Philosophy is rigorous, he claimed, as long as it moves in the sphere of direct intuition. Without mathematics or the apparatus of proofs, intuition leads to a mass of the most rigorous Erkenntnis (cognition, knowledge) by grasping the essence of being. In one of his earliest philosophical papers, ‘Is there intuitive knowledge?’ (1913), M. Schlick, who a decade later was to be the central ﬁgure of the Vienna Circle, criticized Husserl’s claims. According to Schlick, cognition of anything new is ﬁnding therein elements already known and applying procedures of relating, combining and arranging. Intuition fails to be a method not only for rigorous science but for any science at all. In his book Allgemeine Erkenntnislehre (General Theory of Knowledge, 1918), Schlick extended this criticism to embrace what Husserl in his next book (usually referred to, brieﬂy,7 as Ideas called pure phenomenology). With an extensive terminological apparatus and many neologisms, the Ideas put even greater stress on intuition under the name of Wesensschau (intuition of essences). In the second edition of his book (1925), Schilck repeated his criticism, which was sharply rejected by Husserl in the second edition of his Logical Investigations. This controversy set the stage for what was to be a basic antagonism between phenomenology, which in the 1920s and the early 1930s became more and more inﬂuential on philosophy in Germany and the Vienna Circle in Austria. y 5. Also, outside the Philosophical Society, one could in the 1920s frequently hear discussions about scientiﬁc subjects bordering on philosophy. Monism of the Haeckel and Ostwald varieties, which preoccupied many intellectuals before World War I, had made room for the theory of relativity and the foundation of physics and geometry. These and, later, quantum theory were topics of conversation at almost every gathering of members of that numerous, scientiﬁcally interested intelligentsia made up of professional and businessmen. Some of those discussions, which I remember, were on a surprisingly high level. In the spirit of Boltzmann, phenomena in the large were looked upon as the statistical results of events in the small and were considered predictable with very high probability but not with certainty. This point of view was stressed in excellent semi-popular writings by Franz Exner, a professor of physics at the University of Vienna, and highly regarded by Schro¨dinger. Philosophical discussions were also carried on in coffee houses, which had played a great role in Viennese literary life at the turn of the century. The

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few philosophical Cafe` debates that I attended in the 1920s happened to be rather shallow, but, as Hahn and Ph. Frank told me, in the 1910s, it was their frequent meetings with O. Neurath in a coffee house that led to the idea of founding a discussion group such as the Vienna Circle. y 6. Viennese students of mathematics in the early 1920s were intensely interested in fundamental ideas. When initiated to the concepts of limit and continuity in the ﬁrst hours of the beginners’ course in calculus, many freshmen would become greatly excited. Every year at that time, groups of them gathered together in the hall of the mathematical institute heatedly debating the new idea for at least a week. Once in my last student year (1924), at the height of those discussions, I remember Hahn opening the door of his ofﬁce and looking around half distraught. When he saw me he asked whether I could not prevail on the young students to lower their voices somewhat, the noise being such that working in his ofﬁce was an impossibility. I have never seen anything like it elsewhere. Lively discussions also took place between the participants in a seminar in which the physicist F. Ehrenhaft read Duhem’s before mentioned book of physical theories. Like Mach, that great Belgian historian of science rejected models, which he considered to be parasitical elements of explanations.

NOTES 1. [Editorial note. Menger wrote at this point:] ‘Readers primarily interested in the Circle itself may skip Chapters 3 and 4, returning to speciﬁc passages where references to them occur’. 2. A comparatively moderate metaphysician, Herbart also had followers in Austria, among them R. Zimmermann at the University of Vienna in the second half of the 19th century. 3. Indeed, if c is any positive number, however small, and d is a number between 1 and 1, then the value (3þd)2, which equals 9þ6dþd2, differs from 9 by less than c, provided d is between c/7 and c/7. More generally, for any number x, the value (xþd)2 for xþd differs from the value x2 for x by less than c if d is between c/(2xþ1) and c/(2xþ1). 4. His results paralleled work of his contemporary, the great French mathematician A. Cauchy, and made Bolzano also an important forerunner of K. Weierstrass, the founder of the theory of real functions. 5. Galileo had discovered the existence of a one-to-one correspondence between the positive integers 1,2,3, . . . and their squares 1, 4, 9, . . . despite the fact that the latter constitute a part of the former. But both Galileo and Bolzano stopped short of the step that Georg Cantor took when, in founding set theory, he considered two sets as equinumerous if and only if there is a one-to-one correspondence between their elements. By adopting this deﬁnition in the wake of Galileo’s and Bolzano’s

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discoveries, Cantor explicitly disavowed for set theory Euclid’s axiom that the whole is necessarily greater than any of its parts. The segment from 0 to 2 is equinumerous with its part from 0 to 1. 6. Brentano says ‘Vorstellung’, a word that has no perfect counterpart in English and is sometimes translated representation. 7. The full title is Ideen zu einer reinen Pha¨nomelogie und pha¨nomelogischen Philosophie, 1913; translated into English under the title Ideas: General Introduction to Pure Phenomenology.

CHAPTER 2 IMMEDIATE PREDECESSORS OF THE CIRCLE: MACH, BOLTZMANN, MAUTHNER The Vienna Circle and the work of Brentano, Meinong and Husserl were only indirectly related and mainly by opposition. The present chapter is devoted to Mach, Boltzmann and Mauthner, three precursors of the Circle, though quite unequally treated by the group. Mach was extolled, while the other two were practically ignored. y 1. Mach’s antimetaphysical views were closely connected with [the following] methodological principles which, from 1872 on, permeated his scientiﬁc as well as philosophical writings. the call for descriptions rather than explanations interrelating factual material: direct descriptions of phenomena and indirect ones by comparison with phenomena previously described or by analogy. While admitting, in principle, any useful idea as a heuristic tool of research, Mach ultimately proposed to eliminate as superﬂuous all ﬁctitious elements introduced for the sake of explanations. G. Kirchhoff expressed essentially the same idea in his Principles of Mechanics (1874) by deﬁning the object of mechanics to be the complete and simplest possible description of the motions occurring in nature. the program of a combined and uniﬁed study of physical, physiological and psychological material. This view differed not only from Bolzano’s but was in sharp contrast to Comte’s rigid classiﬁcation. In sharp contrast to Comte’s classiﬁcation of the sciences and his rejection of psychology, Mach proposed a combined and uniﬁed study of physical, physiological and psychological material. This idea of Mach’s, diametrically opposed to one of Bolzano’s, was to [an] extent a profound inﬂuence on the Vienna Circle, even though Mach’s prediction that the most important advances in 55

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physics would come from investigations into the psychology of the human sense organs did not materialize. Mach’s rejection of atomism probably explains why he is rarely mentioned in quantum physics. Yet, in the paper of 1925 that marked the foundation of quantum mechanics, W. Heisenberg emphasized that his aim was ‘to develop theoretical quantum mechanics analogous to classical mechanics in which only relations between observable quantities occur’ and to ignore ‘fundamentally unobservable positions and velocities of electrons’, statements that might have been written by Mach. With regard to causality, Mach agreed with Hume’s scepticism. Elaborating this point of view, he proposed to replace all causal explanations by descriptions of functional relations – an attitude nowadays also adopted by some economists. But it should not be overlooked that even the physical sciences abound in descriptions of merely qualitative nature. When we describe thunder as the air waves following a large electric spark, we have come a long way from the explanation involving the blow of an irate god’s hammer, and yet, that road has not been paved with functional relations. y 2. After Mach’s retirement, his chair at the University of Vienna went to the physicist Ludwig Boltzmann (1844–1906). Like Mach, Boltzmann repudiated metaphysical speculation. He was strongly repelled by the writings of Hegel and Schopenhauer, especially Schopenhauer, expressing his aversion in quite uninhibited terms. Like Mach, he rejected some philosophical problems as being empty of content. As an example, both men mentioned instances of the question ‘why?’. It is justiﬁed in certain circumstances, but our habit of asking it overreaches its aim and leads to such pseudo-questions as, why has everything a cause? Why does the world exist? Why is it as it is? ‘My view’, Boltzmann said, addressing the Scientiﬁc Congress in St. Louis in 1904, ‘is totally different from the theory that certain questions fall outside the boundaries of human cognition, because of a defect or an imperfection of man’s cognitive capacity’. ‘I regard’ he added, in accord with Mach and anticipating Mauthner and Wittgenstein, ‘the existence of these questions and problems themselves as an illusion’. Perhaps even more than Mach, Boltzmann stressed the pragmatic aspect of science and philosophy. He deﬁned the task of philosophy to be the formulation of basic concepts in such a way that they yield precepts of the greatest possible precision for inﬂuencing events according to one’s purposes. In three of their most important speciﬁc views, however, Boltzmann and Mach were in basic disagreement. In the ﬁrst place, Boltzmann not only went beyond Mach by calling Berkeley’s negation of a material world a ‘folly’, but

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he went on to defend realism.1 He proceeded according to the three customary steps: by dismissing solipsism as absurd; by admitting the existence of the human recipients of sense impressions and by assuming an objective material world on the grounds that the sensations of various human being, as communicated, basically all agree with each other and with his own. Secondly, Boltzmann was the foremost champion of atomism at the turn of the century while this doctrine was attacked by Mach and rejected by all advocates of phenomenological theories. Boltzmann argued that the assumption of a continuous distribution of matter and the description of phenomena by differential equations also involved hypotheses. He believed that the continuum could only be comprehended as the limiting case of approximating ﬁnite collections of points – collections of ever more points with ever smaller distances between neighbouring points. In his Lectures on the Principles of Mechanics, Boltzmann made an interesting further remark. One cannot rule out the possibility, he wrote, that when reaching a certain stage in the process of approximation, one achieves the best picture of the phenomena and that, from that stage on, the consideration of still more points with still smaller distances between neighbours may make the picture worse again.2 So he began his afore-mentioned lectures with the basic assumption of an (unspeciﬁed) ﬁnite number of material points. Boltzmann further emphasized that the formulae of phenomenological theories were also models of a sort. In fact, in his article ‘Models’, in the older editions of the Encyclopaedia Britannica, he went so far as to refer to the mathematical theories of Kirchhoff’s school as a development of the models proposed by Maxwell, inasmuch as the atoms and ether involved in those models were not meant to be descriptions of existing entities. In keeping with this view, in his Lectures on Gas Theory, Boltzmann did not claim to give an explanation or even a precise description of the true nature of small particles. He rather referred to the ideas of the kinetic theory as mechanical analogies to the processes of nature. Thirdly, Boltzmann placed a greater emphasis than Mach did on deduction. This emphasis was connected with (and in fact is a corollary of) his use of Bernoulli’s model of gases. Whoever starts with a model proceeds by deducing consequences and ends by comparing the inferences with the results of observation. The same procedure is followed by those who propose a system of formulae as the basis of a physical theory, and it was clearly this methodological parallelism that prompted Boltzmann to connect the phenomenological theories of Kirchhoff’s school with Maxwell’s models. On the contrary, he blamed obscurities in the principles of mechanics on the attempt to establish connections with experience at the

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very outset of a theory before clearly formulating the hypotheses and on the tendency to mention hypotheses only at the point where they are inevitable or even to conceal them altogether. Boltzmann admitted that the deductive method fails to reveal the way by which the models have been obtained. But it clearly separates the contributions of thought from those of experience. In the 1890s, much was written about concepts as pictures of external objects. Hertz in his Principles of Mechanics (1894) tried to make the relation between concepts and objects more precise by demanding that the inferences drawn from the pictures be images of consequences in nature of the things pictured. In a similar spirit, Boltzmann in his Lectures on the Principles of Mechanics (1897) called thoughts symbols for objects bearing to them a relationship analogous to that of letters to sounds or of notes to tones. And he proposed to replace the question as to ‘the real nature of things’ by ‘the more modest but clearer question as to which conceptual picture represents our experiences at present in the simplest and the most unambiguous way’. He considered the task of a theory to be the construction in us of a picture of the external world guiding us in our thoughts and experiments. The theory thereby also becomes the quintessence of the practical since the precision of its conclusion cannot be achieved by any routine of estimating or by trial and error. ‘Nothing’, he is reported to have said, ‘is more practical than theory’. But, unlike Hertz, Boltzmann denied that the formation of the symbols had to be controlled by the laws of thought (Denkgesetze). According to him, these laws (which unfortunately he did not precisely deﬁne) are inherited habits of thinking that have undergone an evolution in the sense of Darwin. Gradually, men have become accustomed to use and to combine words and thoughts in such a way that they can successfully inﬂuence the world and communicate with one another. The laws of thought evolved by eliminating faulty methods and by transmitting others to following generations. These laws are now innate, though by no means infallible. In each individual, they certainly are modiﬁed by education and by personal experience, and they may well even be subject to further evolution. y 3. In Berlin, Mauthner worked for three decades as a journalist and belletristic [sic] writer. A satirical book, Totengespra¨che (Dialogues of the Dead), ends with a stab at academic philosophy. Mauthner describes a meeting of the philosophers from Socrates to Schopenhauer, all of them shaken by Homeric laugher, for word had reached them from the living about a planned Congress of Philosophers, which they considered to be the funniest of contradictions in term. Near the end of the past century, Mauthner started a massive critique of language and began the publication of three voluminous

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philosophical works: Contribution to a Critique of Language (1902), Dictionary of Philosophy (1910, 1923) and Atheism and its History in the Western World (1922). [He wrote:] ‘the pitiful poverty of [language] is mistaken for wealth. A word superstition develops: the belief in knowledge of reality through language’. Mauthner even spoke of insolence in language, listing among his examples the words ‘idea’, ‘the best of all words’ and ‘the categorical imperative’. [He claimed that] critique of language must teach liberation from language as the highest aim of self-liberation: ‘if I wish to climb upward in the critique of language, then I must crush every rung of the ladder as I step on it’. In some cases, Mauthner found any critique of language to be inadequately critical in that it tries to say what cannot be said. So, he considered silence to be the culmination of critical attitude (I remember a passage in which Mauthner commends the poet Maeterling for advocating silence while protesting that in his appeal for silence Maeterling talks too much). Austrian philosophy, as Weiler correctly remarks, ‘largely through the inﬂuence of Mach, has shown features which mark it off from the general trend of German philosophy. Mauthner, who chose to be a German over being an Austrian became thus one of the representatives of Austrian spirit in German philosophy’.3 While Mauthner was convinced that his critique of language had deﬁnitely solved all philosophical problems, his impact on philosophical thinking in Germany, especially on academic philosophy, was insigniﬁcant. Also in the English-speaking world, he remained neglected. Weiler’s book and his article on Mauthner in the Encyclopedia of Philosophy will probably make him more widely known and more justly appreciated. For the ﬁrst decade of the 20th century, Mauthner had introduced a new and important element into philosophy. Philosophy as a critique of language was different from all the various views held at that time. For most of them it was destined to survive.

NOTES 1. In a moment of mellowness, Boltzmann was satisﬁed to characterize the language of realism merely as more practical (zweckma¨ssiger) than that of idealism. 2. Boltzmann also suggested the possibility that a certain degree of approximation to differential equations might produce an optimal description of nature – an idea that amounts to the suggestion that differential equations may give way to difference equations. 3. [Editorial note. In Menger’s Reminiscences, this quotation is preceded by ‘Gershon Weieler says: ‘‘the only teacher who was signiﬁcant for Mauthner’s later career was Mach’’ and it is followed by: ‘‘he hardly made any impact on philosophical thinking in Germany’’’ Menger, 1994, pp. 23–24)].

CHAPTER 3 THE VIENNA CIRCLE [In Menger’s Reminiscences this part is Chapter Five (‘Vignettes of the members of the Circle in 1927’), where Moritz Schlick is described as ‘an extremely reﬁned, somewhat introverted man’; Hans Hahn, ‘a strong, extroverted, highly articulate person who always spoke with a loud voice’; Olga Hahn Neurath, ‘always smoking a big cigar’; Otto Neurath, ‘a man of immense energy and curiosity, very fast in grasping new ideas, through an often distorting lens of socialist philosophy’; Rudolf Carnap, ‘systematic, sometimes to the point of pedantry . . . a truly liberal and completely tolerant man’; Victor Kraft ‘[who] like Schlick, Feigl and myself, by no means shared all the political ideas and ideals of Neurath’; Friedrich Waissman, ‘a very clear expositor [who] unfortunately dragged out his studies [of mathematics and philosophy] at the University’; Herbert Feigl, ‘[who] did probably more than anyone else to make some of the Viennese ideas known in America’; Theodor Radakovic, ‘a student of Hahn’s . . . too shy to take part in the discussion of the Circle, although he attended the meetings regularly’; Edgar Zilsel, ‘a militant leftist [who] wanted to be considered only as close to, and not as a member of, the Circle’; and Felix Kaufmann, ‘a philosopher of law, an ardent phenomenologist, the only participant with a true sense of humour’ (Menger, 1994, pp. 55–68). These following parts are those unpublished].

All members of the Circle had a common background of scientiﬁc and mathematical knowledge, which extended into symbolic logic, while many a traditional philosopher in Central Europe at that time was struggling with basic scientiﬁc ideas or was ignorant of mathematics. They had, moreover, been active in research. Each of the members took the scientiﬁc method very seriously and in fact expected a consistent Weltbild (picture of the world) through what in the circle was called Wissenschaftliche Weltauffassung (scientiﬁc conception of the world). They enjoyed graphic and dramatic arts, music and literature including lyric poetry and even some works of mystics. But where cognitive content was concerned, all members fervently believed in clarity and aggressively demanded it. A hazy expression of presumed truths indicated to them at best laziness of the proponent or his incompetence as expositor, at worst the imperfection of his thoughts or even his insincerity. What metaphysicians call depth raised in them a strong instinctive distrust even before logical analysis of the work would perhaps bare the cognitive emptiness of its speculations in a systematic way. If today one frequently reads statements beginning with words ‘In the opinion of the Vienna 61

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Circle . . . ’, then it must be emphasized that there are very few subjects on which there was a consensus in the group. All members were highly individualistic. Neurath was the only one who seemed anxious to forge opinions of the circle, and Carnap was not quite averse to this tendency. The Neuraths lived in an old building located in a workers’ district of Vienna where Otto kept his extensive library. In the years 1923–1925, I went to their picturesque apartment a few times with my friends Otto Schreier and Witold Hurewicz, who were to become distinguished mathematicians. Felix Kaufmann (see later) had introduced us. In these totally informal meetings, we had stimulating conversations with Otto, but all in all, we mathematicians felt more drawn to the quiet Olga. Although strongly individualistic himself, he was, as matter of principle, even more strongly opposed to individualism. He always regarded himself as expressing the ideas and feelings of the masses. His interest in statistics ﬁtted in that general frame and so did of course his literary taste. Once he recommended to me to read Jack London with the words: ‘Great stuff. On proletarian concern’. In 1919, he went to Munich to serve the leftist government in Bavaria; he was imprisoned when the shortlived regime broke down, but soon returned to Vienna. In Austria, however, the powerful social-democratic party did not seem to be disposed to make use of his talents, which included a great gift for organization and, when he served a cause, the skill of politician in every sense of the word. Neurath wrote extensively about the maximization of the happiness of a society. As early as 1910, Neurath was fully aware of the need for precise deﬁnitions to make the problem meaningful – a need not yet felt by many writers on philosophy, ethics and economics at that time, while Neurath himself ignored the marginalists’ results on exchange. He struggled with the problems of interpersonal comparison and measuring of ‘pleasure’ The deeper reason was of course his tacit assumption of an organized society of the future distributing all ‘pleasures’ rather than the free exchange of goods by individuals, each act enhancing the pleasure of both contractors. In 1924 Neurath founded the social and economic Museum of the city of Vienna (Gesellschafs und Wirtschaftsmuseum der Stadt Wien). As its director (1924–1934), he developed what he called Vienna Pictorial Statistics (Wiener Bildstatistik). Another frequent or perhaps even regular visitor of the Circle at the time when I joined was Marcel Natkin, who studied the concept of simplicity, in which I have always been interested. [Menger’s drafts Part I ends at this point. Menger’s drafts and Menger’s edited book overlap quite similarly in those following parts dealing with ‘vignettes’ of some members of the Vienna Circle and with Wittgenstein].

PART II MENGER’S VIENNESE AUTOBIOGRAPHY (BOOK B) [Part II in Menger’s drafts is divided into four chapters. The ﬁrst is mainly taken up with Menger’s description of L. E. J. Brouwer, the Dutch mathematician with whom Menger worked as assistant during his stay in Amsterdam in the mid-1920s; none of this part is included in Menger’s Reminiscences, although the book makes several references to Brouwer. Moreover, Menger’s Selected Papers (Menger, 1979) contain a long and very detailed chapter, ‘My memories of L. E. J. Brouwer’, written in 1978, and therefore at the time of Menger’s drafts. The second chapter deals with Menger’s contacts within the Vienna Circle after his return from Holland (1927) until his sabbatical years spent in the USA (1930–31). In Menger’s Reminiscences, Chapter Four (mainly devoted to his theory of curve and dimension) and the two-page Chapter Eleven cover that period, as do some passages in Menger’s Introduction to Hahn’s Philosophical Papers (Menger, 1980). The third chapter of the drafts is concerned with Menger’s visits to the USA. Chapter Thirteen in Reminiscences deals with this topic, but it omits these following parts. Finally, the fourth chapter of the drafts describes Menger’s years in Vienna between 1931 and 1938, when his Colloquium grew increasingly important. His disagreement with philosophical developments in the Circle became very evident, and the political situation in Austria was then lapsing into the fascist darkness; Menger’s Reminiscences, which end with Schlick’s death in 1936, wholly omit these parts].

CHAPTER 4 BETWEEN VIENNA AND AMSTERDAM (1923–1927)1 In 1923, an issue entered the discussion between Schreier2 and myself, though hardly anyone else in Vienna was interested in it at that time. Historically, the topic was connected with the question of intuition or Wesensschau, but substantially, in my opinion, the two were quite distinct. Elaborating on ideas of the 19th century algebraist L. Kronecker and eloquently supported by Weyl,3 the Dutch mathematician L. E. J. Brouwer developed what he called intuitionistic mathematics, the mathematical controversy centred on existential propositions. But those who closely associated intuitionism in mathematics with the intuition in Husserl’s pure phenomenology or Bergson’s metaphysics were misled by the similarity of the two words. Whatever, if anything, Brouwer’s reconstruction of mathematics and the phenomenology in Husserl’s Ideas had in common, they certainly had opposite effects: Husserl claimed for his Wesensschau (and Bergson for his intuition) insights that empiricists such as Schilck denied or regarded as empty words. Brouwer, to the contrary, rejected statements that everyone else claimed to be solid parts of mathematics, and he denied or regarded as empty words theorems proved by men such as Hilbert. If some domain of individuals is under consideration, then traditional mathematicians acknowledge two ways of providing that in that domain an individual possessing a certain property exists: either by exhibiting in that domain an individual possessing the property or by deriving a contradiction from the assumption that all individuals under consideration lack that property. Intuitionist mathematicians rejected the second indirect proof and only recognized existence proofs of the ﬁrst kind. Consider, for example, Goldbach’s conjecture that even every number is the sum of at least one pair of prime numbers (as 8 ¼ 3þ5 or 10 ¼ 3þ7 ¼ 5þ5). The question was (and as of this writing still is) undecided. But had someone claimed the conjecture to be false, in other words, had he asserted the existence of an even number that is not the sum of two primes, then Brouwer would have demanded to see an 65

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example of such a number or at least to be given a number below, of which he was guaranteed to ﬁnd an instance by testing all smaller numbers. He would not have been satisﬁed with a mere indirect proof to the effect that the assumption ‘each even number is the sum of two primes’ leads to a contradiction. When I ﬁrst studied this theory, in 1928, I was greatly impressed. The insistence in mathematics on the display, or at least the possibility of a display of entities that one claimed to exist, appeared to me to be the counterpart of Mach’s insistence on observables and of the demand for veriﬁability of assertions in science. Was not intuitionism, I asked myself, mathematical empiricism or positivism as opposed to the mathematical metaphysics of formalists, who claimed the existence of unconstructed entities? True, the effects of Brouwer’s demands on the theory of real numbers and on mathematical analysis were devastating. But so were the effects of positivism on metaphysics without causing me undue concern. Since, however, at that time my main interests were point sets, a different aspect of intuitionism was of even greater importance to me: the obscure deﬁnition of the concept of set just then published by Brouwer, which ﬁlled half a page. I failed to understand that tortuous sentence, and all those whom I asked, including Hahn and Schreier, were at a complete loss to explain it to me. So I began to consider the idea that, as soon as my work would permit me to leave for a few weeks, I might go to Amsterdam to get to the bottom of the matter of intuitionistic set theory. Schreier, ﬁrmly formalist, spurned intuitionistic scruples and went to Hamburg.4 Intuitionism recognizes theorems that are proved by the constructive and rejects all other mathematical results. Before discussing the dogmatism inherent in this program, we point out that the basic concept of constructivity has not as yet been deﬁned in a satisfactory manner. The only remark that can be made with certainty is that the ideas concerning constructivity expressed by different mathematicians differ from each other. For example, Brouwer rejects the law of excluded middle but admits certain indenumerable sets, whereas Borel has no objections against the law of excluded middle but rejects indenumerable sets as being inadequately deﬁned. The author of this paper has expressed the opinion that the concept of constructivity, if capable of a precise deﬁnition at all, is likely to be deﬁnable in various ways, especially in various degrees. Even in the most intuitive parts of geometry, there is neither an idea nor a term that would cogently demand a particular deﬁnition and be incapable of deﬁnitions in several different ways. The same is undoubtedly true, only to a higher degree, with regard to the hazy term

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‘constructivity’, which can be deﬁned, if at all, in a wider or less wide fashion. To each of these various concepts of constructivity, one could develop corresponding deductive mathematics. For example, it may be possible to deﬁne so strict a concept of constructivity that only the ﬁnite sets can be obtained constructively, or a somewhat weaker concept yielding denumerable sets. Consistency may be considered as the widest constructivity. So far, intuitionistic theories have proceeded as follows. Each of them has settled on one (not clearly circumscribed) idea of constructivity: the corresponding mathematical developments are called meaningful, all transcending developments [are called] meaningless. For the question in mathematics and logic is not which axioms and methods of concluding one assumes, the question is what can be derived from the axioms by means of the methods of concluding. Whether mathematician A declares the axiom of choice to be ‘admissible’ or ‘believes’ in it, and whether mathematician B calls it ‘unconstructive’ and ‘cannot associate any meeting’ with it, these facts are of interest for the biographies of the mathematicians A and B or perhaps for history, but certainly not for mathematics and logic. There, the question is exclusively what follows from the axiom of choice. What matters is the question as to the propositions into which certain propositions can be transformed by according to certain rules of transformation. Justiﬁcations of the acceptance or rejection of certain propositions or rules of transformation by appeals to intuition are in the last analysis nothing but empty words. What matters in the praxis of mathematics are not evaluations of meaningfulness, acceptances and rejections; what matters are exclusively implications. According to this – one might say – implicationistic – point of view, each intuitionistic program concerning the mathematics of inﬁnity as outlined above is to be rejected, but also, absolute insistence on mere consistency is arbitrary. The mutual relations of the various mathematics corresponding to various concepts of constructivity (once the latter is really made precise) will be interesting, just as already today relations between various classes of sets (which may well correspond to the various concepts of constructivity) are of interest, while no active mathematician would dream of calling some of the classes admissible and others meaningless. Only in one respect, as far as one can guess today, will the widest concept of constructivity, that is, the mere demand for consistency, be distinguished compared to the narrower concepts, namely by the aesthetical properties of the corresponding mathematics. But it must not be overlooked that, logically, this criterium is not more binding than appeals to intuition.

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NOTES 1. [Editorial note. In Menger’s Reminiscences, this unpublished part should have followed a very brief paragraph on Menger’s ‘student days in Vienna (1920–1924)’, inserted in chapter three (‘The philosophical atmosphere in Vienna’, Menger 1994, pp. 31–33)]. 2. [Editorial note. In Menger’s Reminiscences, Otto Schreier is described as Menger’s ‘friend and fellow-student of mathematics’, whose ‘untimely death in 1929 was a great loss for mathematics’ (Menger, 1994, p. 31)]. 3. [Editorial note. In Menger’s Reminiscences, Herman Weyl is described as ‘a great mathematician’, particularly fond of Husserl’s philosophy and too interested in metaphysics, an interest that ‘greatly confused’ the young Menger’s as well as Schlick (Menger, 1994, p. 33)]. 4. His death there at the age of 27 in 1928 was a great loss for mathematics. [Editorial note. Note that there is a discrepancy on the year of his death between Menger’s drafts and what is written at p. 31 in Menger’s Reminiscences. Schreier effectively died in 1929 at 28 years old].

CHAPTER 5 VIENNESE MEETINGS 1927–1930 y 1. My presentation to the Circle. When I presented my theory to the Circle, I found a mixed reception. Schlick, however, slightly shook his head, the mock-smile appeared on his face, and he tried to exchange glances. Only Waismann responded. Kaufmann was too loyal a friend to openly go against me even though he strongly felt that I was wrong. And Carnap was in deep thought. Perhaps, this was the moment when he found the adapt name of ‘Principle of Tolerance’ for my theory. This was the only time that I ever was thus castigated by Schlick, and I was so thoroughly convinced of being right that I did not shorten my talk. In one of the ﬁrst meetings in the fall of 1927, Hahn advanced his own version of a picture theory. It was based on the remark that language maps the states of affairs on verbal symbols in a one-to-many way, that is to say, each Sachverhalt has more than one description. Logic, which is exclusively concerned with our way of talking about the world, states the rules of transformation of complexes of words into others symbolizing the same state of affairs, for example, not-p’ into p’. Similarly, ‘what holds for all holds for any’ is not a property of the world but a result of the multiplicity of our linguistic symbols for the same state of affairs and describes the use of the word ‘all’. Up to this point, The Circle expressed agreement. But opinions were divided on Hahn’s suggestion that the multiplicity of the symbolism represented an indispensable and most useful trait of language. y 2. Several discussions were centred on the axiom of reducibility. This is an assumption introduced by Russell to rescue logic and mathematics from its position under the Damocles sword of the antinomies that had been discovered at the turn of the century. One of them was connected with the set, S, of all sets that are not elements of themselves. The set of all men is not a man and hence not an element of itself. In fact, a set being an element of itself (as would, e.g., the set of all sets) is a rather odd and abnormal

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occurrence. But unfortunately, the sets that are not elements of themselves lead to an antinomy. For either of the assumptions that S is or is not an element of S implies, as one readily veriﬁes, its own negation (i.e. the other assumption). The same dilemma occurs if one asks whether the following sentence is true or false: all sentences on this line are false.

This is a version of the ancient Greek paradox of the man from Crete who says, ‘Whatever I say is a lie’. The class of all properties is another source of antinomies. Russell tried to avoid them by introducing a hierarchy of properties of different types. Consider, for example, the natural numbers 1,2,3 . . . as individuals and their basic properties given by predicates such as being even, being odd, being prime and the like. For each property, its negation (e.g. being nonprime) is a property and so is for any two properties their disjunction as well as their conjunction (e.g. being prime and even). Besides these predicative properties, also called properties of type 1, there are properties of type 3, deﬁned by references to classes of properties of type 2 and so on. An example of a property of type 2 is the property (p2), being the smallest number in the class of all numbers that are prime. In this axiom of reducibility, Russell postulated that for each property of a type 2,3, . . . , there exists a property of type 1 that has the same extent. For example, (p2) has the same extent as the property of being prime and even; for the smallest of all prime numbers is 2, and 2 is also the only number that is both prime and even. Russell’s axiom is intimately connected with Leibnitz’s Principle of the Identity of Indiscernibles, which claims that two entities that have all properties in common are identical or, in other words, that any two (nonidentical) entities disagree in at least one property. To avoid antinomies, Russell took a step that amounted to relativizing [sic] Leibnitz’ Principle to some class P of properties and a hierarchy of properties of class P; identity of type 2 is agreement in all properties of type 2, that is, properties whose deﬁnition refers to the class of all properties of type 1 and so on. To prove that identity of type 1 implies identity of any type, Russell needed the axiom of reducibility. Even more important was Russell’s application of the axiom of reducibility to the theory of real numbers. Since Dedekind, real numbers have been as certain classes of fractions, for example, the real pﬃﬃdeﬁned ﬃ number ( 2) is deﬁned as the class of all fractions having squares less than

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2; the rational real number 3/4 (conceptually to be distinguished from the fraction 3/4) is deﬁned as the class of all fractions less than 3/4. More generally, a class, r, of fractions containing some but not all fractions is called a real number if it has the following property: (*) for each fraction f belonging to r, there exists in r a fraction that is greater than f. Of two real numbers, r and ru, the former is greater if its deﬁning class includes a fraction not contained in the class deﬁning ru. Now consider a class R of real numbers that does not include a largest number and has, say, 3 as un upper bound, that is to say, all numbers in R are less than 3. Of great mathematical importance is the number 1, called the least upper bound of R, deﬁned as an upper bound of R with the following property: (**) for each fraction f that is less than 1, there exists in R a real number that is greater than f. In contrast to (*), where the existential quantiﬁer applies to fractions, in (**), it applies to real numbers (i.e. classes of fractions), whence the least upper bound of a set of real numbers in a number of a higher type than the numbers in the set. The axiom of reducibility seems to be essential in developing a theory of real numbers of a single type. H. Weyl in his book ‘The Continuum’ called the result of ignoring the variety of types of real numbers the circulus vitiosus of analysis. In the Circle, the discussions about the axiom of reducibility centred on two points. The ﬁrst was the nature of the axiom. Is it or is it not a tautology? Russell postulated the axiom contre coeur, as it were, for he did not feel that it was evident. Wittgenstein said (Tr. 6.1233) that a world was imaginable in which that axiom is not valid. Waismann pointed out that since tautologies say nothing about the world, they belong to those sentences that according to Leibnitz hold in every world and that, consequently, even if the axiom should hold in our world, its tautological character would be demonstrated if one could establish the existence of a world in which the axiom did not hold. I must have shaken my head when so much was said about worlds in connection with a logical proposition; for I remember that Schlick asked me why I was sceptical [sic] and what was my view. I said that the axiom most certainly was not a tautology in the sense of the calculus of propositions and that I found it extremely unlikely that one would ever deﬁne tautologies so as to include the axiom of reducibility. But I used the opportunity also to re-emphasize that I never had heard a clear deﬁnition of tautologies beyond the calculus of propositions. Had I still

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hoped to elicit a deﬁnition from the Circle, I should have been disappointed again. The second point of controversy was the question whether one should or should not postulate the axiom. This was, I remember, the ﬁrst occasion on which I expressed a view that I emphasized throughout the years of the circle with increasing insistence: that value judgments about an axiom are of no interest except for the biography or their proponent. All that matters for mathematics and philosophy is what it is that follows if one does, and if one does not, postulate the axiom. I pointed out that this was also the spirit in which the mathematicians in Warsaw had begun to treat Zermelo’s axiom of choice, according to which for each set S, whose elements are disjointed non-empty sets, there exists a set that has exactly one common element with each set belonging to S. y 3. But something good came out of these discussions about the axiom of reducibility, after all. Waismann succeeded in establishing ‘the existence of a world’, as he put it, in which the axiom is not valid, thereby, demonstrating its non-tautological character. He proved, as mathematicians would say, the independence of the axiom of reducibility from the other axioms by constructing a model satisfying all axioms except that of reducibility. Waismann’s idea was very simple indeed. His world consists of individuals endowed with properties of type 1 subject only to the following two conditions: (1) no two individuals have all properties of type 1; (2) Each property of type 1 belongs to more than one individual. Condition (1) is a rudimentary identity of the indistinguishable. Condition (2) does not hold in our world. For in it, if c is some individual, the property (p12 ) of being identical with c is of order 1 and belongs to no individual but c. Because of condition (2), there is no such property (p1c ) of order in Waismann’s world. But also in that world, there exists a property (p2c ) of order 2 that an individual has the property (p2c ), if and only if it has all the properties of type 1 that c has, and this property (p22 ) that belongs to c and to no other. Since each property of type 1 belongs to more than one individual, there is no property of type 1 with an extent equal to that of (p2c ) in violation of the axiom of reducibility. But one may ask, do not conditions (1) and (2) lead to contradictions? They certainly do in any ‘world’ consisting of only a ﬁnite number of individuals with a ﬁnite number of properties of type 1. For under these conditions, the conjunction of all properties of order 1 of a certain individual c is a property of type 1. It is a property of c and, by (1), of no other individual, but this contradicts (2). Waismann actually deﬁned, however, a model with inﬁnitely many individuals, which satisﬁes (1) and (2).

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Let [a,b[ denote the set of all rational numbers r such that arrrb. The individuals in Waismann’s world are the rational numbers in [0, 1[. Each property of type 1 is (after a correction recently communicated to me by Professor A. Sklar) a set of rational numbers that is the union of a ﬁnite number of intervals [r, r1[, where r and r1 are rational numbers in [0, 1[. An individual has a property if, and only if, the rational number belongs to the corresponding set. The disjunction and conjunction of two properties, that is, the union and the intersection of two sets [ill.] themselves properties of type 1, and so it is the negation of any property, that is, the complement of the set; for example, the complement of [1/4, ½[ is the union of [0, 14[ and [1/2, 1[. For any two different rational numbers there clearly exists a property (e.g. an interval) including only one of them, whence condition (1) is satisﬁed, and obviously, each interval includes inﬁnitely many rational numbers. Thus, a model satisfying both conditions exists. In it, if c is any individual, the property (p2c ) of type 2 discussed above, which holds for c only, has an extent that is different from that of any property of type 1. y 4. Much of the discussion about reducibility was devoted to F. P. Ramsey’s paper ‘The Foundation of Mathematics’, which had appeared in 1925 and reached Vienna some time thereafter. Especially Schlick and Hahn were greatly impressed by that paper. Ramsey was of the opinion that Wittgenstein had deﬁned tautologies even beyond the calculus of propositions. What is essential about conjunctions and disjunctions is that their truth or falsity is determined by the truth and falsity of their components. A conjunction is true if, and only if, all components are true; a disjunction is true unless all components are false. Here, the number of components is irrelevant; there may be even inﬁnitely many components. In this case, the compound becomes a quantiﬁed truth function – a conjunction becomes an ‘all’ proposition, a disjunction a ‘some’ or existential proposition. Hence, Ramsey thinks, the idea of tautologies can be extended to quantiﬁed propositional functions. But even if all that be admitted, a precise deﬁnition of tautologies still depends on whether quantiﬁcations are restricted to individuals or also applied to classes and properties; it depends in the latter case on questions concerning the hierarchy of types of sets and properties; it depends on the rules of procedure in the calculi, which Russell had formulated in a rather inaccurate fashion. The axiom of reducibility, according to Ramsey, is not a tautology. Ramsey’s main accomplishment in that paper was the classiﬁcation of the antinomies into those that are purely logical, that is, deal

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exclusively with logico-mathematical concepts of sets and properties, for example, the set of all sets, and those that are epistemological or semantical, which contain references to extra-logical entities such as a lying person, a particular line in a book and the like. Moreover, Ramsey found that in the treatment of really menacing paradoxes, the purely logical antinomies, the axiom of reducibility was not needed. y 5. Schlick, I seem to remember, told me that Ramsey had visited him in Vienna. If he attended meetings of the Circle, that must have been before I joined or at a time while I happened to be out of town. When in 19[31], several years after Ramsey’s untimely death, his collected papers [The Foundations of Mathematics and Other Logical Essays] were edited by R. B. Braithwaite, I was all the more sorry to have missed him in Vienna, for the collection includes an extremely interesting paper Truth and Probability, written in 1926, but not published during Ramsey’s lifetime. In it, Ramsey developed a combined axiomatic theory of probability and economical value, which made him a forerunner of the axiomatic of utility developed by J. von Neumann and O. Morgenstern in the1950s. I should have loved to discuss these matters with Ramsey, for just at the time when he created his general theory connecting probability and value, I myself approached that connection from a different angle about which I reported either in the Circle or to its senior members individually.1 For many years, I had been interested in the so-called St. Petersburg paradox, so named because the game of chance with which it deals was ﬁrst discussed in the Commentarii of the Academy of St. Petersburg.2 A coin is tossed until ‘head’ shows for the ﬁrst time and this will terminate the game. Peter promises that if it is terminated at the nth throw (i.e. if ‘head’ ﬁrst falls at the nth throw), he will pay Paul 2n ducts. What is a fair price for Peter to ask, and for Paul to pay, for the chance that Peter thus offers Paul? The mathematical expectation of a gain of d ducats with the probability p is traditionally deﬁned as the product, pd ducats. If I am promised $ 3 provided a die shows 5, then my mathematical expectation is $ (1/63). Now the game G, which Peter offers Paul, may be said to be the conjunction of a series of simple games, G1, G2, G3 . . . . To be played until Paul for the ﬁrst time wins one of them (all played with the same coin but that is inessential). The nth game consists in tossing a coin n times, and in the event that the ﬁrst n1 times tail and the nth time head turns up (for which the probability is 1/2n), Peter will pay Paul 2n ducats, otherwise nothing. Each game Gn thus creates for Paul an expectation of (1/2n) 2n ducats, that is, 1 ducat. If Peter offered Paul only the ﬁve games, G1 . . . G5 (or, for that

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matter, any ﬁve of the games), then Paul’s mathematical expectation would be 5 ducats. Since Peter does not limit the series of games, it follows that Paul’s mathematical expectation in toto, that is, in the game G, is inﬁnite. Now apart from the fact that Peter’s wealth is limited, obviously Paul cannot pay more than his own entire wealth for the chance offered him by Peter, but, unless he is out of his mind, he will not even risk more than a very small fraction of his fortune in the game. In 1738, Daniel Bernoulli tried to explain this paradox by an ingenious hypothesis, which made him the ﬁrst forerunner of the theory of marginal utility. He assumed that a man attributes to an increase of d ducats to a wealth of w ducats a value proportional to d but inversely proportional to w, which leads to what Bernoulli calls a ‘moral value’ proportional to log(1þd/w). The chance to increase one’s wealth with a certain probability Bernoulli evaluated by the product of the probability and the moral value of the increase. This he called the ‘moral expectation’, for example, Paul’s moral expectation from the game Gn is proportional to (1/2n) log(1þ2n/w). Now it is easy to show that the sum of the inﬁnitely many contributions of G1, G2, G3, . . . to Paul’s total moral expectation from the game G is ﬁnite – a result that Bernoulli and the mathematicians after him, including Poincare´, considered as a solution of the paradox. In a lecture before the Viennese Economical Society in 1927, I showed, however, that Bernoulli’s method, for all the ingenuity of the underlying idea, offers only an ad hoc solution for the particular game G. If the amount that Peter promises to pay for the 2 game Gn is raised from 2n to 22 ducats, then even Paul’s moral expectation turns out to be inﬁnite and yet he would refuse to risk an appreciable part of his wealth in that game. Moreover, I saw the real problem not so much in the inﬁnitude of certain expectations as in the general discrepancy between actual human evaluations (as reﬂected in the behaviour of persons gambling, purchasing chances and the like), on the one hand, and mathematical as well as moral expectation, on the other. For certain probabilities and certain ratios of the gain to the wealth of the evaluating person, the latter attributes to chance a value that is higher than the mathematical expectation; for other probabilities and ratios, a value that is lower than even Bernoulli’s moral expectation. People’s actual expectation thus is not simply proportional to the probability, as it has been assumed to be in all traditional formulae, and no realistic description of the discrepancy between formula and actual behaviour is possible as long as the factor p remains the noli me tangere that it has been since Bernoulli. Nor must p simply be replaced by a function of p. One must rather introduce functions jointly, depending on d, w and p.

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I suggested that economists investigate the form of the dependence varying from person and indicative of certain traits of the person’s character.3 But Professor H. Mayer, the editor of the Zeitschrift fu¨r Nationalo¨konomie (Journal of Economics), explicitly discouraged me from submitting my paper to his journal and it was only published in 1934, when O. Morgenstern had gained inﬂuence in the editorial board. Morgenstern was only one of the very few Austrian economists who were free from prejudices against mathematical methods in economics, but in the mid1930s, he began to be seriously interested in mathematics and logic.4 Especially Carnap listened with great attention to my results in connection with the Petersburg paradox, and more than 20 years later, in his book ‘Logical Foundation of Probability’, he quoted everything that I had reported in the Circle. y 6. In 1926, Carnap had published a very clear and well-written booklet Physikalische Begriffsbildung (The Formation of Physical Concepts). He describes in it three levels of such formations: the qualitative, the quantitative and what he called the abstract, by which he meant essentially descriptions with reference to the coordinatized space-time continuum. The bibliography included works by Mach, Schlick and others, but no mention of Russell or Wittgenstein. The booklet contained in fact only one hint in the direction of the later discussions in the Circle. ‘The formation of a concept’, Carnap said in the Introduction, ‘is the formulation of a law concerning the use (die Verwendung) of a sign (e.g. a word) in representing Sachverhalie’. In the fall of 1926, when Carnap came to Vienna and joined the circle, he brought with him an advanced draft of an important, much longer book Der Logische Aufbau der Welt (The Logical Structure of the World). Several philosophers had claimed that a man’s picture of the world could be built up from his immediate sense data such as Machian ‘elements’. In his book, Carnap tried to carry out this ambitious project in the systematic and thorough manner that was so characteristic of him. In the fall of 1927 in the Circle, he discussed some last details. The book was published in 1928. The propositions of afﬁne geometry correspond to physical situations, for example, incidences of chalk dots and lines on a blackboard. The concepts in Carnap’s construction of the world likewise reﬂect observable phenomena. But Carnap chose an auto-psychological rather than a physical basis. And he did not proceed deductively from postulates to theorems but rather ‘constituted’ (i.e. deﬁned to sketch a conceptual picture of the entire world with its immensely manifold phenomena, the structure of the picture

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reﬂecting the structure of the world). Carnap did not regard Mach’s elements as the most wishful foundation (mainly because of the results of Gestalt psychology developed after Mach’s analysis of sensations) but rather chose some more complex experiences and recollections. More speciﬁcally, the basis of the psychological material consisted of (1) basic elements that Carnap allied to elementary experiences, that is, places in the stream of his or some other subjects’ experiences, where, by a place in the stream he meant a subject’s undivided experience at some moment, which may include visual, acoustic, tactile and other elements; (2) a single basic relation, namely the recollection of similarity (recognized by the comparison of the memory image of one experience with another experience). Carnap developed methods for the analysis of experiences by means of the basic relation, for example, methods for the separation of visual and acoustic components of experiences, and, more generally, for the resolution of experiences in Machian elements. For example, when comparing a c-e-gchord with a previously heard g-sound (or c-sound or e-sound), one perceives a similarity that is lacking in the comparison with, say, a d-sound. Systematically pursuing such comparisons, one can analyze the chord so as to obtain three constituent sounds. All these analyses concern the psychological material. The corresponding theoretical construction operates with undeﬁned terms ‘elementary experience’ and ‘recollection of similarity’. Where there is any danger of confusion, Carnap writes pexperiencep for the psychological experiences and cexperiencec for the undeﬁned term in the construction. The c-terms correspond to the p-terms as far as the assumptions of construction permit. For example, to the ptriad chordp, there corresponds a ctriad chordc. But a ptriad chordp has three pconstituentsp, whereas the ctriad chordc, being an element of the theory and, therefore, unanalyzable therein, has no constituents in it. What corresponds, for example, to the pconstituents g-soundp of the p chordp is a certain class of celementary experiencesc, namely the class of all c sounds recollected to be similar to a g-soundc, which includes the ctriad chordc classes of celementary experiencesc which play in Carnap’s construction of the world the role of constituents of elementary experiences. By such abstractions (i.e. formation of classes), Carnap constituted, on the one hand, Mach’s elements and, on the other, more and more complex entities, for example, quality classes, similarity of qualities, visual places and their arrangement in the (2-dimensional) visual ﬁeld, colours and their arrangement in the colour space. In his sketch of a constitution of the time order, I could not follow Carnap. (I believe that it was later improved).

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He then proceeded to the level of physical objects in the (3-dimensional) visual world including his own body and the bodies of other persons, and, carrying out Mach’s program, he further constituted his own conscious and unconscious mental processes. The totality of these processes is what is called his self. It had not been postulated – it was constituted. Finally on the highest level, there were the constitution of hetero-psychological objects by the association of psychological events to the bodies of others on the basis of their expressions (voices, gestures, etc.) and an outline of the constitution of cultural objects (the objects of history, social sciences, humanities). All constructions, from Mach’s elements to cultural objects, were ultimately based on celementary experiencesc and crecollections of similarityc. Whatever was constituted was introduced as classes of basic elements,5 relations between elements, classes of such classes and relations, relations between such classes and so on. This is expressed by calling the construction extensional. By the extensional view on properties and relations is meant the mere consideration of the classes of entities having the properties and being in the relations (as opposed to a study of the ‘intentional’ meaning of properties and relations). The thesis of extensionality claims that all statements about a propositional function can be transformed into statements about classes. Carnap seemed to consider this thesis as a corollary of the view expressed in the Tractatus that only the truth or falsity of a proposition p, (not its ‘content’) inﬂuences truth or falsity of any proposition containing p. (Whether Wittgenstein himself admitted this connection with the thesis of extensionality I don’t know. To reconcile Wittgenstein’s view with the existence of propositions such as ‘I believe p’, Carnap claimed that ‘I believe p’ actually does not involve p (as does, say, ‘p implies q’) but is rather about my idea of p. The last part of the book sheds light on the theses and systems of traditional philosophy. For example, Carnap believed that even in his system with an auto-psychological basis, real and unreal (i.e. merely imagined) things could be distinguished, namely by Schlick’s criteria for reality: a position in time and the participation in a comprehensive system governed by laws. He called this the empirical problem of reality, in contrast to the metaphysical problem of reality concerning existence independent of cognizing conscience, on which the traditional schools of philosophy diverge. Realists claim the existence of physical as well as hetero-psychological objects; idealists deny the existence of physical and, in the extreme (solipsistic) school, even of hetero-psychological objects. Phenomenalists teach the existence of unrecognizable things-in-themselves of which we recognize only the appearances.

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Carnap points out that all questions answered by the construction theory as well as metaphysical realism, idealism and phenomenalism are uniformly answered. The age-old controversies between the three classical schools concern only the domain of the unconstructable, which Carnap excludes from science. y 7. Every normal child constructs or constitutes his or her world, including hetero-psychological entities, from primitive sense experience, but of course with help from teachers in the widest sense of the word. In importance, the teachers’ communications are second to none of the other components; in fact, as the case of Helen Keller6 demonstrates, they are at least as important as most of the others combined. These communications (again in the widest sense of the word) also create a normal young child’s ability to understand speech and to speak. Through language, they implant into the child’s world in statu nascendi, in addition to countless words that are constructible in Carnap’s system, some words that are not, and they may even create what F. Mauthner called word fetishism. Yet, from a practical point of view, the constitution of those children’s worlds is fairly satisfactory, since people (other than metaphysicians) all in all agree on the world – even people speaking different languages. Carnap’s book seems to imply that in constituting his or her world, a child theoretically could dispense with communications from others even though practically there is no such possibility for human beings as now exist. Another point of discussion was Carnap’s contention that each propositional function, property, relation and so on has domain of meaningfulness. According to Carnap, the propositional function ‘x is a German city’ is true for Berlin, false for Paris and meaningless for the moon. Following Husserl, Felix Kaufmann had always maintained ideas of this kind and was delighted to hear Carnap conﬁrm them, while I was less enthusiastic. Being a German city is a conjunction if lying in German and being city. Paris, lacking only the ﬁrst component, leads to a false but meaningful proposition. I should imagine that the same can be said for the Krupp factory, which lacks only the second component. Thus if ‘x has the property P’, one is allowed to analyze P. But then how many components of P must a subject lack to fall outside of that so-called domain of meaningfulness? It seemed to me that one could analyze ‘a German city’ to the point where ‘material object’ is a component, and this component would also be an attribute of the moon. Russell had banned as meaningless only assertions combining elements of different types such as ‘the class of all colours is white’. Carnap seemed to believe that his own obviously not well-deﬁned domains of meaninglessness

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would somehow result from Russell’s ban. But the moon seems to have as little to do with classes of German cities as Berlin with classes of satellites. Besides, one might conceivably construct classes of classes of elements, which can also be obtained as classes of classes of classes of classes of other elements, and this would make it difﬁcult if not impossible to use Russell’s types in deﬁning meaninglessness of sentences about the world. At best, it would be a formidable task to prove in detail that confusions of types are responsible for all sentences that, more or less arbitrarily, are declared to be meaningless. As far as language is concerned, Carnap mentioned, and used, four variants in his constitution system. But this is not the kind of multiplicity of language alluded to in Chapter IX and to be discussed in subsequent chapters.7 For, as Carnap wrote, the ﬁrst of the four is ‘the’ symbolic language of logic while the three others ‘offer nothing but translation from this ﬁrst basic language’: paraphrases in word language, translation into the realistic language customary in empirical sciences and into something of a facilitating control language. Carnap’s book was of an unprecedented wealth of well-organized scientiﬁc and logical material to which the discussions in the Circle, let alone the preceding brief description, could not possibly do justice. All members of the Circle were greatly impressed, but in particular, Hahn and Neurath: Hahn, because of the synthesis of the two philosophical ideas that had always been uppermost in his mind, Mach’s empiricism and Russell’s logic; Neurath, because he saw in the demand for constructability of concepts the sharpest than available weapon against metaphysics because he thought that it demolished the wall between sciences and humanities, which he regarded as an artiﬁcial separation. He trusted that the unscientiﬁc mysticism on the humanistic-sociological side would fade out now that the way to a scientiﬁc treatment of all meaningful problems had been opened. y 8. Logical Structure was a technical book, written in a fairly conciliatory tone. Carnap quoted all philosophical writings that were in any way parallel to his own ideas and was restrained even in his criticism of metaphysics at the end of the book. In 1927/28, he wrote and presented to the Circle a short and essentially polemical book Pseudoproblems in Philosophy, which was more radical and less technical. In fact, it was addressed to the non specialist. ‘The meaning of a statement’, Carnap said with strong Wittgensteinian overtones, ‘lies in the fact that it expresses a (conceivable, not necessarily existing) state of affairs’. He then proceeded to apply this oversimpliﬁed

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criterion to the theses of philosophers about reality and about heteropsychological concepts. In Logical Structure, after a painstaking positive ‘construction of the world’, he had called the metaphysical theses unconstructible and had on this ground excluded them from science. In ‘Pseudoproblems’, on the basis of his at best very sketchy criterion of meaning, he called them meaningless. y 9. As the academic year went on and Carnap got more radical, Neurath got more excited and aggressive. When the idea of spreading the new insights uppermost in his mind, Neurath suggested that a society (ein Verein) for the promotion and propagation of a scientiﬁc view of the world be founded and named after Mach. ‘We have the Philosophical Society’, Schlick protested, and someone else pointed out that the Viennese Volkshochschule (People’s university) was not only of general excellence but had Zilsel, Feigl and Waismann on its staff. But Neurath easily convinced Hahn and Carnap that this was not enough. Finally, he prevailed on Schlick, as the occupant of Mach’s chair at the University of Vienna, to share the chairmanship of a Verein Ernst Mach with Hahn and the socialdemocrat inspector of the public school system, H. Vokolok, while Neurath himself and Carnap would be the secretaries. The Verein would start its activities with some public lectures in 1928/29. Schlick was not altogether happy. But Neurath was on the war path. y 10. In the ﬁrst semester of my teaching at the University of Vienna in the fall of 1927, I had to offer a course on projective geometry, which used to be referred to as the geometry of joining and intersecting. Indeed, the theory is based on the operations of forming the line that is the join of two points, and the point that is the intersection, also called, the meet, of two lines in the plane, or in space, the plane that is the join of a line and a point, the point that is the meet of a plane and a line, and the line that is the meet of two planes. The situation is similar in higher dimensional projecting and even in afﬁne geometry (i.e. the Geometry of Euclidean spaces in which perpendicularity and congruence are ignored), where planes contain lines having a meet that is vacuous (i.e. parallel lines). Naturally, I wanted to base projective geometry on a system of axioms concerning the two all-important operations. The foundation of the theory should reﬂect the paramount feature of the theory itself. But, to my surprise, I could not ﬁnd such an axiom in the literature. So, I set one up for my course. To make the system absolutely complete, I found it necessary to postulate some utterly simple properties of the operations, for example, that the line

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jointing point P to point Q is the same as the line joining Q to P. No geometer had wanted to bother with such trivia, let alone formulate them as postulates. But the astonishing result was that such simple properties of joining and intersecting were not only necessary postulates but also a sufﬁcient foundation for the entire projective and afﬁne geometries of any dimension. Those postulates were analogous to the axioms of algebra about adding and multiplying numbers, the one about the join of two points to an axiom about the sum of two numbers, namely that the sum of m and n is equal to the sum of n and m, the so-called commutativity of addition. Similarly, both joining and adding are what is called associative operations and so on. I therefore referred to my foundation of geometries on joining and intersecting as algebras of geometry. Moreover, the algebra of projective geometry was similar to Boole’s algebra of classes and, in fact, contained the latter as a special case. Classes in Boolean algebra satisfy the laws of the algebra of geometry and besides also so-called distributive laws analogous to the one postulated in arithmetic for any three numbers a,b,c: a ðb þ cÞ ¼ ab þ ac For example, if the class A is disjointed from the classes B and C, that is, if its common part with either one is the vacuum [missed], then also the common part of A with the union of the classes B and C is [missed], which is the join of [missed] and [missed], just in algebra 0þ0 ¼ 0. In the more general algebra of geometry, non-distributive law is postulated or in general valid. The most important concept that can be based on the operations of joining and intersecting is that of a part relation. A geometric object is a part of an object B if A and B have the join B and the meet A. The relation so deﬁned can be demonstrated to have all the properties that one expects of a geometric part relation, the simplest of which is transitivity: if A is a part of B, and B is a part of C, then A is a part of C. This deﬁnition of parts enabled me to integrate the initial sentence of Euclid’s Elements in a deductive theory, which had never been done before. The famous words ‘A point is that which has no part’ had been written and rewritten for 2000 years without ever being used in any deduction. The want of a basic treatment of the geometric part relation had made such a use impossible. In the algebra of geometry, however, I could deﬁne a point is a non-vacuous object that has no part except itself and the vacuum. The qualiﬁcations of Euclid’s sentence must be included in the deﬁnition just as

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in deﬁning a prime number as an integer that has no factors one must add ‘except itself and 1’. Without these qualiﬁcations, every integer would be a prime number since every integer has itself and 1 as factors, just as every geometric object has itself and the vacuum as parts. Indeed, the result of joining an object to itself or to the vacuum is the object itself. And one must restrict the deﬁnition of prime numbers to integers 6¼1, since otherwise, 1 would be a prime number. Similarly, one must restrict the deﬁnition of a point to objects that are non-vacuous since otherwise the vacuum would be a point. A line can then be deﬁned as an object that is the join of two points, and, from there, one can go on and develop projective and afﬁne geometries systematically. I do not recollect whether I reported this theory to the Circle. I remember, however, presenting it to the Vienna Mathematical Society in a meeting which some members of the Circle attended. But the old Viennese geometers were not much interested in the fact that the geometry of joining and intersecting could be based on assumptions about joining and intersecting, and also Hahn’s reception of the paper was rather cool. Yet, I published it in 1928 in Germany, but at this time, these ideas attracted little attention. One aspect of that algebra which, I felt, might have interested philosophers was its power in the way of synthesizing various chapters of mathematics: branches of geometries of all dimensions; the algebra of classes and aspects of the theories of measure and of probability, the measure of sets and the probability of events displaying important analogies to the dimension of the objects studied in projective and afﬁne geometry. Much more in this direction could be said – and was said several years later in America – while I myself, somewhat disappointed by the general lack of interest in the subject, turned to other mathematical studies. y 11. I met Łukasiewicz, Tarski, Lindenbaum and other young logicians, whose work I found most interesting. Lesniewski, whom someone had described to me as a kind of Polish Wittgenstein, had just published a very verbose account of his ideas in the Fundamenta Mathematicae. Unfortunately, I barely met him for when he saw me, he exclaimed in honest surprise, ‘Are youngsters made professors in Vienna?’ I felt greatly amused and, in a way, ﬂattered since I seemed to have successfully camouﬂaged my receding hairline. But my hosts, the mathematicians, were very angry and apologized in front of Lesniewski for his remark, which they considered as an insult to the University of Vienna and to me, and I could not win them

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over to my way of looking at this trivial matter. I never saw Lesniewski again. For the ﬁrst time, I heard about Łukasiewicz’s multi-valued logic and other achievements including his frontal notation. Instead of symbolizing the phrase, p implies q, in the traditional way by a sandwiched sign, for example, an arrow as in p-q, Łukasiewicz writes C p q. This frontal symbol C makes parentheses unnecessary. The expression p-q-r is ambiguous and parentheses are indispensable to distinguish p ! ðq ! rÞ

and

ðp ! qÞ ! r

Instead, Łukasiewicz writes, unambiguously and without any need for parentheses CpCqr and CCpqr But what I appreciated most was the fact that the Warsaw logicians were interested in philosophical problems not altogether different from those discussed in the Vienna Circle but that they attacked them in connection with, and partly on the basis of, their precise logical studies. They always conﬁned themselves to concrete questions and completely eschewed all vague generalities, which clouded some of the Viennese discussions. I decided to familiarize my friends in the Circle as well as the younger Viennese mathematicians with the logico-philosophical work of the Warsaw school. For this purpose, I invited Tarski to visit my Mathematical Colloquium and to deliver three lectures, for two of which I planned to invite the entire Vienna Circle. Of course, the mathematicians of the Colloquium, including Go¨del, received the lectures with great interest. But in Warsaw, I found what I had always been looking for: philosophical discussions in accord with what I called Pascal’s Principle – everyone begins with a speciﬁcation of the terms admittedly taken as undeﬁned. y 12. Hilbert had introduced what he called metamathematics: the theory of (formalized) mathematics, developed in what became known as the metalanguage. Tarski’s lecture, which also incorporated important results of Lindenbaum, dealt with some general features of (formalized) deductive theories. For the beneﬁt of the philosophers who had missed the preceding lecture, I explained that in his methodological studies, Tarski left ‘meaningful sentence’ and ‘consequence’ undeﬁned, thus making the results valid no matter what idea one might associate with that term and what logic one might use. But in 1930, not only Schlick and Waismann recognized just one logic – ‘the logic’ – but also Carnap and even Hahn.

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NOTES 1. [Editorial note. In 1923, Menger wrote ‘Unsicherheitsmoment in der Wertlehere. Betrachungern in Anchluss an das sogenannte Petersburger Spiel’ (Menger, 1979). Hans Mayer, the then editor of Zeitschrift fu¨r Nationalo¨konomie, refused to publish it because of the excessive use of mathematical formulas. The paper, discussed in 1927 at a meeting of the Viennese Economic Society, was only published in 1934, when Oskar Morgenstern was appointed editor of the Zeitschrift (Becchio, 2008)]. 2. [Editorial note. The paradox is named from Daniel Bernoulli, who, in 1738, presented the problem and his solution in the paper ‘Specimen Theoriae Novae de Mensura Sortis, Commentarii Academiae Scientiarum Imperialis Petropolitanae, Tomus V [Papers of the Imperial Academy of Sciences in Petersburg, Vol. V], 1738, pp. 175–192. It was ﬁrstly translated from Latin to English in 1954 by Louise Sommer as ‘Exposition of a New Theory in the Measurement of Risk’, Econometrica 22 (1): 22–36. In this English translation, Menger himself wrote some remarks, in particular, he composed footnotes 4, 9, 10, and 15 (pp. 28, 31–32 and 34)]. 3. [Editorial note. Editorial note. Since its presentation, Bernoulli’s paradox had been discussed by mathematicians and philosophers. Due to Menger’s paper (regarded as ‘the most extensive study on the literature of the problem, and the problem itself’ (Sommer, 1954, p. 35)) and the following reference in O. Morgenstern and J. von Neumann’s ‘The Axiomatic Treatment of Utility’ (the appendix of the second edition of The Theory of Games and Economic Behaviour, (1947)), it was included also in the ﬁeld of interest of the most important economists of that time. This was the reason for what, in the early 1950s, its English translation urged. An editor’s footnote of ‘Econometrica’ clearly stated: ‘In view of the frequency with which Bernoulli’s famous paper has been referred to in recent economic discussion, it has been thought appropriate to make it more generally available by publishing this English version’ (p. 23). Recent references, mentioned by the editor, were Arrow, K. J., ‘Alternative Approaches to the Theory of Choice in Risk-Taking Situations’, Econometrica, Vol. 19, October, 1951; Mosteller, Frederick, and Philip Nogee, ‘An Experimental Measurement of Utility’, Journal of Political Economy, 5, Oct., 1951; Samuelson, Paul A., ‘Probability, Utility, and the Independence Axiom’, Econometrica, Vol. 20, Oct. 1952; Friedman, M., and Savage, L. J., ‘The Expected Utility-Hypothesis and the Measurability of Utility, Journal of Political Economy, Vol. LX, December, 1952. Alchian, A. A., The Meaning of Utility Measurement, American Economic Review, Vol. XLIII, March, 1953; Herstein, I. N., and John Milnor, An Axiomatic Approach to Measurable Utility, Econometrica, Vol. 21, April, 1953; Marschak, J., Why ‘Should’ Statisticians and Businessmen Maximize ‘Moral Expectation’?, Second Berkeley Symposium on Mathematical Statistics and Probability, 1953. Later, P. Samuelson published ‘The St. Petersburg Paradox as a Divergent Double Limit’ (International Economic Review (Jan 1960), (1): 31–37), where he referred to Ramsey, while Arrow started his paper ‘The use of unbounded utility function in expected-utility maximization’ quoting Karl Menger (Quarterly Journal of Economic, (Feb. 1974),88 (1): 136–138)]. 4. [Editorial note. In addition to Morgenstern, Menger’s interest for economics and social sciences was particularly appreciated by other economists such as Ragnar Frisch. See his letter to Menger (September 18, 1935) in which Frisch wished that Menger

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would keep on dealing with the possible application of mathematics to economics and social sciences (Menger’s papers, Perkins Library, Duke University):

Dear Professor Menger, Thank you for yours of July 26th. I was very pleased to hear from you. It reminded me of the interesting and stimulating time we spent together in Cleveland several years ago when the Econometric Society was founded. I do not think I ever received your paper on the risk element in value theory. I should appreciate very much indeed your sending me a copy. I am very happy to hear that you expect in the future to devote part of your energy to the problems of economics and the social sciences. There is a vast need for people, with a genuine mathematical knowledge and ability, taking up problems in this ﬁeld. I am sure that we may expect highly signiﬁcant contributions from you. If you come into this ﬁeld, may I take this opportunity of asking you to give me as editor of Econometrica a chance of seeing your MSS. if you complete anything about applications of mathematics to economics. I am not the least surprised that you ﬁnd that very difﬁcult mathematics may be needed to clear up questions of economics or social science or the logical or the social-logical questions which you have taken up. I have myself found that frequently one runs into extremely difﬁcult mathematics in the application to economics. If you have a spare copy of your book on social-logic, I should be very glad to have it. (..) Needless to say I should be very happy to see you again and discuss many matters with you. I shall be at the meeting Best regards, Cordially yours, Ragnar Frisch].

5. In arithmetic, real numbers are introduced as certain classes of fractions. In afﬁne geometry, one introduces the class consisting of a line and all parallel lines as an ‘ideal point’. 6. [Editorial note. Helen Adams Keller (1880–1968) was an American writer. She was the ﬁrst deafblind person to graduate from college, thanks to her teacher, Annie Sullivan, who managed to break through the isolation imposed on her by a complete lack of language]. 7. [Editorial note. Maybe Menger had in mind a scheme of his book: in Menger’s Reminiscences, chapter nine deals with ‘the communication of metaphysical ideas. Wittgenstein’s ontology’].

CHAPTER 6 VISITING THE UNITED STATES (1930–1931) [Chapter Thirteen of Menger’s Reminiscences deals with Menger’s visit to the United States in the academic year1930/31, but it is restricted to Menger’s stay at Harvard, where he spent the fall semester and lectured on dimension theory and metric geometry. At Cambridge, Menger met, among others, ‘the outstanding mathematician’ G.D. Birkhoff; the philosopher H. M. Sheffer, who ‘had discovered in 1913 that all particles of the calculus of propositions can be expressed in terms of a single one’, ‘extensive[ly] used’ in Wittgenstein’s Tractatus ‘without mentioning its author’; P. Weiss, N. Wiener and J. Schumpeter, the latter being a visiting scholar as well. But the person who most impressed Menger was P. Bridgman, who ‘appeared to him as a modern reincarnation of Mach’(Menger, 1994, pp. 158–173). The ‘Harvard sections’ of Menger’s Reminiscences are comprised between paragraphs 14 and 15 of the present publication and are not reprinted].

y 1. In April 1930, I had an appointment with Hahn in Athens, where he attended a parapsychological congress, while I was visiting friends.1 On common excursions in cars, I met two or three members of the congress. One conversation turned to astrology. A Viennese, whom I knew as an intelligent man, made speciﬁc claims about the determination of human destinies by the constellations at birth. ‘I wonder how one knows all that’ Hahn said. The man spoke about original revelations followed by millenniums of observation. Apropos observation, Hahn brought up the usual objections about people with totally different lives, though born under the same constellations, in fact on the same day. ‘But constellations change ever within an hour, in fact from minute to minute, and minor changes may have major consequences’ the man said. ‘Thus the exact instant of birth is decisive and that instant is rarely known’. ‘But how can you speak of the exact instant of a child’s birth?’ Hahn asked. ‘Delivery of a child is a lengthy process’. C. Caratheodory, the great Greek mathematician, who spent most of his life working in German universities, happened to be in Athens at that time and invited Hahn and me to dinner in a beautiful seaside place. He told us 87

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about a trip to the United States from which he had recently returned. As I was to ﬁnd out later, Caratheodory’s work was never fully appreciated in America, certainly not in his life time, but he spoke about his experiences with satisfaction and only commented on a certain American chauvinism of the kind usually associated with the French. Of all the places that he had visited, he liked best one of which we had not heard of until then – the Rice Institute in Houston, Texas. A few days later, looking through my mail, upon arrival in Vienna, I found a letter inviting me to lecture at Harvard during the fall semester of 1930, and, a few minutes later, a letter from G. C. Evans invited me to lecture during the spring semester of 1931 at the Rice Institute. y 2. During my last visit before I left, Miss Wittgenstein gave me Ludwig Boltzmann’s pamphlet Reise eines deutschen Professors ins Eldorade (A German professor’s trip to Eldorado), which the physicist published in 1905 when coming home from the United States where he had lectured in California. She said that upon my return she would ask me how my impressions compared with those that Boltzmann had collected just a quarter of a century earlier. Even in preparing my trip, I looked through the booklet for observations that I might utilize. But the only real danger that Boltzmann had to face – contaminated drinking water – did, I was assured, no longer exist; and his preoccupation with Austrian cooking I did not share. After the revolt of July 1927, Monsignor Seipel, the leader of the Christian Socials, publicly asked O[tto] Bauer to dissociate his Social Democrats from the rebels even though it was clear that in July the outraged masses had deﬁed party discipline and that for two days the leadership had completely lost control. Had Bauer ‘drawn the line’, then his party would have undergone a split; on the contrary, since he did not draw it, fascists of various shades developed even more threatening paramilitary organizations for the protection of the country against excesses of radicals. So these organizations multiplied and diversiﬁed and, with their competing parades, made up life in Austria. Given the ever increasing Nazi danger in Germany, I failed to see a way out of the situation. The lovable Austria to which I had returned in 1927 began to disappear. I said goodbye to my friends, among them to the Cassandra who in 1927 had predicted a collapse of the stock market, ‘What do you know about America? What have you read about it?’ she asked. Although I knew most of continental Europe, I had preserved my naivete´ and my ideals about the New World. ‘Well’ I said enthusiastically, ‘America

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is the land of freedom and progress. The government does not interfere with people’s lives. There are no privileges of birth; everyone is judged by, and treated according to, his personal merits and nothing else. America is the land of great inventors and engineers. Immense efﬁciency has created unheard of progress and produced fabulous further opportunities. The cruel Indians about whom I read as a boy in Cooper’s novels are not dangerous any more. The black African slaves whom the English imported and sold in America were set free by Abraham Lincoln. What have I read? In the ﬁrst place, the ‘Leaves of Grass’, to me the greatest modern poetry. I have also read some Mark Twain,2 Jack London3 and Sinclair Lewis with great enjoyment’. On the way to the Canadian steamer that took me to Montreal, I spent two days in Berlin and Hamburg. There was great excitement about the outcome of the elections that had just been held. Nazis had been elected to the Reichstag. During the academic year 1930/1931, I lectured in various parts of the United States. Feigl, who likewise spent that year in America, energetically publicized the Circle, applying himself especially to prepare the way for Carnap. He decided to stay in the United States for good, while Carnap left Vienna to join Philipp Frank in Prague. From there, Carnap visited the circle several times. He no longer talked about the language and, to the displeasure of Waismann, used several languages. He still referred to the logic, though. But I continued to belabour, this point until, after the second or third visit, Carnap gave up these references, too, despite the protests of Schlick as well as Waismann. During my absence, two other points had come to prominence. Schlick, like most epistemologists, wanted to give a description of the foundation of empirical knowledge (das Fundament der Erkenntnis). He saw this foundation in utterly simple Konstatierungen, by which he meant the establishment of comment-free, primitive observations such as here now, the pointer of his instrument coincides with the mark. Philipp Frank had always heralded pointer readings as the ultimate source of our knowledge of physics. Now also Carnap and Neurath propounded a variant of this idea in their theory of so-called protocol sentence. Let F stand for a Konstatierung such as the one above, and let X denote the observer and (P, t) the place and time of F. Then an example of a protocol sentence (the protocol corresponding to F, which may be written or read by any proponent, Y) is at ðP; tÞ; the observer X has observed F.

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Differences between a protocol sentence and a Konstatierung are obvious. Schlick rules out that X lies. But it cannot be ruled out that Y lies. Moreover, Y may be under a delusion about F. But after establishing a oneto-one correspondence between Konstatierungen and protocol sentences (at least for each proponent, Y), a mathematician qua mathematician may well take a less urgent view than philosophers with regard to the problem as to which of the two is the foundation of our knowledge. y 3. In the United States I was fortunate enough to meet almost everyone who at that time was actively interested in questions discussed in the Vienna Circle. I also looked for opportunities to widen my knowledge about pragmatism, although I could not follow the doctrine in all its ramiﬁcations – certainly not into the identiﬁcation of truth with usefulness. (Usefulness for whom? If for society, how is one to know or, rather, to decide what is useful? Usefulness for now or in the long run?) But although the identiﬁcation of truth and usefulness seems to me untenable, no matter how usefulness is deﬁned, the Schlick Kreis in the late 1920s did not in my opinion appreciate enough and heed the basic emphasis of pragmatism, and I was looking forward with great anticipation to learn more about American philosophy and, of course, in particular about studies related to those of the Vienna Circle. Ever since the differences between the climatic conditions, east of the Rockies and north of the Alps, might not to some extent account for differences between typical characteristics of Americans and Central Europeans: the greater ﬂexibility of Americans and the greater steadiness of Central Europeans. Venturing into the ﬁeld of typical American and typical Austrian qualities, I am again reminded of a topic once discussed in the Circle – a comparatively little know paradox or near-paradox. Even though the reader [may] not [be] interested in this section in the sequel, I mention the matter as another example of the pervasiveness of the scientiﬁc conception of the world, which raises questions about topics as harmless as the term ‘a typical Frenchman’; for this is what the paradox is about. Let a quality be said to be typically French if it is shared by more than half of all Frenchmen. Clearly, for any two typically French qualities, there is at least one Frenchman possessing both. A Frenchman is said to be a typical Frenchman if he possesses all typically French qualities. The nearparadoxical difﬁculties can be illustrated in the case of just three typically French qualities, Q1, Q2, Q3 (say, patriotism, articulateness and politeness) and by assuming that the population of France is divided into seven groups

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of roughly equal size, which will be referred to as groups 1, 2 . . . 7. For a quality, Q, to be typically French it is sufﬁcient (though of course not necessary) that Q is shared by all members of four or more of the seven groups. Example 1. The quantities are distributed as follows. Q1 : 1; 2; 3; 4; 5;

Q2 : 1; 2; 3; 4; 6;

Q3 : 1; 2; 3; 4; 7.

All members of the groups 1,2,3,4 possess all three typically French qualities and hence are typical Frenchmen, whereas each member of the groups 5,6,7 lacks some of those qualities and hence is not typically French. The typical French constitute a majority – a fact, which one might think, could be expressed by saying that the quality, T, of being a typical Frenchman is a typically French quality. The quality Tu is not a typical French quality. In venturing to speak about typical members of a population, a logically trained person of course always remains aware of the facts underlying the famous paradox of the typical Frenchman: that the quality of being typical (in the sense of possessing more than two thirds of all qualities that are typically French, i.e., shared by more than two thirds of all Frenchmen) is untypical among Frenchmen (i.e., it is possessed by less than two thirds of all Frenchmen). y 4. The ﬁrst observation that I made upon settling in Cambridge appalled me: the degree of my ignorance about things American. I mentioned this fact because I am sure that I was not much less informed about the United States than were at that time most Austrians. (In the Circle, Schlick, who had been in America and was married to an American, was the only exception.) Secondary education in the Austrian empire put greater stress on the conditions in ancient Egypt, Athens and Rome than on the history and the institutions of the New World. Inasmuch as I had formed a picture of the United States at all it was that of a land with enormous opportunities, without prejudices and with complete personal freedom, where people dressed and wore their hair any way they wanted; in other words, my pale image was at least forty years ahead of its time. I thus was greatly surprised, in some respects, that it was to become the ultimate home of almost all members of the Vienna Circle. y 5. Nothing illustrates my initial ignorance better than surprise at ﬁnding the Constitution to be a living powerful instrument, mentioned almost daily in the newspapers. So, for the ﬁrst time, I read and at once reread the

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document several times with increasing admiration. In the Austrian constitutional monarchy the most widely known passage of the constitution was ‘article fourteen’, which enabled the emperor to rule by decree, without Parliament; and it was on the basis of that notorious ‘article fourteen’ that World War I was declared. I began to see dimly a connection between that document and a developing inner feeling of necessity and inevitability in the essential going-on, a feeling that I had not experienced in any of the European countries in which I had lived. On the contrary, I was amazed to ﬁnd the president accorded a veneration, the like of which I had not seen in Vienna since the death of the Emperor Franz Joseph. Also, although many Americans in 1930 seemed to believe that the United States had something of a monopoly in democracy I found in most of those whom I met an incomparably greater deference to European nobility – at least in theory – than existed in Europe and especially in the Austrian republic. But since very few Americans had ever seen a European aristocrat or former aristocrat, the feelings were practically quite unimportant compared to some other facts. To mention only one minor example, at the end of his day’s work, a waiter (provided his skin was white) would often be waited upon, along with new guests, in the seat of one whom he himself had waited upon a while earlier. This could not happen in Europe for all its democracy. In political and economic matters, despite more than a year of a severe depression, most Americans were completely self-assured. In stinging contrast to this posture, I noticed a marked difﬁdence in most aesthetical matters: hesitant statements of taste and noncommittal judgements about art, music and literature. Often this attitude was a consequence of ignorance; but even more often I found it paired with knowledge and understanding. Absence of dogmatism is, of course, what empiricists regarded as the only sound way to deal with matters of taste and to discuss questions of aesthetics. But in the America of 1930, that frame of mind was conspicuous to a Viennese observer. For this combination of attitudes was the exact opposite of the over-assertiveness on cultural causes combined with feelings of inferiority in economic-political matters that prevailed in Austria in the ﬁrst years after World War I and then changed. Six years later, I would ﬁnd also changes in the American attitudes. y 6. Most Americans seemed to be much kinder and more hospitable to strangers and foreigners than were most Austrians outside the intellectual elite of Vienna. Observing the behaviour of Americans to each other, I was struck by what seemed to be everyone’s perpetual watchfulness that no one else encroached on his own prerogatives. Defensive reactions to

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criticism – even to merely implied criticism – were often very strong. On the street or in buses and restaurants, Americans were more considerate and displayed much better manners than Austrians of the1920s did. Keeping secrets was not exactly the forte of Austrians: but there were rather strict limits of indiscreetness – some of them set by the penal code. In America, I was surprised to see that high-class papers and magazines often published details concerning the health, the love lives and ﬁnancial transactions of individuals as well as reports about goings-on in closed meetings on all levels, each of which seemed to presuppose a breach of secrecy somewhere along the line. The point is, of course, that those items were known, not that they were printed, for the guarantee that the press may print all they know is a fundamental provision of the Bill of Rights. A trait that I found considerably less developed in many Americans than I had expected it to be was curiosity for simple things and circumstances near them: strange shapes and forms in the surrounding landscape, the ﬂora and the fauna near their homes, interesting buildings in the towns they lived in, the meaning and the origin of the names of those places and even of their own names, any historical details. Few Americans seemed to feel that such useless knowledge enriched life, sharpened children’s power of observation and, if only on rare occasions, even turns out to be useful. On the contrary, where usefulness was apparent, I always found people instantly awakened to penetrating curiosity and readiness for action. y 7. Money was much more talked and written about, and it was even more intensely coveted than in Austria. Although most Europeans acquired enough means and simply enjoyed what they had; many Americans, after having made much money, considered it their principal aim to make more. Financial success was also a condition for the recognition of artistic and intellectual achievement. There was no aura of spirituality and no respectability about the starving poet and musician or the freezing painter living in an attic. In fact, hardly any such people existed. The cinemas showed such young men giving up great ideas to embrace lucrative occupations (e.g., leaving a symphony forever unﬁnished and turning to the composition of money-making music), presenting this course as something of a duty to society. All over the world, most professors in 1930 were poorly paid. But in America, because of their low income, they were also denied the prestige that they enjoyed in Europe. This prestige was accorded to medical doctors (most of whom had big incomes), notwithstanding the fact that these men had been trained and awarded their degrees by professors.

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y 8. In Europe, especially in poorer countries, I had observed the constant preoccupation of American travellers with standards of living, their own and their hosts’, often completely disregarding the differences between their scales of values; and I could see on many occasions that the hosts resented this attitude, especially where they were unable to explain themselves. In America, I found references to the standard of living used as explanations and, I thought misexplanations of various phenomena, especially of waste. The ﬁrst car cemetery that I saw right after crossing the border from Canada greatly impressed me. But the two most conspicuous examples during that year that I spent more or less as a tourist were those of soap in hotels and of food in restaurants. Every cake of soap was disposed of after a day’s use; and monumental quantities of excellent food remained unconsumed or were rejected, especially by children – to me who grew up during the Viennese famine an exasperating sight. The explanation was always the same. Our standards of living are so high that some cakes of soap and plates of food do not matter. But certainly, I ventured to remark, unconsumed, wasted goods do not enhance anyone’s standard of living. As a matter of fact, in an indirect way, they do, I was told. Waste stimulates greater production, more employment and greater prosperity – the method that has made America great. In 1930, when I ﬁrst heard this argument, I knew very little about America; nor can I say that I foresaw ecological crises. But even then I wondered whether the same methods that make a nation great necessarily keep it so. And I became convinced that certain processes could not, if only for physical reasons, continue indeﬁnitely. y 9. I have mentioned the odd combination during the ﬁrst years after World War I of exaggerated Austrian feelings of economic-political inferiority and of cultural superiority. In America in 1930, I observed the exact opposite: an all-pervasive belief in economic invulnerability of the United States, where nothing could stop perpetual progress to ever higher levels of prosperity, wealth and power and an overmodest assessment of the nation’s cultural achievements, especially a difﬁdence in artistic matters. Yet a year had passed since the disastrous crash of October 1929 without real economic recovery. On the contrary, to my amazement I found many Americans, who, while extolling the culture of 1930 in Europe (especially , France and Germany), seemed to discount or not to connect with culture their own public libraries, the like of which existed nowhere else; the newspapers, musical performances and theatres on Broadway, second to none of their European counterparts; the New Yorker without an equal; Sinclair Lewis

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and Eugene O’Neill; great works of cinematographic art, including Walt Disney’s; and American architecture, one of the greatest accomplishments ever. y 10. As to architecture, I found a congenial comment in Boltzmann’s reminiscences, a remark about the great impact on him of New York City with its ‘houses like towers’, though in 1905 the city was only at the very beginning of its architectural development. In 1930, the impression that I received was totally overpowering – more than older Americans who had witnessed the gradual growth of the town could appreciate and more than today with the universal proliferation of skyscrapers perhaps even foreigners can imagine. y 11. On a different level, but deeply impressive too, was what on my way west, I saw of the Indians in New Mexico and Arizona, most of all, their art (in 1930 totally unknown in Europe): the mystical birds and conventionalized beasts on their ceramics; their symbolic wooden dolls and sand paintings; and (in exhibitions) the carvings and totem poles of north-western tribes – all this, to my taste, unrivalled by any works by the traditional preabstract American artists of those days that I had seen. With black people I had little contact during that year. But after seeing many negroes in the southern States I could not help indulging in speculations of a type quite contrary to positivistic mentality – rather conforming to the title of Popper-Lynkeus’s book Phantasies of a Realist – namely in thoughts about the question, what if such and such (that did not happen) had actually happened? What if in about the year 800 A.D. the Hellenistic heritage had gone to black natives of Africa rather than to the Nordic conquerors of Southern Europe? What if further on, about the year 1000, those Africans had sent out ships to Italy for the purpose of capturing white slaves and seized, among others, all the ancestors then living of Dante, Columbus and Galileo? Finally, what if these slaves had produced offspring in pairs exactly corresponding to those actually producing offspring in Italy? Would their enslaved descendants have been the Dante, the Columbus, the Galileo whom we revere? One of Boltzmann’s remarks dealt with the extremes of the American weather. I remembered a conversation in the Circle about a meteorological law that is valid in Central Europe north of the Alps, called the law of the constancy of weather. Asked what will be the weather tomorrow, one has in Austria a high probability (I believe, a chance more than 70%) of answering correctly by saying ‘the same as today’. The explanation lies in the fact that the high mountains of the Alps are parallel to the west-east direction of the

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principal winds. In contrast, in America, with the north-south course of the Rocky Mountains, small changes of the wind direction may cause great changes in the weather, which therefore is subject to frequent and enormous ﬂuctuations between extremes. If any meteorological law is noticeable to a layman east of the Rockies, it is one of the inconstancy of weather. I wondered whether this difference might partly account for the allegedly greater emotional stability of Central Europeans. Be that as it may, the weather certainly supplied ample material to Americans indulging in their rather striking preoccupation with record numbers in all quantitative aspects and phases of nature and human life. y 12. Another object of even more intense American preoccupation that I observed was education. The topic was being discussed much more frequently, in a much more animated way and in a much livelier spirit than it was in Central Europe, where the system of secondary and advanced education was being considered as a fairly immutable permanent ﬁxture. Austrian education had always ranged somewhere between the British class system and American democracy, probably nearer to the former until after World War I when the Social Democrats not only greatly improved the elementary schools (especially in Vienna) but tried to move everything somewhat closer to democratization. I enthusiastically admire the American ideal of opening all educational opportunities to everyone, although I strongly believed in a provision: to screen those able and willing to proﬁt from these opportunities at a reasonably early stage in each phase, lest the idea develop that everyone be entitled to a certiﬁcation of having completed that phase (i.e., to a ‘diploma’ or a ‘degree’). Such a misconception obviously could not but depreciate the certiﬁcates and lower the general educational standard. What I found particularly admirable in most of the numerous universities and colleges that I visited was their ﬂexibility in the face of changing needs or sudden opportunities, in marked contrast to the rather petriﬁed organization of Central European universities. In a way, this situation reﬂected the greater mobility and the immensely greater ﬂexibility of Americans in doing a variety of jobs – at any time what was most needed and most proﬁtable – an attitude little known to Austrians of that period once they were somehow settled. All the mathematicians to whom I spoke that year discouraged me from trying to contact John Dewey because they were deeply dissatisﬁed with the inﬂuence of the Columbia School of Education on the pre-college teaching of mathematics. There was one observation that profoundly disturbed me: waste of every kind, from insigniﬁcant machine products to valuable machines; from

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disposable containers and writing paper to slightly damaged refrigerators and automobiles in car cemeteries; and most of all, waste of food. y 13. I went for a couple of weekend to Princeton, the home of three outstanding topologists, all three of whom, however, were known to take a negative view of set-theoretical methods in geometry. The only one of them with positive views about logic and axiomatics and with some interest in philosophy was Oswald Veblen. Veblen was an unusually suave and soft-spoken man of great charm, and I became fond of him – I believed, like every one – as soon as I saw he could be heard expressing a strong distaste for big business and its tactics. But he did give the leaders of the American industry credit for paying the workers high wages. In this spirit, he himself tried to better the lives of young mathematicians, ﬁnancially and otherwise. He repeatedly spoke to me about the conditions under which he and the American mathematicians of his age group had spent their youth. Their talents, he said with a bitter smile, had been misused: essentially, they were assigned to teach calculus in colleges; and he wanted the next generation of mathematicians to fare better. I admired his obvious determination even though his speciﬁc complaints about teaching calculus did not strike a fully sympathetic chord in me. I saw too much room for improvement in the development of that ﬁeld. Because of his antagonism to set-theoretical methods, Veblen conﬁned his studies chieﬂy to combinatorial topology. But earlier in this century he had given what many American mathematicians considered to have been the ﬁrst correct proof of Jordan’s famous theorem. Ironically enough, although his set-theoretical reduction of the general theorem to the special case of polygons was correct, Veblen’s proof of that combinatorial special case was not. A correct combinatorial proof of Jordan’s theorem for polygons was given by the set-theoretician Hahn. In his foundation of projective geometry, Veblen chose as undeﬁned concepts points and certain sets of points, called lines. Planes he deﬁned, notably as sets of points – point sets with certain properties. Thinking that he would be pleased to hear that I had eliminated all inﬁnite sets from the foundation of projective and afﬁne geometry, I told Veblen about the algebra of geometry that I had developed. But I encountered in him the same attitude that I had observed in some algebraists who were opposed to set-theoretical methods: when used for one’s own purpose, set theory is excusable and its elimination of no great merit. So he soon turned the conversation from algebra of geometry to his ﬁrst papers, on the foundation of Euclidean geometry, though remarking that he really should not talk

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about this subject. For, he said smiling, he had developed ‘a complex about that work and the lack of its recognition abroad’. ‘Of course’ he added with some bitterness, ‘until quite recently it was almost impossible for an American mathematician to ﬁnd recognition in Europe at all’. But, as he had in his writings about 20 years earlier, Veblen unfortunately still claimed that he had developed the entire Euclidean geometry (including the theories of perpendicularity and congruence) in terms of points and the sets of points called lines; and this was an error. The error was, however, of some philosophical interest for the sake of which it was later to be discussed in Vienna in the Colloquium as well as the Circle. The situation is as follows. Ignoring congruence and perpendicularity in the Euclidean plane one can develop afﬁne plane geometry in terms of points, lines, and incidence or of points and certain sets of points a` la Veblen or of points and betweenness or in other ways. Having somehow developed the theory of the afﬁne plane one may introduce a) ideal points (‘points of inﬁnity’) as maximal classes of parallel lines, each consisting of a line and all lines parallel to it, and b) an ideal line (‘line at inﬁnity’), incident? with all ideal points. Although these concepts can be completely dispensed with in afﬁne geometry they form a bridge (and in fact the historical approach) to projective geometry. But to develop all of Euclidean geometry, it is necessary to deﬁne perpendicularity; and this is indeed what Veblen tried to do. He proceeded by (1) assuming that the ideal points are somehow grouped in pairs according to certain conditions – in technical terms, that an involution be given n? the ideal line – and (2) by calling two lines perpendicular if their ideal points constitute such a pair. But there are many ways of choosing the grouping of the ideal points in pairs – there are, in other words, many involutions on the ideal line. Perpendicularity thus was not really deﬁned in mere terms of points and lines, but relative to an arbitrarily chosen involution. This procedure amounts of course to using perpendicularity as a primitive concept besides points and lines – a fact that, at least as late as 1931, Veblen did not see. Another conversation led from geometry to logic. I used the opportunity to speak about my brief meeting with Post and my admiration for his Thesis.4 I mentioned his minor position and his state of depression. For a moment, Veblen seemed to be taken aback; then he said he would look into the matter. (As I learned later, Veblen knew that the second part of Post’s Thesis remained unpublished because it had been rejected. When a series of ‘Annals of Mathematics Studies’ was founded in Princeton in [1940], one of the ﬁrst issues was devoted to the second part of Post’s Thesis.5 During this conversation, Veblen took me to visit the logician, A. Church, who had published a paper on set theory connected with the axiom of choice.

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In discussing this work, Veblen, to my surprise, raised the question as to whether some day one might not decide the validity or non-validity of that axiom by experiments or observations – perhaps by testing some of its characteristic consequences in the spirit of Gauss’ attempt to decide the validity of Euclid’s Parallel Postulate by measuring the sum of the angles in triangles. He seemed to think of decisions on the basis of astronomy. I found it hard to follow this line of thought. y 14. The Rice Institute in Houston, where I lectured from February to May, 1931, was everything that Caratheodory had said about it and more. The mathematicians G.C. Evans and L.R. Ford were interested in philosophy and Charles Morris in the Philosophy Department had just begun to do work in semantics and the theory of symbols. I talked with them a great deal about the Vienna Circle while the more traditional philosopher, Tsanoff, listened with scepticism.6 Evans asked me to write three papers for the Rice Pamphlets. One was to be about metric geometry, the second about topology, the third about philosophy. In the last one, I wrote critically about philosophy. I called it the metaphysics of what ‘evinces itself’ and of what is ‘meaningless’, which at that time dominated the discussions of the Vienna circle. But there was an ill star over the project of this publication. The manuscript of the papers was mislaid; and I had no copies. A few years later, when at the invitation of [its] President, I gave three lectures at the Rice Institute (which were published in the Rice Pamphlet, 1938), I chose three mathematical topics and I never found time to rewrite the philosophical essay. y 15. It was at the Rice Institute in February 1931 that I learned about an event which, if anything in the realm of pure ideas can be called world shaking, in the long run has certainly deserved this name. In the Ko¨nigsberg meeting, when the ﬁrst tremors of the event were felt, I did not attend. The news came to me in a letter from No¨beling reporting about a paper that in January 1931 Kurt Go¨del had presented to our Colloquium, which No¨beling conducted while I was in America. Here is the prehistory of the event. Among the many fundamental discoveries of the Swiss mathematician Euler in the middle of the 18th century was the possibility of proving certain number–theoretical propositions by means of analysis that is, theorems about the natural numbers, 1,2,3, . . . and the theory of functions. Euler’s discovery gave rise to a new branch of arithmetic, called analytic number theory, which has ﬂourished ever since. The arithmetical theorems proved by ingenious analytic methods included also arithmetic results that mathematicians were unable to prove by non-analytical (that is purely arithmetic) methods such as induction.

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Although this fact enhanced the admiration for analytic number theory it also created the suspicion that the purely arithmetical methods were not yet sufﬁciently developed and that new ideas had to be conceived in traditional number theory to catch up with the results of analysis. Although many number theorists, anxious only for results, did not much care whether they obtained them arithmetically or analytically, for my part, while greatly appreciating the heuristic value of the latter procedure, I wished that ultimately all arithmetic propositions were proved arithmetically. I often spoke about this desideratum to members of the Colloquium, especially to No¨beling. Now he wrote me that Go¨del, in his talk before the Colloquium, had demonstrated the total impractibility [sic] of this program. Roughly speaking, what Go¨del proved was that arithmetic includes propositions that cannot be decided arithmetically, propositions such that neither they nor their negations can be derived from the axioms of arithmetic. And much more generally, in every w-consistent theory rich enough to include all natural numbers there are propositions which, on the basis of the assumptions of the theory, are undecidable. Here, w-consistency of arithmetic, means that for no property F of natural numbers it is provable both that F holds for each of the numbers 1,2,3, . . . and that there exists an integer for which F does not hold. And richness of a theory means that in it one can express or prove all propositions that are, respectively, expressible or provable on the basis of Peano’s axioms of arithmetic and the restricted calculus of propositional functions. The assumptions of w-consistency and richness are indispensable in Go¨del’s theorem: the former, because if it is not satisﬁed, then every proposition as well as its negation can be proved (whereas in the undecidable case neither the proposition nor its negation can); the latter, because among the less-rich theories, there are some in which each proposition is decidable, for example, the calculus of propositions. From Houston I returned via California and Asia to Vienna. y 16. At the end of the academic year 1930/31, Herbert Feigl decided to remain permanently in the United States following a call, I think, to Iowa. Even though my lectures had met with great interest in all places that I visited I was not offered an opportunity to stay. No doubt at that time I would have accepted a permanent invitation. I was too much attached to the Vienna of the1920s, too loyal to my colloquium which I could not hope to rebuild elsewhere, and I would have missed the Thursdays in the Circle. But I was very anxious to visit the United States again and to learn more about that unique country.

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NOTES 1. [Editorial note. On Hahn’s interest for parapsychology, see also Menger (1980, pp. xv–xvii)]. 2. Whose European fame incidentally started in Vienna. 3. On the recommendation of Neurath, who considered London as one of the most important modern authors because ‘his writings are a proletarian matter’. 4. [Editorial note. Menger met Emil Post on the occasion of his visit to New York from Harvard. But, probably because of the interest for logics of mathematics they shared, Menger had already had contact with the American logician before leaving for the U.S.A as is testiﬁed by this following letter from Post (Menger’s papers, Perkins Library, Duke University). Emil Post to Karl Menger New York, N.Y., January 28, 1930 Dear Prof. Menger, As you were kind enough to request, I am sending you a reprint of my dissertation under separate cover. May I thank you, professor, for having mentioned my work on Logic to the Columbia people. At the time that I wrote my dissertation respect for Mathematical Logic was at a very low effect in this country. This, coupled with the contempt that familiarity breeds, made my work seem most trivial to those who were nearest me. I would have liked to have talked at length with you about my more recent logical ideas. But perhaps it is just as well. Unﬁnished work is best left in the worker’s mind. May I express my admiration for your formulation and development of the dimensionality concept. Your analysis of the analogy between dimensionality and measure is one of those strokes of genius which keeps mathematics alive in the hearts of its devotees. Sincerely yours, Emil L. Post]

5. [Editorial note. Emil Post The Two-Valued Iterative Systems of Mathematical Logic, Annals of Mathematical studies 5, 1941]. 6. [Editorial note. As we know, Morris will become the ‘‘American side’’ of the Vienna Circle in setting the project of the Unity of Science Movement and of the Encyclopaedia. The following letter reveals some details of the very beginning of that project (Menger’s papers, Perkins Library, Duke University). Charles Morris to Karl Menger July 5, 1934 Wien XIX Huschkagasse 18

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Dear Professor Karl Menger, Perhaps you remember me from the time of your visit to the Rice Institute. I was then in the philosophy department. I am in Vienna during the month of July to make connection with members of the Wiener Kreis - going to Prague in August to be with Carnap for some weeks. It would please me very much if you could ﬁnd time for a short talk. Perhaps it would be possible for Professor Hahn to be there too. I have not met him and would very much like to do so. My time is quite ﬁne, and I could come to your ofﬁce or we could meet in the city for coffee, whatever is best for you. With best regards, I am Sincerely yours Charles W. Morris]

CHAPTER 7 RETURN FROM THE U.S. (1931/32) AND IN VIENNA AGAIN (1931–1934) y 1. The ﬁrst piece of news about Vienna that reached me in September 1931, after arriving in Genoa, was quite charming; I mention it and since many later recollections about Vienna that I shall have to record are quite the opposite of charming, Austrians have always been fond of swallows. There was a superstition, especially among peasants, that a house was safe from ﬁre if they had built a nest under its roof. Every year, the Viennese smiled when the ﬁrst swallows returned from the South – an event always recorded in the newspaper because it heralded the advent of summer, and when the swallows were leaving, everyone was a bit sad, because another summer was gone. In September 1931, however, something unprecedented happened. A sudden, premature frost caught the poor birds on their ﬂight to the South. By the hundreds they fell exhausted to the ground, unable to continue their journey. But, as the newspapers reported, the Austrians chartered airplanes to take their little friends to Italy. y 2. In Austria, during the year of my absence, things had taken a serious turn to the worse. Early in 1931, France had proposed some modest steps in the direction of a European federation. Coudenhove’s Paneuropeans rejoiced. But the Austrian government jumped at this opportunity to proclaim an economic union with Germany. Of course, nothing was farther from the wishes of the French than just this coalition, which would have greatly strengthened Germany with its resurging large-scale nationalism. So, France exerted the strongest pressure against this economic Anschluss and backed political steps by withdrawing her credits from Austria’s largest bank, the Creditanstalt. A few days later, that institution was on the verge of collapse. I had read this grave news in American papers early in the year. Upon my return, I could see how the situation was further aggravated by the effects of the American depression, which just then began to be felt in Austria. For the

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ﬁrst time since the introduction of the stable Schilling in 1924, a monetary crisis developed, which led to a slight devaluation of the currency and to the dire forecasts by L. Mises. Actually, during the winter, things somewhat calmed down, but the economical calamity in Austria had ushered in the great Central-European depression, which in turn created large-scale unemployment and ominous political unrest. The near-Nazi paramilitary organizations became ever more powerful. Despite Social Democrat protests, illegal bands arranged provocative meetings and parades. The swift rise of the Nazis in Germany greatly encouraged all fascist forces rightly deserving of support. Yet, even the creation of a single assistantship was completely out of question. In Austria, when there are openings in universities, the faculty submits a list of candidates to the country’s ministry of public instruction, which selects, usually from the submitted list, the professor who subsequently is appointed. Very few Christian social scholars were available or, at any rate, proposed by the more and more nationalistic faculty. So, as far as the candidates were concerned, the ministry, which was run by Christian Democrats, merely watched that no Social Democrat slipped in, unlikely as it was that such a man had even been promised. To all appearances, disinterested in the quality of the universities, the bureaucrats concentrated their efforts on economizing. When a chair became vacant, their ideal would have been to discontinue it or to leave it vacant indeﬁnitely. Only after long and intense pressure did they agree to ﬁll it – usually by appointing an associate in place of a full professor, and an assistant in place of an associate. For example, I myself had been invited in 1927 to the chair of geometry that had been held by associate professors for decades and had been vacant since K. Reidemeister left Vienna in 1925. At long last, the Minister had agreed to ﬁll the chair again. But soon I learned that because of budgetary difﬁculties, I would have to wait about two years as assistant professor. Only then and after considerable pressure, the Minister ofﬁcially appointed me to the chair. In 1931, I was greatly concerned about the excellent young people working with me. My ﬁrst idea was to ask Clara Wittgenstein for advice. So, when in the fall of 1931 she invited me to dinner, I talked to her at length about the differences between the academic situations in America and Austria, describing the splendid group of young people working in my colloquium and the practical difﬁculties of keeping them together. When, however, a few weeks later she invited me again, it soon became clear to me that in the meantime she had made inquiries of her own and that the information obtained from her sources had not borne out my superlatives.

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As I continued thinking, it occurred to me that one might somehow approach the remarkable Viennese intellectual community, though certainly not with appeals for support. They would have to be buyers of a commodity that they were eager to acquire. So, the next time she invited me, I spoke about the colloquium and the difﬁculties. In particular, I tried to describe – naturally in superlatives – the importance of Go¨del’s epochal discovery (although just he was not in need of ﬁnancial support). Miss Wittgenstein listened attentively, especially when realizing that Go¨del worked in the ﬁeld of mathematical logic. A few weeks later she invited me again, but I at once felt a subtle change in her tone. When the colloquium later came up in the conversation, it became clear that she had obtained an outside opinion but that the information had not been favourable. In fact, she intimated that I perhaps exaggerated the signiﬁcance of Go¨del’s work. After that visit, my relations with Miss Wittgenstein lost some of their cordiality and I saw her only a few more times before her terminal illness. In Ludwig Wittgenstein’s conversations with Waismann and Schlick (recorded up to the year 1932 in Waismann’s noted Ludwig Wittgenstein und der Wiener Kreis, edited by B. F. McGuinness1), Go¨del is not even mentioned, though on several occasions, the book deals with Widerspruchsfreitheit (freedom of contradiction). It does not speak well for Waismann’s mathematical insight that he never brought up Go¨del’s discovery while Wittgenstein kept quoting Hilbert’s view on the foundations of arithmetic, which had become obsolete because of Go¨del’s paper. Only years later, in England, Wittgenstein seemly took cognisance of parts of Go¨del’s work. Passages of his posthumously published Remarks on the Foundations of Mathematics, an Appendix to Part I (said to be written in 1937 or 1938), are devoted to unprovable propositions. When reading those pages, I could not help wondering whether Wittgenstein had fully understood Go¨del’s work and I asked Go¨del about the passages dealing with his discovery. Go¨del answered that with regard to his theorem about undecidable propositions, ‘it is indeed clear from these passages that Wittgenstein did not understand it (or pretended not to understand it)’. He interprets it as a kind of logical paradox while in fact it is just the opposite. y 3. In 1932, I had my ﬁrst chance to teach differential and integral calculus and, as it turned out, it was also my last opportunity in Vienna, for calculus was taught alternately by the occupants of the three chairs for mathematics, while I held the chair for geometry, but in 1932, Hahn, tired of the chore,

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asked me whether I would be willing once to switch courses with him and I was delighted to oblige. Calculus courses at the University of Vienna correspond in contents and methods to American courses in so-called advanced calculus, Austrian students having received a basic training in the mechanics of calculus in secondary schools. In my own freshman year as a student at the University, I skipped the calculus course since it was in conﬂict with courses on mechanics and advanced algebra that I was anxious to take. So I was something of an autodidact in the subject. I spent every weekday of that academic year, 1920/21, from 8 to 9 a.m. in the (unheated) library to study De la Valle´e Poussin’s Traite´ d’Analyse, an excellent book in which only the very ﬁrst words deeply puzzled me. In traditional geometry, the study of curvature was totally conﬁned to objects in coordinated spaces, where each point is an n-tuple of numbers – objects deﬁned by functional relations between the coordinates of their points so-called equations. The function involved being differentiable. But in 1931, I had developed a theory of the curvature of curves in general metric space without coordinates, without equations and without differentiability assumptions. When I taught the course in 1932, the students were greatly interested in the limit concept, but, in contrast to American students a few years later, did not ask questions about the notion of variables. Yet, I tried to clarify that obscure subject as much as I could and carefully refrained from using such ill-deﬁned phrases, in particular, I refused to use senseless phrases such as ‘The function x, which assumes the value x for each number x’; ‘The function f(x) of the independent variable x’ or even ‘the function log x’ and similar ill-deﬁned expressions that infested (and still are infesting) the entire literature in the ﬁeld of calculus. One of the repeatedly discussed topics in the ﬁeld of philosophy of science was the role of observables in the development of theories in physics. Still under the impression of Einstein’s elimination of ether, all agreed on the exclusion of unobservables from the principal propositions and theorems. But there was no consensus on the inadmissibility of unobservables in hypotheses and especially in the domain between hypotheses and experience. Several members, including Hahn, favoured such auxiliary concepts. (I could see nothing but advantages where they played a heuristic role). y 4. Undoubtedly, even before the Circle existed, the men who were to be its members used the word ‘meaningless’ (sinnlos) in connection with abstruse philosophy and only in this context. At the time when I joined the group, the

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word was used rather infrequently, hence it came up. When ‘meaningless’ was used with regard to an idea or a sentence, then I usually found that it reﬂected a deﬁnite attitude of mine, which did not care to use that idea nor either to assert or to negate that sentence. But in the discussions of 1929 and 1930, things changed. Waismann began to apply the word ‘meaningless’ more and more frequently – often to ideas that seemed to me at least to deserve further consideration. Schlick, Carnap and students of theirs also used the term freely and seemed to believe that by attaching the stigma meaningless to an idea or a sentence, they were terminating its discussion once and for all. Reﬂecting upon the matter, I remembered ideas relating meaning and context that I formed as a student. One day, I said to Waismann, ‘If you consider a sentence to be meaningless, then don’t assert it. In cases where you feel that more, say explicitly that you will not assert the sentence. But calling it meaningless is a dangerous procedure’. Waismann did not change his ways; but we now often advanced in the Circle as ultima ratio for the rejection of ideas or sentences the fact that Wittgenstein considered them meaningless. Matters reached the point where, even in discussions on the foundations of logic and mathematics, Waismann would declare a view meaningless if it deviated from his own or went beyond Wittegenstein’s. Moreover, he began characterizing whatever he himself propounded as meaningful (sinnvoll) – a claim that prompted me to ask him on several occasions ‘What is it that you add or rather what do you think that you add to a sentence which you assert by calling it meaningful?’. I also pointed out Waismann’s strong antagonism to Husserl, his appeals to meaning resembled the Wesensschau of phenomenology, and that, in the absence of clear criteria concerning meaning, they introduced into the discussion a subjective, uncontrollable and variable element. But Schlick obviously agreed with Waismann. So did, in 1929/30, Carnap, although he seemed at least to listen when I pointed out that only on the basis of criteria for meaningfulness, and relative to them, would I admit rejections of sentences as meaningless. Kraft and even Hahn remained in silence. Neurath’s position in the controversy was somewhat unique. He avoided the word ‘meaningless’ – I never heard him say it. But since he used ‘metaphysical’ instead, he differed from Waismann on this point only in terminology, and for no reason would he never have given up his habit of using ‘metaphysical’ frequently and freely. More signiﬁcant, however, was his critique, expressed on another occasion of Wittgenstein’s command to be silent about what cannot be said. Neurath felt – I thought correctly – that

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here the words ‘about what’ were misleading in that they suggested that there was after all something to be silent about. Yet, it seemed to me that my suggestion to Waismann pointed to an easy escape from the dilemma of either making subjective, ill-deﬁned value statements about the meaning of a sentence or being ‘silent about’ its supposed content. If its simple non-assertion is felt to be inadequate (e.g. in discussions about it, especially if it is being asserted by someone) say explicitly that you will not assert that sentence. In other words, follow the maxim, Assert or don’t assert or say that you will not assert. This maxim not only allows one to distinguish between simple silence (i.e. non-assertion) and emphasized silence; it also permits the speaker to express, in an unassailable way, further gradations of his convictions about meaninglessness. He can say, for example, that he will not assert the sentence at the moment or not until (or unless) certain conditions are satisﬁed or never. The word ‘meaningless’, from this point of view, is conﬁned to the infrequent cases of a violation of some of the rarely formulated explicit rules concerning meaning, while ‘metaphysical’ is restricted to self-acknowledged metaphysical discourse. In the numerous situations where there are no precise conventions about meaning, my evaluation-free maxim appears to me to this day to be the best solution to the difﬁculty. The circle, in the years around 1930, however, suffered an inroad of a mysticism of meaning. Go¨del, in all likelihood, agreed with me. But I do not remember whether he was or was not present at that meeting. Either way, he did not inﬂuence its outcome. In such discussions, Go¨del would make one or two remarks penetrating to the very essence of the matter in question, expressed in a minimum of words, but in a totally dispassionate, casual tone ending many sentence with ‘Don’t you think so?’. It has always been hard for me to imagine him as a ﬁghter for a some. y 5. While Wald was not especially interested in the Circle or in philosophy in general, his mathematical work before long reached a depth that made his results interesting to members of the Kreis. A minor irritant to me was the introduction [in the Circle] (I think, by Neurath) of the battle cry Einheitswissenschaft (literally, Unity-Science) for the entire movement. The term was to express that Natur und Geisteswissenschaften (natural sciences and humanities) form a unity. This point had always seemed to me to be somewhat debatable, even after Hahn’s interesting interpretation of historical assertions as predictions. But even if the unity is granted – in fact, especially in this case – the term unity-science

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does not seem to me to make any sense, nor has it ever taken root in English. But even the more meaningful expression, unity of science seemed to me to erect a wall between scientiﬁc and humanistic scholarship. It seems to me that, by virtue of their aesthetic qualities, certain mathematical theories have a closer afﬁnity in structure and aim to works of art than to various scientiﬁc results. ‘How shall we call ourselves?’ I think, it was Neurath who ﬁrst raised the question. Various suggestions were made; ﬁrst ‘Neo-positivists’. Someone thought ‘Neoempiricists’ would be better. ‘I consider it as essential that the name includes a reference to logic’ said Hahn, and Carnap strongly seconded this idea. ‘Well, then Logical Empiricists’. ‘That expresses what we really are’, he said. As far as I remember, the question was never ofﬁcially settled. Schlick was not fond of sobriquets, and each of the names that had been proposed was occasionally used. [The following part (up to the end of the present paragraph 5) is very similar to Menger 1980, p. xiii] Hahn’s contribution – in my opinion not enough appreciated in the literature – to the Unity of Knowledge discussion was his interpretation of historical statements as predictions. Poincare´ had pointed out that historical facts, such as Caesar having crossed the Rubicon, are uninteresting to the physicists because they will not recur, while physicists deal only with repetitive events about which predictions are possible. But, said Hahn, the basis of our belief in propositions about Roman history is a corpus of documents, old writings, coins and excavated objects; whence the belief itself amounts to the prediction that whatever as yet unknown old writings, documents, coin and excavated objects may become known in the future will conﬁrm the proposition or at least be consistent with it. y 6. The guest lectures of foreign scientists were always crowded, but they were free and open to the public. In my optimistic belief in the Viennese intellectuals at that time, I surmised, however, that high-level (if semipopular) scientiﬁc lectures might well be a commodity for which, notwithstanding the economical depression, professional and businessmen would be willing to pay. Of course, a group of good lecturers was needed. So, I ﬁrst asked Hahn whether he would care to participate in such a series. The lectures should be held in an auditorium of the building of the Mathematical Institute at rather high entrance fees, which would in their entirety go into a fund for the support of young scientists. Hahn accepted with pleasure and we decided to approach Thirring next. He also greatly agreed and went with us to his

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friend, the noted chemist H. Mark, who had just joined the faculty and proved to be very cooperative. Of course, we had to found a catchy title [for the lectures]. I suggested Crisis and Reconstruction in the Exact Sciences, which seemed to please everyone. In consecutive weeks, Mark should speak about ‘Classical physics, shaken by experiments’; Thirring about ‘The crisis of intuition’ and myself about ‘The new logic’. Since we all felt that a ﬁfth would be desiderable [sic], I suggested that we ask No¨beling, who was a good expositor and lecturer, to deliver a talk about ‘The fourth dimension and the curved space’ thereby also giving the public an idea of the type of young scientists that we wanted to further. Since three of the ﬁve lectures dealt with mathematical topics, 60% of the returns would go into a fund for the support of young mathematicians, while the balance would be shared between students of physics and chemistry. We ﬁxed the entrance fee at the order of magnitude of the price of a good seat in the Staatsoper. Mathematics students volunteered, sending out announcements of the lecture series and sold the tickets. Within a week or so the auditorium was sold out, while reservations were still coming it. The series was received with great interest and many listeners suggested that we should offer such lectures every year (which we did). y 7. A few months later, we published the lectures in a booklet to which I wrote a two-page preface describing at the end why the lectures had been arranged and how they were received.2 Favourable reviews appeared in many countries. One of them I found particularly gratifying, not only because its author was P. W. Bridgeman, but because it brought out two ideas that I have always emphasized: the unusual character of the intellectual community of Vienna in those days and the signiﬁcance of modern logic for scientists. Incidentally, Bridgeman also points out the connection of the lectures with the spirit of the Vienna Circle. In the Review of Scientiﬁc Instruments, Bridgman wrote in 1933: ‘As may be inferred from the title, and the known interests of the philosophical group at Vienna, the emphasis is placed on the reactions which recent activities in physics and mathematics bring about in the general philosophical outlook. The discussion goes much more into details and makes much greater demands on the concentrated attention of the audience than would be possible in popular lectures in this country, but the Vienna audience seems to have survived and, in fact, it is stated that every seat was ﬁlled and that the lectures will be made an annual event. No popular exposition in English lays such emphasis on the signiﬁcance of recent developments in mathematics; in view of the greatly increasing use of mathematics in physics I believe that many

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physicists will ﬁnd in these lectures stimulating fresh points of view to ponder. The last lecture, ‘The new logic’, on recent changes in points of view toward logic, is particularly to be recommended’ (my italics). The second series of lectures was offered in the following academic year (1934/35) under the title Old Problem – New solutions in the Exact Sciences. I started it speaking about squaring of the circle and Hahn ended it with a talk on inﬁnity, which has the sad signiﬁcance of being his last public lecture.3 A third series, Recent progress in the exact sciences, offered in 1935/ 36, contained only one mathematical-methodological contribution, namely a lecture of mine,4 Recent progress in the exact treatment of the social sciences.5

NOTES 1. [Editorial note. F. Waismann Wittgenstein und der Wiener Kreis. Aus dem NachlaX herausgegeben von B.F.McGuinness. Oxford. Basil Blackwell. 1967]. 2. [Editorial note. The booklet Menger was talking about is AA.VV. ‘Krise und Neuaufbau in den Exakten Wissenschaften. Fu¨nf Wiener Vortra¨ge’ (Leipzig & Wien, 1933); his ‘Vorwort’ (Preface) was unsigned (pp. i–ii); his paper, ‘Die neue Logik’ (pp. 94–122), was translated in English four years later as ‘The New Logic, Philosophy of Science’, 4, 1937, pp. 299–336]. 3. [Editorial note. Menger was referring to ‘Alte Probleme – Neue Lo¨sungen in den exakten Wissenschaften. Fu¨nf Wiener Vortra¨ge’, Zweiter Zyklus, Leipzig and Wien (1934); Menger’s paper was ‘Ist die Quadratur des Kreises lo¨sbar?’ (pp. 1–28)]. 4. [Editorial note. Menger was referring to ‘Neuere Fortschritte in den exakten Wissenschaften. Fu¨nf Wiener Vortra¨ge’. Dritter Zykius, Leipzig & Wien, 1936, in which Menger wrote ‘Einige neuere Fortschritte in der exakten Behandlung sozialwissenschafﬂicher Probleme’ (pp. 103–132)]. 5. [Editorial note. A year later, in 1937, Menger presented his paper at the Third Annual Conference of the Cowles Commission (‘An Exact Theory of Social Relations and Groups’, in Report of Third Annual Research Conference on Economics and Statistics, Cowles Commission for Research in Economics, Colorado Springs, 1937, pp. 71–73), and the following year, it was published as ‘An Exact Theory of Social Groups and Relations’, in American Journal of Sociology 43, 1939, pp. 790–798. Menger was invited at the conference personally by Alfred Cowles in a letter of December 1, 1936 (Menger’s papers, Perkins Library, Duke University)].

CHAPTER 8 IN VIENNA BETWEEN 1934 AND 1938 y 1. In conclusion, I comment on two features of my mental make-up of which I have been aware since my earliest youth. They concern my memory. Compared to most people of my acquaintance, I have a memory that normally is at most average in breadth and intensity – probably less than average. But during an experience that I feel may be potentially useful to me I resolve to remember what I am seeing or hearing or understanding and then this experience remains engraved in my mind with complete accuracy for decades, perhaps to the end of my life. I don’t know how far I could extend this faculty, which I have never tried to abuse in the sense of trying to remember too much. Even so, occasionally trivia pop up with perfect clarity which, however, for some long forgotten reason, may at one time well have seemed to me worth remembering. But all in all I am well satisﬁed with my past selections. Most of what I resolved to remember I later actually found useful. This usefulness is enhanced by the second feature of my memory. Interesting reminiscences tend to return spontaneously at the very moment when they are relevant, that is, when they are connected with a current experience or the subject of my momentary thinking. ‘Connected’ here is meant in the broadest sense: in space and/or time, by similarity contrast, by analogy or by any of various other relationships. My memory thus is, on the one hand, strongly regulated as to its content and, on the other hand, largely spontaneous as to its reproductions. I don’t know how common these features are, but I suspect that the second is rarer than the ﬁrst. y 2. [This paragraph, footnote excluded, is edited in Menger 1974 p.114]. Although the University of Vienna was closed for the greater part of 1933/ 1934, the Circle held its informal meetings even during that trying time.1 But Neurath, who happened to be abroad during the February events never returned to Austria, and at the time of the July revolt, Hahn and Carnap had gone to Prague in 1931, Feigl had emigrated to the United States;

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and Go¨del had been estranged from the Circle by the rather superﬁcial Carnap-Hahn-Neurath manifesto Wissenschaftliche Weltaufassung (Scientiﬁc View of the World)2 and had stopped attending the meetings even before 1934. Schlick, Waismann and Kraft, the former two, then, completely under Wittstengein’s spell, were the only members of the original Circle left in Vienna. Although I never mentioned my book3 in the circle I discussed it when it was near completion in my Mathematical Colloquium, in which K. Go¨del, A. Wald and F. Alt regularly participated, and with the economist O. Morgenstern, who began to be interested in mathematics and possible new applications to economics. When the manuscript was completed, I showed it to Schlick. To judge from his comments, he read the material carefully. My own search for studies of groups in the literature had only led me to interesting writings of two authors: the 19th century Austrian, G. Ratzen and then the contemporary L[eopold] von Wiese in Germany. Their works did not primarily concern groups and any logico-mathematical treatment of the relations between groups. I asked Schlick whether the positive ideas in my manuscript were new. He suggested that I might be interested in J. M. Guyau’s Morale sans obligation ni sanction. I read that booklet with interest indeed, but found, as Schlick had predicted, only few points of contact with mine and those only with the critical sections of my book. y 3. Whenever in the course of the years of the Circle Waismann was in difﬁculties, he came to see me. He knew of my high regard for him even though I never could hide my irritation at his dogmatic declarations. Actually toward the mid-1930s ‘that is meaningless’ began to give way to the question ‘Whatever does he mean by that?’ (Ja was meint er denn samit?) When other philosophers’ theses were being discussed – expressing an attitude that I fully understood and appreciated, I helped him whenever I could. Now also ﬁnancial problems came up. ‘Why not ask Wittgenstein for help?’ I suggested. ‘Wittgenstein has no money’ Waismann answered. ‘He has given away everything’. ‘But most of it, I have been told, to members of his closest family’, I said. ‘Don’t you think that if he said a single word his sisters would help you?’ Waismann shrugged his shoulders. I tried to ﬁnd a tutorship for him but it was not successful as in the case of Wald.4 In 1933, Waismann began to speak to me about the plan to write an Introduction to Mathematical Thinking. The book should develop the theories of the natural numbers (1,2,3, . . . .) following Peano; of the (positive and negative) integers as pairs of natural numbers; on the fractions as pairs

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of integers; of the real numbers as sequences of fractions; and of the complex numbers as pairs of integers. Thus, he would present arithmetic in the traditional way, but in an at once pedagogically skilful and epistemologically correct manner. For example, he would make it clear that the theory of integers is totally independent of the existence of hot and cold substances or assets and liabilities, though these phenomena lend themselves to applications of the theory. Similarly, Waissman would develop basic ideas of geometry and their relations to the various arithmetical theorists without in any way using the former as the basis of the latter. He would furthermore present the formalist, logicist and intuitionist ideas on the foundations of mathematics hoping also to include some thoughts of Wittgenstein. I found the plan excellent and so did Hahn, and we warmly encouraged Waismann; we did feel, however, that we could subsidize the project from the funds available through our lecture series. Now Waismann showed me a more detailed plan and I was sure that the book would not only be a valuable addition to the mathematical literature but would also bring Waismann ﬁnancial returns.5 [At this point of the his drafts, Menger quoted the following passages of a letter from Waismann]: ‘In view of the long delays’, Waismann wrote me, ‘it is quite understandable that certain doubts have arisen as to the completion of the book. It will be best if I openly tell you the development of the work up to now so that you may form your own opinion’. He continued: at Easter 1933, a large part (about three quarters) of the work was ﬁnished and I gave Wittgenstein the manuscript to read. Though he was in agreement with content and form (he called it a ‘valuable work’) he beseeched Schlick and myself not to publish the book in this form since it would in his opinion immensely gain by following a quite different approach. I had aimed at the clariﬁcation of philosophical problems by a linguistic (sprachlich) investigation of the terms in which the problems were formulated. The solution of the philosophical problems was the main aim, the grammatical investigation the means. Wittgenstein proposed to me, however, to write a book including nothing philosophical but rather systematically developing the grammar. He suggested that I should think over how I could formulate a series of examples leading from the simplest to the most difﬁcult concepts of philosophy. In possession of such a presentation one need not bother with the solution of philosophical questions; they drop in one’s lap like ripe fruit. I said at the time that I liked this idea very much but considered its implementation to be enormously difﬁcult and that I did not feel equal to such a task. Moreover, I felt that my original presentation had certain advantages that would be lost in the rewriting. In brief, I strongly resisted the suggestion. Finally I asked Schlick to defend my cause before Wittgenstein since I did not want to negotiate with him further. But Wittgenstein brought Schlick around to his point of view. I went to Schlick in the middle of the night. Wittgenstein was there too; and both asked me at least to try to rewrite the book in the suggested direction. I ﬁnally agreed under the condition that

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I might withdraw from the work and would not be responsible if the book appeared late or not at all. (As I veriﬁed in July, Schlick notiﬁed the publisher of this agreement). I now tried to ﬁgure out such an outline, but without success; I did not see a way. In July 1933 I told Wittgenstein that I had reached an impasse and showed him my inadequate sketches. He thereupon proposed that he himself would try to work out an outline and he actually went to work. I think it was in winter of that year that he wrote me he could not master the task. He gave me a free hand, but strongly advised me to try further and, if possible, to write the book in the direction he had individuated. After various attempts I ﬁnally thought I saw a way. I did not want to give up my old idea completely but considered the following plan of the work: in an extensive introduction (about one ﬁfth of the book) it would be demonstrated how philosophical problems originate from unclear thinking and how they dissolve if one clearly realizes the meaning of the terms used. After the importance of grammatical investigations had in this way been established, the systematic part should begin. In organizing this part I proﬁted from the fact that I just then began teaching a course on logical grammar . . . I carefully prepared the lectures writing down everything beforehand. In this way I ﬁlled ﬁve thick note- books in shorthand. The attendants of the course prepared a typewritten elaboration including what I added in the lectures, partly in answering their questions. This elaboration is likewise in my possession. Just like the book on mathematics, I am writing the book on philosophy on the basis of the course using the attendants’ elaboration to support my memory. The course lasted until July. In spring I began writing the ﬁnal copy completing six chapter in two months. Here in Ober-Woltersdorf I write two chapters a week since I am not detracted by anything else. So by September 15 six more chapters will be ready. Altogether the book will include eighteen to twenty chapters (at about twelve printed pages each); in addition, there is the introduction which, except for a few gaps, I had written earlier (it will be about sixty printed pages). So what remains to be done is the completion of the introduction and the last six to eight chapters. The whole book will be about 300 printed pages. How I shall work in Vienna will depend on my schedule. Certainly in the ﬁrst weeks I shall have much work with the composition of my courses; and even thereafter I naturally shall progress more slowly than here. As a conservative estimate, however, I’ll say that I can submit the manuscript at the latest by the end of the winter semester. This makes allowance for unforeseen incidents. But to prove to the publisher that this time the matter is serious I promise to turn in the introduction and more than half of the systematic part by the end of October. (In spring, Schlick advised me to have now at least a preliminary part of the book printed; but I wanted ﬁrst to complete the introduction until I would see precisely what it would include so that repetitions would be avoided). Now I have plainly described to you the state of my work. You understand that the main difﬁculty has been the arrangement of the material. This is overcome. Whether I have succeeded I will not judge. I’ll only say that the work can now proceed quietly and steadily according to a ﬁxed plan. Finally, I wish to speak about Schlick’s position to the content of the book. It is my ﬁrm conviction that Schlick would be in complete agreement. During the seven years that I conducted the Pro-seminar I never had even the slightest difference of opinion with him and often found to my surprise that in his course or his seminar (which I did not attend) Schlick presented a topic exactly the way I had (or would have) done. All Schlick read of the present manuscript is the Introduction as far as it was written, and it met with

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his full approval. As to the rest, I communicated some parts to him in conversations and asked in a few cases for his opinion. Other parts I presented in his presence to the preseminar in lecturing or by eliciting them by the Socratic method from the audience. Never was there the least disagreement. Excuse the length of this epistle. But I thought that your very justiﬁed questions could be answered only by a detailed exposition.

y 4. In the course of their semi-fascist measures, the Dollfuss government founded an organization, called Vaterla¨ndische Front (Patriotic Front), and strongly suggested that every ‘good Austrian’ should join and wear the redwhite-red badge he received when enlisting. Government employees outside of the universities had no choice. Of course some people outside as well as in the universities honestly favoured the front and joined it for this reason. One of them was Schlick. He told me that he regarded the front as the last hope so at least Austria would stem Nazism, which was bound to lead to the ruin of the German people (zum Untergang des deutschen Volkes). Even before the organization of the Front, Schlick had once mentioned to me that he planned to write to Cardinal Innitzer proposing a union of all who wished to avert the Nazi menace. (Whether he carried out this plan I don’t know). But the very ﬁrst faculty members to wear the badge, long before it became necessary, also included many of him? I happened to know that they hated the idea – some because they were Nazis, others because of leftist convictions. Among the most insipid ﬁgures at the beginning of the fascist regime are, as I found out, men who not only do what they have to do unless they want to become martyrs, but do things (to which they are opposed) earlier and more ostentatiously than absolutely necessary or do even more than they have to. As time went on, the pressure greatly increased and more and more badges appeared. Finally, there was one in the lapel of every member of the mathematics faculty, except in Hahn’s and mine. I considered myself a good Austrian and sincerely loved the country until it began to change in 1932. And even thereafter, though unsympathetic to the Dollfuss government, I might well have joined the Front voluntarily, in the spirit of Schlick. But I hated the idea of expressing my allegiance to Austria by wearing a badge that was forced on me. So I discussed the matter with Hahn, who considered it to be serious and said he would wait to the last possible moment. Finally, after he too appeared with the badge, I signed up. y 5. In the fall of 1935, W. Heisenberg in Leipzig was invited to deliver a lecture in the series of talks by foreign guests. The German government gave him an exit visa, but permitted him only a very short visit – I believe of two days. His beautiful talk on Fundamental Questions of Modern Physics,

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which was attended by all Viennese members of the Circle, was perfect in form, although content did not go beyond what we all knew. One afternoon, Thirring drove the guest through the Vienna Woods and invited one or two physicists and myself to join them. There was at that time a tacit understanding between scientists: living under different dictatorships never to discuss politics, except perhaps if they were intimate friends. In keeping with this convention, the Austrians carefully avoided any allusion to the political situation. Heisenberg made a couple of remarks with remote implications to current issues that sounded rather neutral. But then the conversation somehow turned to England’s spectacular rise and enormous wealth during the 19th century and Thirring ascribed these facts to British liberalism. This elicited a most unexpected answer from the guest. ‘It would have to be investigated, though’, Heisenberg said, ‘Whether that development was not a belated result of Cromwell’s regime’. Whether this idea was Heisenberg’s or merely quoted by him I don’t know – I never read about it anywhere. At any rate, Heisenberg’s remark has lived in my memory as one of the strongest apologies of dictatorship I have ever heard. At that time I got interested in optative and imperative logic and spoke to Tarski about the matter. In particular, we discussed the question as to whether or not the proposition ‘I wish A and B’ implies ‘I wish A’. We disagreed on the answer though I cannot now recall who took which stand. But I do remember that we defended our respective positions rather fervidly and, it seems, convincingly. For after about ﬁfteen minutes I noticed that in the heat of the debate we had inadvertently exchanged our original positions. I mentioned my observation to Tarski and we terminated the discussion laughing. y 6. In 1935 I met Karl Popper who was about to publish his book Logik der Forschung with which unfortunately I became familiar only much later. But even when I ﬁrst met him, I was greatly impressed by his ideas as well as by the extent of his knowledge, especially his insight into quantum mechanics. In the next meeting of the Circle, I talked to Schlick about Popper with the idea of introducing him into the circle but found to my surprise that Schlick knew him. To my suggestion that Popper be invited to our meetings, however, Schlick offered a strong idiosyncratic resistance totally uncharacteristic of his usual behaviour and I had to give up. But I invited Popper to meetings of the Mathematical Colloquium where Wald clariﬁed R. von Mises’ idea of collective, a task to which Popper had been one of those who made pioneering contributions. Carnap, then in Chicago, told me after thoroughly studying the catalogue that even though the University of Chicago owned copies of most of my

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books he considered the collection as such a gem that he would exert all his inﬂuence to make the University acquire and keep it as a separate unit, if it were not for a desperate lack of space which made this idea impracticable. Late in the spring of 1936, some exhibition was arranged on the premises of the former imperial palace. In view of the political situation I was surprised to receive an invitation to the ofﬁcial opening. When I entered, the hall was teeming with guests, and among them I saw Schlick. We stood in conversation when a path was made through the crowd for President Miklas of the Republic leaving with his party. He was followed by several men whom I had never seen before, and one of them, to my surprise, exchanged friendly nods with Schlick. Assuming that the man had some connections with the semi-fascist government I teasingly asked Schlick, ‘what kind of friends have you?’ But Schlick got very serious and said, ’Oh, this is not a friend. He is a plain clothes’ man who has several times guarded me. For I am being threatened by a crazy person, a former student of mine. Off and on, he is put in an asylum; but whenever they release him he renews his threats over the phone’. Seeing that the plain clothes’ man had now been given another assignment I said, ‘but now you are left in peace, I hope’. ‘By no means’, Schlick sighed. ‘As a matter of fact, the fellow has been released from the asylum just recently and already he has threatened me over the phone again. But since he has never actually carried out any threat I have the impression that the police begin to disbelieve me. So I don’t dare to notify them lest they think it is I who is crazy’. A few weeks later, Schlick was killed on the steps to the university by a shot from the man’s revolver.6 Rechspost, the ofﬁcial daily paper of the Christian Social party and at that time of the government, devoted a leading editorial to the Schlick tragedy. As everyone else, the paper said in effect, we deeply regret what has happened. None is in favour of murder. But . . . and then came the buts alluding to the fact that the demented murderer was a former student of Schlick’s. Is it surprising, the paper asked, that a corrosive philosophy such as Schlick’s had an unsettling effect on a student? And so on. For me, this was the end of the Austria of my youth. All of Vienna was stunned. The newspapers were full of reports of the tragedy and published long obituaries. I was deeply depressed. The next day when I went to the Mathematical Institute and was seen by a colleague who had never liked Schlick, a grin appeared on his face and without a word he went to his ofﬁce while I was entering mine. My grief got mixed with other feelings.

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A tragicomic aftermath brought no relief. Waiting in front of my ofﬁce door was a second man, a mathematical amateur who had visited me off and on for four years after I once had consented to spend one hour with him discussing his problem. I did so against the emphatic advice of Hahn since I have always felt that the public is entitled to be heard by professors of a public university. A detailed discussion during that one hour should have convinced the man of the errors in his reasoning had he been rational. But he was highly neurotic and, as Hahn had predicted, many other visits followed importuning me more and more. During the spring term of 1936, I had not heard from the man so that I hoped to be rid of him. But on the very morning after Schlick’s assassination he reappeared. In 1935, I had spoken about that annoyance to Thirring, who tried to comfort me by describing a neurotic amateur inventor who had been pestering him for the past three years. So after my mathematician left on that terrible morning, I told Thirring that he had turned up again, and just on that day. ‘Yours too?’ Thirring exclaimed ‘Mr inventor stood before my ofﬁce door this morning’. We discussed the subtle threat implied by the choice of the moment of reappearance. I only felt relief in the thought that in a few days I would be leaving to attend the mathematical congress in Oslo and would be absent from Vienna until September. The following Thursday evening, the members of the Circle got together for a brief last meeting before the end of the academic year mainly speaking about the tragedy. y 7. Applying the Nu¨rnberg laws to Austrian universities the Nazis dismissed everyone not meeting their racial requirements (including those Jews who had always faithfully voted with the German nationalists). But also some people perfectly meeting those requirements lost their position: Thirring, his professorship because of leftist leanings; Morgenstern, his directorship to the Institute for Business Cycle because of ‘unbearable’ views; Go¨del, his lectureship. One of the ﬁrst deeds of Morgenstern’s successor was the dismissal of Wald from his modest post; but like Morgenstern himself and Go¨del, Wald as well as F. Alt, found their way to the United States. Schro¨dinger in Graz, known as anti-Nazi, was at once not only dismissed but put in jail. What saved him was an intercession of Pope Pius XII. Though Schro¨dinger was not Catholic, the Pope wrote on his behalf to President De Valera of Ireland, an admirer of the 19th century Irish mathematician and physicist Hamilton, whose work in theoretical mechanics Schro¨dinger had extensively used when developing his wave mechanics. After an adventurous and dangerous ﬂight the Schro¨dingers eventually arrived safely in Dublin.

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Nor was even a pro-Nazi mentality (in addition to the fulﬁlment of the racial requirements) a complete safeguard in those basically lawless days. A professor at the Vienna Polytechnicum, 100% ‘Aryan’ and very nationalistic, jubilated when Hitler came, but presently was dismissed from his professorship. What pretext, if any, was given to the speechless man I don’t know. He had been a very good teacher, but also a demanding examiner with a high percentage of failures in his courses. It is fairly clear what must have happened. One or other of the students whom the professor had failed gained some inﬂuence with the new regime – as did, for example, all who had been illegal members of the Nazi party under the Schuschnigg government – and took revenge on a hated teacher (the latter, soon after his dismissal, committed suicide). And in a supreme irony of history, the economist Spann, who, through his holist philosophy and fanatic teaching, probably had done more for the rise of Nazism in Austria than anyone else in academic life was thrown into a Nazi prison and, I was told, lost an eye as a result of mistreatment in jail. Probably there existed minor differences between Spann’s holism and the ofﬁcial Nazi doctrine; and as is well known, the ﬁrst victims of fanatic tyrants usually include like-minded fanatics who dissent in some details. Karl Schlesinger committed suicide the day when Hitler entered Vienna; and poor Mrs. Dub was sent to a concentration camp, where she perished. The changes that Hitler’s coming brought about in some Austrians were truly startling. Two cases come to my mind. Cardinal Innitzer, who a decade earlier had been a just and courageous rector of the University of Vienna, raised the swastika above the archiepiscopal palace and went out to address a Nazi crowd which, however, threw tomatoes in his face. An eminent algebraist at the University was beseeched by a promising young student to sign a statement merely attesting that the student, who was well known to him, was a mathematician. The young man desperately needed such a statement in order to get some visa but his teacher, whom I knew ten years earlier as a just man free of prejudices, refused to sign. (Fortunately the promising student escaped anyway, thus probably saving the old professor from becoming an accomplice to murder). y 8. After Schlick’s death and the end of the Circle. In later years I have often wondered why I conﬁned myself to the presentation of papers and to conversations with individual members instead of taking a more active part in general discussion of the Circle; for example, why I failed to mention what I must still have vividly remembered of my careful reading of Berkeley less than a decade earlier; why I did not more emphatically bring up

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Mauthner when the critique of language was the principal topic of the discussions; and why I missed several other opportunities. The trouble was my habit of quietly observing situations or people that fascinate me; and the discussions in the Circle gave me ample opportunity to indulge in this habit. The combination of the content and the technique of those debates certainly offered a fascinating spectacle. I was at a complete loss to understand how anyone living in Europe in 1936 could not see impending disaster – disaster not only for Austria but for all of Europe. Yet there were many in Austria as well as in Western Europe who failed to see. These cases come to my mind. An eminent economist who held a high position in the public life of France wrote me at that time asking my permission to include some writings of my father in an economic anthology. Of course I was delighted to oblige but closed my letter with a short if urgent description of the terrible situation in Austria and the grave threat to all of Europe. The economist thanked me most graciously for the granted permission but ignored the last paragraph of my letter. Sometime later I read in newspapers that paciﬁst Bertrand Russell had been asked in London ‘What shall we do when the Nazis invade England?’ and answered ‘Why, treat them as tourists!’ In the fall of 1936, Schro¨dinger visited Vienna to give a lecture and to negotiate with the Minister of Education about a professorship in Graz. He had been successor of Mach, when the Nazis came. Up to the events of 1934 and 1936, I had been a truly loyal Austrian and would have been delighted beyond words with Schro¨dinger’s return to the Austria of 1930 that I loved. But under the circumstances prevailing in the fall of 1936, when I felt to be myself already with one foot abroad counting the days until I would be gone altogether, I said to Schro¨dinger whom in ? this matter I regarded as temporarily confused, ‘You left Berlin because of the Nazis even though there are, after all, also other people and things in Berlin. And now you want to go to the University of Graz, where at present there is practically nothing but Nazis?’ ‘Well, you see’, he said rather vaguely and wistfully, ‘the Alps . . . ’. y 9. [In 1937/38 Menger was lecturing at Notre Dame University, Indiana, where he knew the Nazi troupes had invaded Austria]. In February 1938, I considered the situation as so critical that I went to President O’Hara to tell him about my library stored in Vienna and to get him interested in accepting the books as a loan gift to the University of Notre Dame provided

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that I should succeed in bringing them to the United States. Father O’Hara glanced through the catalogue for a couple of minutes and then said that he would be only too happy to help me save the collection and to place it in a separate section of the Notre Dame library. ‘Let me look who is now working in our consulate in Vienna’, he said turning the pages of some directory. ‘Oh, Mr. A., with whom I worked in Venezuela. I think it would be best if I wrote him a letter today advising him to ask the store house to send you the books to Notre Dame’. ‘I don’t know’. I replied. ‘Terrible things may happen any day now, even before your letter reaches Vienna’. ‘I see’, Father O’Hara said. ‘then I shall send my friend a cable today’. ‘I think’, I said, ‘This might be safer’. The cable arrived in time and on March, 1938, the very day when Hitler’s troops marched into Austria, the crates containing my books left the port of Hamburg on an American ship bound for New York. President O’Hara was an extremely energetic, efﬁcient, and resourceful man. In younger years he had been in the diplomatic service and was particularly well acquainted with Latin American affairs. Ever since the order of the Holy Cross had made him president of the University of Notre Dame, he ran the organization in a completely autocratic way. Even fellow members of the orders felt, as I could observe on several occasions, his iron hand. Personally I had only the most pleasant dealings with Father O’Hara and reading the news about the goings-on in Austria my gratitude for his extending the invitation to join his faculty just at the time when I received it grew from month to month.

NOTES 1. [Editorial note. About the very difﬁcult political situation in Austria see the following excerpt of a letter from Oswald Veblen (April 12, 1933) to Menger in which the American logician seemed to be very trustful about Roosevelt’s new deal: ‘it is easy to understand that the disturbed political situation in Europe does not help anyone. I should be very much interested to hear what you think about events as they are developing. We hear nothing more from Germany on account of the censorship. Here in America the economic situation has been, and is, very bad but the people are cheerful and there have been no real political disturbances. In fact, since the new government came in, there has been a general optimism among the ordinary people which, however, does not seem to be shared by the important ﬁnanciers’ Karl Menger Papers, Perkins Library, Duke University]. 2. [Editorial note. Hahn Hans, Neurath Otto, Carnap Rudoph, Wissenschaftliche Weltauffassung. Der Wiener Kreis, Wien: Arthur Wolf Verlag, 1929].

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3. [Editorial note. Karl Menger Moral, Wille und Weltgestaltung, Wien, Springer, 1934 (English translation in Menger 1974)]. 4. [Editorial note. As it is well known, Menger provided a job for Wald who became a sort of employer in Morgenstern’s Institute for Konjunkturforschung (Businness Cicle Institute)]. 5. [Editorial note. Menger was referring to F. Waissman Einfu¨hrung in das mathematische Denken. Die Begriffsbildung der modernen Mathematik, Wien, 1936 (English translation: Introduction to Mathematical Thinking by T. J. Benach, New York, 1951); Menger wrote the Vorwort (‘Foreword’), pp. iii–iv.]. 6. [Editorial note. The killer was Johann Nelbo¨ck, who had studied philosophy and mathematics with Schlick, who was his supervisor of a thesis on logic in empiricism and positivism; during the trial ‘he managed to persuade the jury that he had killed the freethinker Schlick for ideological reasons’; he was released from prison after the Anschluss ‘having pointed out that his deed, the elimination of a teacher spreading Jewish maxims alien and pernicious to the people, had rendered a service to National Socialism’ (Golland & Sigmund 2000, p. 43)].

AUTHOR INDEX Adler, Fritz, 43 Alchian, A., 85 Alt, Franz, 5, 24 Arrow, Kenneth, 2–3, 34, 85

Dewey, John, 19, 96 Dubislaw, Walter, 7 Duhem, Paul, 43 Enriques, Federigo, 7 Ernst, Schro¨der, 24 Euler, Leonhard, 99 Evans, G.C., 99 Exner, Franz, 51

Bauer, Otto, 88 Bergson, Henri, 65 Berkeley, George, 56, 85, 121, Bernoulli, Daniel, 21–22, 75, 85 Bernoulli, Nicholas, 21 Birkhoff, G.D., 87 Bo¨hm-Bawerk, Eugene, 31 Boltzmann, Ludwig, 56, 88 Bolzano, Bernhard, 46 Boole, George, 24, 82 Borel, Felix, 66 Brentano, Franz, 48 Bridgman, Paul, 87 Bridgman, Percy W., 19 Brouwer, Luitzner E. J., 65

Feigl, Herbert, 10, 14–15, 19–20, 61, 100 Fichte, Johann Gottlieb, 24, 45 Ford, Lester R., 6 Frank, Philipp, 8–9, 19, 43, 89 Franz Joseph (the Emperor), 92 Frege, Gottlob, 24, 50 Friedman, Milton, 16, 85 Frisch, Ragnar, 85–86 Galileo, (Galilei), 52, 95 Georg, No¨beling, 5, 24 Go¨del, Kurt, 5, 14, 24, 99 Goldbach, Christian, 65 Golland, Louise, 2, 15 Gomperz, Theodore, 43 Gottfried Haberler, 41

Cantor, George, 47 Caratheodory, C., 87 Carnap, Rudolph, 8, 19 Cauchy, Augustin-Louis, 52 Church, Alonso, 98 Cohen, Robert, 10 Columbus, (Cristoforo), 95 Comte, August, 55

Hahn, Hans, 12, 61 Hayek, Friedrich, 4, 26, 32–34, 41 Hegel, Georg F. W., 24, 45, 56 Heisenberg, Werner, 56, 117–118 Hempel, Carl, 9 Herbart, J. F., 46 Herstein, I. N., 85

Dante, (Alighieri), 95 Darwin, Charles, 58 De la Valle´e Poussin, Charles Jean, 106 Debreu, Gerhard, 2–3 Dedekind, Richard, 70 125

126 Hertz, Heinrich Rudolf, 58 Heyting, Arend, 25 Hilbert, David, 24–25, 27, 30, 65, 84, 105 Hume, David, 32, 49, 56 Hurewicz, Witold, 62 Husserl, Edmund, 17, 41, 48, 50–51, 55, 65, 68, 79, 107, Innitzer (Cardinal), 117, 121 Joergensen, Joergen, 19 Kakutani, Shizuo, 3 Kaufmann, Felix, 61–62, 79 Keller, Helen, 79 Kelsen, Hans, 4 Kierkegaard, Søren, 4 Kirchhoff, Gustav Robert, 55, 57 Konvitz, Milton, 8 Koppl, Roger, 35 Kraft, Viktor, 10 Kraus, O., 48–49 Kronecker, Leopold, 25, 65 Leibniz, Gottfried, 34 Lenzen, Victor, 19 Lesniewski, Stanislaw, 83–84 Lewis, Sinclair, 89, 94 Lincoln, Abraham, 89 London, Jack, 62, 89 Luce, Duncan, 29, 34 Łukasiewicz, Jan, 85 Mach, Ernst, 17, 20, 81 Machlup, Fritz, 41 Maeterling, Maurice, 59 Mally, Ernst, 48 Mannoury, Gerrit, 7 Mark, H., 110 Marschak, Jakob, 85 Masaryk, Tomas, 48

AUTHOR INDEX Mauthner, Fritz, 17, 41, 55–56, 58–59, 79, 122 Maxwell, James, 57 Mayer, Hans, 23, 85 McGuinness, Brian, 2, 10–11, 15, 20 Meinong, Alexius, 49 Menger, Carl, 1, 4, 23 Menger, Karl, 1–2, 4, 6–7, 10–11, 13, 18–21, 23, 25–27, 29, 31, 33, 35, 41–42, 85, 101–102, 123–124 Mill, John Stuard, 43, 50 Milnor, John, 85 Mises (von), Ludwig, 2–5, 19, 24, 29–31, 33, 74, 85, 111, 114, 118 Moore, George E., 49 Morgenstern, Oskar, 5, 10, 23–24, 26, 29, 33, 85 Morris, Charles, 6, 8, 19, 99, 101 Mosteller, Frederick, 85 Mulder, Henk L., 10 Nagel, Ernest, 7 Natkin, Marcel, 62 Nelbo¨ck, Johann, 124 Neumann (von), John, 2–5, 19, 24, 29–31, 33, 74, 85, 111, 114, 118 Neurath, Hahn, Olga, 61 Neurath, Otto, 6, 19–20, 61 Nogee, Philip, 85 O’Neill, Eugene, 95 Peano, Giuseppe, 24, 100, 114 Pierce, Benjamin, 24 Pious XII (the Pope), 120 Poincare´, Henri, 75, 109 Popper, Karl R., 7, 26, 118 Popper-Lynkeus, Josef, 95 Post, Emil, 101 Radakovic, Theodore, 7, 61 Raiffa, Howard, 29, 34 Ramsey, Frank, 73–74, 85

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Author Index Ratzen, G., 114 Reichenbach, Hans, 19 Reidemeister, Kurt, 104 Rougier, Louis, 19 Russell, Bertrand, 19, 50, 122

Tolstoy, Lev, 4 Twain, Mark, 89 Twardowski, K., 48 Veblen,Oswald, 97, 123 Vokolok, H., 81

Samuelson, Paul, 85 Savage, Leonard, 85 Schelling, Friedrich, 24, 45 Schelsinger, Karl Schlick, Moritz, 4, 13, 20, 24, 61 Schopenhauer, Arthur, 56, 58 Schreier, Otto, 62, 68 Schro¨dinger, Erwin, 51, 120, 122 Schumpeter, Joseph, 35, 87 Schutz, Henry, 8 Seipel, (Mons), 88 Sheffer, H. M., 87 Sklar, Abe, 2, 15 Sommer, Louise, 85 Spann, Othmar, 4, 19 Sto¨hr, A., 43–44

Waismann, Friedrich, 18 Wald, Abraham, 5, 24 Weaver, W., 6 Weieler, Gershon, 59 Weierstrass, Karl, 47, 52 Weiss, P., 87 Weyl, Hermann, 65, 68, 71 Whitehead, Alfred, 24 Wicksell, Knut, 31 Wiener, Norbert, 87 Wiese (von), Leopold, 2–5, 19, 24, 29–31, 33, 74, 85, 111, 114, 118 Wittgenstein, Clara, 104 Wittenstein, Ludwig, 105

Tarski, Alfred, 8, 83–84, 118 Thirring, Walter, 109–110, 118, 120, Tintner, Gerhard, 9

Zermelo, Ernst, 72 Zilsel, Edgar, 61 Zweig, Stefan, xii