Cambridge and Vienna: Frank P. Ramsey and the Vienna Circle

CAMBRIDGE AND VIENNA FRANK P. RAMSEY AND THE VIENNA CIRCLE

VIENNA CIRCLE INSTITUTE YEARBOOK [2004]

12

VIENNA CIRCLE INSTITUTE YEARBOOK [2004] 12 Institut ‘Wiener Kreis’ Society for the Advancement of the Scientific World Conception Series-Editor: Friedrich Stadler Director, Institut ‘Wiener Kreis’ and University of Vienna, Austria Advisory Editorial Board: Rudolf Haller, University of Graz, Austria, Coordinator Nancy Cartwright, London School of Economics, UK Robert S. Cohen, Boston University, USA Wilhelm K. Essler, University of Frankfurt/M., Germany Kurt Rudolf Fischer, University of Vienna, Austria Michael Friedman, University of Indiana, Bloomington, USA Peter Galison, Harvard University, USA Adolf Grünbaum, University of Pittsburgh, USA Rainer Hegselmann, University of Bayreuth, Germany Michael Heidelberger, University of Tübingen, Germany Jaakko Hintikka, Boston University, USA Gerald Holton, Harvard University, USA Don Howard, University of Notre Dame, USA Allan S. Janik, University of Innsbruck, Austria Richard Jeffrey, Princeton University, USA Andreas Kamlah, University of Osnabrück, Germany Eckehart Köhler, University of Vienna, Austria Anne J. Kox, University of Amsterdam, The Netherlands Saul A. Kripke, Princeton University, USA Elisabeth Leinfellner, University of Vienna, Austria Werner Leinfellner, Technical University of Vienna, Austria James G. Lennox, University of Pittsburgh, USA Brian McGuinness, University of Siena, Italy Kevin Mulligan, Université de Genève, Switzerland Elisabeth Nemeth, University of Vienna, Austria Julian Nida-Rümelin, University of Göttingen, Germany Helga Nowotny, ETH Zürich, Switzerland Erhard Oeser, University of Vienna, Austria Joëlle Proust, École Polytechnique CREA Paris, France Alan Richardson, University of British Columbia, CDN Peter Schuster, University of Vienna, Austria Jan Šebestik, CNRS Paris, France Karl Sigmund, University of Vienna, Austria Hans Sluga, University of California at Berkeley, USA Elliott Sober, University of Wisconsin, USA Antonia Soulez, Université de Paris 8, France Wolfgang Spohn, University of Konstanz, Germany Christian Thiel, University of Erlangen, Germany Walter Thirring, University of Vienna, Austria Thomas E. Uebel, University of Manchester, UK Georg Winckler, University of Vienna, Austria Ruth Wodak, University of Vienna, Austria Jan, WoleĔski, Jagiellonian University, Cracow, Poland Anton Zeilinger, University of Vienna, Austria

Honorary Consulting Editors: Kurt E. Baier Francesco Barone C.G. Hempel † Stephan Körner † Henk Mulder † Arne Naess Paul Neurath † Willard Van Orman Quine † Marx W. Wartofsky † Review Editor: Michael Stöltzner Editorial Work/Layout/Production: Hartwig Jobst Camilla R. Nielsen Erich Papp Editorial Address: Institut ‘Wiener Kreis’ Universitätscampus, Hof 1 Spitalgasse 2-4, A-1090 Wien, Austria Tel.: +431/4277 41231 (international) or 01/4277 41231 (national) Fax.: +431/4277 41297 (international) or 01/4277 41297 (national) email: [email protected] homepage: http://univie.ac.at/ivc/

The titles published in this series are listed at the end of this volume.

CAMBRIDGE AND VIENNA FRANK P. RAMSEY AND THE VIENNA CIRCLE

Edited by

MARIA CARLA GALAVOTTI Università di Bologna, Italy

A C.I.P. Catalogue record for this book is available from the Library of Congress.

ISBN-10 ISBN-13 ISBN-10 ISBN-13

1-4020-4100-4 (HB) 978-1-4020-4100-6 (HB) 1-4020-4101-2 ( e-book) 978-1-4020-4101-3 (e-book)

Published by Springer, P.O. Box 17, 3300 AA Dordrecht, The Netherlands. www.springer.com

Printed on acid-free paper

All Rights Reserved © 2006 Springer No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Printed in the Netherlands.

EDITORIAL Frank Plumpton Ramsey, who was born on February 22, 1903 in Cambridge, England, and died in London on the 19th of January 1930, was certainly one of the most important and promising philosophers of the 20th century. Only his early and unexpected death at the age of 26 probably prevented him from becoming one of the leading figures in the philosophy of science and analytic philosophy – perhaps at a par with Ludwig Wittgenstein, his lifelong close friend but also intellectual adversary. It is well known that in his short life Ramsey immensely enriched philosophy and science with his profound and highly topical contributions on the foundation of mathematics, logic, and economics. As a gifted student at Trinity College, Fellow at King’s College and Lecturer at Cambridge University he influenced Wittgenstein, Russell and Keynes as well as the Vienna Circle with his contributions on the foundations of mathematics, logic, and economics. Especially his significance for philosophy with his focus on notions of truth, decision making, belief and probability is worth mentioning. The intellectual context of Ramsey’s thinking can also be illustrated with the famous Bloomsbury Group.1 My perspective of Frank Ramsey’s life and work was shaped by my personal acquaintance with Ramsey’s sister Margaret Paul (whom I met in 1992 when she shared biographical information and research literature on her brother.) Especially the period he spent in Vienna in 1924 and his contacts with the mathematician Hans Hahn, the physicist Felix Ehrenhaft, among others, spurred me to focus on Ramsey’s connection with the early Vienna Circle. I also repeatedly noticed Ramsey’s significance while writing my book on the Vienna Circle:2 Already in 1929, Ramsey was listed in the manifesto of the Vienna Circle and given credit for attempting to further develop Russell’s logicism and cited as an author related to the Vienna Circle. There are references to his articles on “Universals” (1925), “Foundations of Mathematics” (1926), and “Facts and Propositions” (1927). The proceedings of the “First Meeting on the Epistemology of the Exact Sciences in Prague” (September 15-17, 1929) mention Ramsey as one of the “authors closely associated with the speakers and discussions”, together with Albert Einstein, Kurt Gödel, Eino Kaila, Viktor Kraft, Karl Menger, Kurt Reidemeister, Bertrand Russell, Moritz Schlick and Ludwig Wittgenstein.3 1

2

3

Cf. The British Tradition in 20th Century Philosophy. Ed. by Jaakko Hintikka and Klaus Puhl. Vienna: Hölder-Pichler-Temspky 1995. Friedrich Stadler, The Vienna Circle. Studies in the Origins, Development, and Influence of Logical Empiricism. Vienna-New York: Springer 2001. Erkenntnis I, 1930/31, pp. 311 and 329.

VI

EDITORIAL

But looking at the earlier communication of Ramsey with Wittgenstein and the Vienna Circle these references are not really surprising: whereas it is rather well known that Ramsey visited Wittgenstein in 1923 and 1924, his communication with Schlick and his probable participation in the Schlick Circle have not been fully appreciated. Carnap’s notes on the discussion in the Schlick Circle include Ramsey’s definition of identity, the foundations of mathematics and probability: July 7, 1927: “Discussion by Carnap and Hahn about Carnap’s arithmetic and Wittgenstein’s objection to Ramsey’s definition of identity”.4 Accordingly, Carnap reported on an earlier discussion (June 20, 1927) in the Wittgenstein group with Schlick and Waismann, in which the great “genius” also objected to Ramsey’s notion of identity. Precisely this issue was on the agenda again 4 years later when Wittgenstein met Schlick and Waismann alone (December 9, 1931).5 His lifelong dealings with Ramsey is documented later on in Carnap’s Philosophical Foundation of Physics (1966) with its special focus on the Ramsey sentence. Another reference is worth mentioning here. Commenting retrospectively on his article “The Role of Uncertainty in Economics” (1934), the mathematician Karl Menger, a member of the Vienna Circle and the founder of the famous “Mathematical Kolloquium”, recognised the relevance of Ramsey’s paper “Truth and Probability” (1931) – unknown to him at the time – for his own research, while distancing his own contribution from this study: 6 But the von Neumann-Morgenstern axioms as well as Ramsey’s were based on the traditional concept of mathematical expectation and on the assumption that a chance which offers a higher mathematical expectation is always preferred to one for which the mathematical expectation is smaller. My study was not.

In connection with his stay in Vienna, there is another fact of Ramsey’s life that merits attention: he underwent a (supposedly successful) psychoanalytic therapy with the lay psychoanalyst and historian of literature Theodor Reik (1888-1969), who, by the way, also gave him a book by the theoretical physicist Hans Thirring. After studying the influence of Logical Empiricism in the Anglo-Saxon world, I turned to the investigation of the mutual relations and influences between Austrian and British philosophy of Science since 1900 by writing a completion of Herbert Feigl’s famous account “The Wiener Kreis in America”. It complements “The Wiener Kreis in Great Britain”7 and can be seen as a reconstruction of the 4 5 6 7

Stadler, The Vienna Circle, p. 238f. Ibid., p. 441. Karl Menger, Selected Papers 1979, p. 260. Friedrich Stadler, “The Wiener Kreis in Great Britain: Emigration and Interaction in the Philosophy of Science”, in: Edward Timms/Jon Hughes (eds.), Intellectual Migration and Cultural Transformation. Refugees from National Socialism in the English-speaking World. Vienna-New York: Springer 2003, pp. 155-180.

E DITORIAL

VII

“Austro-British Connection”, with Ramsey as one of the intermediaries and innovators. Here, we can write yet another history with regard to the transfer and transition “from Wiener Kreis to Vienna Circle in Great Britain”: it was, above all, the founder and head of the Vienna Circle, Moritz Schlick, who fostered early intellectual contacts with Britain. Schlick visited England at least twice in the late 1920s as his still unpublished correspondence with Ramsey (1927/28) reveals.8 Ramsey, who invited Schlick to the “Moral Sciences Club” in Cambridge, discussed his personal controversy with Wittgenstein which was triggered by his article “The Foundations of Mathematics” (1925): I had a letter the other day from Mr Wittgenstein criticising my paper ‘The Foundations of Mathematics’ and suggesting that I should answer not to him but to you. I should perhaps explain what you have gathered from him, that last time we didn’t part on very friendly terms, at least I thought he was very annoyed with me (for reasons not connected to logic), so that I did not even venture to send him a copy of my paper. I now hope very much that I have exaggerated this, and that he may perhaps be willing to discuss various questions about which I should like to consult him. But from the tone of his letter and the fact that he gave no address I am inclined to doubt it. 9

This description is also confirmed by Wittgenstein’s critical and ambivalent comments on Ramsey in his Diaries (April 26, 1930).10 These contacts continued, and in one of his last letters before his death, Ramsey reported to Schlick on Wittgensteins’s impact on his own philosophy (namely in the sense that it “quite destroyed my notions on the Foundations of Mathematics”) as well as on Cambridge philosophy in general.11 After Ramsey’s premature death Schlick, whose book on Einstein’s relativity theory was immediately translated into English already in 1920, delivered a programmatic paper on “The Future of Philosophy” at the “Seventh International Congress of Philosophy” in Oxford 1930, announcing the linguistic turn in philosophy.12 Here he advocated the dissolution of the classical philosophical canon by drawing a functional distinction between scientific philosophy on the one hand, and related scientific theorizing on the other. Carnap, too, played an important role in this interaction: on the invitation of Susan Stebbing he delivered three lectures at the University of London in October 1934, where he came into contact with Russell, Woodger and Richard 8

9 10

11 12

Cf. Schlick papers at the Vienna Circle Archives located in Haarlem, The Netherlands. The correspondence will be published as part of the Schlick edition project: http://www.univie.ac.at/ivc/Schlick-Projekt/ Ramsey to Schlick, July 22, 1927. Ibid. Ludwig Wittgenstein, Denkbewegungen. Tagebücher 1930-1932, 1936-1937. Hrsg. von Ilse Somavilla.Frankfurt/M.: Fischer 1999, p. 20f. Ramsey to Schlick, Dec.10, year not dated, op.cit. Cf. Moritz Schlick, “The Future in Philosophy”, in: Schlick, Philosophical Papers, Vol II, ed. by Henk L. Mulder and Barbara van de Velde-Schlick. Dordrecht: Reidel, pp. 210-224.

VIII

EDITORIAL

Braithwaite, the friend of Ramsey and editor of his collected papers in 1931. Here he also met the young Max Black, who wrote his Ph.D. thesis on “The Theories of Logical Positivism” and published on The Nature of Mathematics (1933) under the influence of Moore and Ramsey. He also translated Carnap’s books The Unity of Science (1934) as well as Philosophy and Logical Syntax (1935). Later on he described the “Relations between Logical Positivism and the Cambridge School of Analysis” (1938/39) concluding that “there should be room for further fruitful exchange of opinions between the two movements”.13 Therefore, it is no coincidence that Black many years later described Ramsey in the Encyclopedia of Philosophy (ed. by Paul Edwards) as one of the most brilliant men of his generation; his highly original papers on the foundation of mathematics, the nature of scientific theory, probability, and epistemology are still widely studied. He also wrote two studies in economics, the second of which was described by J.M. Keynes as ‘one of the most remarkable contributions to mathematical economics ever made’. Ramsey’s earlier work led to radical criticisms of A.N. Whitehead and Betrand Russell’s Principia Mathematica, some of which were incorporated in the second edition of the Principia. Ramsey was one of the first to expound the early teachings of Wittgenstein, by whom he was greatly influenced. In his last papers he was moving toward a modified and sophisticated pragmatism.14

This was only one aspect of the flourishing bilateral exchange of ideas also on the level of institutions and periodicals, e.g., the journal Analysis and the “Analysis Society” (from 1936), of course, with A.J. Ayer as the most important intermediary with his extremely influential book Language, Truth and Logic (1936). In summary, we can say that in Britain there was a lively scholarly dialogue between Central European and English philosophers – with the focus being mainly analysis, as compared to the turn from Carnap’s “Wissenschaftslogik” (logic of science) to “Philosophy of Science” in the U.S.A. But there had also been mutual contacts since around 1900 which cannot be separated from what has been referred to as the Anglo-Saxon ‘Sea Change’ (H.St. Hughes).15 What we have here is a dynamic network at work on different levels with distinct convergences and divergences of ideas and theories. Moreover, it is a network that reflected an intellectual preoccupation with several philosophical and methodological debates conducted between thinkers from different countries: from the Austro-German Methodenstreit and the Positivism disputes (Lenin vs. Mach, Horkheimer vs. Neurath) to the foundational debates in mathematics and logic since the 1920s. But the style and form of theorizing changed under different social conditions in the countries of immigration triggering off selforganizing processes of innovation and scholarly exchange. This can be 13 14 15

In: Erkenntnis/Journal of Unified Science, Vol. VIII, p. 34. Black, “Frank P. Ramsey”, in: P. Edwards (ed.), The Encyclopedia of Philosophy. Vlm. 7/8, p. 65. H.St. Hughes, The Sea Change. The Migration of Social Thought, 1930-1965. New York: McGraw-Hill Book Company 1975.

EDITORIAL

IX

exemplified by the Bloomsbury Group, Wittgenstein’s Cambridge, Neurath’s Oxford and last but not least Hayek’s and Popper’s London.16 In 2003 we already witnessed two centenary conferences dedicated to Frank P. Ramsey in Cambridge, UK (June 30 to July 2, 2003) and Paris (October 24-25, 2003). The last one in Vienna (November 28-29, 2003), organized by the Institut Wiener Kreis / Institute Vienna Circle together with the University of Vienna, was deliberately entitled “Cambridge and Vienna” to indicate the focus on the exchange and influence of ideas, as exemplified by Ramsey and the Vienna Circle. This appeared necessary to us because we still lack a profound understanding of Ramsey’s life and work in the German-speaking scientific community. This cannot be compensated by the fact that there is a German translation of Ramsey’s Foundations: Essays in Philosophy, Logic, Mathematics and Economics (ed. by D.H. Mellor in 1978) by the publisher Frommann-Holzboog (Stuttgart-Bad Cannstatt 1980). We are still waiting for the intellectual biography on Ramsey which was already planned by his sister Margaret, who in the meantime has also passed away. Maybe our proceedings will offer another incentive for such a valuable and necessary book in English and German. The organizers and speakers mourned the passing of two friends and extraordinary scholars: Dick Jeffrey had already agreed to come before he died.17 Unfortunately, we also had to commemorate the unexpected death of Donald Davidson (19172003), who had readily accepted our invitation to participate in our Ramsey conference with a paper on “Ramsey and Russell on Subject and Predicate”. This paper was also planned as our distinguished 11th Vienna Circle Lecture 2003. I personally had the privilege and pleasure to meet Donald Davidson once several years ago on the occasion of a dinner with his wife Marcia Cavell and Kurt Fischer here in Vienna and I was very much impressed by his sober and intellectual personality. We then invited him to our conference which he was looking forward to as he expressed in one of his e-mails to me. When we contacted his widow Marcia Cavell to ask her for Donald’s finished manuscript of this conference, we agreed to organise a memorial session for Donald Davidson which friends and colleagues were to attend. And it was a great honor that Patrick Suppes and Michael Dummett agreed to contribute to the memory of their common friend. Although Michael Dummett was not able to

16

17

Cf. D.J. Edmonds and J.A. Eidinow, Wittgenstein’s Poker. The Story of a Ten Minute Argument between two Great Philosophers. London: Faber and Faber 2001. Hans Veigl, Wittgenstein in Cambridge. Eine Spurensuche in Sachen Lebensform Wien: Holzhausen 2004. Cf. Maria Carla Galavotti, “Remembering Dick Jeffrey (1926-2002)”, in: Induction and Deduction in the Sciences. Ed. by Friedrich Stadler. Dordrecht-Boston-London: Kluwer, p. 353f.

X

EDITORIAL

attend our conference for health reasons, we were happy that he sent us his paper together with his memories and an obituary of Donald Davidson, so that all these commemorative pieces by Dummett and Suppes are now included in this volume. Let me express my sincere thanks to all of our speakers for having participated in the conference and contributed to the proceedings. Special thanks go to my colleagues on the program committee: especially to Maria Carla Galavotti, who initiated and chaired the conference and served as editor of its proceedings, and to Eckehart (Kay) Köhler for his help.

Vienna, October 2004

Friedrich Stadler (University of Vienna, and Vienna Circle Institute)

TABLE OF CONTENTS

A. CAMBRIDGE AND VIENNA. FRANK P. RAMSEY AND THE VIENNA CIRCLE

GABRIELE TAYLOR: Frank Ramsey – A Biographical Sketch .............................. 1 BRIAN MCGUINNESS: Wittgenstein and Ramsey ................................................ 19 MICHAEL DUMMETT: The Vicious Circle Principle ............................................ 29 PATRICK SUPPES: Ramsey’s Psychological Theory of Belief ............................. 35 BRIAN SKYRMS: Discovering “Weight, or the Value of Knowledge” ................ 55 STATHIS PSILLOS: Ramsey’s Ramsey-sentences ................................................ 67 ECKEHART KÖHLER: Ramsey and the Vienna Circle on Logicism ..................... 91 J. W. DEGEN: Logical Problems Suggested by Logicism ................................. 123 WERNER LEINFELLNER: The Foundation of Human Evaluation in Democracies from Ramsey to Damasio ...................................................... 139 MARIA CARLA GALAVOTTI: Ramsey’s “Note On Time” .................................. 155

B. GENERAL PART

R EPORT – D OCUMENTATION HELEN E. LONGINO: Philosophy of Science after the Social Turn .................... 167 ALLAN JANIK: Notes on the Origins of Fleck’s Concept of “Denkstil” ........... 179 YAMAN ÖRS: Hans Reichenbach and Logical Empiricism in Turkey .............. 189

XII

C ONTENTS

R EVIEWS Steve Awodey & Carsten Klein (eds.), Carnap Brought Home:The View from Jena. Full Circle: Publications of the Archive of Scientific Philosophy. Volume 2. Chicago: Open Court, 2004. (Christopher Pincock) ................................................................................. 213 Bergmann, Gustav, Collected Works Vol. I:Selected Papers I, edited by E. Tegtmeier, Frankfurt/Lancaster: Ontos-Verlag, 2003. (Daniel von Wachter) ................................................................................. 219 Ferrari, Massimo, Ernst Cassirer – Stationen einer philosophischen Biographie. Von der Marburger Schule zur Kulturphilosophie, Meiner: Hamburg, 2003 (German translation of Cassirer. Dalla Scuola di Marburgo alla filosofia della cultura, Florence: Olschki, 1996) (Gabriele Mras) ........................................................................................... 223 Richard C. Jeffrey, Subjective Probability:The Real Thing, Cambridge University Press, 2004. Richard C. Jeffrey, After Logical Empiricism/Depois do Empirismo Lógico, English edition with Portuguese translation by António Zilão, Lisbon: Colibri, 2002. (Matthias Hild) ........................................................................................... 228 Patrick Suppes, Representation and Invariance of Scientific Structures, CSLI publications, Stanford, California (distributed by Chicago University Press). (Claudia Arrighi / Viola Schiaffonati) ........................................................ 231

A CTIVITIES OF THE I NSTITUTE V IENNA C IRCLE

Activities 2004 ................................................................................................. 237 Preview 2005 .................................................................................................... 241

C ONTENTS

XIII

O BITUARIES

Donald Davidson: A Brief Memoir (Michael Dummett) ..................................................................................... 243 Obituary of Professor Donald Davidson (1917–2003) (Michael Dummett)...................................................................................... 245 Memories of Donald Davidson (Patrick Suppes) .......................................................................................... 251

Index of Names ................................................................................................ 253

G ABRIELE T AYLOR

FRANK RAMSEY – A BIOGRAPHICAL SKETCH

I. Over ten years ago, in September 1992, Margaret Paul and I spent a fortnight in Vienna, in the footsteps of Frank Ramsey. Margaret was Frank’s much younger sister. She was a Fellow of Lady Margaret Hall, Oxford, and a tutor in Economics. She was married to George Paul, Philosophy tutor and Fellow of University College, Oxford. They had four daughters. After her retirement in 1983 she began writing a memoir of her brother Frank, a task she found totally absorbing until her illness prevented her from continuing this kind of work. She died last year. The manuscript she left behind is practically complete, and I want to try and give a picture of Frank and his short life as it emerges from those pages. Being so much younger, and Frank dying so young, she cannot be said to have shared his life to any great extent, but of course they shared the same background: both were born into a distinguished academic family in Cambridge. Their father, Arthur Stanley Ramsey, a mathematician, was successively Fellow and Tutor, Bursar and finally President (ViceMaster) of Magdalene College, Cambridge, from 1897 until 1934. Her mother, Agnes, a graduate of Oxford, had a passionate interest in social work and feminism. She was killed in a car accident when Margaret was only 10. There were four children: Frank, the eldest; Michael, the only one of the children not to reject the religion of their parents and grandparents. He was archbishop of Canterbury from 1961-74; Bridget, who became a doctor, and finally Margaret. Obviously, quite apart from having easy access to letters and diaries, she was in a unique position to tell Frank’s story from an insider’s point of view. Frank was 14 when Margaret was born, a scholar at Winchester College. He took a keen interest in the new arrival. His letters home show him giving thought to what sort of book would be suitable to give his baby sister for her christening. He had some suggestions to make: What book? I think the Just-So-Stories nicely bound for 5/- Perhaps Esmond or Ivanhoe or Pickwick or Martin Chuzzlewit or David Copperfield would be better. What do you think? I can easily afford 5/- by eating less at school shop.

Eventually he decided on Thackeray’s Esmond.

1 M. C. Galavotti (ed.), Cambridge and Vienna: Frank P. Ramsey and the Vienna Circle, 1–18. © 2006 Springer. Printed in the Netherlands.

2

GABRIELE TAYLOR

The same question occupied him at the time of Margaret’s first birthday: should he give her Shakespeare’s Tragedies, the shorter poems of Browning or perhaps some Carlyle. But again he settled for Thackeray, this time Vanity Fair. Margaret kept these two leather-bound copies, inscribed ‘Elizabeth Margaret Ramsey from her brother Frank’ throughout her life. These early letters from Winchester, quite apart from the charm of their particular subject-matter, indicate characteristics and concerns that were typical of Frank throughout his life. They show, firstly, the amount and range of his reading. He clearly did not neglect English literature while also working at mathematics ahead of his form, and having to cope with a great deal of classics, for which he was said not to have had a natural aptitude. He was very keen to ‘get on’: ‘I ought to get some prizes this term’, he wrote, ‘You see, I could get my classical-form prize, my maths prize, a problem paper prize and a prize for all round…’.1 But particularly, the letters show his interest in his small sister to be clearly focused on her mental development rather than on her present state or desires. This concern for the intellectual well-being of his siblings is expressed in a number of letters sent to his mother over the years. So, for example, he seemed to worry about Bridget’s involvement with and success in various sporting activities, which he thought might interfere with her academic advancement. Again, while in Vienna he was told that Margaret, aged 7, would be sent to the Perse school, where her sister Bridget had already been for some years. Frank sounds quite upset: Gug (Bridget) says you are sending Margie to the Perse; if so, for god’s sake don’t leave her long in that outrageous institution but send her somewhere where she’ll learn something like the amount she is capable of. If she goes on like Gug it will be criminal of you.

The Perse school, incidentally, was the best regarded girls’ school in Cambridge. His brother Michael also gave cause for concern: he got only a II.1 in the first part of the classical tripos, largely because, his father thought, he found the debates at the Union an irresistible attraction; and a General Election in which Liberals hoped for the re-establishment of the Liberal Party as a dominating force in politics soon engrossed much of his time and thoughts.

Frank wrote sympathetically: It is a pity about Mick’s Mays. It looks as if he were in the wrong tack in doing classics; odd seeing how interested he is in them.

The number and the content of Frank’s letters sent home when away from Cambridge show him to have been on very close and affectionate terms with all the members of his family, but particularly so with his mother. He seems to have reported to her everything he thought of importance in his life, all the pleasures and the worries of the moment. His letters written from Vienna are characteristic

FRANK RAMSEY – A BIOGRAPHICAL SKETCH

3

in this respect, and so convey a vivid impression of what he did, felt and thought during that period. Frank went to Vienna on two occasions: first in September 1923 to visit Wittgenstein and discuss the Tractatus with him. On this occasion he only passed through Vienna, on his way to and from Puchberg, where Wittgenstein was teaching in the village school. He went the second time a few months later, in March 1924, to see Wittgenstein again if possible, but mainly for personal reasons. When Margaret and I visited Vienna nearly 70 years later she was deeply involved with her research into Frank’s life. And to this research she devoted her stay in Vienna. That was the point of our visit. So we went to the house where Frank had had lodgings when in Vienna for the second time: in the Mahler str. 7, practically next door to the Opera house, which he visited frequently and where he discovered his love of opera. We went to the house where Wittgenstein’s sister Gretl lived: the Palais Schönborn in the Renngasse. Gretl was married to the American Stonborough, and she and her family occupied the first floor. We got a glimpse of the splendid wide marble staircase but could not explore any further as the house had been taken over by offices. Frank saw Gretl frequently and got very attached to her: he went to call on her shortly after his arrival in Vienna and wrote home: She lives in a baroque palace of the time of Maria Theresa, with a vast staircase and innumerable reception rooms very beautiful. She must be colossally wealthy. She was out, but I gave her secretary my address and I got a message asking me to dinner that day. I went and had an evening tête-à-tête with her. – She is 42, handsome and intelligent, and I enjoyed talking to her very much. And I had a very good dinner. She asked me if I should like to come and visit her regularly, and I said I should love to. So she asked me to a dinner party with music on Saturday and said she would then fix a day of the week for me to dine with her every week.

When, a few months previously, Frank had visited Wittgenstein in Puchberg he saw him living in poverty: He has one TINY room, containing a bed, washstand, small table and one hard chair and that is all there is room for. His evening meal which I shared last night is rather unpleasant coarse bread and butter and cocoa.

After this experience of Wittgenstein’s way of life the discovery of the wealth of the family came as a shock to Frank. His two visits were a contrast in every respect. The first one, in September 1923, just after Frank had taken his final examinations, was undertaken and dedicated entirely to a discussion of the Tractatus. He spent a fortnight with Wittgenstein in Puchberg, the longest time they had together until Wittgenstein returned to Cambridge in 1929. Frank, in his second undergraduate year at Trinity College, aged 18, had translated the Tractatus. There is an account (initially given by I.A. Richards)

4

GABRIELE TAYLOR

according to which Frank learned to read German to a sufficiently high standard in just over a week. This is not so: he had German lessons at Winchester at least during his last year there. But the teaching of modern languages was not taken very seriously, and as usual Frank thought he was not getting on fast enough. So he consulted Ogden – a friend of the family- as to how best to learn the language. Ogden gave him what turned out to be excellent advice, viz. to read German books side by side with their English translation. He also sent a list of suggestions, among them Brentano’s Vom Ursprung Sittlicher Erkenntnis (The Origins of our Knowledge of Right and Wrong) and Mach’s Analyse der Empfindungen (The Analysis of Sensations). Frank studied these, but he also turned the advice on its head and read works by English philosophers together with their translations into German: he read Berkeley and Hume in German. The method was successful: he won the school prize in German, an event of which he wrote to Ogden: It’s disgraceful that there is no one in the school who knows more German than I do, but there isn’t, and I have won the German prize….

Much more important, of course, than winning any prizes, was that his combined study of German and Philosophy enabled him to translate the Tractatus. As a result of this translation Wittgenstein asked Frank to come and visit him. By the time the visit took place Frank, in his last Lent-term as an undergraduate, had also begun work on a review of the Tractatus for the periodical Mind which was published in October 1923, after his talks with Wittgenstein. These talks were clearly very intense. Frank wrote home: He is prepared to give 4 or 5 hours a day to explaining his book. I have had two days and have got through 7 out of 80 pages + incidental forward references… He has already answered my chief difficulty which I have puzzled over for a year and given up in despair myself and decided he had not seen…. He is great. I used to think Moore a great man, but beside W!

And in another letter: It is terrible when he says “is that clear?” and I say “no” and he says “Damn. It is HORRID to go through all that again”….

Altogether Frank found the fortnight both healthy and intellectually extremely profitable; a pleasant and inexpensive life: in the morning I walk for 3 hours in the mountains…. In the afternoon I listen to W from lunch to dinner. Then I read Gibbon…

After the Puchberg visit Frank and Wittgenstein wrote to each other, on Frank’s side (only his letters survive) full of affection and friendship, referring to personal problems as well as worries about e.g. the axiom of infinity. Both he and

FRANK RAMSEY – A BIOGRAPHICAL SKETCH

5

Russell tried to persuade Wittgenstein to come to Cambridge, but they were not successful. Frank’s second visit to Vienna, a few months later, was certainly also pleasant and no doubt also intellectually stimulating in some way, but it was hardly as healthy and certainly not as inexpensive as the previous one. Nor was it focussed on Wittgenstein or indeed primarily on philosophy. Frank had certainly intended to see Wittgenstein again in Puchberg, but in the end, apart from a brief farewell visit in September, he only saw him on two weekends during the six months of his stay. The reason was at least partly that he had gone to Vienna to be psychoanalysed, and his psychoanalytic sessions occupied much of his time. They also made it harder for him to think about philosophy or about work in general. ‘I am going down to Puchberg’, he wrote in March, ‘though I don’t want to talk about work as I have forgotten almost all about it.’ When he did go, the visit seems to have been a disappointment: I stayed a night at Puchberg last weekend. Wittgenstein seemed to me to be tired, though not ill; but it isn’t really any good talking to him about work, he won’t listen. If you suggest a question he won’t listen to your answer but starts thinking of one for himself. And it is such hard work for him like pushing something too heavy uphill.

The second visit, a couple of months later, was no more successful. He found Wittgenstein more cheerful, but, he adds, ‘he is no good for my work.’ He thought Wittgenstein quite exhausted by his teaching job and frugal way of living, and thought it absurd of him to refuse all financial help from his family, who were so anxious to help him in any way they could. But ‘he rejects all their advances, even Christmas presents or invalid’s food, when he is ill, he sends back.’ This is from a letter to Maynard Keynes. He and Keynes were again discussing the possibility of getting Wittgenstein to Cambridge, and Frank points out that any offer of financial assistance would certainly be refused. He adds that the only thing that would persuade Wittgenstein to come to England would be an invitation from Keynes himself to stay with him in the country, ‘in which case he would come.’ He adds: I’m afraid I think you would find it difficult and exhausting. Though I like him very much I doubt if I could enjoy him for more than a day to two, unless I had my great interest in his work, which provides the mainstay of our conversation.

Keynes evidently shared the view that it would be difficult and exhausting to have this guest: there was no invitation for Wittgenstein that year. Rather than battling with philosophical problems with Wittgenstein, he enjoyed the society of Wittgenstein’s sister Gretl and other members of Wittgenstein’s family; he listened to music, went to the opera and saw the Cambridge friends who were in Vienna at the time. There was briefly Richard Braithwaite, and throughout the time Lionel Penrose, (later the distinguished geneticist), both very close friends for the rest of his life. He also had some contact with some members of the Vienna Circle, at least shortly before leaving Vienna he met

6

GABRIELE TAYLOR

Moritz Schlick at a dinner at Gretl’s, whom he thought ‘a very nice man’. He was invited to tea by Hans Hahn, where he was rather overawed but also flattered by being included in a discussion entirely in German about atomic theory. But the framework of his daily life was provided by analysis with Theodor Reik, whom he saw every weekday from 12 to 1. II. After the ’14-’18 war psychoanalysis had boomed in England, but there were as yet few British psychoanalysts, and for those wishing to be analysed Vienna was the obvious place to go to. Among the people who influenced Frank in this enterprise were James and Alix Strachey, who did much to spread an interest in this new science in England, and whom Frank knew personally. Both James and Alix had been psychoanalysed by Freud, and Freud had asked them to translate his works – a translation which became their lives’ work and resulted in the 1950ies publication of the English Standard edition. There were other followers of Freud in Cambridge whom Frank knew, e.g. John Rickman, who later became one of the leading figures in the London Institute of Psychoanalysis. Rickman had been at school with Lionel Penrose, and they all met at Cambridge. Another of Frank’s close friends, Sebastian Sprott, had visited Freud in 1922, so Frank would have been well informed about psychoanalytic practice in Vienna. It is then unsurprising that Frank was interested in psychoanalysis, and that, wanting to be psychoanalysed, his thoughts should turn to Vienna, especially since Lionel Primrose, encouraged by Rickman, had gone to Vienna and had a flat there, which Frank could share. But why this desire for psychoanalysis? Partly it seems to have been in the air, the thing to do, in Frank’s particular circle. The idea appealed to Frank particularly because at the time when the thought first came to him, his third year at Trinity, he was in a state of depression. In a rare diary entry during that year he speaks of his feeling of loneliness: I feel lonely…. I need some satisfactory human relationship and have none; I feel this more than ever before. Before I have felt keenly the unsatisfactoriness of some particular relation; but this is more; it is a general feeling of isolation….

The rest of the entry, quite a long one, goes through the list of his friends, all male, and the respects in which they are unsatisfactory. The ones free of such drawbacks are not around, and in any case he doubts that his affection for them is matched by theirs for him. He was in this state of mind when he met, and fell in love with, a young married woman, Margaret Pyke, and his relation with Margaret became his preoccupation throughout that year and beyond. Both she and her husband accepted him as a friend, he visited them often and they seemed very glad to see him. But her feelings for him were nothing like his feelings for her, and for much of the time Frank was tortured by her apparent coldness and

FRANK RAMSEY – A BIOGRAPHICAL SKETCH

7

neglect of him. This of course added to his depression and feeling of isolation. His father became worried about him and wrote, looking back to that time: As his Tripos drew near he suffered a good deal of mental unrest and slept very badly and I began to be afraid whether he would get it over without a breakdown. He was very well prepared in Mathematics and it was not, I think, anxiety about the Examination that worried him but deeper problems about the meaning of life and his relation to other people. (In 80 Years and More.)

Preoccupation with Margaret and work on the Tractatus did not of course prevent Frank from doing brilliantly in Part II of the mathematical Tripos. Although Frank was on very affectionate terms with his mother and reported to her everything of interest in his Viennese life, they did not, naturally, always see eye to eye. There was, firstly, the question of the use of being psychoanalysed. At some stage of his analysis Frank wrote to his mother asking her about events in his early childhood and about children’s books he had liked, which she was to send so that Reik could read them. Agnes was not impressed: if, she suggested, analysis just consisted of putting together facts about his early childhood, Frank could have asked her in the first place without bothering to go to Vienna. Frank replied that information about a person’s childhood was not analysis.2 Agnes was worried also about the analysis being so time-consuming as to prevent Frank from doing much work, and about the expense of the whole enterprise. Both worries were up to a point shared by Frank. Even in Winchester Frank had kept account of his income and expenses, and these figure in his letters from Vienna as well. Shortly after arriving in Vienna he heard that he had won the Allen scholarship, (£250 p.a.) but, Frank thought, even with this addition his financial problems were not solved: the Allen scholarship was meant to help support him while working on his thesis, and while holding it he was not allowed to teach to fill the gap between receipts and expenses. So his immediate financial future did not look rosy. Agnes suggested that he could make a little money, and get himself known, by reviewing for the New Statesman. He replied: ‘it is rot to think you get known (except to the ignorant public) as an authority by reviewing in the NS.’ Shortly after this correspondence Frank had the unexpected and happy news that he had been elected to a Fellowship at King’s College, Cambridge, and his money problems were solved. The question of whether Frank was doing enough work worried Agnes even more persistently than that of the expense. Was not psychoanalysis taking up too much of his time and energy, and could it be right to use some of his scholarship money to support him in Vienna when he was doing so little work? Frank found this attitude rather irritating. He replied, perhaps somewhat defensively: I had a letter from the Registrar saying my residence here is approved. It seems to me perfectly proper to spend a scholarship being analysed, as it is likely to make me cleverer in the future, and discoveries are made by remarkable people not by remarkable diligence. …

8

GABRIELE TAYLOR

Later in his course of analysis he puts his case more strongly: Psychoanalysis is very important even I think to one’s work. You see obscure unconscious things may decide your attitude about certain things, especially personal factors in a controversial subject. Lots of work on the Foundations of Mathematics is emotionally determined by such things as: 1) Love of mathematics and a desire to save it from those (villainous and silly) philosophers. 2) Whether our interest in mathematics is like that in a game a science or an art. 3) General Bolshevism against authority. The opposite; timidity. Laziness or the desire to get rid of difficulties by not mentioning them. If you can see these in other people you must be careful and take stock of yourself.

These observations don’t perhaps altogether settle the question as to whether psychoanalysis is so important for one’s work, but Frank seemed to be convinced. However, he himself every now and then expressed feelings of guilt about his doing so little work. But he shared his parents’ work ethic and his expectations of himself were extremely high. He was hardly idle: while in Vienna he wrote the greater part of ‘The Foundations of Mathematics.’ Frank’s psychoanalytic sessions seemed to be going well. Reik was a clever and distinguished man, and Frank respected his intellect. Not that he altogether enjoyed the experience: ‘It is surprisingly exhausting and unpleasant’, he wrote to Sebastian Sprott : For about 2 times I said what came into my head, but then it appeared that I was avoiding talking about Margaret, so that was stopped and I was made to give an orderly account of my relations with her… I rather like him, but he annoyed me by asking me to lend him Wittgenstein’s book and saying, when he returned it, that it was an intelligent book but the author must have some compulsive neurosis.

In all his letters from Vienna to his parents and friends Frank says nothing about how, if at all, he thought psychoanalysis had changed him in his attitudes to personal relationships. When he returned to Cambridge in October ’24 he was cured of his infatuation for Margaret Pyke, but of course that might have passed anyway. There is a comment on Frank’s sessions from Alix Strachey, written in November 1924 from Berlin, where she was being psychoanalysed, to her husband James in London. She had met Reik at a conference in Würzburg: (Reik) was enthusiastic about Frank Ramsey’s beautiful character, and seemed to think, analytically, that all was for the best.

And again: He (Reik) said to me that he had done all he could to Frank in the short time at his disposal-that the analysis had gone very well owing to Frank’s crystal clear mind and soulwas enthusiastic about him; and wound up by saying that there’d never been much wrong with him. All of which seems fairly reasonable.

FRANK RAMSEY – A BIOGRAPHICAL SKETCH

9

It is in a later letter, written back in Cambridge at the end of the year to his wifeto-be that Frank indicates how helpful he had found psychoanalysis as far as his personal relationships were concerned: I wrote a long letter to my psychoanalyst saying how happy I was & how grateful I felt to him. Because he did make it possible though you may not see how….

Frank’s six months in Vienna was his first long period away from Cambridge and his home since leaving Winchester nearly four years previously. In some ways his letters to his mother did not change much between Winchester and Vienna: rather like a child he took it for granted that his parents would do all manner of menial tasks for him. They arranged for his belongings to be moved from Trinity to King’s, forwarded his letters, posted books, acted as his banker, bought furniture, ordered taxis, bought clothes for him. They treated him as a mixture between a very special person for whom too much could not be done, and as a rather helpless child. This reliance on his mother prompted some of his more malicious friends to think that Frank needed psychoanalysis not to be cured of his passion for Margaret Pyke, but rather to be cured of his dependence on his mother. Perhaps more surprising than his attachment to his mother is that he also got on so well with his father. Arthur Ramsey was not an easy man to live with. He was absorbed in College affairs and this did not, perhaps, contribute to ease and happiness at home. Margaret Paul describes him sitting ‘at the end of the table at Howfield3 meals, often silent, occasionally telling stories about the latest iniquities at Magdalene, and sometimes roaring abuse at Agnes, or a servant, about some feature of the meal that displeased him.’ Frank appears to be the only one of his children who actively sought his society, asked his advice on books and lectures, went for walks with him and went on holidays with him. But of course Arthur was extremely proud of a son who showed his brilliance very early in life, and whose chosen subject was his own. In many ways Arthur was wholly admirable, a man of integrity, enormously hard working, a good and devoted teacher. Colleagues at College remarked on how intolerant he was of any opinion which differed from his own, but as far as his children were concerned he showed surprising tolerance: ‘Though his children becoming atheists must have been a great sorrow to him,’ his daughter writes, ‘he never protested or even tried to argue with any of them about it. And again, when it came to their marrying, though he was not always pleased by the timing or choice of partner, he accepted their wishes in a good humoured way.’ III. Frank got married to Lettice Baker a year after leaving Vienna. Beyond the statement of this fact there is hardly any comment on either Lettice herself or on the marriage in his father’s (unpublished) Memoirs, written when Arthur was in

10

GABRIELE TAYLOR

his eighties. He only adds that, having taken the Moral Science Tripos, Lettice was able to some extent to share Frank’s interests. But Michael, at the time of their marriage a third year undergraduate in Cambridge, thought that his parents exercised a ‘kindly tolerance’ towards Lettice: I don’t think, they felt much in common with her probably, but they were entirely friendly as far as I know…. Lettice and they had very different temperaments. You see Frank was different from them and Lettice was still more Frankish than Frank was.

Lettice had enjoyed great freedom as a child and remembered her childhood as an entirely happy one. In 1911, aged 13, she went to Bedales, which was then recently founded, a progressive, co-educational school, and much freer and more friendly than most boarding-schools at the time. There she met and became a close friend of Frances Partridge, the writer and longest surviving member of the Bloomsbury group, and after Lettice’s marriage to Frank a close friend of his as well. Lettice was a graduate of Newnham College, Cambridge, and at the time of meeting Frank in 1924 she was doing some research in the Psychological Laboratory at Cambridge for the Industrial Training Board. She was 5 years older than Frank, considerably more experienced and held unconventional views. It seemed natural to her that she and Frank should live together. Agnes may have shown ‘kindly tolerance’ towards Lettice, (and Lettice herself agrees that she was friendly and generous) but she was so horrified at the discovery of the nature of their relationship, that Frank felt compelled to cancel a weekend away which he had planned with Lettice. He wrote to her at the end of the letter cancelling the weekend: I don’t really feel as if I had enough moral courage to go on living with you; but I can’t say for certain. What about marrying? It is risky, but what do you think? I don’t feel sure of myself… It seems so absurd not to go away as we had planned; but I can’t. I should be worrying about mother and possibly being caught. It is a shame for you. …

Frank’s own view of the situation is not clear. Lettice assumed that of course he had no sympathy at all with his mother’s attitude; and indeed he wrote to her: I had a long argument with mother yesterday about free love, and she maintained that it threatened the order of society and the security of women, and said she was sure my bark was worse than my bite or she would be in a perpetual state of anxiety about me.

But at about the same time he read a paper to the Apostles, the select debating society which Frank attended regularly and with enthusiasm. In it he said that with the decline of religion the old ideas of marriage were collapsing, and we ought to consider whether this movement is a good one, and if so, what, if anything, we should attempt to substitute for the old morality… I think the institution of marriage is a great benefit to the female sex especially if we suppose, as seems reasonable, that apart from it the care and maintenance of children would fall on their mothers.

FRANK RAMSEY – A BIOGRAPHICAL SKETCH

11

This rather seems to echo his mother’s views. It is true, of course, that papers read to the Apostles did not always express the speaker’s own opinions but were put forward merely to elicit views and discussions. Be that as it may, Frank seems to have thought his mother’s case against free love at least worth considering. The year preceding his marriage was a very busy one for Frank. His teaching-obligations were considerable. Each week he lectured three times and gave a dozen or more supervisions. After a year or so of carrying this teaching load he wrote to Maynard Keynes, asking his advice as to whether anything could be done to lessen it. ‘It is not that I dislike teaching’, he said, ‘but that doing so much seems to interfere much too seriously with what I mainly want to do.’ Since, he said, his main interest was in philosophical questions nearly the whole of his teaching was quite disconnected from his own work, and did not involve reading or thinking anything useful for it. And he expressed some envy of tutors at his undergraduate College, Trinity, where the teachingload was much lighter. It seems that nothing could be done about it, but it seems also that a year or so later when, as Frank put it, he was ‘in the swing of it’, teaching was not felt by him as being quite so burdensome. His lectures on the Foundations of Mathematics were a great success, judging by comments of some of those who attended them. So for instance one member of his audience, Sir Douglas Elphinstone wrote to Margaret Paul many years later (1990): … my recollection of him is of a big and rather clumsy man. He generally had his hair in tufts all over the place. I remember him in a brown tweed suit, much more countrified than the conventional grey suits normally worn by other lecturers. And I think he wore his clothes untidily… Shining through all this was a round cheerful face, and his style of lecturing was also cheerful; he imparted an enjoyment of his subject, and spiced a clear exposition with little touches of humour.

He speaks of Frank introducing his audience to rigorous logical methods in analysis, which, he wrote, almost set his (Sir Douglas’s) mind on fire. And he concludes: There were other good lecturers who prepared their work properly and had the gift of clear exposition (for Ramsey’s thoughts and expressions were always clear); there were others who lectured not like a version of a printed text book, but who threw something of themselves into their work. But Ramsey exuded some sort of personal charm into his lectures. It was like going into a friend ’s house to go into his lecture room.

Reactions to Frank’s supervisions were not quite as favourable. His sister writes: ‘He impressed his pupils by his friendliness, but not all profited from his explanations.’ One former pupil wrote: I must honestly admit, that his intelligence, knowledge and teaching was miles above my head and I understood nothing of what he tried to teach me.

12

GABRIELE TAYLOR

Another one, Alister Watson, who became a friend, described him as an idiosyncratic teacher. So for instance Frank frequently tackled questions in applied mathematics by working out the answer from Newton’s laws, and then roaring with laughter at his success. As usual, Frank worried about not doing enough work, by which of course he meant his own work. But during the first vacation as a Fellow of King’s he completed his paper on the Foundations of Mathematics, on which he had worked during his stay in Vienna. He submitted it for the University Smith Prize, but was not awarded it. His father thought that the reason for this lack of success was that it had baffled the awarders: ‘it is very unlikely that any of them understood it’ he said. Frank himself seems to have been rather downcast by his failure to be given the prize: ‘After it was refused a Smith’s prize I didn’t think I could publish it’ he wrote to Keynes. But he was cheered by praise from Russell, and of course the paper was published. He was still working on what he felt were problems in it he had not solved, and he had a contract with Ogden for a book on the Foundations of Mathematics. But that was never written. Frank also worked on the problem of Universals, a paper which he read to the Moral Science Club and which was published in Mind in October ’25. After reading the paper he was too stimulated to sleep. Instead he wrote a note to Lettice: … The discussion was a very pleasant surprise. It was almost only with Moore who was very reasonable and intelligent…. I wasn’t at all discomfited as I feared, and to one of my arguments against a theory of his he admitted he could see no answer….

So it can hardly be said that during his first year at King’s he neglected his own work altogether. During that year Frank also kept up his interest in psychoanalytic theory. Among his unpublished papers are detailed notes on Freud’s ‘Papers on Metapsychology’, which he greatly admired and thought ‘illuminating’. He helped found a small society which was to meet monthly to discuss psychoanalytic topics. James Strachey, who came from London to attend these meetings, commented after the first one. ‘I was crushed by the unaccustomed intellectual level – especially of Ramsey.’ He also said that Frank thought psychoanalytic theory very muddled: He is thinking of devoting himself to laying down the foundations of Psychology. All I can say is that if he does we shan’t understand them. He seems quite to contemplate, in his curious way, playing the Newton to Freud’s Copernicus.

And finally, during that year, he also greatly enjoyed the social life King’s had to offer. He found good friends of his at the College: Richard Braithwaite, who had also recently been elected to a Fellowship, Alex Penrose and John Maynard Keynes among others. He enjoyed the occasional Feast, but also ordinary, everyday dinners in the company of his colleagues. Richard Braithwaite said of him:

FRANK RAMSEY – A BIOGRAPHICAL SKETCH

13

He had an excellent effect on every company. He was very appreciative of people – modest, added to the pleasure of life- he wasn’t a spoil sport. The least malicious person I have known. … On the other hand, he had a gift for avoiding people he thought of as morons, but if you do it discreetly you can’t be savagely condemned.

IV. Frank’s first year of marriage was a relatively tranquil one, free of disasters and upsets. His brother Michael, then a third year undergraduate at Magdalene College, speaks of him as being very happy, marvellously settling down with a most lovely happy marriage and being a Fellow of King’s, and finding his feet intellectually in these different ways, and having an awfully good circle of friends, and all very happy, friendly with the rest of the family, too….

But the following year, 1926-7, brought many changes. From the point of view of Frank’s work it was fertile: he wrote two philosophical articles: ‘Truth and Probability’ and ‘Facts and Propositions’, and also one of his two economics articles: ‘A Contribution to the Theory of Taxation’. In the domestic sphere it was the year in which his daughter Jane was born. But it was also the year in which Agnes was killed in a car accident, and the year in which he fell in love with a friend of Lettice’s, Elizabeth Denby. Frank’s interest in Economics was long-standing, in at any rate his last year at Winchester he had read enormously widely in Economics – as well as in Philosophy and Politics. By that time he was at least as interested in these subjects as he was in Mathematics. Keynes, in a short biography of Frank, refers to the early age, about 16, at which his precocious mind was intensely interested in economic problems. Economists living in Cambridge have been accustomed from his undergraduate days to try their theories on the keen edge of his critical and logical faculties….

Keynes was editor of The Economic Journal and so received from his Cambridge colleagues many drafts for discussion and publication, and frequently the writer mentioned having shown the article to Ramsey. In June ’28 Frank himself sent to Keynes the draft of the article ‘A Mathematical Theory of Saving’. Keynes described it as ‘one of the most remarkable contributions to mathematical economics ever made,’ but Frank enclosed a letter with the draft in which he said: Of course the whole thing is a waste of time as I’m mainly occupied on a book about logic, from which this distracts me so that I am glad to have done it. But it’s much easier to concentrate on than philosophy and the difficulties that arise rather obsess me.

14

GABRIELE TAYLOR

It seems that this important economics paper was written in a few weeks as a more or less light relief from his work on logic. His work on logic, however, brought him again in touch with Wittgenstein, who wrote to him in July ’27, commenting on ‘The Foundations of Mathematics’, which Schlick had sent him. The letter broke a two-year silence. In the summer of 1925 Wittgenstein had approached Keynes to suggest a visit, and this time Keynes responded positively. Wittgenstein arrived at the place in Sussex where Keynes was then living in the middle of August. The visit could not have been timed more badly: Keynes himself had just got married, to the ballerina Lydia Lopokova; he had business in London and was preparing for a visit to Russia to meet his wife’s relatives. So he asked Frank to come and help entertain Wittgenstein. It was only days before Frank’s own wedding. Lettice had gone to Dublin to stay with her mother, and had left Frank, perhaps not the most practical of men, to see to the flat they were to live in and to deal with preparations for the wedding, send out invitations and so on. So Frank had his hands full. Still, he went to Sussex for a couple of days. It was during that stay that there was a quarrel – quite untypically as far as Frank was concerned. But he can hardly have been in a state of mind which allowed him to concentrate on Wittgenstein, and Wittgenstein was no doubt disappointed and hurt that the visit did not at all live up to expectations, owing to Keynes’ many other preoccupations. It is not clear what the quarrel was about, except that Frank stated it was not about logic. At any rate, they did not correspond for two years, though in his letters to others Frank, while still annoyed with Wittgenstein, always emphasised his affection for and admiration of him. The renewed correspondence may well have encouraged Wittgenstein to return to Cambridge at the beginning of 1929. Agnes’ death in August ’27 was of course devastating for every member of the family. One of the consequences was that Frank and Lettice decided to leave their flat and move to Howfield, the Ramseys’ house in Huntington Road, so that his father and young sister should not be left alone with just servants. So they, with baby Jane and a nurse, moved in and made it their home-base for about a year, after which they found their own house in Cambridge, where Frank lived for the last 15 months of his life, and Lettice for the remaining 57 years of hers. Perhaps the decision to move to Howfield had been taken too hastily: it was an unhappy time and an unhappy arrangement, since Arthur’s way of life and his expectations of members of his household was not theirs, especially not Lettice’s, who was used to an easy-going social life with friends dropping in unannounced at all hours. So after some months it seemed best for Arthur’s sister Lucy to come and run the household, and for Frank and Lettice to have their own establishment. They had a so-called open marriage. They had also agreed to have no secrets from each other. Both of them seemed to interpret this to mean that they had to tell each other not only of other lovers appearing in their lives, but also to describe in minute detail their affairs, their feelings for the relevant person, and that person’s overwhelming attractions. Not surprisingly, Lettice managed to cope

FRANK RAMSEY – A BIOGRAPHICAL SKETCH

15

better with this state of affairs than Frank did, but even she could not have been pleased with a birthday- letter he sent her to Dublin where she and Jane were staying, which after brief congratulations dealt entirely with the time he was spending with Elizabeth. Christmas ’27 he also spent with Elizabeth on a holiday in France. While there he had a letter from Lettice in Dublin informing him that she had fallen in love with an Irish writer called Liam Flaherty. Frank was outraged by this piece of information and accused her of destroying their happy peace together to which he was looking forward to returning: Frankly your letter gave me an awful shock, I can’t see how you could imagine it wouldn’t. I felt quite furious and still after a lot of reflection it seems to me very sickening…. I feel this is such an unfortunate time. I had decided to give up Elizabeth, and you know how important she is to me….

Lettice, in her turn, was amazed and hurt by his reaction: I do think you rather a pot to be calling me a black kettle…. I can’t help feeling hurt at your seeming unfairness….

There is a fairly hectic exchange of letters over the next few weeks, some angry and accusatory, others apologising for having been angry and accusatory. Frank to Lettice: … I am sorry I am so unreasonable. The truth is since I got your letter… I have been in a frenzy of anger unlike anything I’ve ever in my life experienced before… I have never been one to feel malice for more than a moment, but now I’m simply consumed by anger, hatred and all uncharitableness….

The letter ends: … Really, of course, I must blame myself for not knowing my own mind. I began it with Elizabeth, but I find that in fact I can’t stand the strain of this sort of polygamy and I want to go back to monogamy, but it’s now too late.

Lettice’s affair with Liam fizzled out after a few weeks, and she faithfully reported her consequent misery and depression. Frank, of course, was much relieved by this turn of events, and, with equal honesty, explained his improved state of mind to her. She replied: Well, it’s something that my being unhappy has helped you! What people we are! Let me cheer you by saying I’m still very gloomy and depressed.

Elizabeth, however, remained in Frank’s life. They do not seem to have met very often, but they corresponded regularly. Frank’s letters to her, on her instructions, were burnt after her death in 1965. She never married. Like Frank’ s mother, she was a social reformer, primarily interested in improving the lot of working-class people, and she devoted her life trying to improve working-class housing.

16

GABRIELE TAYLOR

V. The pen-ultimate chapter in Margaret’s manuscript, with the title ‘Lettice, Elizabeth and Economics’, deals with the correspondence from which I have just quoted, as well as with his work in Economics and Keynes’ reaction to it. In her last chapter, ‘Death’, there are naturally few quotations from Frank himself; they are restricted to a cheerful account of a walking tour in Ireland in August ’29, a couple of notes to G.E. Moore, with whom he had planned weekly discussions which had to be cancelled because of Frank’s illness, and his last letter to Lettice. Of course, there were other important events in his life: he and Lettice had a second daughter, Sarah, and Wittgenstein returned to Cambridge and stayed with them in their house in Mortimer Road for a fortnight, and after that visited them frequently. He seems to have got on well with Lettice, and Frank’s worry, expressed to Keynes, that Wittgenstein might not want to see him again, turned out to be quite unfounded. Wittgenstein went to see Frank after his operation in Guy’s Hospital, London. Since Frank had not come round after the operation he spent some time with Lettice and her close friend Frances Partridge, who writes of the event: When I came back in the evening and went into the cubby-hole again I was surprised to see the remarkable head of Wittgenstein hoist itself over the back of a chair. Everyone reacts differently when the white face of death looks in at the window. Wittgenstein and I were brought closer together than ever before by our intense sympathy for Lettice … Wittgenstein’s kindness, and also his personal grief, were somehow apparent beneath a light, almost jocose tone….

There are other comments from Frances Partridges which, I think, catch particularly well some of Frank’s chief characteristics. She finds him difficult to describe, partly, she says because of his great simplicity. He had a rather quiet voice, he would consider things, I mean he wasn’t one of these volatile talkers at all, he was somebody who listened to other people and responded… In any ordinary sort of thing in life, like going for a walk and looking at the view, sitting on a lawn, eating and drinking, and so on, he was just a very normal person, and totally unselfconscious… I would recognise the quality of his voice more than anything, which had this rather measured and gentle nature to it…. He was not an ambitious man… he was following his thought, he had no desire to make a splash in the world. He was a simple character in many ways… he had a tremendous sense of humour; his whole face cracked when he laughed with a rather hee-hawing noise…it was delightfully easy to get him laughing….

She adds:

FRANK RAMSEY – A BIOGRAPHICAL SKETCH

17

I don’t think Frank ever made conversation. He would talk of what was interesting him but he would probably rather remain silent than talk for the sake of making things go – he was always himself, not always very good socially. (Interview 1982)

Frank’s simplicity and lack of self-consciousness are often remarked upon by his friends, as are his modesty, lack of aggression, and friendliness. Margaret Paul remarks that while his early death may have led to some idealisation, the view of him as exceptionally fortunate in his nature was held by many people before his death. As Frank himself said in the letter to Lettice I quoted: he rarely experienced malice, and when he did then only for a moment or two. He seems to have been free of all the destructive emotions Nor, as far as one can tell, did he arouse them in others. Many of his after all very clever friends (e.g. Richard Braithwaite and Lionel Penrose) thought Frank much more brilliant than they were themselves, but none seems to have felt the least resentment on that account. Perhaps the last word should be left to his brother Michael: when they were both in Cambridge Frank and Michael had a great many conversations together on a wide range of topics, sometimes about religion, about which, of course, they differed entirely, Frank but not Michael thinking Reason the highest court of appeal. Michael thought his brother the cleverest person he knew, and felt himself to be intellectually on a much lower level. ‘But’, he said, ‘there was a total lack of uppishness about him – he never made me feel inferior…. That was the wonderful joy of it.’

N OTES 1.

2.

3.

He found life at Winchester pretty austere and strenuous, and at any rate initially was not happy there. His sister suggests that he tried to cope with his unhappiness by reading and working very hard and becoming competitive, aiming not only at being top but also, at least in mathematics, being top by a considerable margin. Peter Pan, apparently, was a book Frank had been fond of as a child and which was sent to Vienna. A book he had not liked so much was Alice in Wonderland. Agnes, while at St. Hugh’s College, Oxford, had been friendly with Lewis Carroll, who gave her a presentation copy of Alice. As a child Frank was so frightened by some of the strange pictures in it that Agnes cut them all out. The name of their house in Huntingdon Road.

R EFERENCES Margaret Paul: Frank Ramsey Unpublished. Arthur Stanley Ramsey: 80 Years and More. Unpublished. ‘Better than the Stars’. A Radio Portrait of Frank Ramsey, BBC 1978. Ed. Perry Meisel and Walter Kendrick: The Letters of James and Alix Strachey, Basic Books, Inc. Publishers New York 1985.

18

GABRIELE TAYLOR

John Maynard Keynes: Essays in Biography, The Royal Economic Society 1972. Frances Partridge: Memories, Phoenix Paperback 1981.

Department of Philosophy University of Oxford 33 Templar Rd. Oxford U.K. [email protected]

B RIAN M C G UINNESS

WITTGENSTEIN AND RAMSEY This is inevitably a tale of two cities – and it is fitting that (as I understand) a parallel celebration has been held in Cambridge. They were the two cities of Wittgenstein obviously enough, but in a measure of Ramsey too. Later than Wittgenstein (by the interval of their difference in age) he came to Vienna as a pilgrim, just as Wittgenstein had gone to Cambridge. He to learn from Wittgenstein as Wittgenstein to learn from Russell. But they were to find other things also in those cities – Wittgenstein the whole ambience of Bloomsbury, Ramsey the home of psychoanalysis, the family of Wittgenstein (like many visitors – and even later biographers – he seem to have fallen in love with Wittgenstein’s powerful sister) and the seeds of the Vienna Circle. My purpose is to see how the two men interacted intellectually and what that tells us about the two cities as intellectual centres. I would not propose a comparative evaluation of the two, for one obvious reason and for one less so – Ramsey died before developing all his powers, while Wittgenstein could die content that he had made his contribution. So much is obvious, but an overlap in the themes they treated has often obscured the fact that they were trying to do quite different things. Needless to detail here how before the First War Russell helped Wittgenstein to make the existential choice between being an aviator (in those days also a constructor of planes) and a logician, largely by bringing him into a group where he could make free use of his intellect. To be surrounded by Moore, Keynes, the Stracheys and even the younger Apostles (then practically the Cambridge branch of Bloomsbury) was a new experience for him. His family background was one of wealth and high culture but not intellectual to the degree cultivated in this new environment. Naturally he wanted to change them – for one, he maintained that mathematics would improve people’s taste because taste comes of thinking honestly. They were all against him. He even attempted to resign from their Society (the Apostles), thinking that the younger members “had not yet made their toilets”. The brittle arguments of the Society, where the paradoxical or the scandalous would be defended for sheer love of argument seemed to him intolerable. And there was another thing: all, even the older members, lacked what he called reverence: even Russell (whom, at that period, he still respected) was so Philistine as to appreciate the advantages of their age as opposed to previous ones. Still between them the members of this group set him on the way to writing his first and in some ways his greatest work. He was to repair Russell’s logic, he was to deal with Keynes’s probability in two or three paragraphs, and he was to

19 M. C. Galavotti (ed.), Cambridge and Vienna: Frank P. Ramsey and the Vienna Circle, 19–28. © 2006 Springer. Printed in the Netherlands.

20

BRIAN M CGUINNESS

show that ethics, Moore’s field, did not consist of propositions at all. And perhaps this is what they wanted from him: they “looked to him for the next big step in philosophy”, as Russell told Wittgenstein’s sister. The original Abhandlung, whose completion he announced to Russell in 1915, was the product of this Cambridge period, but the additions he made to it in 1916-18 (the passages on God, freedom and the mystical) issued rather from the next two phases in his life. Tolstoy’s religion had taken hold of him in the war and the circle of young disciples of Kraus and Loos whom he met in Olmütz acted as midwives to the utterance of what he had previously and, as he thought, necessarily left unsaid. Russell was shocked by the mysticism that thus entered in, while as for religion the least hint of it was enough to exile one from the drawing rooms of Bloomsbury. “We have lost Tom”, was Virginia Woolf’s comment on T.S. Eliot’s conversion. Still, when the manuscript turned up in Cambridge it made an immediate impression at least on one Trinity undergraduate. C.K. Ogden, as Hugh Mellor recounts, had helped Ramsey to learn German (from Ernst Mach’s Analyse der Empfindungen) while still at Winchester, for he won the German prize there. Later Ramsey undertook to translate Wittgenstein’s newly arrived manuscript when many, even Moore, doubted that this was possible. The translation he dictated in Miss Pate’s office – the typescript still exists and was then worked on by Ogden in correspondence with Wittgenstein – it gives the atmosphere of the work very well, though for a textbook (as it became) less Pathos was needed, as Geach pointed out. Perhaps this atmosphere led to Broad’s quip about his “younger colleagues’ (notice not his own) dancing to the highly syncopated pipings of Herr (if you please) Wittgenstein’s flute” but Ramsey’s review written in the year he graduated as a Wrangler is a model of clarity. Syncopation or complexity where necessary was no barrier to him and it remains one of the best introductions to the Tractatus. The young Apostle then went to see his elder brother, recommended by a host of common acquaintances and by the merit of his own translation. He brought an extraordinary quickness of mind and perhaps equally important a most open manner: I quote Frances Partridge’s diary from a few years later: As with many great men (and I am sure he is one) Frank is outwardly simple and unselfconscious. His tall ungainly frame becomes somewhat thicker at the hips; his broad Slavonic face always seems ready to break into a wide smile and his fine rapidly vanishing hair floats in wayward strands around his impressive cranium. He’s intensely musical etc.1

The last point we shall return to: it is of some importance. The qualification “outwardly simple” is well chosen. Mrs Partridge will have been aware of the inner tensions that worried an admiring father when Ramsey was an undergraduate and the emotional crisis that led him to want analysis in 1924 (it cured him, he said, at any rate of the wish to talk about himself).

WITTGENSTEIN AND RAMSEY

21

In discussion Ramsey and Wittgenstein went through the Tractatus together. This issued incidentally in a few extra propositions that were to be added in a second printing if any (there have been scores of them) but haven’t been except in our “philologische Ausgabe” (mine and Schulte’s, the publisher is Suhrkamp). I hope they will be more widely published. However, the discussions were principally devoted to a project that Ramsey describes in a letter to Moore (6.2.1924 – he is explaining his application for a scholarship): I am working on the basis of Wittgenstein’s work, which seems to me to show that Principia is wrong not merely in detail but fundamentally. I have got Russell’s manuscript of the stuff he is inserting into the new edition and it seems to me to take no account of Wittgenstein’s work at all. There is a new Theory of Types without the axiom of reducibility, on which however Russell hasn’t succeeded improving a lot of ordinary mathematics, whose truth, he concludes, remains doubtful. But I have got on Wittgenstein’s principles a new theory of types without any doubtful axioms, which gives all the results of Russell’s one and solves all the contradictions. But Wittgenstein and I think it wrong to suppose with Russell that mathematics is more complicated formal logic (tautologies) and I am trying to make definite the vague idea Wittgenstein has of what it does consist of. If I am successful I think it will illuminate not only mathematics but physics also because a successful theory of mathematics will help one to separate and give a true account of the a priori element in physics. (This certainly exists for “this is not both red and blue” is a priori.)2

(It is interesting that exactly this example occurs in Wittgenstein’s discussions with Schlick and Waismann in 1929 (22 December): it was from the Ramsey discussions, not the Vienna Circle ones that these thoughts took their origin.) Later in the year 1924 Ramsey was to try to induce Wittgenstein to return to England and work there: Keynes would have provided the means, but Wittgenstein declined saying the well of his scientific inspiration had dried up. None the less he seems to have done some work with Ramsey, who kept him in touch with mathematical developments during these years, and let his name be known – Becker mentions it as that of a semi-intuitionist. In the biography of Wittgenstein this is quite an important point, since we otherwise know little of what he was doing intellectually during those years as a schoolmaster. Perhaps it needed Ramsey to provoke this re-direction of attention towards the foundations of mathematics, though the original function of the Tractatus had been to replace the first eleven chapters of Principia. Otherwise, for the early and mid-20’s we only have an indication that he involved himself in biography (perhaps autobiography is meant – a fragment remains) and psychology (we know of his exchange of dreams with the sister already mentioned). When Ramsey came to write the entry on Mathematical Logic (s.v. Mathematics) for the Encyclopaedia Britannica 13th edition his faith in Wittgenstein’s system was still entire, except that he now thought it more plausible to maintain that mathematics was reducible to logic. He constructs his article round the Tractatus and concludes:

22

BRIAN M CGUINNESS

By using the work of Wittgenstein a solution has been constructed…, on which the theory of types is so modified that all need for an axiom of reducibility disappears and mathematics consists entirely of tautologies in Wittgenstein’s sense.3

This entry was published in 1926, but there can be little doubt that it was written before the paper on “The Foundations of Mathematics” read to the London Mathematical Society on 12 November 1925 and published in all collections of his work. To the content of that paper I shall return shortly, only saying that I assume Wittgenstein cannot have known it when he met Ramsey in August 1925. That meeting came about with the all the interlacing of intellectual and personal life characteristic of Bloomsbury. Keynes was newly married to the lovable but eccentric Lydia: indeed these days were to have been their honeymoon. Ramsey was on the eve of his marriage but had to leave his bride and come down to Sussex so that Wittgenstein would have someone to talk to. Geoffrey Keynes and his wife were invited to make up the party and Virginia and Leonard Woolf came over from their house nearby. We know the sort of things that went on from other occasions – Lydia’s boutades taken literally by the Bloomsburyites, Wittgenstein discontented when, instead of his being allowed to prevail in an argument, the subject was blithely changed, Keynes and Wittgenstein talking so fast that no one else could get a word in, then for Ramsey long walks with Wittgenstein, which gave them an opportunity to quarrel about psychoanalysis. This last particular Ramsey recounts in a touching letter to his bartered bride, whom he misses so much. Wittgenstein who must have had half an eye on the possibility of returning to England (he communicated with his friend Eccles in Manchester also) decided to stay in Vienna and take over the building of a house for his sister. He seemed to have broken with Ramsey. Contact was resumed only and in a very chilly manner when Wittgenstein had finished the house and was meeting occasionally with a number of members of the Vienna Circle – Carnap, Waismann, and Feigl in particular. A passage in Ramsey’s paper just mentioned now caught his eye and he dictated a letter, typed by Carnap, which denounced the device by which Ramsey had defined identity, in breach of the Tractatus’s exclusion of that concept. Ramsey, in the end, had not followed Wittgenstein’s principles. A handwritten opening of the letter, “Dear Mr Ramsey”, suggested that he might like to reply to Professor Schlick (for whose thinking in fact Ramsey had not great respect). In a draft reply to Schlick Ramsey explains that he has quarrelled with Wittgenstein though not over this issue. (All the same this did become a contested issue to which Wittgenstein returned in his notes and no doubt his conversations several times.) Such conversations there were to be, because Wittgenstein returned to Cambridge at the beginning of 1929. I have called this a tale of two cities but Vienna was not a place for his work: there he would only be a wealthy amateur. When he did come it was enough for Keynes to tell him that he thought he would find Ramsey worth talking to about logic and other things. And in fact their discussions were a joy:

WITTGENSTEIN AND RAMSEY

23

They’re like some energetic sport and are conducted, I think, in a good spirit. There is something erotic and chivalrous about them. They educate me into a degree of courage in thinking. In science I only like to [quaere only reluctantly] go for a walk on my own.4

The appended sentence is puzzling as written – unless Wittgenstein is thinking of how he otherwise was. Amended (as by Wolfgang Kienzler or myself) it makes more sense but perhaps the most significant point is that Wittgenstein here uses the term “science” (German Wissenschaft) for his own activity and with a positive tone. He had done so in the past, in an enthusiastic letter to Russell of 15.12.1913, and in a significant passage on solipsism: Only from the consciousness of the uniqueness of my life does religion – science – and art arise.5

And his sister echoes this in her reflections on the impossibility of religion, science and ethics being one for them (the family) since religion was missing. Schlick and Frege of course used the word for his and their work, but after his discussions with Ramsey, Wittgenstein no longer did and limited the word to natural science and, with a difference, mathematics. One example from many: My aim is thus other than that of the scientists and my train of thought different from theirs.6

We shall see that Ramsey was partly responsible for this. The topics of the discussions (cut short by Ramsey’s illness and death within a year) are in some cases known to us from papers among the remains of Ramsey edited by Maria Carla Galavotti, others from comments on Ramsey in Wittgenstein’s preliminary or semi-edited notebooks and typescripts. The first twenty or so sections in her book show a number of areas in which Ramsey mulls over, not without criticism of Wittgenstein, problems which appear in the development of a semi-systematic re-writing of the Tractatus such as Waismann was about to begin back in Vienna. Visual space, the nature of meaning, the idea that logic must take care of itself and so on. Later in the selection we find Ramsey talking about the foundations, if any, of physics and mathematics, and above all about the infinite. It seems from later notes of Wittgenstein’s that Ramsey defended an extensional interpretation: I said on one occasion that no extensional infinite existed. Ramsey replied, Can’t one imagine a man living for ever, i.e. simply never dying, and isn’t that extensional infinity? And, to be sure, I can imagine a wheel turning and never stopping. There is a strange difficulty here: it seems to me nonsense to say that there are in a room an infinite number of bodies, as it were by accident. On the other hand I can think in an intentional manner of an infinite law (or an infinite rule) that always produces something new – ad infinitum – but naturally only what a rule can produce, i.e. constructions.7

24

BRIAN M CGUINNESS

The topic recurs in other manuscripts of Wittgenstein’s. Note that “The Infinite” was precisely the topic Wittgenstein chose for the talk actually delivered to the Aristotelian Society in Nottingham in July 1929. Now in fact among Ramsey’s papers there is one in German (with some examples and sentences in English), to which my attention was drawn by Maria-Carla Galavotti and which she entitled “Ist die primäre Zeit unendlich?”, such being its first words. The initial paragraph does indeed appear in the published Philosophische Bemerkungen, as Professor Galavotti remarks, but the remaining paragraphs are drawn from Wittgenstein’s large notebook no. 2 (MS 106), with a few comments or summaries in English and a small number of spelling mistakes not of the usual Wittgensteinian kind. (This point and a good discussion of the content of the paper occur in Wolfgang Kienzler’s Wittgensteins Wende zu seiner Spätphiliosophie 19301932.) This particular selection of Wittgenstein’s remarks does not occur elsewhere, so it is probably not drawn from a prepared document. It seems to me not unlikely that it was dictated to Ramsey as a draft for the Aristotelian Society talk, perhaps for translation. The paper discusses various problems of interest to Wittgenstein and Ramsey at the time (I shall mention one shortly) but ends up (to show its general nature), as follows: Infinite possibility is represented by a variable whose place can be filled in infinitely many ways: and the infinite should not occur in a proposition in any other way.8

At Nottingham Wittgenstein left undiscussed his paper on “Logical Form”, which had been presented in advance and is in fact printed in the Proceedings. Now this paper is also one in which the hand of Ramsey appears. Indeed it is only natural if the paper or papers presented at the Joint Session represented Wittgenstein’s chief preoccupations during the year and hence also his conversations with Ramsey, which were so important to him. This is pre-eminently true of the “Logical Form” paper, which presents, as is well known, a revision of the system of the Tractatus. As Wittgenstein explained to his friends in Vienna, he no longer thought that an elementary proposition was itself confronted with reality. It was now a propositional system that was laid against reality – and this had the consequence not that there were an infinite number of elementary propositions but that there were none, as is indeed said in the paper on the infinite we have just been discussing. Given that this topic is one raised at the very beginning of Ramsey’s contact with Wittgenstein I think there is little doubt that he helped Wittgenstein to solve (sit venia verbi) the problem of colour incompatibility (and more generally that of the synthetic a priori), which had been a trouble since Tractatus 6.3751 and which was a subject of preoccupation in Vienna. The direction of influence between the two thinkers has been a subject of some discussion, Kienzler and Eva Picardi and Rosaria Egidi representing Wittgenstein as making the larger contribution, whereas Ulrich Maier and Mathieu Marion for example think Ramsey taught Wittgenstein to view mathematics in an intuitionist and even finitist way. I cannot enter into all these topics today but it seems to me that influence is not the right word: we might better remember

WITTGENSTEIN AND RAMSEY

25

Gilbert Ryle’s reply when asked whether he had been influenced by Wittgenstein: “I learnt a lot from him.” Now Wittgenstein clearly learnt a lot from Ramsey and came back to philosophy with a knowledge of the thought of Weyl, Brouwer and Hilbert that he would not have had otherwise. But he certainly did not adopt a position near to intuitionism under Ramsey’s influence – Ramsey’s conversion (if such it was) occurred after their meeting in 1925 and Wittgenstein’s enthusiasm for Brouwer did not result from but was the reason for going to the 1928 lecture. It was not a Cambridge product. Ramsey reports in a letter to Fraenkel in 1928 that [in 1924] Wittgenstein did not accept his (Ramsey’s) solution which avoided the need for an axiom of reducibility but rejected all those parts of mathematics that depended on it: “his conclusions were nearly those of the moderate intuitionists. What he now thinks I do not know.” This position of Wittgenstein’s became known (probably through Ramsey) and, as we have seen, Wittgenstein is quoted as a semi-intuitionist alongside Chwistek in Becker’s book on mathematical existence. Least of all did Wittgenstein owe his finitism to Ramsey: in his passages on the matter (taking the line we have seen) Ramsey is always presented as the believer in an actual infinite whom it is important to refute. The general thesis that Ramsey was the chief inspirer of Wittgenstein’s second philosophy seems to me mistaken, unless by the second philosophy is meant that intermediate phase in which a revised dogmatism still seemed possible. The real change came, and this is indicated even by the tribute to Ramsey in the preface to Philosophical Investigations, with the abandonment of the search for the essence of language, which was inspired by Sraffa and by the reading of that least Viennese of figures, Spengler, or, in other words, with the move away from dogmatism, as Wittgenstein called it in his conversation with Waismann in December 1930. Looking back, in the rough notebook (Ms 157b) used when he was making a determined effort to write the definitive account of his changed view (his 1936-37 Ms 142, most of which survives in the opening sections of Philosophical Investigations), Wittgenstein says that the idea of the family [i.e. family resemblance, by inference and by other references that of Spengler] and [the realization that] understanding was not a pneumatic process were two axe strokes against [his previous doctrine – of the crystal clarity of logic in itself]. Sraffa had shown him that he had to accept as a sign something for which he could not give the rules and grammar. From this point of view it is not surprising that in the original version of his well known list of influences on himself Wittgenstein includes just four – Frege and Russell, Spengler and Sraffa – the muses respectively of his first and of his later philosophy. Ramsey is not even added later (as Hertz, Kraus and others are). Ramsey indeed was (almost) the enemy, though no doubt the enemy within – note that one of Ramsey’s last papers shows that he thinks philosophy consists of definitions, precisely what Wittgenstein wanted to get away from. Ramsey’s

26

BRIAN M CGUINNESS

contribution perhaps consisted in showing the difficulties that arose from Wittgenstein’s earlier position. Of course we do not know how Ramsey would have developed had he lived, nor how this would have affected Wittgenstein. I suspect that Ramsey did not have the willpower to control Wittgenstein nor Wittgenstein the wit to convince Ramsey. Their paths would probably have diverged in any case. I will give a few instances of the passages where Wittgenstein, characteristically unsparing, anticipates this in describing this difference between Ramsey and himself on philosophical matters. They should be set against an awareness of the love and concern that Frances Partridge saw when he accompanied her to Ramsey’s deathbed. Ramsey’s death roughly coincided with Wittgenstein’s gradually distancing himself from Bloomsbury and the circle of the Apostles, to which he had formally returned in 1929 (a dinner was held on the occasion). For a while he interested himself in the literary, dramatic and musical activities of the young, above all the privileged young – “all those Wykehamists”, as Leavis scornfully described them (Wittgenstein’s own phrase was “all those Julian Bells”). He took Dadie Rylands round the College garden explaining how Shakespeare should be produced. He analysed the symbols in the poems of William Empson’s circle. He criticized John Hare’s (the later Lord Listowel’s) singing and commented on the paintings of Julian Trevelyan. Something – more than one thing probably – changed him. His views were perhaps not given the attention they deserved. Rylands smiled at the advice that was given, Julian Bell wrote a poetic epistle, addressed to Braithwaite, protesting against the cultural hegemony claimed by Wittgenstein. John Cornford’s scorn for his teachers may have been directed against, for it was certainly resented by, Wittgenstein. Wittgenstein began to find friends and disciples in less privileged and more earnest circles, who were primarily intent on personal improvement: King, Lee and Townsend, who have published their notes on his lectures; the circle round Skinner; and particularly Drury, Smythies and Rhees, who remained close to him till the end of his life. Each group is worthy of description, but none is remotely to be thought of in connexion with Bloomsbury. They were prepared, however, for the difficult task of discipleship: it meant that they had to get the essential things right and yet be prepared to disagree with Wittgenstein: above all they could not play with ideas, or indeed with much else. He found also friends of his own age and on his own level and, by a social law that I have observed operate at Oxford, these tended to be foreigners who (more than was necessary but not more than was natural) felt themselves outside the cosy world of the colleges. Piccoli, the professor of Italian, was one example. But the chief figure of this kind was undoubtedly Sraffa and here Wittgenstein was confronted with willpower almost equal to his own. If Sraffa made him feel like a tree stripped of its branches, Sraffa in the end found their conversations too much – “I won’t be bullied by you, Wittgenstein”, Smythies (who, you might

WITTGENSTEIN AND RAMSEY

27

say, had been bullied by both) heard him say. Sraffa resembled Wittgenstein even in some of the methods and aims of his scientific work. He too could use their common friends Ramsey and Alister Watson to help him with the mathematics he needed, but he took strictly what he needed from them. And I am struck by a summary judgement of Amartya Sen9: [Sraffa’s] later work did not take the form of finding different answers to the standard questions in mainstream economics, but that of altering – and in some ways broadening – the nature of the inquiries in which mainstream economics was engaged.

Sen adds that it would be surprising if Sraffa had not been influenced by his own philosophical position but had stayed within “the rather limited boundaries of positivist or representational reasoning commonly invoked in contemporary mainstream economics.” Instead he addressed (according to Sen) foundational economic issues of general social and political interest (some of which have been discussed for over two hundred years). Sen has some valuable suggestions for the influence of Sraffa’s philosophical position on Wittgenstein, but I will not go into those here. I want instead to quote one Cambridge contemporary who felt that Wittgenstein also went, or wanted to go, outside the recognized borders of his subject. No great figure but a thoughtful friend of the Bloomsbury group, Sydney Waterlow, wrote to Moore as follows: [On reading Ramsey] contrast between his quite extraordinary powers and his immense vitality on the one hand and on the other the poverty of his Weltanschauung. Wrong that there should be such a contrast; something has gone terribly wrong. His drift towards stating everything in “pragmatic” terms could not, however arguable, put the wrong right. [Ought we to accept only a limited circle of beliefs that are not nonsense] My own belief is that this simply cannot be the case and nothing that a Ramsey can say to the contrary can affect me in the least. For one thing there is a cocksureness in his attitude which I feel to be cosmically inappropriate. A Russell or a Keynes can never grow out of that pertness – there is no principle of growth in them – but Ramsey is so good that he might have if he had lived. [The unsatisfactoriness of Principia Ethica] But what is satisfactory I haven’t the faintest idea. I rather think Wittgenstein knows and I believe one has got to find out.10

How Moore replied we do not know. He will not have mocked Waterlow as Virginia Woolf does in her diaries, where, however, she tells us something about him that brings him into connexion with Wittgenstein: Waterlow at this time had discovered the other inspirer of Wittgenstein’s Wende, Spengler (the word Weltanschauung betrays it), and this had changed the world for him. It is true the Wende took Wittgenstein in directions not envisaged by Spengler, Waterlow, or followers of Wittgenstein such as Paul Engelmann, but that his tendency was to break the boundaries, to change the donne is undeniable. It is far from clear that Ramsey would have wanted any such thing.

28

BRIAN M CGUINNESS

N OTES 1. 2. 3. 4.

Frances Partridge, Memories, Gollancz 1981 p.129. CUL: Papers of G.E. Moore (letter from F.P. Ramsey 6.2.1924). Encyclopaedia Britannica 13th edition (New Volume 2, p.831). Wittgenstein Papers 105 4 15 Feb.1929. Ich habe sehr genußreiche Diskussionen mit Ramsey über Logik etc. Sie haben etwas von einem kräftigen Sport und sind glaube ich in einem guten Geist geführt. Es ist etwas Erotisches und Ritterliches darin. Ich werde dabei auch zu einem gewissen Mut im Denken erzogen. Es kann mir beinahe nichts Angenehmeres geschehen als wenn mir jemand meine Gedanken gleichsam aus dem Mund nimmt und sie gleichsam im Freien aufrollt. Natürlich ist alles das mit viel Eitelkeit gemischt, aber es ist nicht pure Eitelkeit. Ich gehe in der Wissenschaft nur gern [quaere nur ungern or nicht gern] allein spazieren. 5. Noteboooks 1914-1916 p.79 1.8.1916. Nur aus dem Bewußtsein der Einzigkeit meines Lebens entspringt Religion – Wissenschaft – und Kunst. 6. Wittgenstein papers 109 207 6 Nov.1930. Mein Ziel ist also ein anderes als das der Wissenschaftler und meine Denkbewegung von der ihrigen verschieden. 7. Wittgenstein Papers 105 23 (February 1929). Ich sagte einmal es gäbe keine extensionale Unendlichkeit. Ramsey sagt darauf kann man sich nicht vorstellen daß ein Mensch ewig lebt d.h. einfach nie stirbt, und ist das nicht extensionale Unendlichkeit? Ich kann mir doch gewiß denken daß ein Rad sich dreht und nie stehenbleibt. Hier liegt eine merkwürdige Schwierigkeit: Es scheint mir unsinnig zu sagen daß in einem Raum unendlich viele Körper sind gleichsam als etwas Zufälliges. Dagegen kann ich mir ja intentional ein unendliches Gesetz denken (oder eine unendliche Regel) durch die immer neues produziert wird – ad infinitum – aber natürlich nur was eine Regel produzieren kann, nämlich Konstruktionen. 8. Ramsey Papers 004-23-01. Die unendliche Möglichkeit ist durch eine Variable vertreten die eine unbegrenzte Möglichkeit der Besetzung hat: und auf andere Art darf das Unendliche nicht im Satz vorkommen. 9. Amartya Sen “Piero Sraffa: a student’s perspective”, Atti dei convegni Lincei, vol. 200, Rome 2004, pp.23-60. 10. CUL: Papers of G.E. Moore (letters from Sydney Waterlow 6 & 23 Jul. 1931).

Dipartimento di filosofia e scienze sociali Università degli studi di Siena Via Roma 47 53100 Siena Italy [email protected]

M ICHAEL D UMMETT

THE VICIOUS CIRCLE PRINCIPLE According to Frank Ramsey, and likewise to Gödel, the validity of Russell’s Vicious Circle Principle depends on whether mathematical objects exist independently of us in an abstract realm, or whether they are human creations, brought into being by intellectual constructions we have effected and sustained in being by our ability to repeat those constructions. (A work of fiction or a poem is a human creation, but, once written down or printed, is not sustained in being by human intellectual activity.) The example always given to illustrate the former alternative is that of picking out a particular man in a room as the tallest man in the room. This specification involves quantifying over the set of men in the room, a set of which the individual thus picked out is a member. This apparently violates the Vicious Circle Principle, but is quite evidently legitimate. On the diagnosis of Ramsey and of Gödel, its legitimacy derives from the fact that the men in the room exist independently of any observer or commentator. It is at first sight paradoxical to hold that, in order to determine the validity of a logical principle such as the Vicious Circle Principle, we should first have to settle a grand metaphysical question such as the ontological status of mathematical objects. Surely the validity of the Vicious Circle Principle for any given domain of quantification depends, not on the solution of so large a metaphysical problem, but on how that domain is to be specified. If we try to specify it by appeal to quantification over that very domain, we shall have violated the Vicious Circle Principle: we shall have committed a genuinely vicious circle. Quantification over a domain assumes a prior conception of what belongs to that domain: by trying to specify what belongs to the domain by using quantification over that same domain, we assume as already known what we are attempting to specify. But if we can specify the extent of the domain in some manner which determines the scope of quantification over it without assuming that as already given and available for use in our specification, we shall be free to pick out a particular element of the domain by quantifying over it. Suppose, for example, that we start with some first-order theory, our individual variables ranging over some domain, such as the real numbers, which we take to be well defined. We want to extend our theory by expanding the language to admit a new sort of variables for classes of elements of the original domain. If we explain the notion of such a class predicatively, that is, as associated with a formula of the original unextended theory with one free variable, membership of the associated class being determined by satisfaction of the corresponding formula, there will be no circularity in our manner of specifying the domain of the

29 M. C. Galavotti (ed.), Cambridge and Vienna: Frank P. Ramsey and the Vienna Circle, 29–33. © 2006 Springer. Printed in the Netherlands.

30

MICHAEL DUMMETT

new class-variables. But if we were to make the classes correspond in the same sense to formulas of the expanded language, we should by this impredicative specification have violated the Vicious Circle Principle, since formulas of the expanded language will include ones involving quantification over classes; our explanation will indeed be viciously circular. If, on the other hand, we specify that the class-variables are to range over every possible function from elements of the base domain – real numbers, in the example – to the truth-values true and false, we escape this difficulty: we may specify a particular class by quantification over classes. There is much room for doubt whether the intended sense of “possible” is a legitimate or truly comprehensible one, namely, when we speak of all possible functions from real numbers to truth-values, that is, of all possible determinations, for every real number, of whether it belongs to the associated class or not. But there can be no doubt that, if it is legitimate, we shall in this way have evaded the Vicious Circle Principle. Why, then, do Ramsey and Gödel insist that the validity of the Vicious Circle Principle depends upon whether mathematical objects exist independently of us or are created by us in thought? Their idea is surely this. When we wish to specify a domain of quantification consisting of physical objects, all we need to do is to select a suitable concept, in Frege’s sense of “concept”. A Fregean concept is determined by a precise condition for an arbitrary object to fall under the concept. If we wish to quantify over the mammals in the London Zoo, we shall select the concept mammal in the London Zoo; if we wish our domain of quantification to comprise just the elephants in the London Zoo, we shall select the concept elephant in the London Zoo. Making a choice of the relevant concept is all that we need to do in order completely to determine the domain of quantification. We do not need, in addition to selecting the concept that fixes the condition for membership of the domain, to lay down what objects there are which fall under the concept, or how many of them exist: external reality does that for us. Sometimes, indeed, it is like this in the mathematical realm. If, for some reason, we wanted to quantify over prime numbers of the form 2n - 1, we should need simply to specify that our domain was to consist of numbers falling under the concept prime number equal to 2n - 1 for some n. Reality would, as it were, run through the numbers 2n -1 for n = 0, 1, 2, 3, ..., checking each one to see whether it was prime or not, and consigning each such prime number to our domain. We can leave reality to separate out the elements of our domain because we are forming that domain from within a larger, already determinate, domain, that of the natural numbers. The natural numbers form the zoo from among whose denizens we pick out the elements of our new, smaller, domain. If, however, we want our domain to consist of the real numbers, or of the ordinal numbers, the matter appears quite differently to most of us. Associated with the expressions “real number” and “ordinal number” are concepts, just as there are with the words “mammal” and “elephant”. The concept real number is determined by the condition that a mathematical object that is given to us by

THE VICIOUS C IRCLE P RINCIPLE

31

means of some definition or construction rollst satisfy if we are to recognise it as being a real number; we might take this condition to be that of having a determinate position with respect to the rationals, categorising each rational number as being less than or equal to it or greater than it, and some as one and some as the other. (The final phrase is to exclude infinitary end-points of the rational line.) Similarly, the concept ordinal number is determined by the condition for a given mathematical object to be recognised by us as being an ordinal number; we might take this condition to be that of being the order-type of some well-ordered sequence. Such concepts do not, however, appear to most of us to be adequate to determine a domain of quantification. We do not seem to be in a position to require reality to run through all the mathematical objects that there are in order to decide of each one whether it falls under the concept real number or not, or under the concept ordinal number or not. If we are not in that position, then we must do more to fix a definite domain of quantification than to specify a concept under which the elements of our domain must fall. We must also stipulate a criterion for the existence of objects that are to be elements of our domain, or at least for their belonging to the domain over which we intend to quantify. Why this difference? Why do we think that we need to do more than simply to specify the concept under which the elements of our domain are to fall? The reason is that only very few of us are full-fledged realists about the mathematical realm. If we were, we should assume that there is an absolutely determinate totality of mathematical objects, as determinate as the totality of molecules in a particular glass of water or of monkeys in a certain jungle at a given time. In such a case, it would be enough, in order to specify a domain of quantification, to select a concept under which some mathematical objects fall and others do not; for then reality could decide whether any given object fell under the concept as well as we can, once the object is given to us. If we were total or full-fledged realists about the mathematical realm, that is to say, if we interpreted it in just the same way as realists about the physical world interpret that world, we should think that we could specify a domain of quantification simply as consisting of those mathematical objects that fall under the concept real number. It would consist of just those objects that effect a Dedekind cut in the rational line and do nothing else. Whether or not it would contain an element effecting every possible such cut, in the sense intended by platonists who speak of “every possible Dedekind cut”, would depend upon just which mathematical objects reality comprises – which mathematical objects the Creator has chosen to bring into existence. It would be unnecessary, from this uncompromisingly realist standpoint, to specify the domain by speaking of every possible Dedekind cut: rather, it would consist of every mathematical object that there actually was which effected such a cut. Similarly, we could specify a domain as consisting of all the ordinal numbers there are. There need be no fear that this would lead to contradiction via the Burali-Forti paradox. We could modify the concept ordinal number so as to apply only to those order-types of

32

MICHAEL DUMMETT

well-ordered sequences that had a successor. There would then be just one wellordered sequence that had no ordinal number as its order-type – the sequence of all ordinal numbers; or we might hold that this sequence has no order-type. But this would involve no contradiction. We – or at any rate, all but a small minority of us – are not normally disposed to think of the existence of mathematical objects in this resolutely full-fledged realist manner. Think, for example, of the odd impression made on us by the question whether there exist large cardinals of some particular type. The question may be intended as asking whether it is consistent to assume the existence of such large cardinals. But, if more is being asked than this, what can the question mean? If it is allowed that there may be no contradiction in postulating the existence of large cardinals of that type, but held that there still remains a question whether there really are such cardinals, what could determine the answer to such a question? It is not, surely, a matter of whether God chose to create such cardinals. Kronecker told us that the existence of any mathematical objects other than the natural numbers is the work of man. The existence of monkeys of one or another kind is a contingent matter, a matter of what God has chosen to create. But the existence of mathematical objects should surely be a matter of necessity, of what we could not have found to be otherwise. How should we determine what mathematical objects exist of necessity save by fastening upon same criterion of our own for their existence? Hilbert held that those mathematical objects exist whose existence may be consistently assumed. The mathematical realm, on this view, is maximally full: there are all the mathematical objects that there can be. One who believes that all mathematical theories can be captured in a first-order formalisation may appeal to the completeness theorem in support of this. But this consistency criterion of existence is itself inconsistent. The existence of objects of each of two types may be consistently postulated, and yet a contradiction may result from postulating the existence of both together. It is presumably consistent to assume the existence of the sets which constitute a model of Quine’s NF, but that cannot be combined with the existence of the sets required by ZF, that is, if both are to be sets in the same sense. If we do not think that a first-order axiomatisation can embody every one of our mathematical conceptions, we must allow that there are consistent theories which hold good only if there are only finitely many objects altogether, and others which have only infinite models. Theories which are individually consistent may not be collectively so. I think that full-fledged realism about mathematical objects is a view that is barred to us. We cannot explain what, on such a view, determines which mathematical objects exist; we cannot treat their existence as a matter of what reality holds. I cannot claim that no one is a full-fledged mathematical realist. Professor Timothy Williamson has explained to me in personal correspondence that he hesitates between structuralism and full-fledged realism. In his view, quantification over the elements of a domain has always to he understood as quantification over every object there is, restricted in the standard way by some suitable

THE VICIOUS C IRCLE P RINCIPLE

33

predicate (“is an elephant” or “is a real number”, etc.). Thus if mathematical realism is correct, it must be full-fledged realism. This is certainly a minority view, and I think it untenable. If I am right, we are in no position to declare the Vicious Circle Principle generally irrelevant, as it is when we are quantifying over physical objects The first diagnosis was the right one The Vicious Circle Principle may prohibit us from specifying a domain of quantification in certain ways; it may not apply when we specify the domain in other ways. It is not a matter of whether the elements of the domain exist independently of us or are created by us in thought. It is a matter of the means we adopt to specify that domain and lay down what its elements are to be.

Sir Michael Dummett 54 Park Town Oxford OX2 6XJ UK

P ATRICK S UPPES

RAMSEY’S PSYCHOLOGICAL THEORY OF BELIEF

1. R EJECTION OF F REQUENCY T HEORY AND K EYNES ’ T HEORY OF P ROBABILITY In my analysis of Ramsey’s theory of belief, I shall, in the first part, closely follow the development of his own ideas in his well-known essay, “Truth and Probability”, written in 1926 and published posthumously in 19311. The pagination of my quotations shall be from the 1931 version, edited by Braithwaite. Ramsey begins by admitting some reasons for favoring the frequency theory. More importantly, he endorses it as a proper theory for use in physics, but he immediately goes on to give his reasons for why it is not the proper theory for the logic of partial belief. He doesn’t say a great deal more, but turns immediately to his main object of criticism, Keynes’ Theory of Probability 2. I will not spend much time on these criticisms, but I think it is important to give some sense of what Ramsey has to say about what is wrong with Keynes’ views. His reaction to them shaped many of his own ideas about partial belief. Here is his first point: But let us now return to a more fundamental criticism of Mr. Keynes’ views, which is the obvious one that there really do not seem to be any such things as the probability relations he describes. He supposes that, at any rate in certain cases, they can be perceived; but speaking for myself I feel confident that this is not true. I do not perceive them, and if I am to be persuaded that they exist it must be by argument; moreover I shrewdly suspect that others do not perceive them either, because they are able to come to so very little agreement as to which of them relates any two given propositions. (p. 161)

Notice the line of argument. It is psychological in character. Ramsey simply does not perceive that probability relations are a species of logical relations and, therefore, have full objective validity. The kind of view that Keynes argued for is scarcely defended by anyone today, so I move on to another point. Ramsey states a second view of probability that is also logical in character, but, as he puts it, is ‘more plausible’ than Mr. Keynes’. Here is his summary: This second view of probability as depending on logical relations but not itself a new logical relation seems to me more plausible than Mr. Keynes’ usual theory; but this does not mean that I feel at all inclined to agree with it. It requires the somewhat obscure idea of a logical relation justifying a degree of belief, which I should not like to accept as

35 M. C. Galavotti (ed.), Cambridge and Vienna: Frank P. Ramsey and the Vienna Circle, 35–53. © 2006 Springer. Printed in the Netherlands.

36

P ATRICK S UPPES

indefinable because it does not seem to be at all a clear or simple notion. Also it is hard to say what logical relations justify what degrees of belief, and why; any decision as to this would be arbitrary, and would lead to a logic of probability consisting of a host of socalled ‘necessary’ facts. (p. 165)

The essence of this proposal is to take a more conservative approach and to make probability depend on logical relations, but not be itself a primitive logical concept. This, too, he finds unsatisfactory. Again, this second view is not one that really has, in such bald form, any serious advocacy today. Ramsey returns in several places to other criticisms of Keynes. He seems unable to resist and, I suppose, for good reason. It is hard to think of another book in the history of probability, as badly thought out as Keynes’, which has had so much attention. 2. T HE M EASUREMENT OF P ARTIAL B ELIEF AS S UBJECTIVE P ROBABILITY Ramsey next moves directly to his own theory of the logic of partial belief, and he concentrates on problems of measurement. His section is entitled ‘Degrees of Belief’. At the very beginning, Ramsey starts with the following passage on the importance of measuring partial belief. The subject of our inquiry is the logic of partial belief, and I do not think we can carry it far unless we have at least an approximate notion of what partial belief is, and how, if at all, it can be measured. It will not be very enlightening to be told that in such circumstances it would be rational to believe a proposition to the extent of Ҁ, unless we know what sort of a belief in it that means. We must therefore try to develop a purely psychological method of measuring belief. It is not enough to measure probability; in order to apportion correctly our belief to the probability we must also be able to measure our belief. (p. 166)

We may begin analysis of Ramsey’s theory of measurement of beliefs by his initial paragraph. The important point is his emphasis on the necessity of having a psychological method of measuring belief, in order to have a usable measurement of subjective probability. He does not here use the term ‘subjective probability’, but, for ready reference, it is the term now more current than ‘degree of belief ’. In the next passage, Ramsey summarizes what should be the main ingredients of a procedure for measuring partial belief. Let us then consider what is implied in the measurement of beliefs. A satisfactory system must in the first place assign to any belief a magnitude or degree having a definite position in an order of magnitudes; beliefs which are of the same degree as the same belief must be of the same degree as one another, and so on. Of course this cannot be accomplished without introducing a certain amount of hypothesis or fiction. Even in physics we cannot maintain that things that are equal to the same thing are equal to one another unless we take ‘equal’ not as meaning ‘sensibly equal’ but a fictitious or hypothetical relation. I

R AMSEY ’S P SYCHOLOGICAL THEORY OF B ELIEF

37

do not want to discuss the metaphysics or epistemology of this process, but merely to remark that if it is allowable in physics it is allowable in psychology also. (p. 168)

In the subsequent paragraph, which I shall not quote, Ramsey emphasizes a point that he makes in several other places, namely, that the measurement of beliefs will be approximate and will sometimes be more accurate than others, but this is not surprising, for exactly the same thing happens in physics. This is perhaps a good point to mention a reference that is surprisingly missing from Ramsey’s discussion of the measurement of belief, which he properly judges as a difficult problem. He does not mention the remarkable efforts at measuring psychological quantities, with a theory of approximation or threshold incorporated, by Norbert Wiener3, published just four years before Ramsey’s manuscript was written. Norbert Wiener was earlier a student in Cambridge studying with G. H. Hardy and Bertrand Russell. Less likely is that Ramsey would have known of the treatment of measurement by the German mathematician Otto Hölder4, which was much more sophisticated in its approach, especially from a mathematical standpoint, than the rather naive approach by Norman Campbell5 that he does occasionally cite. My point is that it is surprising that someone with Ramsey’s combination of interests and knowledge did not know more about the already rather extensive and detailed literature on the theory of measurement. He is, as in some other things, much too caught up in the writings of those at or close to Cambridge. In any case, after a sentence or two I have left out, he continues on the same page in the following way: But to construct such an ordered series of degrees is not the whole of our task; we have also to assign numbers to these degrees in some intelligible manner. We can, of course, easily explain that we denote full belief by 1, full belief in the contradictory by 0, and equal beliefs in the proposition and its contradictory by ½. But it is not so easy to say what is meant by belief Ҁ of certainty, or a belief in the proposition being twice as strong as that in its contradictory. This is the harder part of the task; but it is absolutely necessary; for we do calculate numerical probabilities, and if they are to correspond to degrees of belief we must discover some definite way of attaching numbers to degrees of belief. (p. 168)

What Ramsey recognizes in this passage and what follows is the necessity of finding arithmetical operations corresponding to the physical process of addition, so familiar in physics, so this is what he has to say on the following page: Such is our problem; how are we to solve it? There are, I think, two ways in which we can begin. We can, in the first place, suppose that the degree of a belief is something perceptible by its owner; for instance that beliefs differ in the intensity of a feeling by which they are accompanied, which might be called a belief-feeling or feeling of conviction, and that by the degree of belief we mean the intensity of this feeling. This view would be very inconvenient, for it is not easy to ascribe numbers to the intensities of feelings; but apart from this it seems to me observably false, for the beliefs which we hold most strongly are

38

P ATRICK S UPPES

often accompanied by practically no feeling at all; no one feels strongly about things he takes for granted. (p. 169)

Again, it is appropriate to refer to Wiener, for it is exactly the measurement of intensity of feeling, for example, of the loudness of a sound or comparable sensory phenomena that Wiener’s theory of 1921 was concerned with. In this connection, as I could have mentioned earlier, it is also surprising that Ramsey does not mention the elaborate theory of measurement which Wiener built on, and which occupies a large part of the third volume of Principia Mathematica 6. The earlier volumes of which, and the first part of this volume, were of importance in Ramsey’s own work in the foundations of mathematics. But on this last point, I do not really blame Ramsey. Hardly anyone beyond Wiener, as far as I know, has made extensive use of the theory of measurement developed in Part VI of the third volume. It is difficult to read, from a notational standpoint, and does not seem to have really new ideas concerning measurement itself. The treatment of thresholds by Wiener, in contrast, is a genuine new mathematical development, really essentially the first, from a mathematical standpoint, even though the concept of thresholds was introduced much earlier in psychology by Fechner7. (For a detailed survey of the literature, see Suppes, Krantz, Luce and Tversky8.) In any case, Ramsey rejects the use of degree of feeling generated by a belief as a way of measuring the degree of belief. He goes on in the next passage, immediately following, to opt for the degree of belief as a causal property. We are driven therefore to the second supposition that the degree of a belief is a causal property of it, which we can express vaguely as the extent to which we are prepared to act on it. This is a generalization of the well-known view, that the differentia of belief lies in its causal efficacy. (p. 169)

Immediately after this quotation, Ramsey acknowledges that this idea is discussed by Russell in his Analysis of Mind 9. Russell dismisses the causal theory of belief, but Ramsey defends it in a subsequent paragraph. But what Ramsey has to say in its defense is unsatisfactory in terms of a theory of identifying our beliefs. He does make the point, much agreed to by philosophers of many different persuasions, that beliefs play a role of a substantive kind in determining our actions. He states very clearly, on page 173, his firm support of what has come to be called the standard belief-desire model of action, much defended and popularized in more recent years by a number of philosophers. Here is how he summarizes his thought on page 173: I mean the theory that we act in the way we think most likely to realize the objects of our desires, so that a person’s actions are completely determined by his desires and opinions.

A simple part of Ramsey’s defense of this ‘Let us look to the effects’ theory of belief is that he defends that, just as in physics, we can study beliefs without knowing what they are. He uses the excellent example of the attitude toward

R AMSEY ’S P SYCHOLOGICAL THEORY OF B ELIEF

39

electricity in the nineteenth century. One could, it was recognized, study the effects of electricity and, yet, not really be able to say what electrical current is. James Clerk Maxwell, the most important and original Cambridge scientist of the nineteenth century, strongly supports this view about electricity in the following passage from Volume II of his Treatise on Electricity and Magnetism 10: It appears to me, however, that while we derive great advantage from the recognition of the many analogies between the electric current and a current of material fluid, we must carefully avoid making any assumption not warranted by experimental evidence, and that there is, as yet, no experimental evidence to shew whether the electric current is really a current of a material substance, or a double current, or whether its velocity is great or small as measured in feet per second. A knowledge of these things would amount to at least the beginnings of a complete dynamical theory of electricity, in which we should regard electrical action, not, as in this treatise, as a phenomenon due to an unknown cause, subject only to the general laws of dynamics, but as the result of known motions of known portions of matter, in which not only the total effects and final results, but the whole intermediate mechanism and details of the motion, are taken as the objects of study. (p. 218)

It seems to me that Maxwell gives the right analysis, the one that Ramsey should have used quite directly, in the case of belief, namely, that if all we know are the effects of belief, then, like electricity, beliefs remain an unknown cause. I am belaboring this point, because I think it is often a failure of modern philosophers not to recognize the difficulty of characterizing, in a psychologically and, at the same time, scientifically satisfactory way, the nature of belief, or, to put it bluntly, how do we identify beliefs? It seems to me there are positive arguments of several kinds. And yet we do not want to accept that we can just be satisfied, as Maxwell was not, in the case of electricity, with what we can infer about beliefs as unknown causes. So, Ramsey, we might say, was right in one important aspect of belief, that there is a causal importance of beliefs, but restricting this causal account to their effects is scientifically unsatisfactory. Ramsey returns in several different passages to questioning the theory that beliefs are known by “introspectible” feelings, as he puts it, of varying degrees of belief. In doing so, he implicitly recognizes the attractiveness of this theory. Somewhat surprisingly, given Ramsey’s admiration of Hume, he does not refer to Hume’s strong claim that the intensity of feeling is the mark of a belief. I quote here just the single most famous passage from Hume’s Treatise: Thus it appears, that the belief or assent, which always attends the memory and senses, is nothing but the vivacity of those perceptions they present; and that this alone distinguishes them from the imagination. To believe is in this case to feel an immediate impression of the senses, or a repetition of that impression in the memory. ’Tis merely the force and liveliness of the perception, which constitutes the first act of the judgment, and lays the foundation of that reasoning, which we build upon it, when we trace the relation of cause and effect. (p. 86) 11

40

P ATRICK S UPPES

As in many things, Hume puts his finger on the central problem, how indeed do we distinguish a belief from a fancy or an idle thought? In fact, the fallibility of memory makes this one of the great conundrums of forensic epistemology, and raises the problem of the various cases we have to distinguish, now much discussed at some length in both the literature of psychology and of the law. Well, it is not possible to go into the nuances of all this later literature, but it does seem to me it is a criticism of Ramsey that he is much too casual about the problem of identifying or recognizing beliefs and how to separate them from the products of imagination. I come back to this point from a different angle and in somewhat more detail later. 3. R AMSEY ’ S P ROPOSED M ETHOD FOR M EASURING B ELIEFS I now turn to the part of the theory of belief for which Ramsey is justly most famous, namely, his analysis of the problem of measuring beliefs and his proposed solution. He begins this discussion on page 172 and it continues for the next 12 pages. He stresses, as I have already emphasized, his concern to find a method of measuring beliefs as a basis of possible actions, not to develop a general system of beliefs unfocused on any practical or pragmatic applications. Before turning to detailed consideration of Ramsey’s positive theory of measurement, I do want to comment, as an application of an earlier remark I made, on his rejection, in too simple a fashion (p. 171), of the measurement of intensity of feeling based on just perceptual differences. As I mentioned earlier, he makes no reference to the work of Wiener, written earlier and published in 1921, which is more sophisticated, both from a psychological and a mathematical standpoint, than Ramsey’s own development. I quote here just the single passage, which is not sufficiently thought out. This does not mean that what Ramsey has to say about betting, which I will turn to in more detail, is incorrect. It is just that his outright rejection of measuring intensity by thresholds is mistaken and not well thought out. Here is what he says: Suppose, however, I am wrong about this and that we can decide by introspection the nature of belief, and measure its degree; still, I shall argue, the kind of measurement of belief with which probability is concerned is not this kind but is a measurement of belief qua basis of action. This can I think be shown in two ways. First, by considering the scale of probabilities between 0 and 1, and the sort of way we use it, we shall find that it is very appropriate to the measurement of belief as a basis of action, but in no way related to the measurement of an introspected feeling. For the units in terms of which such feelings or sensations are measured are always, I think, differences which are just perceptible: there is no other way of obtaining units. But I see no ground for supposing that the interval between a belief of degree ѿ and one of degree ½ consists of as many just perceptible changes as does that between one of Ҁ and one of 5/6, or that a scale based on just perceptible differences would have any simple relation to the theory of probability. On the other hand the probability of ѿ is clearly related to the kind of belief which would lead to

R AMSEY ’S P SYCHOLOGICAL THEORY OF B ELIEF

41

a bet of 2 to 1, and it will be shown below how to generalize this relation so as to apply to action in general. (p. 171)

A much later detailed theory of measuring subjective probability with thresholds is to be found in Domotor and Stelzer12. I examine the developments from page 172 to 187 in numbered remarks, in which I try to lay out clearly the elements of Ramsey’s detailed proposal, because it is of great historical importance, even though it is possible to criticize it from a later perspective that is crowded with many subsequent formal and empirical developments. 1. Betting. Ramsey begins with this clear endorsement: The old-established way of measuring a person’s belief is to propose a bet, and see what are the lowest odds which he will accept. This method I regard as fundamentally sound; but it suffers from being insufficiently general, and from being necessarily inexact. (p. 172)

Ramsey mentions in the next few sentences a classical problem with betting, namely, the diminishing marginal utility of money, as confounding the interpretation of the odds ratio. 2. Beliefs and desires. He then goes on to say that the way to success is to ‘take as a basis a general psychological theory, which is now universally discarded, but nevertheless comes, I think, fairly close to the truth in the sort of cases with which we are most concerned. I mean the theory that we act in the way we think most likely to realize the objects of our desires, so that a person’s actions are completely determined by his desires and opinions.’ Then Ramsey remarks that he regards this theory as a useful approximation, even if not exact, and it being a somewhat artificial system of psychology, but one that can, as he puts it, ‘like Newtonian mechanics ... still be profitably used, even though it is proved to be false.’ (p. 173) 3. Goods not pleasures. The theory of belief and desires should not be confused with utilitarianism. He distinguishes pleasures from “goods”, but he also says that, to start with, before developing a theory, he will assume that goods are numerically measurable and additive. He returns to the detailed theory later. 4. Good and bad, not ethical. “It should be emphasized that in this essay good and bad are never to be understood in any ethical sense but simply as denoting that to which a given person feels desire and aversion.” (p. 174) 5. Taking account of uncertainty. How are we ‘to modify this simple system to take account of varying degrees of certainty in his beliefs’. (p. 174)

42

P ATRICK S UPPES

This concern for uncertainty, as I would put it, is, of course, central to Ramsey’s theory of partial belief. In these passages Ramsey is just outlining in a summary way the ingredients he will bring together in his axioms. 6. Mathematical expectation. He now asserts the following important proposal: I suggest that we introduce a law of psychology that his [a person’s] behaviour is governed by what is called the mathematical expectation. (p. 174)

He makes use of the ordinary notion, but I’ll use his terminology because it has this unusual plural usage of goods and bads. So, mathematical expectation is this: ... if p is a proposition about which he is doubtful, any goods or bads for whose realization p is in his view a necessary and sufficient condition enter into his calculations multiplied by the same fraction, which is called the ‘degree of his belief in p’. We thus define degree of belief in a way which presupposes the use of the mathematical expectation. (p. 174)

Of course, later, he converts this explicitly into the standard formula for maximizing the expected utility with respect to the utility function and the subjective probability function. I comment on this again. 7. A simple example of computations. Ramsey now gives on pages 175-176 a simple example of how computations of goods and bads with degrees of belief should be made. It’s a rather nice example about the costs of seeking directions when going on an unknown road. I will not go through the details, but it is useful, pedagogically, to make clear how he is going to tackle the question of measurement from the standpoint of how things are working in this intuitive example. 8. Minimal assumptions about measurement. He now says that we do not want to assume that goods are additive. So, the details of how we can measure goods and bads need to be worked out. This leads up to his treatment of utility. What is critical, as he already makes clear (p. 176), is that we need to get a way of judging that the distance between any two goods or bads can be equal to or greater than, or less than, the difference between any two other goods or bads. Or, as I would put it in more current terminology, we will use the formulation of qualitative utility differences governed by an ordering principle. 9. A difficulty solved. Ramsey points out there is a difficulty with his formulation of how to go about measuring these utility differences. We need an ethically neutral proposition. He says this means we want to find a proposition p, which we may call ‘ethically neutral if two possible worlds differing only in regard to the truth of p are always of equal value’. (p. 177) He notes in a footnote that he is assuming Wittgenstein’s theory of propositions – not that he is using very much of that theory in what is going on here.

R AMSEY ’S P SYCHOLOGICAL THEORY OF B ELIEF

43

10. Defining degree of belief ½. Ramsey now makes the important step, which he describes as follows: We begin by defining belief of degree ½ in an ethically neutral proposition. The subject is said to have belief of degree ½ in such a proposition p if he has no preference between the options (1) Į if p is true, ȕ if p is false, and (2) Į if p is false, ȕ if p is true, but has a preference between Į and ȕ simply. (p. 177)

He does not give a sustained argument for there existing such an ethically neutral proposition p, but makes the existence of such a proposition his first axiom on the next page. 11. Operational definition of utility differences. He now gives the belief of degree ½, – but I will not go through the details –, an operational definition of what is meant by the measured difference in value between two goods or bads being to the difference in two others (for details see the equivalence (1) in the next remark). What Ramsey is doing here is very much what is still done, even though the language has changed and, certainly, the sort of casual talk about goods and bads is no longer current. Davidson and I used Ramsey’s idea extensively in our 1950s theoretical and experimental studies 13,14. 12. Axioms. On pages 178-179, Ramsey states his axioms. What is important to note is that he is not giving axioms for the measurement of the belief directly, but for measuring utility. The following point is relevant, particularly for readers familiar with de Finetti15 and Savage16, especially de Finetti, who does not directly and formally use utility or value at all in his approach to quantifying subjective probability. Ramsey’s measurement of subjective probability, as we shall see after we finish with the axioms, is based on first measuring utility, having available in doing so only events of probability ½. By the way, it is useful to make clear how this difference works. If we have the event ½ and we think of having the utility of four goods or bads Į, ȕ, Ȗ, į, we can write the following simple equivalent equations, with ½ being the only probability that enters the expectation calculation: ½ u(Į) + ½ u(ȕ) = ½(Ȗ) + ½(į) if and only if u(Į) - u(į) = u(Ȗ) - u(ȕ) . Ramsey does not work out the detailed consequences of his axioms, but it is apparent that they are essentially correct and there are many modern discussions of such axioms for the measurement of utility or value, as the modern terminology tends to put the matter. For an extensive review of the literature up to the beginning of the 1970s, see Krantz, Luce, Suppes and Tversky17.

44

P ATRICK S UPPES

13. Measurement of belief. Having laid down axioms for the measurement of utility, as discussed, Ramsey is now in a position to define the measurement of belief without additional axioms, simply by using the quantitative measure of utility to do so. Here is the way he describes the matter in words. If the option of Į for certain is indifferent with that of ȕ if p is true and Ȗ if p is false, we can define the subject’s degree of belief in p as the ratio of the difference between Į and Ȗ to that between ȕ and Ȗ; which we must suppose the same for all Į’s, ȕ’s and Ȗ’s that satisfy the conditions. (p. 179)

This is standard and quite familiar in the literature. 14. Conditional probability. Ramsey now takes the additional important step of introducing conditional beliefs, which lead to conditional subjective probabilities. He spends some time with this topic, including stating various laws that hold for conditional probability, which I shall not review in detail, but refer the reader to what he terms the ‘fundamental laws of probable belief’ on page 181. These laws are all elementary as laws of probability, but essential for a coherent theory of belief, as is emphasized in the next point. 15. Belief and consistency. This is what Ramsey says about the elementary laws of probability he lists: Any definite set of degrees of belief which broke them would be inconsistent in the sense that it violated the laws of preference between options, such as that preferability is a transitive asymmetrical relation, and that if Į is preferable to ȕ, ȕ for certain cannot be preferable to Į if p, ȕ if not-p. (p. 182)

16. Probability laws as laws of consistency. We can see from this last remark their importance. We extend the logical notion of consistency to include the satisfaction of these elementary laws of probability. So, from Ramsey’s standpoint the laws of subjective probability are ‘laws of consistency, an extension to partial beliefs of formal logic, the logic of consistency’. (p. 182) 17. Three final remarks. Ramsey concludes these developments by three remarks. The first is that the developments are based fundamentally on the idea of betting, an idea which de Finetti also took as fundamental. Second, the developments are based throughout on the idea of mathematical expectation. This is the way of computing what is to be maximized in choice. To move in some other direction is to make a radical change in the standard theory of maximizing expected utility. His third remark is that he has said nothing about how to deal with matters when the number of alternatives is infinite. It is possible to make a pun here and say that, by now, the number of papers dealing with having the set of alternatives be infinite is nearly infinite. This part of the theory has been

R AMSEY ’S P SYCHOLOGICAL THEORY OF B ELIEF

45

developed thoroughly in subsequent mathematical publications by others. Again, an extensive review is to be found in the reference cited above18. I just want to make one concluding conceptual remark of my own about Ramsey’s important stress upon mathematical expectation. It was the original hope – and attempt – of de Finetti to simply give axioms on subjective belief alone, so de Finetti’s strategy was to introduce an ordering of the form event A is at least as probable as event B. This ordering was assumed to be transitive and connected. Other axioms were introduced to express additivity of subjective probability of disjoint events, and so forth. The problem was that what started out to be a very simple set of axioms did not have a simple result. The axioms de Finetti stressed in earlier work19 were shown by Kraft, Pratt and Seidenberg20 to be inadequate. Namely, a counterexample could be given which showed that the standard representation theorem, to be proved about the quantitative nature of subjective probability, could not be given without further, stronger axioms. There is a tale of many attempts in this direction that I will not review here. My point is just to mention that probably the simplest set of axioms that are necessary and sufficient, expressed just in terms of probabilistic concepts without any introduction of utility, were those given by Mario Zanotti and myself 21. The important point, and the reason for this remark, is that those axioms found it necessary to use as primitive the concept of qualitative expectation, rather than the concept of subjective probability. Perhaps the most important technical reason for requiring this shift is that mathematical expectation is additive in an unrestricted way, whereas the addition of event probabilities requires that the events be mutually exclusive. This restriction causes many formal difficulties. 4. T HE L OGIC OF C ONSISTENCY After completing the section on the measurement of partial belief, that is, the measurement of subjective probability, the next section of Ramsey’s essay is devoted to the logic of consistency. Here, he makes a number of significant and useful remarks about the relation between logic and probability and between our view of the consistency of each of them. Again, he focuses an immediate criticism on Keynes’ theory, saying that Keynes tries to reduce probability to formal logic, but that this is mistaken in several different ways, which I will not examine in detail. What he does do, in terms of his own view, going back to some of the ideas of Peirce, is make a clear distinction between inductive and deductive logic. He insists on the point, on the vast difference between the consistency of logic and the consistency of a person’s set of partial beliefs. The vast difference between these two is exemplified by the elementary fundamental laws of probability, including conditional probability, that he derived from axioms in the previous section.

46

P ATRICK S UPPES

He next discusses, in perhaps the most useful topic in this section, the relation between the calculus of consistent partial belief and the interpretation of the laws of probability in terms of frequencies. He mentions the usual connection between the two via Bernoulli’s Theorem. He does mention one idea that is not often found in more recent discussions. This is: ... the very idea of partial belief involves reference to a hypothetical or ideal frequency; supposing goods to be additive, belief of degree m/n is the sort of belief which leads to the action which would be best if repeated n times in m of which the proposition is true; or we can say more briefly that it is the kind of belief most appropriate to a number of hypothetical occasions otherwise identical in a proportion m/n of which the proposition in question is true. (p. 188)

It is worth mentioning that this device of idealizing a large number of possible observations is also supported by de Finetti22 as a method of evaluating subjective probabilities, especially ones that are very small. The second important remark in this section is Ramsey’s dismissal of the Principle of Indifference as a fundamental part of probability theory. Again, he mentions that ‘... it is fairly clearly impossible to lay down purely logical conditions for its validity, as is attempted by Mr. Keynes.’ (p. 189). Interestingly enough, he does not mention Laplace, who is the one who made this principle so famous in the history of probability. The third and final problem discussed in this section, once again, is centered on a Keynesian problem, but it is interesting from a general standpoint, so I quote Ramsey’s remark. A third difficulty which is removed by our theory is the one which is presented to Mr. Keynes’ theory by the following case. I think I perceive or remember something but am not sure; this would seem to give me some ground for believing it, contrary to Mr. Keynes’ theory, by which the degree of belief in it which it would be rational for me to have is that given by the probability relation between the proposition in question and the things I know for certain. He cannot justify a probably belief founded not on argument but on direct inspection. (p. 190)

What he is doing here, of course, is showing how devastatingly incorrect any attempt to settle questions of the probability of uncertain propositions, by knowledge of certain ones, is, as any kind of general strategy. The argument really is pointing out that Keynes seems to have no way in his system to accept what would amount to conditionalization, based upon possibly fallible memory or perception. 5. T HE L OGIC OF T RUTH The fifth and final section of the 1926 essay, whose title I have also used, is, we might think, in terms of the current patois of philosophy and logic, about

R AMSEY ’S P SYCHOLOGICAL THEORY OF B ELIEF

47

Tarski’s and later definitions of truth. But Ramsey does not have this in mind at all. What he has in mind is, as he calls it, the human logic of truth. Let us therefore go back to the general conception of logic as the science of rational thought. We found that the most generally accepted parts of logic, namely, formal logic, mathematics and the calculus of probabilities, are all concerned simply to ensure that our beliefs are not self-contradictory. We put before ourselves the standard of consistency and construct these elaborate rules to ensure its observance. But this is obviously not enough; we want our beliefs to be consistent not merely with one another but also with the facts. (p. 191)

His central concern is to give a clearer meaning to the question of ‘What is reasonable for a person to have as a given degree of belief in a given proposition?’ Put another way, ‘When is it reasonable to hold a particular subjective probability concerning the occurrence, for example, of some future event?’ Ramsey’s answer to this question is, to my mind, one of the most interesting parts of the essay from a psychological standpoint. In giving the answer that I want to consider, he acknowledges his indebtedness to the writings of Peirce. I will not try to document the relevant parts of Peirce, but will look at the arguments that Ramsey himself gives. Ramsey’s central idea is that The human mind works essentially according to general rules or habits; a process of thought not proceeding according to some rule would simply be a random sequence of ideas; whenever we infer A from B we do so in virtue of some relation between them. We can therefore state the problem of the ideal as “What habits in a general sense would it be best for the human mind to have?” This is a large and vague question which could hardly be answered unless the possibilities were first limited by a fairly definite conception of human nature. (p. 194)

The first point that Ramsey then makes about habits is that he is not restricting habits to just ordinary ones that we learn as children, or sometimes as adults, but is including any rule or law of behavior, as well as instinct. In my own view, this is an excellent generalization of what is often too narrow a use of the concept of habit. He does not expand upon this idea in a way that deals with many of our current controversies about nature versus nurture, but his stand is, for me, very much on the side of the angels. Habits can be found on both sides of that controversy. I also like the fact that he not only emphasizes rules, but also laws. The emphasis on laws and instinct means that we are not caught in some explicit and mistaken rule-type formulation. Consequently, as I discuss later, laws of association or of conditioning can be included in the discussion of habits, as can the kind of instinctive behavior of many lower species, for example, of insects having habits which are patterns of behavior that are not learned, but strongly embedded in the DNA of a given species.

48

P ATRICK S UPPES

He focuses this discussion on degrees of belief. He talks about observing a number of occurrences of common phenomena, such as inedible toadstools or the habit of expecting thunder after lightning. He summarizes nicely his view. Thus given a single opinion, we can only praise or blame it on the ground of truth or falsity: given a habit of a certain form, we can praise or blame it accordingly as the degree of belief it produces is near or far from the actual proportion in which the habit leads to truth. We can then praise or blame opinions derivatively from our praise or blame of the habits that produce them. (p. 196)

He then goes on to apply this not only to habits of observation and inference, but also to habits of memory, and raises the question of what is the best degree of confidence to place, for example, in specific memories. Or, to use his language more exactly, ‘a specific memory feeling’. Our confidence in such a memory feeling should depend on ‘how often when that feeling occurs the event whose image it attaches to has actually taken place’. (p. 196) Following a brief discussion of Hume, Ramsey next asks the question ‘What do we have to say about the person who would make no inductions?’ And he replies that we would think that he ‘had not got a very useful habit, without which he would be very much worse off, in the sense of being much less likely to have true opinions.’ (p. 197). He remarks of this that his view here is a kind of pragmatism. We judge mental habits by whether they work and, so, to adopt induction is a useful habit. Notice he has not spelled out, in really technical detail, what he means by induction, but he intends it to be in terms of what he has laid down as the fundamental laws of partial belief in the earlier formal treatment. Finally, in the last paragraph of this section, he extends the search for inductive or human logic to general methods of thought, or what we might now call methods of scientific inference, but his remarks here do not go very far beyond his casual mention of Mill’s methods and of Hume’s general rules in the chapter ‘Of unphilosophical probability’ in The Treatise 23. My problem with this section, which closes the 1926 essay on truth and probability, is not what he says about habits and the human logic of induction, but what he does not say. He almost takes a turn toward a deeper psychological approach, but does not explore it with any care, and turns back to familiar remarks and rather general remarks customary in the subject and already present in the writings of Hume, Mill and Peirce. In other words, in this section, as opposed to his treatment of the measurement of partial beliefs, and also of utility, where he makes a new and quite original contribution, matters are otherwise. 6. W HAT IS M ISSING IN R AMSEY : P SYCHOLOGICAL M ECHANISMS OF B ELIEF F ORMATION Ramsey writes about many different topics beyond what are included in the essay on truth and probability. These topics range from the foundations of

R AMSEY ’S P SYCHOLOGICAL THEORY OF B ELIEF

49

mathematics to statistical mechanics. His own contributions, of course, greatly vary in depth. He had important and original ideas about the foundations of mathematics, which were early known to be significant contributions. In contrast, although he seemed to know a great deal about physics and many of his remarks about physical ideas and laws are interesting, he, in fact, has no sustained treatment, as far as I know, of any part of the foundations of physics. Matters are more complicated in the case of psychology. In one sense, what he did in his careful analysis and surrounding discussion about the theory of partial belief, and the measurement of such beliefs, is a genuine contribution, not only to philosophy, but also to the systematic science of measurement within psychology, a subject that has a large and controversial history. Indeed, what Ramsey has to say is important and interesting, just in terms of the history of psychology alone, quite apart from its philosophical import. But in this section, I want to examine the extent to which Ramsey probed deeper into looking for a psychological theory of mechanisms to account for partial beliefs. There is not much systematic evidence, as I have, in various ways, already indicated, but there is a useful early piece24, unpublished until the appearance of the book of notes by Ramsey, edited by Galavotti (1991). What is a particularly striking piece in this volume is Ramsey’s imaginary conversation with John Stuart Mill, dated 26 January 1924. I quote two passages: [Mill] ‘I knew that all mental and moral feelings were the results of association, that we love one thing and hate another, take pleasure in one sort of action or contemplation, and pain in another sort, through the clinging of pleasurable or painful ideas to those things from the effect of education or experience. As a corollary to this it is one of the objects of education to form the strongest possible associations of the salutary class; associations of pleasure with all things beneficial to the great whole, and pain with all things hurtful to it. But it seems to me that teachers occupy themselves but superficially with the means of forming and keeping up these salutary associations. They seem to trust altogether to the old familiar instruments, praise and blame, punishment and reward. Now there is no doubt that by these means intense associations of pain and pleasure may be created and produce desires and aversions capable of lasting undiminished to the end of life. But there must always be something artificial and casual in associations thus produced.’ ... Here he [Mill] paused and I broke in at once ‘But you know psychology has advanced since your day, yours is very out of date.’ ‘Has it?’ he answered ‘I don’t think so. You have advanced in philosophy in a way that excites my profound admiration but in psychology hardly at all. Perhaps you are thinking of the followers of Freud, who seem to regard the analysis of the mind as a panacea.’ ‘Yes’ said I, ‘I am thinking of them; you are probably put off by their absurd metaphysics, and forget that they are also scientists describing observed facts and inventing theories to fit them.’ (pp. 305-306)

The next two pages contain Ramsey’s response. [Ramsey] ‘But of course he would dispute your psychology; he [Freud] would say, that the most important associations in determining your desires were those formed very early in life and no longer accessible to consciousness. So that your explanation of your depression must be entirely illusory, in his terminology a ‘rationalization’. The relevant associa-

50

P ATRICK S UPPES

tions could not possibly be dissolved by your own unaided introspection. If you suffer from claustrophobia you may perceive that there is no real danger in closed spaces but nevertheless you cannot bear to be in one.’ (Ramsey, 1991, pp. 306-307)

There is more. First is the imaginary response by Mill, followed by Ramsey’s response to that. ‘I don’t believe it’ he [Mill] answered ‘about the ordinary man; his ‘unconscious’ if he has one is of trivial importance. Freud’s theory was developed from observations not of the normal but of the abnormal, who came to him for treatment. It is not in the least clear that it applies to the ordinary man.’ [Ramsey] ‘But surely the laws of psychology should apply to all cases, normal and abnormal alike, and must be obtained from observations of all kinds of men. The psychoanalysts analyse not only patients but also their pupils who are fairly normal. And their work throws doubt on your psychology as being on much too simple lines; desires and aversions are not generally developed by the simple process of associations of pleasure and pain but by far more complicated laws and mechanisms.’ (Ramsey, 1991, p. 307)

This is fascinating, – and especially fascinating, written by such a very young man –, but it is also evident that there is no move in the direction of psychology as a systematic science. Moreover, this is well reinforced by the kind of statement about these matters one would expect from Ramsey. In the last papers at the end of the 1931 volume, there is one entitled “Probability and Partial Belief ” 25. The first two sentences of this fragment are certainly worth quoting. The defect of my paper on probability was that it took partial belief as a psychological phenomenon to be defined and measured by a psychologist. But this sort of psychology goes a very little way and would be quite unacceptable in a developed science. (p. 256)

In other words, Ramsey recognized, in an unblinkered way, that what he had to say about psychology did not amount to anything like a serious detailed foray into an analysis of the psychological mechanisms producing partial beliefs. The very last quotation, Ramsey’s statement about the work of psychoanalysts that throws doubt on Mill’s psychology of association, begins a line of theory that actually did not have much scientific development in Ramsey’s time, that is, the search for ‘far more complicated laws and mechanisms’ than the ‘simple’ processes of associations. The laws of association had been the dominant approach to cognitive and other psychological mechanisms in philosophy and psychology at least since Hume’s time. His Treatise is not the first, but it is certainly the most significant philosophical treatise to claim priority of place for association above all other mechanisms of the mind. I just recall for you that Hume thought of the mechanism of association doing for the laws of human nature what gravitation did for the laws of nature. In his essay “Truth and Probability”, and elsewhere, Ramsey mentions approvingly, Hume, Mill and Peirce, but he does not discuss, except in the passage quoted above, as far as I know, the theories of association adopted by these three

R AMSEY ’S P SYCHOLOGICAL THEORY OF B ELIEF

51

philosophers as fundamental foundational theories for psychology. At least, I do not know of other passages where he systematically examines their ideas in this respect. From several angles, this is surprising. Book II of Hume’s Treatise, which deals with the passions, or, as we might say now, the emotions, is, in many respects, the scientifically deepest and most clever part of the Treatise. As far as I know, Ramsey does not deal anywhere with the main themes of this book, especially such matters as the subtle use of the concept of association to explain the nature of pride in the first part of Book II. Nor does he examine Mill’s systematic account of the laws of association as a part of his conception of a system of logic. For readers who have forgotten, I remind them that Book VI of Mill’s System of Logic 26 is on the logic of the moral sciences. Chapter IV of this book is on the laws of the mind. Here, Mill certainly gives pride of place to the laws of association. He commends the more extensive work of Alexander Bain and, at the same time, laments the extent to which in philosophy the application of the laws of association in psychology have been neglected. (Mill, 1843/1936, p. 561.) Mill’s chapter IV of this book on the laws of the mind is clearly and systematically written. Without agreeing with everything that he has to say there, I think he presents a very substantial challenge to anyone who wants to deny the central place of the laws of association. Given how well Ramsey knew the writings, in many ways, of Hume and Mill, as well as those of Peirce, it is surprising that there seems to be no extensive confrontation in which he develops systematically and carefully his opposition to the concept of association and the laws associated with this concept as a fundamental basis of psychological theory. Casually, he was skeptical about the laws of association, as expressed in the last 1924 passage above, but that is about as far as it goes. He simply did not enter into any systematic attempt to go deeper into an explanation of the psychological mechanisms that generate partial beliefs. What I said about Ramsey is also true of those two other important and significant forefathers of the twentieth century, Bruno de Finetti and Jimmie Savage. In fact, if anything, they have less to say about psychological mechanisms for generating partial beliefs than Ramsey does. Casually, one might contrast the apparent depth of the work in the foundations of mathematics by Frege, Hilbert, Brower and others to the less developed foundational literature about beliefs and desires. But, in some ways, this would be a mistake in the wrong direction. If we ask for a psychological account of mathematical thinking, for example, an analysis of the mechanisms used in verifying the correctness of a mathematical proof, we will find the literature just as thin as in the case on which I have been dwelling. This is not an idle request. Essentially no deep mathematical theorems of the sort considered fundamental in current research are verified formally. So, what are the psychological mechanisms of cognition and perception that are actually used in checking for correctness? So, let me end on a note which is controversial. It is certainly not agreed to by most of those who have written the modern canon of logic. The recognition

52

P ATRICK S UPPES

that there is something directly psychological about subjective probability is rather widespread and is certainly given a place in Ramsey’s ideas. Indeed, he regards his method of measuring subjective probability and partial beliefs as being, in one sense, a contribution to psychology. Frege, in contrast, to take the beginnings of the modern canon, desires to rid himself of any taint of psychology in logic. I am far from accepting Frege’s view, but there are many aspects that separate the degree of need for psychological mechanisms to account for partial beliefs and psychological mechanisms to account for our claims about the objective correctness of mathematical proofs. This is not the place to enter into a detailed analysis of this difference, – this I have begun in a closely related article27 –, but there is one obvious reason for thinking that the demand for explanation of the psychological mechanisms of partial belief has a more immediate and natural place than a similar demand in the mathematical case. This is that in the mathematical case, all the emphasis is on moving outside individuals to a completely agreed upon objective result – objective in the sense of being the same for all persons. It is exactly the opposite, in some sense, in the case of partial belief. We accept, in the beginning, individual differences in partial beliefs and in the particular assignments of subjective probabilities. It is then natural, in a way it is not, in mathematics, to ask for the psychological mechanisms that account for the differences in partial beliefs. I do not think what I have said is fully satisfactory, but it is, at least, perhaps, a kind of justification of my emphasis on the need for a theory of the psychological mechanisms generating partial beliefs, without, at the same time, asking for a psychological theory justifying the wide agreement about the truth of many mathematical statements and the correctness of many mathematical proofs. But where there is not agreement, as, for example, between formalists and intuitionists, psychological theories of cognitive and perceptual mechanisms become relevant to mathematics, or, at least, so I would claim. Ramsey, in his important article on the foundations of mathematics (1931), did not venture into this territory. N OTES 1. 2. 3. 4. 5. 6. 7.

Frank Plumpton Ramsey, “Truth and Probability”, in: Richard Braithwaite, (Ed.), The Foundations of Mathematics. London: Kegan Paul, Trench, Trubner & Co., 1926/1931. John Maynard Keynes, A Treatise on Probability. London: Macmillan 1921. Norbert Wiener, “A New Theory of Measurement: A Study in the Logic of Mathematics”, in: Proceedings of the London Mathematical Society, 19, 1921, pp. 181-205. Otto Hölder, “Die Axiome der Quantität und die Lehre vom Mass”, in: Ber. Verh. Kgl. Sächsis. Ges. Wiss. Leipzig, Math.-Phys. Classe, 53, 1901, pp. 1-64. Norman Campbell, Physics The Elements. Cambridge: Cambridge University Press 1920. Alfred N. Whitehead/Bertrand Russell, Principia Mathematica. Cambridge: Cambridge University Press 1913. Gustav Theodor Fechner, Elemente der Psychophysik. Leipzig: Druck und Verlag von Breitkopfs Härtel [Elements of Psychophysics (Vol. 1). New York: Holt, Rinehart & Winston 1966] (1860/1966).

R AMSEY ’S P SYCHOLOGICAL THEORY OF B ELIEF

8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27.

53

Patrick Suppes/David H. Krantz/R. Duncan Luce/Amos Tversky, Foundations of Measurement, Vol. II. San Diego, CA: Academic Press 1989, p. 303. Bertrand Russell, Analysis of Mind. London: G. Allen & Unwin 1921. James Clerk Maxwell, A Treatise on Electricity and Magnetism, Vol. II. Third edition. London: Geoffrey Cumberlege. Oxford University Press 1872/1892. David Hume, A Treatise of Human Nature. London: John Noon 1739. Quotations from L.A. Selby-Bigge's edition, London: Oxford University Press 1888. Zoltan Domotor/John Herbert Stelzer, “Representation of Finitely Additive Semiordered Quantitative Probability Structures”, in: Journal of Mathematical Psychology, 8, 1971, pp. 145158. Donald Davidson/Patrick Suppes, “A Finitistic Axiomatization of Subjective Probability and Utility”, in: Econometrica, 24, 1956, pp. 264-275. Donald Davidson/Patrick Suppes/Sidney Siegel, Decision Making: An Experimental Approach. Stanford, CA: Stanford University Press 1957. Bruno de Finetti, “La Prevision: Ses Lois Logiques, Ses Sources Subjectives”, in: Ann. Inst. H. Poincare, 7, 1937, pp. 1-68. Translated into English in H. E. Kyburg, Jr./H. E. Smokler, (Eds.), Studies in Subjective Probability. New York: Wiley 1964. L. Jimmie Savage, The Foundations of Statistics. New York: Wiley 1954. David H. Krantz/R. Duncan Luce/Patrick Suppes/Amos Tversky, Foundations of Measurement, Vol. I. New York: Academic Press 1971. Ibid. de Finetti, “La Prevision: Ses Lois Logiques, Ses Sources Subjectives”, op. cit. Charles Hall Kraft/J .W. Pratt/A. Seidenberg, “Intuitive Probability on Finite Sets”, in: The Annals of Mathematical Statistics, 30, 1959, pp. 408-419. Patrick Suppes/Mario Zanotti, “Necessary and Sufficient Conditions for Existence of a Unique Measure Strictly Agreeing with a Qualitative Probability Ordering”, in: Journal of Philosophical Logic, 5, 1976, pp. 431-438. Bruno de Finetti, Theory of Probability, Vol. I. New York: Wiley. 1974, p. 310. Translated by A. Machi and A. Smith. Hume, A Treatise of Human Nature, op.cit. Frank Plumpton Ramsey, “An Imaginary Conversation with John Stuart Mill”, in: Maria Carla Galavotti, (Ed.), Frank Plumpton Ramsey: Notes on Philosophy, Probability and Mathematics. Naples, Italy: Bibliopolis 1924/1991. Ramsey, “Probability and Partial Belief”, in: The Foundations of Mathematics, op..cit., pp. 256257. John Stuart Mill, System of Logic, London: Longmans, Green and Co. 1843/1936. Patrick Suppes, “Where Do Bayesian Priors Come From?” (In Press)

Center for the Study of Language and Information Stanford University 220 Panama Street Stanford, CA 94305 - 4115 U.S.A. [email protected]

B RIAN S KYRMS

DISCOVERING “WEIGHT, OR THE VALUE OF KNOWLEDGE”

I. I NTRODUCTION In the summer of 1986, I went to Cambridge to look through Ramsey’s unpublished papers. It turned out that the Ramsey archives existed only on microfilm at Cambridge – the originals having been purchased at auction by Nicholas Rescher for the Archives of Logical Positivism at the University of Pittsburgh. Hugh Mellor kindly arranged for me to be allowed to study the microfilm. I still have my temporary library card as a memento. Later, I was able to see the originals at the University of Pittsburgh. I was looking for something, but found something different. What I found of most importance consisted of two manuscript pages, the first of which was entitled “Weight, or the Value of Knowledge.” They were not consecutive in the numbering that had been given to these unpublished manuscripts, but they clearly went together. A few years later, Nils Eric Sahlin published a transcription of these in the British Journal for Philosophy of Science [Ramsey (1990)]. There is also a transcription contained in Maria Carla Galavotti’s collection of Ramsey’s papers [Ramsey (1991)]. I included a facsimile of the second manuscript page in my book, The Dynamics of Rational Deliberation. Today I will tell you something about what I was looking for, what I found, and the relationship between the two. This is an old story, but I will add a few new twists. II. W HAT I WAS L OOKING F OR I was looking to see whether Ramsey had any discussion of coherence of beliefs across time – diachronic coherence –, or of the related question of coherent rules for updating belief. These questions were the focus of intense philosophical discussion at the time (and to some extent still are.) Ramsey says just enough in “Truth and Probability” to whet the imagination, and to hold out the promise of something more. Ramsey introduced the question of coherence to the discussion of degrees of belief, noting that violation of the laws of probability allows a Dutch book:

55 M. C. Galavotti (ed.), Cambridge and Vienna: Frank P. Ramsey and the Vienna Circle, 55–65. © 2006 Springer. Printed in the Netherlands.

56

B RIAN S KYRMS

If anyone’s mental condition violated these laws, his choice would depend on the precise form in which the options were offered to him, which would be absurd. He could have a book made against him by a cunning bettor and would then stand to lose in any event.

The converse, that coherent degrees of belief preclude a Dutch book, is stated two paragraphs later: Having any definite degree of belief implies a certain measure of consistency, namely willingness to bet on a given proposition at the same odds for any stake, the stakes being measured in terms of ultimate values. Having degrees of belief obeying the laws of probability implies a further measure of consistency, namely such a consistency between the odds acceptable on different propositions as shall prevent a book being made against you.

Ramsey was writing a paper for a philosophy club, and did not provide the mathematical details, but they were soon independently supplied by de Finetti. Regarding coherent change in degrees of belief, Ramsey has only this to say: Obviously, if p is the fact observed, my degree of belief in q after the observation should be equal to my degree of belief in p given q before, or by the multiplication law to the quotient of my degree of belief in p&q by my degree of belief in p. When my degrees of belief change in this way we can say that they have been changed consistently by my observation.

He is saying that belief change by conditioning on “the fact observed” is coherent belief change. Again, proofs were only later supplied by others. [See Freedman and Purves (1969), Putnam (1975), Teller (1973), and the review in Lane and Sudderth (1984).] But what about the case of uncertain evidence, where there is no clear-cut “fact observed” within the domain of an individual’s probability function? [Jeffrey (1968), Armendt (1980), Diaconis and Zabell (1982)] Did Ramsey even consider the possibility? And what exactly is the coherence claim in terms of which conditioning on the fact observed constitutes coherent belief change in the cases where it does apply? III. W HAT I F OUND In those two manuscript pages I found an account of the Value of Knowledge – that is to say, the expected utility of pure, cost-free information. The word “Weight” in the heading of the first page is a clear reference to Keynes’ Treatise on Probability. Keynes thought that degrees-of-belief needed two dimensions: probability, which showed how likely an event was judged to be, and weight, which measured quantity of evidential support behind the probability judgement. The following passages indicate both the general nature of Keynes’ concerns and the degree to which he has misconceived the problem:

DISCOVERING “W EIGHT , OR THE V ALUE OF K NOWLEDGE ”

57

The second difficulty... is the neglect of the ‘weights’ in the conception of ‘mathematical expectation.’ ... if two probabilities are equal in degree, ought we, in choosing our course of action, to prefer that one that is based on the greater body of knowledge? The question appears to me to be highly perplexing, and it is difficult to say much that is useful about it. ... Bernoulli’s maxim that in reckoning a probability we must take into account all the information which we have, even when reinforced by Locke’s maxim that we must get all the information that we can, does not seem completely to meet the case. [Keynes (1921) p. 313]

Ramsey clearly saw the answer. Thirty years after Keynes, Bernoulli’s maxim was restated as Carnap’s “total evidence condition” [Carnap (1950)], and the correct analysis was presented to the philosophical community by I. J. Good. (1967). Good cites the treatment of the expected value of new information in Raiffa and Schlaifer (1961). The basic case of pure, cost-free information is already treated by Savage (1954) in Ch. 7 and appendix 2, but Ramsey’s unpublished note anticipates Savage by three decades. The basic principles at work are easy to illustrate in the case of two acts. Suppose that you are going to either buy, or not buy an item on E-bay. You are inclined to buy it. But you have the option of postponing your decision for a minute and reading others’ reports of past transactions with this individual. This information costs nothing more than a mouse click. This could bring negative information about the reliability of the seller that would cause you to forego the purchase, or it might bring information that would confirm your predisposition to buy. To simplify the exposition, we suppose that there are only two possible pieces of information that may come up if you look, one positive and one negative. Then the expected value of buying and of not buying, can be plotted as a function of the probability that the information is positive – as shown in figure 1.

Figure 1

58

B RIAN S KYRMS

If you look and get positive information you move to the right side; if you look and get negative information you move to the left side; before looking you are somewhere in the middle. The lines plotting the expected values of the acts are straight lines because they are averages. Which act is optimal depends on the probabilities of good and bad information. If we take the optimal act at every value and plot it we get the expected value of the Bayes (optimal) act as a function of the probability of good information, as shown in bold in figure 2. You can see that it’s shape is that of a convex function ( It dishes down in the middle). At any point, your expected value of choosing the act that looks best to you in that informational state is just the value of this function.

Figure 2

What is the expected value now, of clicking the mouse, getting the additional information and then deciding? It is a point falling on the dashed line connecting the value of buying with good information on the right, and not buying with bad information on the left in figure 3. It is obviously greater than the value of acting now without the additional information, because the dashed line falls above the bold line except at the endpoints. That is because the dashed line, being an average, is straight, while the bold line – as noted previously – is convex. One can even measure the expected value of information at any point, as the difference between the two curves. The principles illustrated in this simple example hold in more complex cases. There might, for instance, be an infinite number of possible acts, resulting from the setting of some continuously variable control parameter. That would make no real difference in the argument. The expected utility of the Bayes act would still be convex, and the argument would still work. This is the case addressed by

DISCOVERING “W EIGHT , OR THE V ALUE OF K NOWLEDGE ”

59

Ramsey in his two pages of notes. The second page contains a diagram that you could immediately recognize by its resemblance to figure 3.

Figure 3

Ramsey goes a bit further. He supposes that what is to be learned does not take us all the way to the left or right side of the diagram but only part way in each direction. This might be thought of as a model of less than ideal evidence, but the evidence learned is still modeled as evidence that is learned with certainty. The effect of learning the imperfect evidence is assumed to be a shift to a new probability by conditioning on what was learned. [See my (1990) book for a fuller discussion.] IV. D IACHRONIC C OHERENCE We can, by now, say quite a bit about the topic I was looking for but didn’t find – the question of diachronic coherence of degrees of belief. I will take an indirect, but scenic route to the central result. There is a close connection between Bayesian coherence arguments and the theory of arbitrage. [See Shin (1992)] Suppose we have a market in which a finite number of assets are bought and sold. Assets can be anything – stocks and bonds, pigs and chickens, apples and oranges. The market determines a unit price for each asset, and this information is encoded in a price vector x =. You may trade these assets today in any (finite) quantity. You are allowed to take a short position in an asset, that is to say that you sell it today for delivery tomorrow. Tomorrow, the assets may have different prices, y1, ...,ym. To keep things simple, we initially suppose that there are a finite number of possibilities for tomorrow’s price vector. A portfolio,

60

B RIAN S KYRMS

p, is a vector of real numbers that specifies the amount of each asset you hold. Negative numbers correspond to short positions. You would like to arbitrage the market, that is to construct a portfolio today whose cost is negative (you can take out money) and such that tomorrow its value is non-negative (you are left with no net loss), no matter which of the possible price vectors is realized. The fundamental theorem of asset pricing states that you can arbitrage the market if and only if the price vector today falls outside the convex cone spanned by the possible price vectors tomorrow. [If we were to allow an infinite number of states tomorrow we would have to substitute the closed convex cone generated by the possible future price vectors.] The value of a portfolio, p, according to a price vector, y, is the sum over the assets of quantity times price, that is the dot product of the two vectors. If the vectors are orthogonal the value is zero. If they make an acute angle, the value is positive; if they make an obtuse angle, the value is negative. An arbitrage portfolio, p , is one such that p•x is negative and p•yi is non-negative for each possible yi ; p makes an obtuse angle with today’s price vector and is orthogonal or makes an acute angle with each of the possible price vectors tomorrow. If p is outside the convex cone spanned by the yis, then there is a hyperplane which separates p from that cone. An arbitrage portfolio can be found as a vector normal to the hyperplane. It has zero value according to a price vector on the hyperplane, negative value according to today’s prices and non-negative value according to each possible price tomorrow. On the other hand, if today’s price vector in the convex cone is spanned by tomorrow’s possible price vectors, then (by Farkas’ lemma) no arbitrage portfolio is possible. The matter is easy to understand visually in simple cases. Suppose the market deals in only two goods, apples and oranges. One possible price vector tomorrow is $1 for an apple, $1 for an orange. Another is an apple will cost $2, while an orange is $1. These two possibilities generate a convex cone, as shown in figure 4. (We could add lots of intermediate possibilities, but that wouldn’t make any difference to what follows.)

Figure 4

DISCOVERING “W EIGHT , OR THE V ALUE OF K NOWLEDGE ”

61

Let’s suppose that today’s price vector lies outside the convex cone, say apples at $1, oranges at $3. Then it can be separated from the cone by a hyperplane (in 2 dimensions, a line), for example the line oranges = 2 apples, as shown in figure 5.

Figure 5

Normal to that hyperplane we find the vector , as in figure 6.

Figure 6

This should be an arbitrage portfolio, so we sell one orange short and use the proceeds to buy 2 apples. But at today’s prices, an orange is worth $3, so we can pocket a dollar, or – if you prefer – buy 3 apples and eat one. Tomorrow we have to deliver an orange. If tomorrow’s prices were to fall exactly on the hyperplane, we would be covered. We could sell our two apples and use the proceeds to buy the orange. But in our example, things are even better. The worst that can happen tomorrow is that apples and oranges trade 1-to-1, so we might as well eat another apple and use the remaining one to cover our obligation for an orange.

62

B RIAN S KYRMS

In the foregoing, assets could be anything. As a special case they could be tickets paying $1 if p, nothing otherwise, for various contingent propositions, p. The price of such a ticket can be thought of as the market’s collective degree-orbelief or subjective probability for p. We have not said anything about the market except that it will trade arbitrary quantities at the market price. The market could be implemented by a single individual – the bookie of the familiar Bayesian metaphor. Without yet any commitment to the mathematical structure of degrees of belief, or to the nature of belief revision, we can say that arbitrage-free degrees of belief today must fall within the convex cone of degrees of belief tomorrow. This is the fundamental diachronic coherence requirement. We might go further and suppose that tomorrow we learn the truth. In that case a ticket worth $1 if p; nothing otherwise, would be worth either $1 or $0 depending on whether we learn whether p is true or not. By itself this does not tell us a great deal, only that arbitrage-free prior degrees of belief must be nonnegative. Now suppose that we have three assets being traded which have a Boolean logical structure. There are tickets worth $1 if p; nothing otherwise, $1 if q; nothing otherwise, and $1 if p or q; nothing otherwise. Furthermore, p and q are incompatible. This additional structure constrains the possible price vectors tomorrow, so that the convex cone becomes the two dimensional object: z = x + y, x, y, non-negative, as shown in figure 7. Arbitrage-free degrees of belief must be additive. Additivity of subjective probability comes from the additivity of truth value and the fact that additivity is preserved under convex combination. One can then complete the coherence argument for probability by noting that coherence requires a ticket that pays $1 if a tautology is true to have the value $1.

Figure 7

DISCOVERING “W EIGHT , OR THE V ALUE OF K NOWLEDGE”

63

Notice that from this point of view the synchronic Dutch books are really special cases of diachronic arguments. After all, you really do need the time when you find out the truth for the synchronic argument to be complete. (This point has been raised by some as an objection to the application of subjective probability to the confirmation of scientific laws.) If anything, the assumption that there is a time when the truth is revealed is a much stronger assumption than anything that preceded it in this development. (One who rejects this assumption might reject additivity, but still require degrees-of-belief today to fall within the convex cone spanned by degrees of belief tomorrow.) Today the market trades tickets that pay $1 if pi; nothing otherwise, where the pis are some assertions about the world. All sorts of news comes in and tomorrow the price vector may realize a number of different possibilities. (We have not, at this point, imposed any model of belief change.) The price vector for these tickets tomorrow is itself a fact about the world, and there is no reason why we could not have trade in tickets that pay off $1 if tomorrow’s price vector is p, or if tomorrow’s price vector is in some set of possible price vectors, for the original set of propositions. The prices of these tickets represent subjective probabilities about its subjective probabilities tomorrow. Some philosophers have been suspicious about such entities, but they arise quite naturally. And in fact, they may be less problematic than the first-order probabilities over which they are defined. The first-order propositions, pi, could be such that their truth value might or might not ever be settled. But the question of tomorrow’s price vector for unit wagers over them is settled tomorrow. Coherent probabilities of tomorrow’s probabilities should be additive no matter what. Let us restrict ourselves to the case where we eventually do find out the truth about everything [perhaps on Judgment Day], so degrees of belief today and tomorrow are genuine probabilities. We can now consider tickets that are worth $1 if the probability tomorrow of p = a and p; nothing otherwise, as well as tickets that are worth $1 if probability tomorrow of p =1. These tickets are logically related. Projecting to the 2 dimensions that represent these tickets, we find that there are only two possible price vectors tomorrow. Either the probability tomorrow of p is not equal to a, in which case both tickets are worth nothing tomorrow, or probability tomorrow of p is equal to a, in which case the former ticket is has a price of $a and the latter has a price of $1. The cone spanned by these two vectors is just a ray as shown in figure 8. So today, the ratio of these two probabilities (provided they are well-defined) is a. In other words, today the conditional probability of p, given probability tomorrow of p = a, is a. It then follows that to avoid a Dutch book, probability today must be the expectation of probability tomorrow. [See Goldstein (1983) and van Fraassen (1984)]

64

B RIAN S KYRMS

Figure 8

V. D IACHRONIC C OHERENCE AND THE V ALUE OF K NOWLEDGE The traditional setting for the Value of Knowledge theorem is one in which it is assumed that experience delivers up an evidence statement, one of a set of possible evidence statements which partition the space of possibilities. The decision maker then adopts new degree of belief equal to the old degrees of belief conditional on the evidence. If a decision maker believes that the impending learning situation answers to this description, then his probability today will be his expectation of his probability tomorrow. But probability today can well be the expectation of probability tomorrow in a far less structured learning situation. Following Dick Jeffrey [1968], we should pay attention to the fact that evidence may not arrive with certainty. The decision maker may not have in her grasp the observation sentences required to implement that classical model – or even if she has them, she may lack the probabilities conditional on them to implement probability change by conditioning on the evidence. In these cases too, diachronic coherence still has a bite. Probability today must still be the current expectation of probability tomorrow. This happens to be all that is required to prove the theorem that pure, costfree information has non-negative expected value. This was first shown by Paul Graves (1989) in the context of a discussion of Jeffrey’s probability kinematics. The following proof highlights the essential features of our example, and generalizes smoothly to more complicated cases. Let B(p) be the expected utility of the Bayes act according to probability p We write E for current expectation. Because of the convexity of B, (by Jensen’s inequality): E[B(probability after learning)] >= B[E(probability after learning)] By diachronic coherence we can replace E(probability after learning) with (probability before learning). So, E[B(probability after learning)] >= B[probability before learning]

DISCOVERING “W EIGHT , OR THE V ALUE OF K NOWLEDGE”

65

In other words, ex ante an informed decision is surely at least as good, and perhaps better than, an uninformed one. Diachronic coherence implies the value of knowledge. R EFERENCES Armendt, B. (1980) “Is There a Dutch Book Theorem for Probability Kinematics?” Philosophy of Science 47: 583-588. Carnap, R. (1950) Logical Foundations of Probability Chicago: University of Chicago Press. de Finetti, B. (1970) Teoria Della Probabilità v. I. Giulio Einaudi editori: Torino, tr. as Theory of Probability by Antonio Machi and Adrian Smith (1974) Wiley: New York. Diaconis, P. and Zabell, S. (1982) “Updating Subjective Probability” Journal of the American Statistical Association 77:822-830. Freedman, D.A. and R.A. Purves (1969) “Bayes’ Method for Bookies” Annals of Mathematical Statistics 40: 1177-1186. Graves, P. (1989) “The Total Evidence Principle for Probability Kinematics” Philosophy of Science 56, 317-324. Goldstein, M. (1983) “The Prevision of a Prevision” Journal of the American Statistical Association 78: 817-819. Good, I.J. (1967) “On the Principle of Total Evidence” British Journal for the Philosophy of Scence 17, 319-321. Jeffrey, R. (1968) “Probable Knowledge” In The Problem of Inductive Logic ed. I. Lakatos. Amsterdam: North Holland. Keynes, J.M. (1921) A Treatise on Probability. Harper Torchbook edition (1962). New York: Harper and Row. Lane, D.A. and Sudderth, W. (1984) “Coherent Predictive Inference” Sankhya, ser. A, 46: 166-185. Levi, I. (2002) “Money Pumps and Diachronic Dutch Books” Philosophy of Science 69 [PSA 2000 ed. J.A. Barrett and J.M. Alexander] S235-S264. Putnam, H. (1975) “Probability Theory and Confirmation” in Mathematics, Matter and Method. Cambridge: Cambridge University Press. Raiffa, H. and R. Schlaifer (1961) Applied Statistical Decision Theory. Boston: Harvard School of Business Administration. Ramsey, F.P. (1990) “Weight or the Value of Knowledge” Transcribed by N.-E. Sahlin. The British Journal for the Philosophy of Science, 41, (1990), 1-3. Ramsey, F.P. (1991) Notes on Philosophy, Probability and Mathematics. Edited by Maria Carla Galavotti. Bibliopolis: Napoli. Shin, H.S. (1992) “Review of The Dynamics of Rational Deliberation” Economics and Philosophy 8: 176-183. Skyrms, B. (1990) The Dynamics of Rational Deliberation Cambridge, Mass.: Harvard University Press. Teller, P. (1973) “Conditionalization and Observation” Synthese 26, 218-258. van Fraassen, B. (1984) “Belief and the Will” Journal of Philosophy 81: 235-256.

Logic and Philosophy of Science University of California, Irvine 3151 Social Science Plaza Irvine, California U.S.A. [email protected]

S TATHIS P SILLOS

RAMSEY’S RAMSEY-SENTENCES ∗

1. C ONTEXT AND A IMS Frank Ramsey’s posthumously published Theories has become one of the classics of the 20th century philosophy of science. The paper was written in 1929 and was first published in 1931, in a collection of Ramsey’s papers edited by Richard Braithwaite. Theories was mostly ignored until the 1950s, though Ramsey’s reputation was growing fast, especially in relation to his work on the foundations of mathematics. Braithwaite made some use of it in his Scientific Explanation, which appeared in 1953. It was Carl Hempel’s The Theoretician’s Dilemma, published in 1958, which paid Ramsey’s paper its philosophical dues. Hempel coined the now famous expression ‘Ramsey-sentence’. When Rudolf Carnap read a draft of Hempel’s piece in 1956, he realised that he had recently re-invented Ramsey-sentences. Indeed, Carnap had developed an “existentialised” form of scientific theory. In the protocol of a conference in Los Angeles, organised by Herbert Feigl in 1955, Carnap is reported to have extended the results obtained by William Craig to “type theory, (involving introducing theoretical terms as auxiliary constants standing for existentially generalised functional variables in ‘long’ sentences containing only observational terms as true constants)” (Feigl Archive, 04-172-02, 14). I have told this philosophical story in some detail elsewhere (1999, chapter 3). I have also discussed Carnap’s use of Ramsey-sentences and its problems (see my 1999 chapter 3; 2000a; 2000b). In the present paper I want to do two things. First, I want to discuss Ramsey’s own views of Ramsey-sentences. This, it seems to me, is an important issue not just because of its historical interest. It has a deep philosophical significance. Addressing it will enable us to see what Ramsey’s lasting contribution to the philosophy of science was as well as its relevance to today’s problems. Since the 1950s, when the interest in Ramsey’s views mushroomed, there have been a number of different ways to read Ramsey’s views and to reconstruct Ramsey’s project. The second aim of the present paper is to discuss the most significant and controversial of these reconstructions, i.e., structuralism. After some discussion of the problems of structuralism in the philosophy of science, as this was exemplified in Bertrand Russell’s and Grover Maxwell’s views and has reappeared in Elie Zahar’s and John Worrall’s thought, I will argue that, for good 67 M. C. Galavotti (ed.), Cambridge and Vienna: Frank P. Ramsey and the Vienna Circle, 67–90. © 2006 Springer. Printed in the Netherlands.

68

S TATHIS P SILLOS

reasons, Ramsey did not see his Ramsey-sentences as part of some sort of structuralist programme. I will close with an image of scientific theories that Ramsey might have found congenial. I will call it Ramseyan humility.1 2. R AMSEY ’ S T HEORIES Theories is a deep and dense paper. There is very little in it by way of stagesetting. Ramsey’s views are presented in a compact way and are not contrasted with, or compared to, other views. In this section, I will offer a brief presentation of the main argumentative strategy of Theories.2 Ramsey’s starting point is that theories are meant to explain facts, those that can be captured within a “primary system” (1931, p. 212). As an approximation, we can think of it as the set of all singular observational facts and laws. The “secondary system” is the theoretical construction; that part of the theory which is meant to explain the primary system. It is a set of axioms and a “dictionary”, that is “a series of definitions of the functions of the primary system (…) in terms of those of the secondary system” (1931, p. 215). So conceived, theories entail general propositions of the primary system (“laws”), as well as singular statements, (“consequences”), given suitable initial conditions. The “totality” of these laws and consequences is what “our theory asserts to be true” (ibid.). This is a pretty standard hypothetico-deductive account of theories. Ramsey then goes on to raise three philosophical questions. Here are the first two: (1) Can we say anything in the language of this theory that we could not say without it? (1931, p. 219) (2) Can we reproduce the structure of our theory by means of explicit definitions within the primary system? (1931, p. 220)

The answer to the first question is negative (cf. 1931, p. 219). The secondary system can be eliminated in the sense that one could simply choose to stick to the primary system without devising a secondary system in the first place. The answer to the second question is positive (cf. 1931, p. 229). But Ramsey is careful to note that this business of explicit definitions is not very straightforward. They are indeed possible, but only if one does not care about the complexity or arbitrariness of these definitions. The joint message of Ramsey’s answers is that theories need not be seen as having excess content over their primary systems. These answers point to two different ways in which this broadly anti-realist line can be developed. The first points to an eliminative instrumentalist way, pretty much like the one associated with the implementation of theories of Craig’s theorem. Theoretical expressions are eliminated en masse syntactically and hence the problem of their significance does not arise. The second points to a reductive empiricist way, pretty much like the one associated with the early work of the Logical Empiricists – before their semantic liberalisation. Theoretical

R AMSEY ’S RAMSEY -SENTENCES

69

expressions are not eliminated; nor do they become meaningless. Yet, they are admitted at no extra ontological cost. So far, Ramsey has shown that the standard hypothetico-deductive view of theories is consistent with certain anti-realist attitudes towards their theoretical part. He then raises a third question: (3) [Are explicit definitions] necessary for the legitimate use of the theory? (1931, p. 229)

This is a crucial question. If the answer is positive, then some form of anti-realism will be mandatory: the necessary bedfellow of the hypothetico-deductive view. But Ramsey’s answer is negative: “To this the answer seems clear that it cannot be necessary, or a theory would be no use at all” (1931, p. 230). Ramsey offers an important methodological argument against explicit definitions. A theory of meaning based on explicit definitions does not do justice to the fact that theoretical concepts in science are open-ended: they are capable of applying to new situations. In order to accommodate this feature, one should adopt a more flexible theory of meaning, in particular, a theory which is consistent with the fact that a term can be meaningfully applied to new situations without a change of meaning (cf. 1931, p. 230). The important corollary of the third answer is that hypothetico-deductivism is also consistent with the view that theories have excess content over their primary systems. So the possibility of some form of realism is open. It is significant that Ramsey arrived at this conclusion by a methodological argument: the legitimate use of theories makes explicit definitions unnecessary. The next issue then is what this excess content consists of. That is, what is it that one can be a realist about? 3 This, I suggest, is the problem that motivates Ramsey when he writes: The best way to write our theory seems to be this (∃ α, β, γ) : dictionary . axioms (1931, p. 231).

Ramsey introduces this idea with a fourth question: (4) Taking it then that explicit definitions are not necessary, how are we to explain the functioning of the theory without them?

Here is his reply: Clearly in such a theory judgement is involved, and the judgement in question could be given by the laws and consequences, the theory being simply a language in which they are clothed, and which we can use without working out the laws and consequences (1931, p. 231).

Judgements have content: they can be assessed in terms of truth and falsity. Theories express judgements and hence they can be assessed in terms of truth and falsity. Now, note the could in the above quotation. It is not there by accident, I suggest. Ramsey admits that the content of theory could be equated

70

S TATHIS P SILLOS

with the content of its primary system. Since the latter is truth-evaluable, it can express a judgement. But this is not the only way. There is also the way (“the best way”) he suggests: write the theory with existential quantifiers in the front. 3. E XISTENTIAL J UDGEMENTS Ramsey’s observation is simple but critical: the excess content of the theory is seen when the theory is formulated as expressing an existential judgement. In his Causal Qualities, Ramsey wrote: “I think perhaps it is true that the theory of general and existential judgements is the clue to everything” (1931, p. 261). In his Mathematical Logic, he (1931, p. 67ff.) spent quite some time criticising Weyl’s and Hilbert’s views of existential claims. Both of them, though for different reasons, took it that existential claims do not express judgements. Being an intuitionist, Weyl took it that existential claims are meaningless unless we possess a method of constructing one of their instances. Hilbert, on the other hand, took them to be ideal constructions which, involving as they do the notion of an infinite logical sum, are meaningless. Ramsey subjected both views to severe criticism. Its thrust is that existential propositions can, and occasionally do, express all that one does, or might ever be able to, know about a situation. This, Ramsey said, is typical in mathematics, as well as in science and in ordinary life. As he says: “(…) it might be sufficient to know that there is a bull somewhere in a certain field, and there may be no further advantage in knowing that it is this bull and here in the field, instead of merely a bull somewhere” (1931, p. 73). Ramsey criticised Hilbert’s programme in mathematics sharply because he did not agree with the idea that mathematics was symbol-manipulation. He did not deny that Hilbert’s programme was partly true, but stressed that this could not be the “whole truth” about mathematics (1931, p. 68). He was even more critical of an extension of Hilbert’s programme concerning “knowledge in general” (that is, to scientific theories as well) (cf. 1931, p. 71). As we have seen, Ramsey took theories to be meaningful existential constructions (judgements), which could be evaluated in terms of truth and falsity. The extension of Hilbert’s programme to apply to science had been attempted by Moritz Schlick (1918/1925, pp. 33-4). He saw theories as formal deductive systems, where the axioms implicitly define the basic concepts. He thought implicit definitions divorce the theory from reality altogether: theories “float freely”; “none of the concepts that occur in the theory designate anything real (…)” (1918/1925, p. 37). Consequently, he accepted the view that the “construction of a strict deductive science has only the significance of a game with symbols” (ibid.). Schlick was partly wrong, of course. An implicit definition is a kind of indefinite description: it defines a whole class of objects which can realise the formal structure, as defined by a set of axioms. Schlick did not see this very clearly. But he did encounter a problem. Theories express judgements; judgements designate facts (1918/1925, p. 42); a true judgement designates a set of

R AMSEY ’S RAMSEY -SENTENCES

71

facts uniquely (1918/1925, p. 60); but the implicit definitions fail to designate anything uniquely; so a theory, if seen as a network of implicit definitions of concepts, fails to have any factual content. This is an intolerable consequence. Schlick thought that it is avoided at the point of application of the theory to reality. This application was taken to be partly a matter of observations and partly a matter of convention (cf. 1918/1925, p. 71). In his (1932), he came back to this view and called it the geometrisation of physics: by disregarding the meaning of the symbols we can change the concepts into variables, and the result is a system of propositional functions which represent the pure structure of science, leaving out its content, separating it altogether from reality (1932, p. 330).

Seen in this structuralist light, the predicate letters and other constants that feature in the axioms should really be taken to be genuine variables. What matters is not the meaning of these non-logical constants, but rather the deductive – hence structural – relations among them. Scientific theories are then presented as logical structures, logical implication being the generating relation. The hypothetical part comes in when we ask how, if at all, this system relates to the world. Schlick’s answer is that when one presents a theory, one makes a hypothetical claim: if there are entities in the world which satisfy the axioms of the theory, then the theory describes these entities (cf. 1932, p. 330-1). Against the backdrop of Schlick’s approach, we can now see Ramsey’s insight clearly. We need not divorce the theory from its content, nor restrict it to whatever can be said within the primary system, provided that we treat a theory as an existential judgement. Like Schlick, Ramsey does treat the propositional functions of the secondary system as variables. But, in opposition to Schlick, he thinks that advocating an empirical theory carries with it a claim of realisation (and not just an if-then claim): there are entities which satisfy the theory. This is captured by the existential quantifiers with which the theory is prefixed. They turn the axiom-system from a set of open formulas into a set of sentences. Being a set of sentences, the resulting construction is truth-valuable. It carries the commitment that not all statements such as ‘α, β, γ stand to the elements of the primary system in the relations specified by the dictionary and the axioms’ are false. But of course, this ineliminable general commitment does not imply any specific commitment to the values of α, β, γ. (This last point is not entirely accurate, I think. But because it’s crucial to see in what sense it is inaccurate, I shall discuss it in some detail in section 7). 4. R AMSEY - SENTENCES As the issue is currently seen, in order to get the Ramsey-sentence RTC of a (finitely axiomatisable) theory TC, we conjoin the axioms of TC in a single sentence, replace all theoretical predicates with distinct variables ui, and then

72

S TATHIS P SILLOS

bind these variables by placing an equal number of existential quantifiers ∃ui in front of the resulting formula. Suppose that the theory TC is represented as TC (t1,...,tn; o1,...,om), where TC is a purely logical m+n-predicate. The Ramseysentence RTC of TC is: ∃u1∃u2...∃unTC (u1,...,un; o1,...,om). For simplicity let us say that the T-terms of TC form an n-tuple t=, and the O-terms of TC form an m-tuple o=. Then, RTC takes the more convenient form: ∃uTC(u,o). I will follow customary usage and call Ramsey’s existential-judgements Ramsey-sentences. This is, I think, partly misleading. I don’t think Ramsey thought of these existential judgements as replacements of existing theories or as capturing their proper content (as if there were an improper content, which was dispensable). Be that as it may, Ramsey-sentences have a number of important properties. Here they are: RTC is a logical consequence of TC. RTC mirrors the deductive structure of TC. RTC has exactly the same first-order observational consequences as TC. So RTC is empirically adequate iff TC is empirically adequate. TC1 and TC2 have incompatible observational consequences iff RTC1 and RTC2 are incompatible (Rozeboom 1960, p. 371). TC1 and TC2 may make incompatible theoretical assertions and yet RTC1 and RTC2 be compatible (cf. English 1973, p. 458). If RTC1 and RTC2 are compatible with the same observational truths, then they are compatible with each other (cf. English 1973, p. 460; Demopoulos 2003a, p. 380). Let me sum up Ramsey’s insights. First, a theory need not be seen as a summary of what can be said in the primary system. Second, theories, qua hypothetico-deductive structures, have excess content over their primary systems, and this excess content is seen when the theory is formulated as expressing an existential judgement. Third, a theory need not use names in order to refer to anything (in the secondary system). Existentially bound variables can do this job perfectly well.4, 5 Fourth, a theory need not be a definite description to be a) truth-valuable, b) ontically committing, and c) useful. So uniqueness of realisation (or satisfaction) is not necessary for the above. Fifth, if we take a theory as a dynamic entity (something that can be improved upon, refined, modified, changed, enlarged), we are better off if we see it as a growing existential sentence. This last point is particularly important for two reasons. The first is this. A typical case of scientific reasoning occurs when two theories TC1 and TC2 are conjoined (TC1 & TC2=TC) in order to account for some phenomena. But if we take their Ramsey-sentences, then ∃uTC1(u, o) and ∃uTC2(u, o), they do not entail ∃uTC(u, o). Ramsey was aware of this problem. He solved it by taking scientific theories to be growing existential sentences. That is to say, the theory is already in the form ∃uTC(u, o) and all further addi-

R AMSEY ’S RAMSEY -SENTENCES

73

tions to it are done within the scope of the original quantifiers. To illustrate the point, Ramsey uses the metaphor of a fairy tale. Theories tell stories about the form: “Once upon a time there were entities such that ...”. When these stories are modified, or when new assertions are added, they concern the original entities, and hence they take place within the scope of the original “once upon a time”. The second reason is this: Ramsey never said that the distinction between primary and secondary system was static and fixed. So there is nothing to prevent us from replacing an existentially bound variable by a name or by a constant (thereby moving it into the primary system), if we come to believe that we know what its value is. His Causal Qualities is, in a sense, a sequel to his Theories. There, Ramsey characterises the secondary system as “fictitious” and gives the impression that its interest lies in its being a mere systematiser of the content of the primary system. But he ends the paper by saying this: Of course, causal, fictitious, or ‘occult’ qualities may cease to be so as science progresses. E.g., heat, the fictitious cause of certain phenomena (…) is discovered to consist of the motion of small particles. So perhaps with bacteria and Mendelian characters or genes. This means, of course, that in later theory these parametric functions are replaced by functions of the given system (1931, p. 262).

In effect, Ramsey says that there is no principled distinction between fictitious and non-fictitious qualities. If we view the theory as a growing existential sentence, then this point can be accommodated in the following way. As our knowledge of the world grows, the propositional functions that expressed ‘fictitious’ qualities (and were replaced by existentially bound variables) might well be taken to characterise known quantities and hence be re-introduced in the growing theory as names (or constants).6 Viewing theories as existential judgements solves another problem that Ramsey faced. He did not see causal laws (what he called “variable hypotheticals”) as proper propositions. As he famously stated: “Variable hypotheticals are not judgements but rules for judging ‘If I meet a φ, I shall regard it as a ψ’” (1931, p. 241). Yet, he also took the secondary system to comprise variable hypotheticals (cf. 1931, p. 260). Taking the theory as an existential judgement allows Ramsey to show how the theory as a whole can express a judgement, though the variable hypotheticals it consists of, if taken in isolation from the theory, do not express judgements. The corollary of this is a certain wholism of meaning. The existential quantifiers render the hypothetico-deductive structure truth-valuable, but the consequence is that no ‘proposition’ of this structure has meaning apart from the structure.7

74

S TATHIS P SILLOS

5. R USSELL ’ S S TRUCTURALISM In this section, I want to examine the link, if any, between Russell’s structuralism and Ramsey’s existential view of theories. This issue has been discussed quite extensively and has given rise to the view called ‘structural realism’. In The Analysis of Matter, Russell aimed to reconcile the abstract character of modern physics, and of the knowledge of the world that this offers, with the fact that all evidence there is for its truth comes from experience. To this end, he advanced a structuralist account of our knowledge of the world. According to this, only the structure, i.e., the totality of formal, logico-mathematical properties, of the external world can be known, while all of its intrinsic properties are inherently unknown. This logico-mathematical structure, he argued, can be legitimately inferred from the structure of the perceived phenomena (the world of percepts) (cf. 1927, pp. 226-7). Indeed, what is striking about Russell’s view is this claim to inferential knowledge of the structure of the world (of the stimuli), since the latter can be shown to be isomorphic to the structure of the percepts. He was quite clear on this: (...) whenever we infer from perceptions, it is only structure that we can validly infer; and structure is what can be expressed by mathematical logic, which includes mathematics (1927, p. 254).

Russell capitalised on the notion of structural similarity he himself had earlier introduced. Two structures M and M' are isomorphic iff there is an 1-1 mapping f (a bijection) of the domain of M onto the domain of M' such that for any relation R in M there is a relation R' in M' such that R(x1…xn) iff R'(fx1…fxn). A structure (“relation-number”) is then characterised by means of its isomorphism class. Two isomorphic structures have identical logical properties (cf. 1927, p. 251). How is Russell’s inference possible? Russell relied on the causal theory of perception: physical objects are the causes of perceptions.8 This gives him the first assumption that he needs, i.e., that there are physical objects which cause perceptions. Russell used two more assumptions. The second is what I (2001) have called the ‘Helmholtz-Weyl’ principle, i.e., that different percepts are caused by different physical stimuli (cf. 1927, p. 226, p. 252, p. 400). Hence, to each type of percept there corresponds a type of stimuli. The third assumption is a principle of “spatio-temporal continuity” (that the cause is spatio-temporally continuous with the effect). From these Russell concluded that we can have “a great deal of knowledge as to the structure of stimuli”. This knowledge is that there is a roughly one-one relation between stimulus and percepts, [which] enables us to infer certain mathematical properties of the stimulus when we know the percept, and conversely enables us to infer the percept when we know these mathematical properties of the stimulus (1927, p. 227).

R AMSEY ’S RAMSEY -SENTENCES

75

The “intrinsic character” of the stimuli (i.e., the nature of the causes) will remain unknown. The structural isomorphism between the world of percepts and the world of stimuli isn’t enough to reveal it. But, for Russell, this is just as well. For as he claims: “(...) nothing in physical science ever depends upon the actual qualities” (1927, p. 227). Still, he insists, we can know something about the structure of the world (cf. 1927, p. 254). Here is an example he uses: Suppose that we hear a series of notes of different pitches. The structure of stimuli that causes us to hear these notes must be such that it also forms a series “in respect to some character which corresponds causally with pitch” (1927, p. 227). The three assumptions that Russell uses are already quite strong but, actually, something more is needed for the inference to go through. The establishment of isomorphism requires also the converse of the Helmholtz-Weyl principle – viz., different stimuli cause different percepts. Hence, to each type of stimuli there corresponds a type of percept. If the converse of the Helmholtz-Weyl principle is not assumed, then the isomorphism between the two structures cannot be inferred, for the required 1-1 correspondence between the domains of the two structures is not shown.9 The notion of structural similarity is purely logical and hence we need not assume any kind of (Kantian) intuitive knowledge of it. So an empiricist can legitimately appeal to it. It is equally obvious that the assumptions necessary to establish the structural similarity between the two structures are not logical but substantive. I am not going to question these assumptions here (see my 2001).10 Let us grant them. An empiricist need not quarrel with them. Hence, since Russell’s inference is legitimate from an empiricist perspective, its intended conclusion, viz., that the unperceived (or unobservable) world has a certain knowable structure, will be acceptable too. With it comes the idea that of the physical objects (the causes, the stimuli) we can only know their formal, logico-mathematical properties. This inference, as Russell says, “determines only certain logical properties of the stimuli” (1927, p. 253). Russell’s structuralism has met with a fatal objection, due to M.H.A. Newman (1928): the structuralist claim is trivial in the sense that it follows logically from the claim that a set can have any structure whatever, consistent with its cardinality. So the actual content of Russell’s thesis, viz., that the structure of the physical world can be known, is exhausted by his first assumption, viz., by positing a set of physical objects with the right cardinality. The supposed extra substantive point, viz. that of this set it is also known that it has structure W, is wholly insubstantial. The set of objects that comprise the physical world cannot possibly fail to possess structure W because, if seen just as a set, it possesses all structures which are consistent with its cardinality. Intuitively, the elements of this set can be arranged in ordered n-tuples so that set exhibits structure W.11 Newman sums this up by saying:

76

S TATHIS P SILLOS

Hence the doctrine that only structure is known involves the doctrine that nothing can be known that is not logically deducible from the mere fact of existence, except (‘theoretically’) the number of constituting objects (1928, p. 144).

Newman’s argument has an obvious corollary: the redundancy of the substantive and powerful assumptions that Russell used in his argument that the structure of the world can be known inferentially. These assumptions give the impression that there is a substantive proof available. But this is not so. 6. M AXWELL ’ S S TRUCTURALISM Russell’s thesis was revamped by Maxwell – with a twist. Maxwell took the Ramsey-sentence to exemplify the proper structuralist commitments. He advanced “structural realism” as a form of representative realism, which suggests that i) scientific theories issue in existential commitments to unobservable entities and ii) all non-observational knowledge of unobservables is structural knowledge, i.e., knowledge not of their first-order (or intrinsic) properties, but rather of their higher-order (or structural) properties (cf. 1970a; 1970b). The key idea here is that a Ramsey-sentence satisfies both conditions (i) and (ii) above. If we know the Ramsey-sentence we know that there are properties that satisfy it (because of the existentially bound quantifiers), but of these properties we know only their “structural properties”. Maxwell’s association of Russell’s structuralism with Ramsey’s views (cf. 1970b, p. 182) is, at least partly, wrong. To see this, recall that Russell’s structuralism attempted to provide some inferential knowledge of the structure of the world: the structure of the world is isomorphic to the structure of the appearances. I think it obvious that Ramsey-sentences cannot offer this. The structure of the world, as depicted in a Ramsey-sentence, is not isomorphic to, nor can it be inferred from, the structure of the phenomena which the Ramsey-sentence accommodates. Now, the distinctive feature of the Ramsey-sentence RTC of a theory TC is that it preserves the logical structure of the original theory. We may say then that when one accepts RTC, one is committed to a) the observable consequences of the original theory TC; b) a certain logico-mathematical structure in which (descriptions of) the observable phenomena are deduced; and c) certain abstract existential claims to the effect that there are (non-empty classes of) entities which satisfy the (non-observational part of the) deductive structure of the theory. In this sense, we might say that the Ramsey-sentence, if true, gives us knowledge of the structure of the world: there is a certain structure which satisfies the Ramsey-sentence and the structure of the world (or of the relevant worldly domain) is isomorphic to this structure.12 I suppose this is what Maxwell really wanted to stress when he brought together Russell and Ramsey. The problem with Maxwell’s move is that it falls prey to a Newman-type objection. The existential claim italicised above follows logically from the fact that the Ramsey-sentence is empirically adequate, subject to certain cardinality

R AMSEY ’S RAMSEY -SENTENCES

77

constraints. In other words, subject to cardinality constraints, if the Ramsey-sentence is empirically adequate, it is true. The proof of this has been given in different versions by several people.13 Its thrust is this: Take RTC to be the Ramsey-sentence of theory TC. Suppose RTC is empirically adequate. Since RTC is consistent, it has a model. Call it M. Take W to be the ‘intended’ model of TC and assume that the cardinality of M is equal to the cardinality of W. Since RTC is empirically adequate, the observational sub-model of M will be identical to the observational sub-model of W. That is, both the theory TC and its Ramseysentence RTC will ‘save the (same) phenomena’. Now, since M and W have the same cardinality, we can construct an 1-1 correspondence f between the domains of M and W and define relations R' in W such that for any theoretical relation R in M, R(x1…xn) iff R'(fx1…fxn). We have induced a structure-preserving mapping of M on to W; hence, M and W are isomorphic and W becomes a model of RTC. Another way to see the problem is to look at Carnap’s assimilation of Ramsey’s sentences (see my 2000a). Carnap noted that a theory TC is logically equivalent to the following conjunction: RTC & (RTCÆTC), where the conditional RTCÆTC says that if there is some class of entities that satisfy the Ramsey-sentence, then the theoretical terms of the theory denote the members of this class. For Carnap, the Ramsey-sentence of the theory captured its factual content, and the conditional RTCÆTC captured its analytic content (it is a meaning postulate). This is so because the conditional RTCÆTC has no factual content: its own Ramsey-sentence, which would express its factual content if it had any, is logically true. As Winnie (1970, p. 294) observed, under the assumption that RTCÆTC – which is known as the Carnap sentence – is a meaning postulate, it follows that RTCÅÆTC, i.e., that the theory is equivalent to its Ramseysentence.14 In practice, this means that the Carnap sentence poses a certain restriction on the class of models that satisfy the theory: it excludes from it all models in which the Carnap-sentence fails. In particular, the models that are excluded are exactly those in which the Ramsey-sentence is true but the theory false. So if the Ramsey-sentence is true, the theory must be true: it cannot fail to be true. Is there a sense in which RTC can be false? Of course, a Ramseysentence may be empirically inadequate. Then it is false. But if it is empirically adequate (if, that is, the structure of observable phenomena is embedded in one of its models), then it is bound to be true. For, as we have seen, given some cardinality constraints, it is guaranteed that there is an interpretation of the variables of RTC in the theory’s intended domain. We can see why this result might not have bothered Carnap. If being empirically adequate is enough for a theory to be true, then there is no extra issue of the truth of the theory to be reckoned with – apart of course of positing an extra domain of entities. Empiricism can thus accommodate the claim that theories are true, without going a lot beyond empirical adequacy.15 Indeed, as I have argued elsewhere (1999 chapter 3, 2000a), Carnap took this a step further. In his own account of Ramsey-sentences, he deflated the issue of the possible existential commitment to physical unobservable entities by taking the existentially bound

78

S TATHIS P SILLOS

Ramsey-variables to extend beyond mathematical entities. Of the Ramseysentences he said: the observable events in the world are such that there are numbers, classes of such etc., which are correlated with the events in a prescribed way and which have among themselves certain relations; and this assertion is clearly a factual statement about the world (1963, p. 963).

Carnap’s thought, surely, was not that the values of the variable are literally numbers, classes of them etc. How possibly can a number be correlated with an observable event? Rather, his thought was that a) the use of Ramsey-sentences does not commit someone to the existence of physical unobservable entities (let alone, to for instance, electrons in particular); and b) the things that matter are the observable consequences of the Ramsey-sentence, its logical form, and its abstract claim of realisation. Let me grant that this equation of truth with empirical adequacy is quite acceptable for an empiricist, though I should say in passing that it reduces much of science to nonsense and trivialises the search for truth.16 But reducing truth to empirical adequacy is a problem for those who want to be realists, even if just about structure. For, it is no longer clear what has been left for someone to be a realist about. Perhaps, the structural realist will insist that the range of the Ramsey-variables comprises unobservable entities and properties. It’s not clear what the reason for this assertion is. What is it, in other words, that excludes all other interpretations of the range of Ramsey-variables? But let’s assume that, somehow, the range of Ramsey-variables is physical unobservable entities. It might be thought that, consistently with structuralism, the excess content of theories is given in the form of non-formal structural properties of the unobservables. Maxwell, for instance, didn’t take all of the so-called structural properties to be purely formal (cf. 1970b, p. 188). In his (1970a, p. 17), he took “temporal succession, simultaneity, and causal connection” to be among the structural properties. But his argument for this is hardly conclusive: “for it is by virtue of them that the unobservables interact with one another and with observables and, thus, that Ramsey sentences have observable consequences”. Hearing this, Ramsey would have raised his eyebrow. In Theories, he had noted: “This causation is, of course, in the second system and must be laid out in the theory” (1931, p. 235).17 The point, of course, is that we are in need of an independent argument for classifying some relations, e.g., causation, as ‘structural’ and hence as knowable. When it comes to causation, in particular, a number of issues need to be dealt with. First, what are its structural properties? Is causation irreflexive? Not if causation is persistence. Is causation asymmetric? Not if there is backward causation. Is causation transitive? Perhaps yes – but even this can be denied (in the case of probabilistic causation, for instance). Second, suppose that the structural properties of causation are irreflexivity, asymmetry and transitivity. If these properties constitute all that can be known of the relevant relation, what is there

R AMSEY ’S RAMSEY -SENTENCES

79

to distinguish causation from another relation with the same formal properties, e.g., a relation of temporal ordering? Third, even if causation were a non-formal structural relation, why should it be the case that only its structural properties could be known? Note a certain irony here. Suppose that since causation is a relation among events in the primary system, one assumes that it is the same relation that holds between unobservable events and between unobservable and observable events. This seems to be Maxwell’s view above. Now, causal knowledge in the primary system (that is causal knowledge concerning observables) is not purely structural. The (intrinsic) properties of events (or objects) by virtue of which they are causally related to one another are knowable. If causation is the very same relation irrespective of whether the relata are observable or unobservable, why should one assume that the (intrinsic) properties of unobservable events (or objects) by virtue of which they are causally related to one another are not knowable? There seems to be no ground for this asymmetry. In both cases, it may be argued, it is by virtue of their (intrinsic) properties that entities are causally related to each other. In both cases, it might be added, causal relations supervene on (or are determined by) the intrinsic properties of observable or unobservable entities.18 Indeed, these last points are fully consistent with Russell’s (and Maxwell’s) views. Recall that according to the causal theory of perception, which Maxwell also endorses, our percepts are causally affected by the external objects (the stimuli, the causes), which must be in virtue of these objects’ intrinsic properties. (Surely, it is not by their formal properties.) The Helmholtz-Weyl principle (that different percepts are caused by different stimuli) implies that the different stimuli must differ in their intrinsic properties. So the latter are causally active and their causal activity is manifested in the different percepts they cause. In what sense then are they unknowable?19 The general point is that Maxwell’s Ramsey-sentence approach to structuralism faces a sticky dilemma. Either there is nothing left to be known except formal properties of the unobservable and observable properties or there are some knowable non-formal properties of the unobservable. In the former case, the Ramsey-sentence leaves it (almost) entirely open what we are talking about. In the latter case, we know a lot more about what the Ramsey-sentence refers to, but we thereby abandon pure structuralism. 7. W ORRALL AND Z AHAR ’ S S TRUCTURALISM The points raised in the last section are particularly relevant to the Zahar-Worrall (2001) view that adopting the Ramsey-sentence of the theory is enough to be a realist about this theory. Indeed, they are aware of the problems raised so far. They do deal with Putnam’s model-theoretic argument against realism and admit that if this argument is cogent, then the Ramsey-sentence of a(n) (epistemically ideal) theory is true. Note that a theory’s being epistemically ideal includes its

80

S TATHIS P SILLOS

being empirically adequate. But it’s not hard to see that Putnam’s argument is a version of the Newman challenge.20 Zahar and Worrall (2001, p. 248) ask: “should the structural realist be worried about these results?” And they answer: “(…) the answer is decidedly negative”. So Zahar and Worrall do accept the equation of the truth of the theory with the truth of its Ramsey-sentence. In fact, they want to capitalise on this in order to claim that truth is achievable. They claim that two seemingly incompatible empirically adequate theories will have compatible Ramsey-sentences and hence they can both be true of the world (cf. 2001, pp. 248-9). We have already seen the price that needs to be paid for truth being achievable this way: truth is a priori ascertainable, given empirical adequacy and cardinality. But it is interesting to note that Zahar and Worrall are not entirely happy with this equation. They carry on to stress that “the more demanding structural realist” can say a bit more. He (they?) can distinguish between different empirically adequate Ramsey-sentences using the usual theoretical virtues (simplicity, unification etc.), opt for the one that better exemplifies these virtues (e.g., it is more unified than the other) and claim that it is this one that should be taken “to reflect – if only approximately – the real structure of W [the world]” (2001, p. 249). But I doubt that this, otherwise sensible, strategy will work in this case. For one, given that the theoretical virtues are meant to capture the explanatory power of a theory, it is not clear in what sense the truth of the Ramsey-sentence explains anything. If its truth is the same as its empirical adequacy, then the former cannot explain the latter. Further, there is something even more puzzling in the Zahar-Worrall claim. If two theories have compatible Ramsey-sentences, and if truth reduces to empirical adequacy, in what sense can the theoretical virtues help us deem one theory true and the other false? Clearly, there could be a straightforward sense, if truth and empirical adequacy were distinct. But this is exactly what the Zahar-Worrall line denies. Could they simply say that there is a sense in which one Ramsey-sentence is truer than the other? They could, but only if the truth of the theory answered to something different from its empirical adequacy. If, for instance, it was claimed that a theory is true if, on top of its being empirically adequate, it captures the natural structure of the world, then it is clear that a) one theory can be empirically adequate and yet false; and b) one of two empirically adequate theories can be truer than the other. 21 Now, there could be another sense in which an appeal to the theoretical virtues could distinguish between the claim that a theory is true and the claim that its Ramsey-sentence is empirically adequate. This is by conceptually equating truth with empirical adequacy plus the theoretical virtues. If this equation went ahead, then someone could claim that a theory could be empirically adequate and false, if the theory lacked in theoretical virtues. Or, someone could claim that among two empirically adequate theories, one was truer than the other if the first had more theoretical virtues than the second; or if the first fared better vis-à-vis the theoretical virtues than the second. But these claims would amount to an endorsement of an epistemic account of truth. In particular, they would amount

R AMSEY ’S RAMSEY -SENTENCES

81

to forging an a priori (conceptual) link between the truth of the theory and its possession of theoretical virtues. I will not criticise this move now. Suffice it to say that such a move has disputed realist credentials. So it is not open to those who want to be realists.22 In his reply to Russell’s structuralism, Newman pointed to a way in which Russell’s claim would not be trivial, viz., if the relation that generated the required structure W was “definite”, that is if we knew (or claimed that we knew) more about what it is than that it exists and has certain formal properties. Couldn’t we distinguish between “important” and “unimportant” relations and stay within structuralism? Not really. The whole point is precisely that the notion of an ‘important relation’ cannot be part of a purely structuralist understanding. Newman saw this point very clearly (see 1928, p. 147). In order to pick as important one among the many relations which generate the same structure on a domain, we have to go beyond structure and talk about what these relations are, and why some of them are more important than others. It’s not hard to see that the very same objection can be raised against a Maxwell- or a Zahar-Worrall structural realism. And it is equally obvious what the remedy could be. Structural realists should have a richer understanding of the relations that structure the world. Suppose there is indeed some definite relation (or a network, thereof) that generates the structure of the world. If this is the case, then the claim that the structure W of the physical world is isomorphic to the structure W' that satisfies an empirically adequate Ramsey-sentence would be far from trivial. It would require, and follow from, a comparison of the structures of two independently given relations, say R and R'. But structural realists as well as Russell deny any independent characterisation of the relation R that generates the structure of the physical world. On the contrary, structural realists and Russell insist that we can get at this relation R only by knowing the structure of another relation R', which is deemed isomorphic to R. We saw that the existence of R (and hence of W) follows logically from some fact about cardinality. It goes without saying that treating these relations as “definite” would amount to an abandonment (or a strong modification) of structuralism.23 The natural suggestion here is that among all those relations-in-extension which generate the same structure, only those which express real relations should be considered. But specifying which relations are real requires knowing something beyond structure, viz., which extensions are ‘natural’, i.e., which subsets of the power set of the domain of discourse correspond to natural properties and relations. Having specified these natural relations, one may abstract away their content and study their structure. But if one begins with the structure, then one is in no position to tell which relations one studies and whether they are natural or not.

82

S TATHIS P SILLOS

8. R AMSEY AND N EWMAN ’ S P ROBLEM As noted above, Ramsey’s crucial observation was that the excess content of the theory is seen when the theory is formulated as expressing an existential judgement. If, on top of that, Ramsey meant to assert something akin to the structural realist position, i.e., that this excess content, so far as it is knowable, is purely structural, then he would have landed squarely on the Newman problem. So should this view be attributed to Ramsey? Before I canvass a negative answer, let me go back to Russell once more. Russell (1927) took theories to be hypothetico-deductive systems and raised the issue of their interpretation. Among the many “different sets of objects [that] are abstractly available as fulfilling the hypotheses”, he distinguished those that offer an “important” interpretation (1927, pp. 4-5), that is an interpretation which connects the theory (as an abstract logico-mathematical system) to the empirical world. This was important, he thought, because all the evidence there is for physics comes from perceptions. He then went on to raise the central question that was meant to occupy the body of his book: when are physical theories true? As he (1927, pp. 8-9) put it, there is a wider sense in which physics is true: Given physics as a deductive system, derived from certain hypotheses as to undefined terms, do there exist particulars, or logical structures composed of particulars, which satisfy these hypotheses?

“If ”, he added, “the answer is in the affirmative, then physics is completely ‘true’”. Actually, he took it that his subsequent structuralist account, based on the causal theory of perception, was meant to answer the above question affirmatively. Now, Russell’s view has an obvious similarity to Ramsey’s: theories as hypothetico-deductive structures should be committed to an existential claim that there is an interpretation of them. But there is an interesting dissimilarity between Russell and Ramsey. Russell thought that some interpretation was important (or more important than others), whereas Ramsey was not committed to this view. Russell might well identify the theory with a definite description: there is a unique (important) interpretation such that the axioms of the theory are true of it. But, as we have seen, one of Ramsey’s insights is that there is no reason to think of theories as definite descriptions – i.e., as requiring uniqueness. It seems likely that it was this Russellian question that inspired Ramsey to formulate his own view of theories as existential judgements. In fact, there is some evidence for it. In a note on Russell’s The Analysis of Matter, Ramsey (1991, p. 251) said: Physics says = is true if (∃ α, β, … R, S): F(α, β, … R, S…) (1).

He immediately added a reference – “Russell p. 8” – to The Analysis of Matter.24

R AMSEY ’S RAMSEY -SENTENCES

83

(1) looks very much like a Ramsey-sentence. But unlike Russell, Ramsey did not adopt a structuralist view of the content of theories. This may be seen by what he goes on to say: the propositional functions α and R should be “nonformal”. And he adds: “Further, F must not be tautological as it is on Eddington’s view”. As it is clear from another note (1991, pp. 246-50), Ramsey refers to Eddington’s The Nature of the Physical World. In his review of this book, Braithwaite criticised Eddington severely for trying to turn physics “from an inductive science to a branch of mathematics” (1929, p. 427). According to Braithwaite, Eddington tried to show how the laws of physics reduce to mathematical identities, which are derivable from very general mathematical assumptions. This must be wrong, Braithwaite thought, for in mathematics “we never know what we are talking about”, whereas in physics “we do know (or assume we know) something of what we are talking about – that the relata have certain properties and relations – without which knowledge we should have no reason for asserting the field laws (even without reference to observed quantities)” (1929, p. 428). The point might not be as clear as it ought to have been, but, in effect, Braithwaite argued against Eddington that natural science would be trivialised if it was taken to aim at achieving only knowledge of structure.25 I don’t know whether Ramsey discussed Eddington’s book with Braithwaite or whether he had read Braithwaite’s review of it (though he had read Eddington’s book – see 1991, pp. 246-50). It is nonetheless plausible to say that he shared Braithwaite’s view when he said of the relation F that generates the structure of a theory that it should not be tautological “as it is on Eddington’s view”. In fact, in the very same note, Ramsey claims that in order to fix some interpretation of the theory we need “some restrictions on the interpretation of the other variables. i.e., all we know about β, S is not that they satisfy (1)”. So I don’t think Ramsey thought that viewing theories as existential judgements entailed that only structure (plus propositions of the primary system) could be known. It’s plausible to argue that Ramsey took Ramsey-sentences (in his sense) to require the existence of definite relations, whose nature might not be fully determined, but which is nonetheless constrained by some theoretical and observational properties. To judge the plausibility of this interpretation, let’s look into some of his other papers. In The Foundations of Mathematics, Ramsey insisted on the distinction between classes and relations-in-extension, on the one hand, and real or actual properties and relations, on the other. The former are identified extensionally, either as classes of objects or as ordered n-tuples of objects. The latter are identified by means of predicates. Ramsey agreed that an extensional understanding of classes and relations is necessary for mathematics. Take, for instance, Cantor’s concept of class-similarity. Two classes are similar (that is, they have the same cardinality) iff there is an one-one correspondence (relation) between their domains. This relation, Ramsey (1931, p. 15) says, is a relation-in-extension: there needn’t be any actual (or real) relation correlating the two classes. The class of male angels may have the same cardinality with the class of female

84

S TATHIS P SILLOS

angels, so that the two classes can be paired off completely, without there being some real relation (“such as marriage”) correlating them (1931, p. 23). But this is not all there is to relations. For it may well be the case that two classes have the same cardinality because there is a “real relation or function f (x, y) correlating them term by term” (ibid.). He took it that the real propositional functions are determined “by a description of their senses or imports” (1931, p. 37). In fact, he thought that appealing to the meaning of propositional functions is particularly important when we want to talk of functions of functions (Ramsey’s f(φx)), that is (higher-level) propositional functions f whose values are other propositional functions (φx). He wrote: “The problem is ultimately to fix as values of f(φx) some definite set of propositions so that we can assert their logical sum or product” (1931, p. 37). And he took it that the best way to determine the range of the values of f(φx) is to appeal to the meanings of the lower-level propositional functions (φx) (1931, pp. 36-7). Recall Ramsey’s Ramsey-sentence (∃ α, β, γ) : dictionary . axioms. The open formula dictionary . axioms (α, β, γ ) is a higher-level propositional function, whereas the values of α, β, γ are lower-level propositional functions. The Ramsey-sentence itself expresses the logical sum of the propositions that result when specific values are given to α, β, γ. This situation is exactly analogous to the one discussed by Ramsey above. So, it’s plausible to think that the values of α, β, γ are some definite properties and relations. That is, they are not any class or relation-in-extension that can be defined on the domain of discourse of the Ramsey-sentence. This point can be reinforced if we look at Ramsey’s Universals. Among other things, Ramsey argues that the extensional character of mathematics “is responsible for that great muddle the theory of universals”, because it has tended to obscure the important distinction between those propositional functions that are names and those that are incomplete symbols (cf. 1931, pp. 130-1 & p. 134). The mathematical logician is interested only in classes and relations-in-extension. The difference between names and incomplete symbols won’t be reflected in any difference in the classes they define. So the mathematician disregards this difference, though, as Ramsey says, it is “all important to philosophy” (1931, p. 131). The fact that some functions cannot stand alone (that is, they are incomplete symbols) does not mean that “all cannot” (ibid.). Ramsey takes it that propositional functions that are names might well name “qualities” of individuals (cf. 1931, p. 132). Now, Ramsey puts this idea to use in his famous argument that there is no difference between particulars and universals.26 But the point relevant to our discussion is that propositional functions can be names. Given a) Ramsey’s view that the propositional functions of physics should be non-formal, b) his insistence on real or actual properties and relations, and c) his view that at least some relations can be named by propositional functions, it seems plausible to think that he took the variables of his Ramsey-sentence to extend beyond real properties and relations – some of which could be named. I am not aware of a passage in his writings which says explicitly that the variables

R AMSEY ’S RAMSEY -SENTENCES

85

of the Ramsey-sentence extend beyond real or actual properties and relations. But his contrasting of mathematics (in which the variables are purely extensional) to science suggests that he might well have taken the view described above. Now, the other claim, i.e., that some of the Ramsey-variables can be names, also follows from his view, seen in section 4, that some propositional functions can give way to names of properties, as science grows. If I am right, the Newman challenge cannot be raised against Ramsey’s views. Ramsey takes theories to imply the existence of definite (or real) relations and properties. Hence, it’s no longer trivial (in the sense explained above) that if the theory is empirically adequate, it is true. His Ramsey-sentences can be seen as saying that there are real properties and relations such that …. Note that, in line with Ramsey’s denial of a distinction between universals and particulars, the existentially bound variables should be taken to quantify over properties and relations in a metaphysically non-committal way: they quantify over properties and relations which are not universals in the traditional sense, which renders them fundamentally different from particulars.27 The corollary is that Ramsey’s views cannot be described as pure structuralism. The claim that there are real properties and relations is not structural because, to say the least, it specifies the types of structure that one is interested in. Besides, Ramsey does not claim that only the structure (or the structural properties) of these relations can be known. Well, it might. Or it might not. This is certainly a contingent matter. If my interpretation is right, I have a hurdle to jump. It comes from Ramsey’s comment on “the best way to write our theory”. He says: “Here it is evident that α, β, γ are to be taken purely extensionally. Their extensions may be filled with intensions or not, but this is irrelevant to what can be deduced in the primary system” (1931, p. 231). But this comment is consistent with my reading of his views. The propositional variables may range over real properties and relations, but when it comes to what can be deduced in the primary system, what matters is that they are of a certain logical type, which the Ramsey-sentence preserves anyway. Indeed, deduction cuts through content and that’s why it is important. In any case, the comment above would block my interpretation only if what really mattered for theories was what could be deduced in the primary system. I have already said enough, I hope, to suggest that this view was not Ramsey’s. 9. R AMSEYAN H UMILITY Let me end by sketching an image of scientific theories to which the above interpretation of Ramsey’s Ramsey-sentences might conform. As already noted, I call it Ramseyan humility. We treat our theory of the world as a growing existential statement. We do that because we want our theory to express a judgement: to be truth-valuable. In writing the theory, we commit ourselves to the existence of things that make our

86

S TATHIS P SILLOS

theory true and, in particular, to the existence of unobservable things that cause or explain the observable phenomena. We don’t have to do this. But we think we are better off doing it, for theoretical, methodological and practical reasons. So we are bold. Our boldness extends a bit more. We take the world to have a certain structure (to have natural joints). We have independent reasons to think of it, but in any case, we want to make our theory’s claim to truth or falsity substantive. The theoretical superstructure of our theory is not just an idle wheel. We don’t want our theory to be true just in case it is empirically adequate. We want the structure of the world to act as an external constraint on the truth or falsity of our theory. So we posit the existence of a natural structure of the world (with its natural properties and relations). We come to realise that this move is not optional once we have made the first bold step of positing a domain of unobservable entities. These entities are powerless without properties and relations, and the substantive truth of our theories requires that these are real (or natural) properties and relations.28 That’s, more or less, where our boldness ends. We don’t want to push our (epistemic) luck too hard. We want to be humble too. We don’t foreclose the possibility that our theory might not be uniquely realised. So we don’t require uniqueness: we don’t turn our growing existential statement into a definite description. In a sense, if we did, we would no longer consider it as growing. We allow a certain amount of indeterminacy and hope that it will narrow down as we progress. Equally, we don’t foreclose the possibility that what the things (properties) we posited are might not be found out. Some things (properties) must exist if our theory is to be true and these things (properties) must have a natural structure if this truth is substantive. Humility teaches us that there are many ways in which these commitments can be spelled out. It also teaches us that, in the end, we might not be lucky. We don’t, however, draw a sharp and principled distinction between what can and what cannot be known. We are not lured into thinking that only the structure of the unobservable world can be known, or that only the structural properties of the entities we posited are knowable or that we are cognitively shut off from their intrinsic properties. These, we claim, are imposed epistemic dichotomies on perfect epistemic continua. We are reflective beings after all, and realise that dichotomous claims such as the above need independent argument to be plausible. We read Kant, we read Russell, Schlick, Maxwell, Redhead and Lewis, but we have not yet been persuaded that there is a sound independent argument for pushing humility too far (though we admit that we have been shaken). So we choose to be open-minded about this issue. The sole arbiter we admit is our give-and-take with the world. A further sign of our humility, however, is that we treat what appear to be names of theoretical entities as variables. We refer to their values indefinitely, but we are committed to their being some values that make the theory true. As science grows, and as we acquire some knowledge of the furniture of the world, we modify our growing existential statement. We are free to replace a variable with a name. We are free to add some new findings within our growing

R AMSEY ’S RAMSEY -SENTENCES

87

existential statement. We thereby change our theory of the world, but we had anticipated this need. That’s why we wrote the theory as a growing existential statement. We can bring continuity and change under the same roof. The continuity is secured by the bound Ramsey-variables and the change is accommodated by adding or deleting things within their scope. In the meantime, we can accommodate substantial disagreement of two sorts. Scientific disagreement: what exactly are the entities posited? In fostering this kind of disagreement, we are still able to use the theory to draw testable predictions about the observable world. But we do not thereby treat the theoretical part of the theory simply as an aid to prediction. For, we have not conceded that all that can possibly be known of the entities posited is that they exist. We can also accommodate metaphysical disagreement: what is the metaphysical status of the entities posited? Are they classes? Universals? Tropes? Some kind of entity which is neutral? Still, in fostering this kind of disagreement, we have taken a metaphysical stance: whatever else these entities are, they should be natural. To be conciliatory, I could describe Ramseyan humility as modified structuralism. Structuralism emerges as a humble philosophical thesis, which rests, however, on a bold assumption – without which it verges on vacuity – viz., that the world has a natural structure that acts as an external constraint on the truth or falsity of theories. I don’t claim that the image sketched is Ramsey’s. But he might have liked it. In any case, I take it that something like it is true. It’s not attractive to someone who is not a realist of some sort. But it is flexible enough to accommodate realisms of all sorts.

N OTES ∗

An earlier version of this paper was presented in Vienna in November 2003 at the Vienna Circle Institute Ramsey Conference. I would like to thank the organisers (Maria Carla Galavotti, Eckehart Koehler and Friedrich Stadler) for their kind invitation to talk about Ramsey’s philosophy of science. They, Patrick Suppes and Brian Skyrms should also be thanked for their excellent comments. I should also thank William Demopoulos and D H Mellor for their encouragement, and Nils-Eric Sahlin and Robert Nola for many important written comments on an earlier draft.

1.

The name was inspired by the title of Rae Langton’s splendid book Kantian Humility. Langton’s Kant was epistemically humble because he thought that the intrinsic properties of thingsin-themselves were unknowable. I am not claiming that Ramsey was humble in the same way. After I presented this paper in Vienna, D H Mellor told me that there was an unpublished paper by the late David Lewis with the title “Ramseyan Humility”. Stephanie Lewis has kindly provided me with a copy of it. Langton’s book is obviously the common source for Lewis’s and my Ramseyan Humility. But Lewis’s Ramseyan Humility is different, stronger and more interesting, than mine. Still the best overall account of Ramsey’s philosophy of science is Sahlin’s (1990, chapter 5). Here, I disagree with Sahlin’s view that Ramsey was an instrumentalist. As he says, in a slightly different context, “though I cannot name particular things of such kinds I can think of there being such things” (1991, p.193).

2. 3. 4.

88

5.

6. 7. 8. 9. 10. 11. 12.

13. 14.

15. 16. 17. 18. 19.

20. 21.

22.

S TATHIS P SILLOS

This is a point made famous by Quine. As he put it: “Variables can be thought of as ambiguous names of their values. This notion of ambiguous names is not as mysterious as it first appears, for it is essentially the notion of a pronoun; the variable ‘x’ is a relative pronoun used in connection with a quantifier ‘(x)’ or ‘(∃x)’. Here, then, are five ways of saying the same thing: ‘There is such a thing as appendicitis’ (…) ‘The word ‘appendicitis’ is a name’ (…) ‘The disease appendicitis is a value of a variable’ (…)” (quoted by Alex Orenstein (2002, pp.25-6). Compare what Ramsey says of the blind man who is about to see: “part of his future thinking lies in his present secondary system” (1931, p.261). One of Carnap’s lasting, if neglected, contributions in this area is his use of Hilbert’s ε-operator as a means to restore some form of semantic atomism compatible with Ramsey-sentences. See my (2000b) for details. This is an inductively established assumption, as Russell took pains to explain (cf. 1927, chapter 20). Russell agonises a lot about this. He knows that the relation between percepts and stimuli is one-many and not one-one. See (1927, pp.255-6). See also Mark Sainsbury’s excellent (1979, pp.200-11). More formally, we need a theorem from second-order logic: that every set A determines a full structure, i.e., one which contains all subsets of A, and hence every relation-in-extension on A. For an elegant and informative presentation of all the relevant proofs, see Ketland (2004). This is not, however, generally true. For every theory has a Ramsey-sentence and there are cases of theories whose Ramsey-sentence does not give the isomorphism-class of the models that satisfy the theory. This has been recently highlighted by Demopoulos (2003b, pp.395-6). For some relevant technical results, see van Benthem (1978, p.324 & p.329). Winnie (1967, pp.226-227); Demopoulos & Friedman (1985); Demopoulos (2003a, p.387); Ketland (2004). In a joint paper (see appendix IV of Zahar 2001, p.243), Zahar and Worrall call the Carnap-sentence “metaphysical” because it is untestable. What they mean is actually equivalent to what Carnap thought, viz., that the Carnap-sentence has no factual content. They may well disagree with Carnap that it is a meaning postulate. Be that as it may, the Carnap-sentence is part of the content of the original theory TC. So Zahar and Worrall are not entitled to simply excise it from the theory on the grounds that it is metaphysical. The claim that the variables of the Ramseysentence range over physical unobservable entities is no less metaphysical and yet it is admitted as part of the content of the Ramsey-sentence. This point is defended by Rozeboom (1960). I am not saying that striving for empirical adequacy is a trivial aim. By no means. It is a very demanding – and perhaps utopian – aim. What becomes trivial is searching for truth over and above empirical adequacy, since the former comes for free, if the latter holds. For some similar thoughts, see Russell (1927, pp.216-7). This, however, is what Langton’s Kant denies. See her (1998). This whole issue has been haunted by a claim made by Russell, Schlick, Maxwell and others that intrinsic properties should be directly perceived, intuited, picturable etc. I see no motivation for this, at least any more. Note that this is not Lewis’s motivation for the thesis that the intrinsic properties of substances are unknowable. For Lewis’s reasons see his “Ramseyan Humility”. For a more detailed defence of the claim that pure structuralism cannot accommodate causation, see my ‘The Structure, the Whole Structure and Nothing but the Structure?’, presented at the Austin PSA meeting in November 2004. http://philsci-archive.pitt.edu/archive/00002068 For more on this, see Demopoulos 2003a. Maxwell (1970a; 1970b) as well as Zahar and Worrall (2001) take Ramsey to have argued that the knowledge of the unobservable is knowledge by description as opposed to knowledge by acquaintance. This, as we have seen, is true. But note that though they go on to argue that this knowledge is purely structural, and that the intrinsic properties of the unobservable are unknowable, this further thesis is independent of the descriptivist claim. So, it requires an independent argument. It is perfectly consistent for someone to think that the unobservable is knowable only by means of descriptions and that this knowledge describes its intrinsic properties as well. For an excellent descriptivist account of Ramsey-sentences, see David Papineau (1996). For more on this see my “Scientific Realism and Metaphysics” (2005).

R AMSEY ’S RAMSEY -SENTENCES

89

23. A similar point has been made by Demopoulos (2003b, p.398). It is also made by James van Cleve (1999, p.157), who has an excellent discussion of how the problem we have discussed appears in Kant, and in particular in an interpretation of Kant’s thought as imposing an isomorphism between the structure of the phenomena and the structure of the noumena. 24. In some notes on theories that Ramsey made in August 1929, he seems not to have yet the idea of the theory as an existential judgement. He writes: “We simply say our primary system can be consistently constructed as part of a wider scheme of the following kind. Here follows dictionary, laws, axioms etc.” (1991, p.229). 25. Braithwaite came back to this issue in a critical notice of Eddington’s The Philosophy of Physical Science. He (1940) argued against Eddington’s structuralism based on Newman’s point against Russell. He noted characteristically: “If Newman’s conclusive criticism had received proper attention from philosophers, less nonsense would have been written during the last twelve years on the epistemological virtue of pure structure” (1940, p.463). Eddington replied in his (1941). For a critical discussion of this exchange, see Solomon (1989). 26. Propositional functions can name objects no less than ordinary names, which are normally the subjects of propositions. Hence, ultimately, Ramsey denies any substantive distinction between individuals and qualities: “all we are talking about is two different types of objects, such that two objects, one of each type, could be sole constituents of an atomic fact” (1931, p.132). These two types of objects are “symmetrical” and there is no point in calling one of them qualities and the other individuals. 27. This might address worries that the Ramsey-sentence involves second-order quantification. For more on this, see Sahlin (1990, p.157). 28. I think this is the central message of Lewis (1984) devastating critique of Putnam’s model-theoretic argument against realism

R EFERENCES R. B. Braithwaite, “Professor Eddington’s Gifford Lectures”, in: Mind, 38, 1929, pp.409-35. R. B. Braithwaite, “Critical Notice: The Philosophy of Physical Science”, in: Mind, 49, 1940, pp.45566. Rudolf Carnap, “Replies and Systematic Expositions”, in: P. Schilpp (Ed.), The Philosophy of Rudolf Carnap, La Salle IL: Open Court, 1963, pp.859-1013. William Demopoulos & Michael Friedman, “Critical Notice: Bertrand Russell’s The Analysis of Matter : its Historical Context and Contemporary Interest”, in: Philosophy of Science, 52, 1985 pp.621-639. William Demopoulos, “On the Rational Reconstruction of Our Theoretical Knowledge”, in: The British Journal for the Philosophy of Science, 54, 2003a, pp.371-403. William Demopoulos, “Russell’s Structuralism and the Absolute Description of the World”, in: N. Grifin (Ed.), The Cambridge Companion to Bertrand Russell, Cambridge: Cambridge University Press, 2003b. Arthur S. Eddington, “Group Structure in Physical Science”, in: Mind, 50, 1941, pp.268-79. Jane English, “Underdetermination: Craig and Ramsey”, in: Journal of Philosophy, 70, 1973, pp.45362. Jeff Ketland, “Empirical Adequacy and Ramsification”, in: The British Journal for the Philosophy of Science, 55, 2004, pp.287-300. Rae Langton, Kantian Humility. Oxford: Clarendon Press 1998. David Lewis, “Putnam’s Paradox”, in: Australasian Journal of Philosophy, 62, 1984, pp.221-236. Grover Maxwell, “Theories, Perception and Structural Realism”, in: R. Colodny (Ed.), The Nature and Function of Scientific Theories, Pittsburgh: University of Pittsburgh Press, 1970a, pp.3-34. Grover, Maxwell, “Structural Realism and the Meaning of Theoretical Terms”, in: Minnesota Studies in the Philosophy of Science, IV, Minneapolis: University of Minnesota Press, 1970b, pp.181192. M. H. A. Newman, “Mr. Russell’s ‘Causal Theory of Perception’”, in: Mind, 37, 1928, pp.137-148. Alex Orenstein, W V Quine. Chesham: Acumen 2002.

90

S TATHIS P SILLOS

David Papineau, “Theory-Dependent Terms”, in: Philosophy of Science, 63, 1996, pp.1-20. Stathis Psillos, Scientific Realism: How Science Tracks Truth. London & New York: Routledge 1999. Stathis Psillos, “Carnap, the Ramsey-Sentence and Realistic Empiricism”’, in: Erkenntnis, 52, 2000a, pp.253-79. Stathis Psillos, “An Introduction to Carnap’s ‘Theoretical Concepts in Science’” (together with Carnap’s: “Theoretical Concepts in Science”), in Studies in History and Philosophy of Science, 31, 2000b, pp.151-72. Stathis Psillos, “Is Structural Realism Possible?”, in: Philosophy of Science (Supplement), 68, 2001, pp.S13-24. Stathis Psillos, “Scientific Realism & Metaphysics”, in Ratio, 18, 2005. Frank Ramsey, The Foundations of Mathematics and Other Essays. (Edited by R. B. Braithwaite). London: Routledge and Kegan Paul 1931. Frank Ramsey, Notes on Philosophy, Probability and Mathematics. (Edited by M. C. Galavotti). Bibliopolis 1991. William W. Rozeboom, “Studies in the Empiricist Theory of Scientific Meaning”, in: Philosophy of Science, 27, 1960, pp.359-373. Bertrand Russell, The Analysis of Matter. London: Routledge and Kegan Paul 1927. Mark Sainsbury, Russell. London: Routledge and Kegan Paul 1979. Nils-Eric Sahlin, The Philosophy of F P Ramsey. Cambridge: Cambridge University Press 1990. Moritz Schlick, General Theory of Knowledge. (2nd German Edition, A. E. Blumberg trans.). Wien & New York: Springer-Verlag 1918/1925. Moritz Schlick, “Form and Content: An Introduction to Philosophical Thinking”, 1932, in: Moritz Schlick’s Philosophical Papers, Vol. II, Dordrecht: D. Reidel P. C., 1979, pp.285-369. Graham Solomon, “An Addendum to Demopoulos and Friedman (1985)”, in: Philosophy of Science, 56, 1989, pp.497-501. J. F. A. K. van Benthem, “Ramsey Eliminability”, in: Studia Logica, 37, 1978, pp.321-36. James van Cleve, Themes From Kant. Oxford: Oxford University Press, 1999. John Winnie, “The Implicit Definition of Theoretical Terms”, in: The British Journal for the Philosophy of Science, 18, 1967, pp.223 –9. John Winnie, “Theoretical Analyticity”, in: R. Cohen and M. Wartofsky (Eds.), Boston Studies in the Philosophy of Science, Vol.8, Dordrecht: Reidel, 1970, pp.289-305. Elie Zahar, Poincaré’s Philosophy: From Conventionalism to Phenomenology. La Salle IL: Open Court 2001.

Department of Philosophy and History of Science University of Athens Panepistimioupolis (University Campus) Athens 15771 Greece [email protected]

E CKEHART K ÖHLER

RAMSEY AND THE VIENNA CIRCLE ON LOGICISM

1. S CANDAL OF P HILOSOPHY AND M ATHEMATICS I think it is a scandal of philosophy that Logicism – the reducibility of mathematics to logic – receives so little attention anymore, as if it were dead. It is not dead. Instead it has been shamelessly abandoned. Issues for which some of the greatest thinkers such as Frege, Russell and Ramsey fought passionately now barely elicit yawns from “sophisticated” philosophical logicians, who seem to assume that the issue was long ago “settled” in the twenties. Oddly enough, the issue of what relation logic has to mathematics has in the meantime been settled by default in another way: mathematics departments have simply requisitioned logic from philosophy as one of the specialties of mathematics, on the one hand – philosophy having given up the mathematized beast they never had much patience with anyway. But as such a specialty, on the other hand, the mathematicians have, by way of compensation, at least given set theory back to logic, where it has always belonged.1 I first got acquainted with Logicism in Stegmüller’s philosophical institute at the University of Munich in 1960, when Frege was still virtually unknown. Once I read his analysis of the natural number concept, especially the concept of the ancestrals using his new theory of relations, and his sophisticated treatment of measurement,2 I became completely convinced that mathematics was reducible to logic and that the main thesis of Logicism was proven. When I later read more literature on foundations of mathematics, I was quite distressed to discover that, in the meantime, Logicism was widely regarded as a failure.3 What had happened? Well, the antinomies had happened, stopping Frege, and Russell had taken over the torch. Logicism had become a complete hostage of special, newly discovered thorny problems in the logic of Russell which I never thought were so immediately relevant to the reduction of mathematics. Russell created Type Theory, with all its special complexities and artificialities, getting caught in thickets of controversies and blind alleys. What Russell did or did not achieve is still a central question concerning Logicism even now – all the more so because Carnap’s logic (in particular the Logical Syntax of Language) uses a variant of Russell’s Type Theory. Gödel (1944) wrote the severest and most exacting critique of Russell’s contributions, including comments on and praise for Ramsey’s 91 M. C. Galavotti (ed.), Cambridge and Vienna: Frank P. Ramsey and the Vienna Circle, 91–121. © 2006 Springer. Printed in the Netherlands.

92

ECKEHART KÖHLER

refinements. Nevertheless, the complexities of Type Theory unfairly draw attention away from the central claim of Logicism, which should be judged in partial isolation – a standpoint shared by Boolos (1998) shortly before his death, and by Clark (2004). Part of the problem lies simply in Frege’s having raised expectations of rigor to a very high degree in his effort to demonstrate that no intuitions extraneous to logic, no Anschauungen were surreptitiously slipping in to spoil the reduction of mathematics to pure logic. If he had merely kept to the lower level of rigor which Cantor, Dedekind and Zermelo applied, he would perhaps have escaped much of the onus he suffered, as Cantor did with his comparatively informal approach. Instead, Frege (1879) developed his famous Begriffschrift, the world’s first “logistic system” (Church), comparable in rigor to a programming language and only surpassed by Gödel when proving the Incompleteness Theorems.4 Russell in his Type Theory backslid considerably from Frege’s level of rigor back to that of Peano, who had been the “discovery of his life” (not Frege!). But even that level of rigor was sufficient to quickly entangle him in many distracting thorny issues. As everybody knows, Ramsey played a major role in rescuing Type Theory from some of these thorns, but unfortunately his yeoman work was insufficient to persuade either the philosophical community or mathematicians of the virtues of Logicism. One of the worst aspects of Russell’s original Type Theory involved “Ramified Types” or orders, used to solve semantic or intensional antinomies such as Richard’s paradox. Ramsey simply pointed out that the reduction of mathematics to logic doesn’t require any intensional principles, so ramification could be eliminated. In retrospect, we may say that Russell was simply trying to do too many things at once: in his zeal to make logic do all things for all seasons, attempting to realize Leibniz’s dream of a characteristica universalis by creating a universal language, Russell crammed machinery into his formalism to deal with a few semantic issues which we nowadays treat much better in semantic metatheories.5 Instead, he should have ignored some of his ambitions and concentrated on the most important issue in the grand British tradition of dividing and conquering: that of reducing mathematics to a more streamlined logic. And in the same years that Russell was developing his baroque Ramified Type Theory, more practical-minded mathematicians such as Zermelo were “streamlining” Cantor’s jungle of set theory, coming up with a axiom systems (Z, ZF, NB) paradigmatic for the majority of mathematicians today. Bingo! Classical mathematics is easily reducible to set theory! It is peculiar that practically every mathematician is roughly familiar with the fact that, in both major variants of set theory, Zermelo–Fraenkel (ZF) and von Neumann–Bernays–Gödel (NBG),6 as well as in the systems of Quine (NF and ML), natural number theory and analysis are straightforwardly derivable, once suitable models for progressions and continuous point sets are set up. And yet, although set theory has meanwhile been conventionally classified as a branch of logic, one would think the issue of Logicism ought to be regarded as settled positively – open-and-shut case! Even if one still regarded set theory as contro-

R AMSEY AND THE VIENNA C IRCLE ON LOGICISM

93

versial (which “good” logic is not supposed to be!), it is so primarily with respect to exotic questions of large cardinality axioms of uncertain relevance for classical mathematics, and with respect to the fact that no axioms have been found which imply the Continuum Hypothesis (CH), whose solution will always remain the ultimate goal of set theory and all of mathematics since Cantor and Hilbert. But we are farther away from this goal than we ever seemed to be before. There is a patent injustice in the inconsistent ways in which mathematicians judge the success of the Bolzano–Weierstrass reduction program for analysis and the closely related Frege–Russell reduction program for arithmetic. Since the old Bolzano–Weierstrass program of “arithmetizing” analysis has met with very widespread approval (except perhaps among the lovers of the recently revived infinitesimal theory, and among the minority of constructivists – and even Brouwer was a radical arithmetizer), and since this program uses the same set theory sometimes regarded as too dubious for the reduction of arithmetic, it would seem patently unjust to withhold approval of the “remaining” reduction of cardinal-number arithmetic to set theory alone. It seems two significantly different criteria are being applied here, for if Frege–Russell Logicism is rejected because of doubts about Type Theory (a variety of set theory), then the Bolzano– Weierstrass arithmetization of analysis should also be rejected for the same doubts. Conversely, if set theory is sound enough for Bolzano–Weierstrass, it is sound enough for Frege–Russell. So there is a strange reluctance to concede victory to Logicists – as if mathematics were to lose its freedom and integrity if hordes of impertinent logicians and proof theoreticians were suddenly to invade analysis and topology seminars and begin inquisitions on proofs and axioms! (Come to think of it, maybe that would be good at times!) At the same time that mathematicians were being unfair to logic, logicians were being eccentric, too. Even the great disciple of Russell and Carnap, Van Orman Quine, ultimately put set theory outside of logic!7 It is not supposed to be logic’s business to make such contentual, ontologically committing claims; it should rather remain just “structural”; but set theory is famous particularly for claiming existence of infinite sets – most notoriously for power sets. Kneale & Kneale (1962) take the same position as Quine in restricting logic to first-order predicate theory (only second-order and higher-order theories explicitly assume concepts or sets). This seems absurd to me. Logic always dealt with concepts and concept formation, with their intensions and extensions, in addition to propositions and inference; and sets are extensions of concepts (Frege).8 Indeed, as the early Russell saw, doing set theory doesn’t even require sets, all it needs is concepts (his intensional “propositional functions”). But of course, the pioneer of set theory, Cantor, was officially a mathematician, not a philosopher, and his formulations obscured the connection with the logic of concepts. His ally, the mathematician Dedekind, was fairer: he proclaimed that set theory definitely belongs to logic and fell under the “laws of reasoning”. The way Frege separated his symbols into those designating apparently intensional concepts9 and

94

ECKEHART KÖHLER

extensional objects (Gegenstände), including sets (Wertverläufe), made it seem as if they were fundamentally different, and Russell inherited this view. But Ramsey saw the light: there is no fundamental difference between concepts and objects, both depend equally on their role in propositions. This is close to the position Carnap (1947) came to, in which all signs in the language have both extensions and intensions assigned to them, where individual objects have “individual concepts” assigned to them. The membership relationship of set theory ∈ is really identical to the satisfaction or fulfilment relationship between properties and the objects falling under them, and in this sense, it belongs to logic if ever anything does. Since all existence claims of set theory may be regarded as statements about ∈ , (hence about satisfaction), they are statements about logic. Furthermore, existence claims for sets can all be formulated syncategorematically, i.e. without using empirical concepts. This makes them all candidates – acceptable or not – for logical postulates. We see the main problem was unclarity on the nature of logic in general, and in particular whether certain axioms, principles or rules were logically valid, such as the Axiom of Infinity; and a secondary problem was finding an acceptable boundary line between logic and mathematics. I shall say more about these issues in the course of this paper. It should of course never be forgotten how blindingly new modern logic really is, considering the considerable age of logic; how long it took philosophy to recover from the cultural devastation of the Dark Ages; how late the great publications of Frege came; and how long the great predecessors of Frege and Russell, Leibniz and Bolzano, remained largely unpublished or unread in old journals, virtually unknown among academic philosophers. People’s judgments concerning mathematics and logic are dominated by intuitions which stubbornly resist change and remain anchored in the half-knowledge and prejudices of their age. 2. T HE U NCERTAIN B OUNDARY BETWEEN L OGIC AND M ATHEMATICS At the beginning of the twentieth century, when the Logicist claims of Dedekind, Peano, and especially Frege and Russell first attracted attention, Henri Poincaré, always somewhat conservative, doubted the new doctrine. To be sure, there is a certain justice to one of Poincaré’s objections to Logicism: that it illicitly hijacks mathematical motifs into its new logic, and only by doing this can it get the Principle of Complete Induction, for example. Frege’s theory of relations was “brand new”, not accepted in the classical cannon of logic, because this was monomaniacally fixated on the (monadic) subject-predicate forms of Syllogistics, due to Aristotle’s domination of Scholasticism. Hence Poincaré, with seeming plausibility, could argue that Frege’s treatment of concepts as functions (enthusiastically taken over by Russell as “propositional functions”) simply incorporates mathematical functions right into the basis of the new logic. Thus, Frege’s

R AMSEY AND THE VIENNA C IRCLE ON LOGICISM

95

(1879) reduction of complete mathematical induction to ancestrals of relations could be seen as a reduction of mathematics to mathematics, so far as relation theory is regarded as a theory of (mathematical) functions in disguise. Despite Poincaré, most people nevertheless agreed that relation theory naturally did belong to logic, simply because relations treated as two-placed (dyadic) predicates so obviously seem to be extended from the notion of one-placed predicates, used in Syllogistics, as the grammar of ordinary language persuades us. In addition, despite the “second-rank” status of the logic of relations, various principles belonging to it are repeatedly mentioned or used in traditional logical literature, where it seemed to take a natural place, prominently even in Aristotle. In retrospect it seems amazing to us now how hard a struggle Leibniz, perhaps the most brilliant philosopher in history, had with relations and orderings.10 Much more deeply perplexing was the status of Cantor’s set theory, to which Whitehead & Russell devoted major attention in the Principia Mathematica. Set theory was in a long tradition of generalizations in algebra, especially applications of function theory using the notion of series. Sets obviously fit into the mathematical zoo together with groups, fields and series. Russell saw no problem reinterpreting set theory as logic, since sets were easily identifiable with classes interpreted as extensions of one-place attributes or predicates.11 Here Russell was influenced by Frege’s notion of “range of values” (Wertverläufe); Russell already had clearly enough spelled out his view independently of Frege in his Principles of Mathematics (1903), following traditional lines established by Venn, Boole and Dodgson (Lewis Carroll) on the way in which extensional classes are related to intensional properties: Just as Frege had characterized his “Sinn” as “die Art des Gegebenseins eines Gegenstands” (sense is the manner in which objects are presented), Russell originally thought of classes as being generated by intensional procedures.12 That is, every (intensional) property has a procedure which discriminates objects satisfying the property into a class obtained by running through the procedure. Russell naturally supposed that intensions were mental, in the tradition of Idealism, whereas classes (or sets) were thought to be extra-mental, quasi-physical (Gödel).13 Russell’s approach seemed to crash into a dead end when Zermelo (1903, 1908) published his famous proof of the Well-Ordering Theorem using his newly discovered Axiom of Choice (AC). The Axiom of Choice immediately became a great bone of contention between Constructivists and Platonists (or Realists).14 Russell had by this time, after his pioneering study on Leibniz (1900), become a Platonist and was perfectly willing to grant the validity of the AC (which Whitehead had discovered independently of Zermelo and had called the Multiplicative Axiom), especially since the AC was crucial for several famous mathematical theorems, as Zermelo had made clear. But what did worry Russell quite a bit was that the AC seemed to guarantee the existence of sets independently of generation by any intensional classification procedure. If the AC somehow was a characteristic set theoretical axiom, this fact seemed to imply that sets were sui generis and existed independently of being generated by any (mental or logical)

96

ECKEHART KÖHLER

classifying procedure; or similarly, that they were “quasi-physical” in Gödel’s term. Consequently set theory now seemed to Russell to be independent of logic! Frege also came to just this conclusion after Russell sent him news of his famous Antinomy in 1902, and Frege abandoned the disgraced Grundgesetz V which converted equivalences of intensional properties into identities of extensional sets. Apparently as a result of breaking the previously assumed bond with the mental, most mathematicians began regarding sets as quasi-physical entities sui generis, typically beginning with the famous “purely mathematical” notion of the null set and iterating all other sets from that narrow starting point. I don’t see what the fuss is about. Russell should have interpreted the AC not as divorcing choice sets from intensional classifications (i.e., concepts), but rather as simply making the admittedly rather powerful claim that, even in the infinite, some selection procedure can always be devised by a sufficiently strong mind to generate any choice class. (The “strong mind” simply runs through all subsets, randomly selects a member of each and assigns it to the choice set.) This is perfectly compatible with Idealism; the problem is merely that it accentuates the theological, perhaps Hegelian, side of Idealism; in particular, the strong mind generating the choice class would be outside the realm of the elements. Mathematics routinely makes many extreme and transfinite rationality assumptions like this, e.g. that ʌ can be uniquely determined by the completion of an infinite calculation. Mathematicians accept transcendental numbers which can’t even be calculated like ʌ . So then what’s so special about the AC? I say: not much! Ramsey (1926a, §V) himself was sure that under his interpretation the AC was “the most evident tautology” (albeit, under Russell’s original interpretation, the AC was “really doubtful”). Moreover, all sets, just like the classes in von Neumann–Bernays set theory, can be regarded as extensions of properties; and indeed all sets, whether Zermelo-Fraenkel’s or von Neumann–Bernays’s, can be reduced to intensions, as Carnap, Church and others later showed from the forties onward: Set theory can always be embedded within concept theory. I will return to this later when I discuss Quine’s Set Theory and Its Logic. 3. T ROUBLES WITH T YPE T HEORY , R AMSEY ’ S R ESCUE , AND V IENNA By the twenties, opinions on the success or failure of Logicism focused on Russell’s Type Theory, since Russell was the most prominent Logicist.15 It was a serious mistake to restrict such opinions to Type Theory without also considering Zermelo’s axiomatized set theory of 1908, later extended by Fraenkel, since set theory competes with Type Theory as a logic. Frege could or even should have taken the path of using Zermelo’s comprehension axioms instead of his Grundgesetz V. Frege’s Logicist program thus became unfortunately and unnecessarily tied to the misfortunes of Russell’s procrustean Type Theory. A balanced assessment of Logicism’s fate taking set theory into consideration would have resulted in a much more favorable reception by the ’30s. To be sure, Type

R AMSEY AND THE VIENNA C IRCLE ON LOGICISM

97

Theory is to an extent completely natural the way it forms hierarchies; but the restrictions it poses on theory formation makes it at the same time distressingly unnatural, forcing us into contortions when approaching even elementary mathematical problems. For example, in analysis, it leads to rather artificial difficulties with the concepts of limits and accumulation points. Such problems provided Quine (1940) with his principal motivation for using set theory rather than Type Theory. Thus attention was fixated on the special problems which Type Theory faced, which the main alternative, Zermelo’s set theory, did not. Briefly, those problems included 1) the Byzantine labyrinths of ramified types and its horridly ad hoc Reducibility Axiom, and 2) the Axiom of Infinity, needed in order to obtain number theory on the third type level. In addition, 3) the Axiom of Choice (AC) was regarded as somehow more dubious in Type Theory than in set theory, because it seemed less plausible to consider it to be logically valid than as mathematically useful. At this point, the young and promising Ramsey entered the picture, amazingly gifted and solid both in his technical mathematical ability as well as in his philosophical judgment. And he was a glowing adherent of the idea of Logicism, so he was very set on revising Type Theory sufficiently to persuade philosophers and mathematicians of the virtues of Logicism. This effort early on caught everyone’s attention in the Vienna Circle, from Hahn’s to Carnap’s and the younger students like Gödel’s. Most attractive was Ramsey’s elimination of ramified types and the concomitant Reducibility Axiom,16 immediately making Type Theory much more presentable. This was Ramsey’s main influence on Carnap’s Logical Syntax of Language (1934), duly honored by Carnap in §60a and elsewhere in that work. Ramsey’s approach was guided by his famous classification of antinomies into two classes, which we may call intensional and extensional, or perhaps better semantical and object-theoretical; Ramsey found that Ramified Type Theory was needed to deal with semantical antinomies like Richard’s Paradox, but not needed to deal with object-theoretical antinomies like Russell’s Paradox. Ramsey proposed ridding Type Theory of intensions (whose semantic nature was only implicit before Tarski) and moving to a purely extensional theory. This move ironically contravened a prior move of Russell’s for which Russell had become famous, viz. the so-called “no-class” approach, whereby an entire ontological category was subjected to Occam’s razor, leaving only intensional propositions and propositional functions. (Tarski furthermore pointed out that only one category is needed here, since propositions could be construed as zero-argument propositional functions.) Ramsey preferred wielding the razor in the other direction, and the extensionalists generally approved. That included Carnap at the time. In fact, Carnap’s extensionalist formulation of Type Theory in his Abriss der Logistik (1929, §30), the first textbook of mathematical logic for philosophers, has long been regarded as the standard in the field. Carnap was personally directly indebted to Russell, who had written out in longhand for him the main definitions and theorems of

98

ECKEHART KÖHLER

Principia Mathematica in the early twenties when it was impossible for Carnap to purchase any books. Church (1940) finally provided the definitive formalization, later (1951) extending this to a sophisticated formalization of intensions and extensions, close to Russell’s original motive, although directly aimed at explicating Frege. But as we all know, Carnap (1947) himself became a famous intensional logician, placing meaning at the center of logic again – this time using Tarski’s quasi-Hilbertian metatheoretical approach. It is appropriate at this place to insert an anachronistic remark concerning Quine’s much later, didactically brilliant treatment of Type Theory and set theory within one homologous setting in his Set Theory and Its Logic (1963). Usually this book of Quine’s is taken to imply that Type Theory is just a variety of set theory, in a way pushing Ramsey’s approach to its natural conclusion. This then suggests, however, that Type Theory is really as “mathematical” as set theory is usually held to be, whereas the older, intensionally-oriented view was that Type Theory was “logical” because it explicitly dealt with concepts (“propositional functions”), whose intensionality implied semantics – and semantics had been naïvely but perceptively (and correctly!) called “psychological” by both Ramsey and Carnap in the ’20s. But au contraire , Quine could have drawn quite the opposite conclusion from his homologization, namely that set theory is at least as logical as Type Theory – and as a follower of Russell he really ought to have done so. However Quine was perhaps constrained by his famous prejudice against intensional theories, which he didn’t like and preferred to avoid, no matter how inherently logical and traditional they were. Therefore, although Quine’s main program originally was to improve on Type Theory in deriving arithmetic, he was curiously poorly motivated to rescue Logicism. 4. R AMSEY AND C ARNAP ON THE A XIOM OF I NFINITY Once the main problem of Russell’s original Type Theory was removed by Ramsey, the Axiom of Infinity (AI) took center stage as the remaining serious bone of contention. Ordinary Type Theory cannot be revised to avoid this axiom, however, without sacrificing much of mathematics, which requires the existence of infinite sets. The problem is that, looking at type level 0, we aren’t sure if the universe has infinitely many particles in it.17 But we need that many to obtain infinite sets at any level above 0. Thus is AI usually interpreted to be an empirical claim. Ramsey (1925), at the end of his “Foundations of Mathematics”, gave a nice argument showing that, however many individuals n there may exist at type level 0, the proposition that it has exactly n individuals will be tautological, and that it has > n individuals is contradictory; from which he concludes … the Axiom of Infinity in the logic of the whole world, if it is a tautology, cannot be proved, but must be taken as a primitive proposition. And this is the course which we must adopt, unless we prefer the view that all analysis is self-contradictory and

R AMSEY AND THE VIENNA C IRCLE ON LOGICISM

99

meaningless. We do not have to assume that any particular set of things, e.g. atoms, is infinite, but merely that there is some infinite type which we can take to be the type of individuals.

By “some infinite type” Ramsey means “some type level with infinitely many classes”. But we might ask immediately: must all types be < Ȧ ? What about a type levels beyond the infinite? We will always have all the mathematics we want at levels • Ȧ ! And, without AI, we don’t care that number theory and analysis are invalid at levels < Ȧ . Hilbert (1926, p. 184) already proposed transfinite types, and Gödel (1931) suggested them concerning attempts at completeness. Wolfgang Degen (1993, 1999, 2000) has worked with great success on exactly this idea, “vindicating” Logicism. Of course, conservative theoreticians may protest that transfinite types are egregious monsters. But set theory also is exceedingly generous – no one blinks at ℘(Ȧ) or at 2ℵ3 . It just has betterstreamlined axioms. Those prejudiced against any reduction of arithmetic to logic such as Poincaré would no doubt find something to complain about, come what may.18 Ramsey considered the Axiom of Infinity an extra-logical proposition which could be tautological, depending on whether the universe contained infinitely many bodies. Russell before him was more straightforward: he conceded that it was even an empirical proposition; most others followed him in this. As much as others, Carnap of course disliked this mixing of logic with empirical considerations when the derivation of arithmetic depended on it, and he figured out in his Logical Syntax of Language (1934, §§3, 15) a way of verifying the Axiom of Infinity in ordinary Type Theory without assuming any knowledge of the number of individuals: let the level-0 objects simply be positions in dimensions (whose metric properties are left unspecified, e.g. the units used).19 This avoids making specific empirical existence claims, yet makes the Axiom of Infinity logically plausible – because our scales of measurement can have infinitely extended dimensions without the existence of infinitely many or infinitely large bodies. Be that as it may, Carnap had little influence with his idea, although it really deserved consideration. Thus, primarily because the Axiom of Infinity was regarded as not logically valid, Logicism was held to have failed. Gödel flatly regarded it as a failure, AI being partly responsible, in addition to Gödel’s strict interpretation of Logicism as requiring completeness. Wittgenstein also had become very negative towards Logicism, but for quite different, possibly ideological or aesthetic reasons which moved him towards ultrafinitism in logic and opposition to higher types and quantifiers, a position I find hard to accept as a serious one for scientific research because it utterly weakens logic. Other mathematicians, in particular those in the Hilbert school, shared Gödel’s view.

100

ECKEHART KÖHLER

5. C ARNAP THE L OGICIST? By the mid-thirties, after Carnap’s famous Logical Syntax of Language appeared in Vienna in 1934, a reassessment of Logicism seemed to be called for even by its most enthusiastic partisans. I can’t go into much detail here, but fortunately Herbert Bohnert (1975) has written a profound and fascinating review of the development of Carnap’s thinking about Logicism, so this is not necessary. Suffice it to say that Carnap (1934, §84) unfortunately withheld the core claim of classical Logicism: that of reducing numbers to concepts – despite his effort in making the Axiom of Infinity a logically plausible proposition.20 Carnap followed Hilbert’s idea – which was also Gödel’s practice in his Incompleteness Theorem (1931) – of putting natural numbers at type level 0, and so all number theory automatically became available at level 1 of Carnap’s version of Type Theory already, and Peano’s axioms followed straight out of the syntactic metatheory. This made a Frege–Russell definition of numbers superfluous, which is only mentioned in passing (in the context of a discussion of Russell’s elimination of classes as entities separate from concepts in § 38). As Bohnert has it, Logicism then shriveled to the mere statement that logic and mathematics simply belong together. In a sense, this confirms Poincaré’s suspicion that Logicism hijacked mathematics; but Poincaré was an old-fashioned Kantian and insisted that mathematics differed essentially from logic by having content and being based on intuition, whereas logic was empty, as Kant thought. Carnap’s rejection of intuition in his Syntax made it easy for him to put logic and mathematics together. It should be stressed, as Hao Wang has done, that Frege (like Bolzano and Leibniz before him) already emphasized that the basic reason why mathematics is logical is the simple fact that its laws, like the laws of logic, apply to everything: they are in a precise sense super-universal. Mathematics shares with logic the semantic function of theoretically representing reality; both equally involve truth. They are super-universal: they are more universal than physical laws because they are valid no matter what the empirical universe is like. (In Leibniz’s classically baroque image, they hold in all possible worlds!) Wittgenstein totally deflated this hyperbole and persuaded Schlick, Hahn and Carnap that both mathematics and logic are universal because they have no empirical content, i.e. are tautologies. I would like to point out that this claim, which in the first instance seems rather trivial, in fact amounts to a crucial correction of a bad error of traditional Platonism: this often was made to say that mathematics (e.g. geometry21) only applied (strictly) to “ideal objects”, whereas “real objects” never satisfy mathematics exactly. Despite Frege, and despite protests from Occam to Berkeley against the illegitimate dualism of both ideal and real objects, very many naïve people22 say that mathematics deals with a special domain of abstract (“ideal”)

R AMSEY AND THE VIENNA C IRCLE ON LOGICISM

101

objects like sets and numbers, often called “structures”, in distinction from empirical science, which deals with concrete (“real”) objects. This is all nonsense, and Berkeley’s (and Neurath’s) scathing criticism of Platonist Verdopplung is quite correct. Plato himself knew this criticism perfectly well, as the second part of Parmenides makes clear; but he had no suitable answer. Frege did. To see the point, just consider that all the allegedly non-empirical (“abstract”) sets and numbers magically gain empirical content the minute mathematics is applied: in a mathematically formulated physics, all functions and all sets used are physical concepts. For example, the n-tuple assigning magnetic field strength to points in space is the magnetic field itself. Classic authorities have claimed logical validity, or analyticity, to be characterized by what I have labeled “super-universality”. Leibniz of course did so; Bolzano as well. More recently Schröder (1890) did so, and Skolem followed him. Carnap also tried very hard to define Allgemeingültigkeit as the culmination of his Syntax-program, being helped by Gödel in his effort. Unhappily the definition failed, in my opinion, because he failed to realize how unlimitedly powerful the theoretical apparatus he needed for his syntax language. Two remarks are apropos. First, there is an even more fundamental reason why Carnap’s characterization misses the essence of logical validity, and why his approach opened him to the criticism of Quine against the analytic /synthetic dichotomy. Universality plainly and simply cannot be used to distinguish between logic and mathematics, because mere universality does not give us any modal distinction between logical and physical necessity. Second, it is an interesting and deep historical fact that the Leibniz–Schröder–Skolem concept of universal validity in all subject domains is identical with Aristotle’s main characterization of metaphysics! Therefore, to find out what logic is, it will pay to look at Platonic Dualism in more detail, this time avoiding the errors of Verdopplung which even Plato made fun of in his dialogues (e.g. the second part of the Parmenides). 6. T HE R EAL N ATURE OF L OGICAL V ALIDITY The distinction underlying Platonist Dualism which really makes sense is the modal distinction for which Hume became famous in his analysis of “Is” and “Ought”. (Concrete) Reality is Hume’s “Is”, (Abstract) Ideality is Hume’s “Ought”. No “Is” implies an “Ought” and vice versa : the Naturalistic Fallacy is to be avoided. The Abstract and Concrete by themselves are not the basis for Platonic Dualism, since both always appear together anyway; as Aristotle observed for matter and form, they’re inseparable. Example: we have electromagnetic fields all over Nature, they are part of empirical reality. Yet if we look carefully at the definition of such fields, we cannot escape that they involve highly theory-laden structures associating vector forces with positions in space in a complex, multi-dimensional manifold.

102

ECKEHART KÖHLER

This distinction between Ought and Is (or between the Ideal and the Real) quickly sets the stage for clarifying the famous analytic /synthetic distinction. As we all know, discussions of the true nature of logic typically devolved to a search for some acceptable characterization of analyticity. The Vienna Circle thought it lay in Wittgenstein’s Tautology. Carnap struggled to extend the domain of tautologies to all mathematics in his Logical Syntax program, an effort which famously earned the negative judgment of Quine (1951), who complained that Carnap had only promulgated a Dogma of Empiricism without providing a plausible common feature of analyticity in all the various systems allowed by the Principle of Tolerance. There is no general, non-arbitrary non-modal characterization of logical truth for all of the infinitely many language systems. Quine’s critique was justified up until 1965, when Carnap finally revealed to an astounded group of top philosophers of science that (rational) intuition was what justified belief in logical principles and rules. Alas, Bohnert, whose papers on Carnap’s views on both Logicism in particular as well as on the analytic/synthetic distinction in general are wonderfully perceptive, almost completely missed this epoch-making event in the history of Logical Empiricism. Bohnert refers to intuition mainly to affirm the well-known fact that it was “officially” rejected among orthodox Logical Empiricists ever since the twenties, when Hahn and Schlick determined that Kant’s Anschauung was too broken to fix, a view almost everyone followed, from Richard von Mises to Hans Reichenbach (except the phenomenologist Kaufmann and the Platonist Gödel). Nevertheless Bohnert has several quite interesting things to say regarding intuition. He expresses an interesting opinion, although rather non-commitally, that Carnap had a “significantly ambivalent attitude toward the power of man’s intuition, both in general and as applied to matters of logic and mathematics”.23 Bohnert is exonerated from not paying attention to a paper Carnap (1968) held under the heading of inductive logic, because Carnap apparently didn’t mention intuition to Bohnert during the several-day-long discussions they had on logicism in 1968, two years before Carnap’s death. In connection with learning logic, Bohnert does stress the pragmatic aspects of learning meanings – the foundation for judgments of logical validity: first that they begin in childhood, and they “seep” gradually wider and wider, interlocking more and more. Bohnert discovers that this very pragmatic view was already hinted at in the Logical Syntax of Language §38a, p. 142. This insight is important, since it captures exactly how intuition develops from vague gropings to gradually increasing certainty and precision. The history of mathematics, logic and statistics – to say nothing of that of law, accounting, measurement standards and many other areas of norms – confirms that at every stage in their development, intuitions are refined, corrected, polished, extended more and more. What’s more, pragmatic criteria for acceptance of principles and rules were always admitted by Carnap throughout his career, even in the medium-rigorous, apparently anti-semantical period of the Syntax program. The best discussion of this key aspect is by Bryan Norton (1977), who argues (correctly in my opinion),

R AMSEY AND THE VIENNA C IRCLE ON LOGICISM

103

that it is precisely with reference to pragmatic motives and interests which lend objectivity to formal abstractions, grounding discriminations of validity of logic. 7. A FFINITIES BETWEEN R AMSEY AND C ARNAP In the 1950s, Carnap came much closer to Ramsey’s philosophy after reading Shimony (1953, 1955), who first discovered extremely important connections between Carnap’s confirmation theory and the subjective probability theories of de Finetti (1937) and Ramsey (1926), particularly between Carnap’s concept of regularity and the Ramsey–de Finetti concept of coherence. The culmination of that work was Carnap (1962). Another area where Carnap was deeply influenced by Ramsey was empirical theory construction, for which Carnap (1958, 1966) made famous use of so-called “Ramsey-sentences”, first introduced in Ramsey (1929). But Carnap had something much more deeply in common with Ramsey, and that was his general approach to philosophy and its relation to human concerns. Ramsey (1929, pp. 263, 269) wrote In philosophy we take the propositions we make in science and everyday life, and try to exhibit them in a logical system with primitive terms and definitions, etc. Essentially a philosophy is a system of definitions … The chief danger to our philosophy, apart from laziness and woolliness, is scholasticism, the essence of which is treating what is vague as if it were precise and trying to fit it into an exact logical category. A typical piece of scholasticism is Wittgenstein’s view that all our everyday propositions are completely in order and that it is impossible to think illogically.

Carnap (1950, Ch. 1) wrote a famous chapter on the methodology of explication, which is central to all good philosophy, and nothing could be closer to the spirit of Ramsey (1929) than that! In particular, although both were strongly influenced by Wittgenstein, both opposed him on the exactly the same issue: both participated in the ideals of the Enlightenment of using reason constructively to improve knowledge and thereby to improve man; whereas Wittgenstein was a pessimist in the tradition of Schopenhauer and Spengler, cynical about improvements through reason (something Wittgenstein and Heidegger have in common). Ramsey was no doubt much more gifted than Carnap in mathematics and in original theorizing, but their principles were in remarkable harmony. One may say the underlying principle behind this harmony was applying a strong faith in the Testability Criterion of meaning to solve philosophical problems, strongly emphasizing the scientific nature of philosophy: Neurath’s Unity of Science !? The central question always was: What is the “cash value” of a concept? What kind of data can we find to decide an issue? This implied a strong motivation to look for technical tools and willingness to legislate new norms (Carnap’s explications and “language engineering”) and assuming responsibility for solid craftsmanship. It was the Testability Criterion that was behind their

104

ECKEHART KÖHLER

common interest in betting quotients to measure strength of belief; simultaneously both gladly imposed norms of rationality on how beliefs should evolve. Similarly, both worked hard to find the occult power of theoretical concepts, both fully appreciating the key role they play in science and its evolution. It’s unfortunate they didn’t meet in Vienna, but Ramsey was too much involved with the always troubled Wittgenstein, as well as with his own intensive psychoanalysis. (Apropos psychoanalysis: Ramsey, Carnap and Gödel all got deeply involved in extended psychoanalyses of dubious value.) A major difference between them was their divergent attitudes towards Realism, which Carnap (1928; 1934, § 17), the ultra-Conventionalist, was famous for opposing, whereas Ramsey was quite at home with Russell’s post-Idealistic Platonist Realism.24 For example, Ramsey (1925; 1931, p. 21) insisted that Richard’s paradox is not merely linguistic, as Peano claimed, but constitutes a violation of logical rationality. Calling it a mere linguistic faux pas amounts to an invitation to both logicians and linguists to dismiss it; it absolutely requires solution, and The only solution which has ever been given, that in Principia Mathematica, definitely attributed the contradictions to bad logic, and it is up to opponents of this view to show clearly the fault in what Peano calls linguistics, but what I should prefer to call epistemology, to which these contradictions are due. (1931, p. 21)

Thus Ramsey, the champion of Realism. (No wonder he became disenchanted with Wittgenstein.) Carnap’s Conventionalism of the Syntax period would seem to put him opposite Ramsey here, but in fact he was much closer than it appears. First of all, Ramsey’s Realism is strongly qualified by what may be called Cognitivist Behaviorism – to be seen in Ramsey’s explication of knowledge in terms of reliable causal chains, where observation and inference “behave” correctly; also to be seen in Ramsey’s (1929a) “explication away” of theoretical concepts as otherwise unspecified existentially quantified variables.25 Conversely, Carnap was passionately interested in logical legislation just like Ramsey. Carnap was a language engineer, but for him language was the tool of epistemology subject to pragmatic criteria which were not nearly as arbitrary as Carnap pretended at the time. Like Ramsey, Carnap was willing to put ordinary language and “conventional” philosophical traditions far behind. In a specific sense, Carnap was a closet Platonist all along. Prime evidence for this is his insistence on separating all logical sentences in the Syntax (and throughout his later research) into analytic and synthetic ones, tertium non datur, leading to his debate with Quine in defense of just this distinction. Carnap’s Kantian Apriorism was holding tough (and Kant was a Platonist in this sense). Surely Ramsey would have been on Carnap’s side here. Carnap himself ultimately solved his confusion by acknowledging fallibilistic intuition, a position whose consequences he had no time left to explore.

R AMSEY AND THE VIENNA C IRCLE ON LOGICISM

105

Both Ramsey and Carnap side with Realism on the issue of permitting nonconstructive concepts, the logical core of the issue concerning Mach’s banishment of theoretical concepts. Ramsey (1925; 1931, p. 37) wrote My method, on the other hand, is to disregard how we would construct them, and to determine them by a description of their senses or imports; and in so doing we may be able to include in the set propositions which we have no way of constructing, just as we include in the range of values of ij x propositions which we cannot express from lack of names for the individuals concerned.

This is exactly the approach Carnap takes to his “indefinite” Language II in his Syntax (1934, Ch. III), where e.g. “indefinite” (impredicative) concepts are admitted, and the non-constructive notion of logical consequence (Folgerung) is admitted, in contrast to the constructive notion of proof (Beweis) of his Language I. Similarly, Carnap (1956) freely allowed all logical methods in his work on empirical concept formation. Incidentally, Carnap was encouraged by Gödel to use just such concepts in his Syntax; and once, after a talk Carnap gave at Neurath’s apartment on psychological (psychoanalytic) concept formation, Gödel reminded Carnap that he (Gödel) had recommended dropping the usually imposed restriction on empirical concepts that they be directly (constructively) definable. As Russell had pointed out during the heyday of Platonistic Logicism, a “robust sense of Realism” forms an essential component of the motivation for successful research. Judging by the logical conventions he actually made, Carnap was behaving for all the world like a robust Realist. 8. G ÖDEL ’ S I NCOMPLETENESS , AND H IS E QUATION OF S ET T HEORY WITH P ROOF Finally, I’d like to close with a comment on Gödel’s position. I mentioned that Gödel thought Logicism was a failure because of problems with Type Theory. But there has arisen the widely held view that Gödel’s Incompleteness Theorems refute Logicism as much as they refuted Hilbert’s Program; the reason being that no specifiable theory can ever provably contain all mathematical truths. Indeed, Gödel showed earlier than Tarski that truth for any theory is not definable within that theory; this follows easily from the first Incompleteness Theorem. Many decades later, for example, Harris and Parrington (1977) first showed that combinatorial complexity theorems of the sort first discovered by Ramsey (1928), including Ramsey’s Theorem itself, although patently number theoretical, cannot be derived in second-order Peano arithmetic. Even certain much weaker Diophantine equations require assumptions from higher analysis to prove. Wilder’s recent proof of Fermat’s Last Theorem also used, apparently essentially, some very exotic higher analysis. Hence it seems the Frege–Russell program of formalizing a logic and then deriving mathematics from that formalization is doomed to failure.

106

ECKEHART KÖHLER

I disagree. Logicism is not inalienably anchored to any specific formalization of logic, nor is it necessarily forced to derive all possible mathematics. The Harris–Parrington proof did not show that Peano arithmetic failed tout court, it showed rather that Peano arithmetic needs extension: “generally accepted” proof rules or logical axioms need strengthening – whereby it is a pragmatic matter whether to strengthen rules or axioms, which Frege emphasized. Although Frege strove of course for high rigor, he made no absolute claims that his axioms and proof rules were definitive. He did not even specify any absolute criterion for formalization; in particular, he did not demand finite axiomatizability, an assumption made by those who think Gödel’s Incompleteness destroys Logicism. After experimenting with exactly one proof rule in (1879, §6), namely modus ponens, Frege later settled for a small number of finitary rules in (1893, § 48) of a sort which Hilbert regarded as models for his finitary approach. However, Frege had no clear idea whether his list of rules were “complete”, only that they seemed strong enough to prove the theorems he aimed for. Frege (1879, §6) explicitly declares it a practical matter how many proof rules should be used and what strength they should have: In logic, one counts a whole series of inference rules (Schlussarten), according to Aristotle; I use just this one – at least in all cases where, from more than one theorem (Urtheil), a new one is derived –. … Thus, an inference following any kind of inference rule can be reduced to our case. Since it is accordingly possible to make do with a single inference rule [modus ponens], the need for perspicuity [Übersichtlichkeit] therefore calls for doing so. Moreover it may be added that there would otherwise be no reason to stop with Aristotelian inference rules, but that we could indefinitely continue adding new ones: out of every theorem expressed in a formula in §§ 13 to 22, a special inference rule could be made.

The demand for perspicuity could subsequently be contravened by the contrary demand for brevity, therefore this does not rule out, writes Frege (1879, p. xiii), that, at some later time, transitions from several theorems to a new one which are possible only indirectly with this single rule could be transformed into a direct one by taking a shortcut. This may indeed be recommendable in later applications. New inference rules would thereby come about.

Thus, Frege (1893, §48) lists eight inference rules, four substitution rules and six rules for parentheses. It is clear that, in general, Frege allowed theorems to be transformed into proof rules. But then what can stop a follower of Frege from converting any transfinite axiom or theorem into a corresponding transfinite rule? Now we face the issue of finitary methodology in the sense of Hilbert, which was definitively formulated only in Gödel (1931). (NB: Hilbert never gave an exact characterization of his “finitary standpoint”; that “homework” waited for Gödel to accomplish – thus Bernays.) Frege famously demanded that all assumptions be made explicit and all proofs be without gaps, in order to guarantee

R AMSEY AND THE VIENNA C IRCLE ON LOGICISM

107

that nothing extraneous slips in. But Frege was not as much fixated, as Hilbert was, on the Phenomenalist ideal of concretely surveying proofs. In particular, I can nowhere find wording in Frege which explicitly excludes transfinite rules. As far as I can see, Frege would merely require that rules have to be explicitly stated, just like transfinite axioms – the position Carnap (1934, §43; 1935) takes, where Carnap employs transfinite rules to divide all logical statements into analytic and contradictory classes. More specifically, I think Frege would have agreed with Ramsey (1925; 1931, p. 37), quoted in the previous section, that “indefinite” concept formation, and hence transfinite axiomatizations, are permissible. The arch-finitist Hilbert (1931) himself was the first to publish such a transfinite rule, now called the Ȧ-rule, claiming it was finitary; Carnap was the first to put transfinite (“indefinite”) rules to work, although he (1934, §48) denied Hilbert’s claim, backed up by Tarski, who had already broached a similar transfinite rule in 1927. Methodologically, Hilbert’s Program was officially restricted to “concrete intuition” (roughly Kant’s sinnliche Anschauung), but as the perceptive Bernays pointed out, there is nothing much concrete left of even second-order Peano arithmetic, which Hilbert needed to arithmetize his desired completeness proof for mathematics. But Frege was a Platonist who rejected psychologistic / phenomenalistic restrictions to concrete observability and who believed that logical insight enables us to grasp logical structures beyond that which our mere empirical senses give us. So despite the prominence in Frege of what looks like a strictly finitary syntax (Begriffschrift), I think it more likely than not that Frege would have supported Carnap’s tolerant attitude toward transfinite syntax rules – so long as they are clearly specified, so that their applicability is definite. But this utterly changes the picture for those who think Logicism requires finite axiomatizability. It is true that Frege wanted his logic to cover all of classical arithmetic. Frege excluded geometry from this reduction, which he thought to be based on intuition and for this reason not logical – although he did mention geometry as a candidate for formalization using a Begriffschrift, together with physics and chemistry. It is hence not clear after all whether Frege wanted to reduce all analysis to logic, for analysis might contain some geometrical intuition. (Brouwer took a radically different path from Frege’s: whereas Frege “displaced” arithmetic intuition by logical insight (Einsicht, the same term Bolzano used) but preserved a special place for geometrical intuition, Brouwer discounted geometrical intuition as well as logical insight and gave arithmetic intuition the dominant role for all analysis! This is why the Intuitionists’ analysis contradicts most people’s geometric intuition, with its highly unexpected propositions about the continuum). Russell was famously quite cavalier about rigor compared to Frege; but on the other hand, Russell was doubtless much more insistent on reducing everything that mathematics could ever contain to logic – e.g. Cantor’s transfinite numbers. Frege had seen no need for a consistency proof, which would have required completeness. For this reason as well, it is rather unclear “how complete” Logicism is required to be. Of course philosophers of mathematics are well-

108

ECKEHART KÖHLER

acquainted with the famous debate between Frege and Hilbert on whether mathematics had to specify content: Hilbert’s Formalism “officially” ignored content, so Hilbert shifted the responsibility of mathematics to merely checking for consistency. But as Gödel (1953) emphasized, if one is convinced of the correctness of one’s theory based on one’s intuition of some content, a consistency proof is not crucial. Moreover, a misconception should be cleared up which is frequently overlooked (but not by Gödel): so long as we don’t have a proof of the contrary, a set of axioms in what Gödel called “subjective mathematics” can very well be complete despite the Incompleteness Proof ; we simply can’t prove it’s complete! There are after all, as Gödel (1951, p. 313) emphasized, other ways to “know” mathematics than by deduction, namely by induction or, to avoid equivocation, by probable reasoning, which can after all attain what Gödel calls “empirical certainty” – not to be sneezed at! We can gradually become convinced that a theory is complete if we use it for a time and find it covers everything we can imagine. This is what proponents of a “theory of everything” in physics hope for – spiting Gödel’s first Incompleteness Theorem (which after all holds for any physical theory containing arithmetic as well). There is something very deep in a certain observation of Gödel’s. He pointed out that every particular set theory is equivalent to some proof theory of a particular strength, and vice versa. Ultimately this insight must go back to Hilbert’s (Leibniz’s!) original vision of proof theory as a kind of arithmetic, in order to accomplish his consistency proof – which is what “metamathematics” is for. But Göttingen had to wait for Vienna to embarrass it with Gödel’s negative result before it was clear how the proof concept is to be mathematized: with Gödel-numbering! In the following sense Gödel equates mathematics with logic, although he still distinguishes them: Gödelization actually shows, through coding, how arithmetic is equivalent to a particular proof system – essentially the one Frege had in his more-or-less finitary Begriffschrift. Once Hilbert (1931) began to consider stronger proof rules, we can understand how theories stronger than arithmetic – e.g. various stages of analysis and on to higher set theories – would be equivalent to transfinite proof systems. Such an equivalence by encoding should not be misunderstood: Although proofs can be interpreted as set theoretical calculations, proofs still remain different from sets and are governed by different intuitions. But the equivalence by encoding allows enterprising mathematicians to go from one area to the other to double-check results, or to find results at all, just as Descartes’ discovery of analytic geometry allowed a deeper interchange and mutual enrichment of algebra and geometry. Now Gödel shared with Carnap and many others the idea that logical truth is what follows from the meaning of concepts alone. But meaning is inherently intensional and thus mental – like proofs.26 At the same time, as mentioned before, sets are taken to be non-mental, and indeed quasi-physical, as Gödel repeatedly remarked! 27 But if Gödel’s encoding equivalence is valid, we thereby discover that something mental is equated with something physical. This tantalizing problem I will leave the reader to ponder.

R AMSEY AND THE VIENNA C IRCLE ON LOGICISM

109

A PPENDIX : A L ETTER FROM R AMSEY TO F RAENKEL [The following manuscript is contained in Carnap’s papers at the University of Pittsburgh Library (where Ramsey’s papers also are). The formulas are in Carnap’s handwriting. Fraenkel had influenced Carnap to clarify the foundations of mathematics, which ultimately led to Carnap’s Logical Syntax (1934). According to his diary, Carnap met with Fraenkel in Vienna on 14, 15 and 25 March 1928, and Fraenkel presumably allowed him to copy some of the text at that time. We can be rather sure Carnap showed the text to Gödel by the following fall, who studied with him in the winter semester 1928/29 in a seminar on “Philosophische Grundlagen der Arithmetik”, where just the themes of the letter were treated, and regularly met with him in cafés. Others whom Carnap may have shown it would have been Waismann and Kaufmann, but also Schlick, who told Carnap about his correspondence and visits with Ramsey.]

081-38-01 Abschrift Ramsey an Fraenkel. Jan. 26 1928. Howfield, Buckingham Road, Cambridge. ... the fact that the so called "non-predicative processes" were of several essentially different kinds. It has always seemed to me very unfortunate that Russell's use of his Vicious-Circle Principle tended to conceal the fact that the circles he wished to eliminate were of two quite distinct kinds. I think that in a general discussion of non-predicative processes there are three things which should be clearly distinguished. First there is the entirely harmless process of describing an object by reference to a totality of which it is a member; an instance of this is "the tallest man in the room". To this process I do not see that objection can reasonably taken; certainly Russell has no such objection, and it seems to me that perhaps some slight alteration ... [X-d over by Carnap] Secondly there is the process of forming a class which is a member of itself. It seems to me that the objection to this is not that it is circular, since (if the Theory of Types be sound) it is equally wrong to suppose that the class is not a member of itself, but simply that it is nonsense. .. (It has always seemed to me that the arguments by which Russell deduced this part of the theory of types from his vicious circle principle were fallacious, but that the theory was nevertheless right in spite of the reasoning being wrong). Thirdly there is the formation of the non-elementary property of having all properties of a certain sort. It is this that makes the real difficulty, because it does seem as if the property arises subsequently to the collection of properties involved in its definition and so cannot be a member of it; whereas in the first or harmless kind of non-

110

ECKEHART KÖHLER

predicative process the object described is evidently independent of that way of describing it. Your first example of "the maximum value of a function" is, of course, the harmless kind. In Russell's symbolism it is

( x) (( ∃ y ) ƒ x = f (y) ƒ;;ƒ (z) ƒ x • f (z)) This is a ij1 (x) first-order [formula] and no sort of circle can arise, because only individual and not functional apparent variables are involved. The third or serious non-predicative process only comes in in the proof that the maximum value or upper bound exists, not in the notion of such a value. In Russell's symbolism we can say that if a function ȥ x is defined thus

ȥ x ƒ;;ƒ = ƒ;;ƒ (ij) ƒ f (ij ˆ;;z , x)

Df

we get a danger of a circle. But a definition of the form

a=

{(

x) (ij x)

Df;; the ij

is not circular whatever the form of

ijx. ———————————— ... I thought that by using Wittgenstein's work the need for the axiom of reducibility could be avoided, but he had no such idea and thought that all those parts of analysis which use the axiom of reducibility were unsound. His conclusions were more nearly those of the moderate intuitionists; what he thinks now I do not know. ... I am very glad to find that you still regard Russell as a possible alternative to Hilbert and Brouwer. I had the impression that in Germany he was regarded as entirely superseded. ... Sheffer's Theory of Notational Relativity was a manuscript which he sent to Russell; he said he was going to publish a book but has not yet done so. This manuscript dealt with various problems of combinatory analysis, from which he promised to make most remarkable applications to logic. Neither Russell nor I could in the least understand how the applications were to be made.

———————————— 8th March 1928. P.S. I ought to confess that what I said in my paper about the Axiom of Infinity doesn't now seem to me satisfactory, nor do I know what ought to be said. But the Multiplicative Axiom I don't feel to be so difficult.

C OMMENT ON R AMSEY : I NTUITING I MPREDICATIVE P ROPERTIES It may be possible to understand Impredicativity better using some ideas discussed between Gödel and Wang (1974, Ch. VI; 1996) concerning “idealized intuition”. Boolos (1971) and Parsons (1977) later took up the topic. The main idea is that of “running through” an infinite set somehow in finite time and making a judgment based on the result of applying some operation. I would like to suggest various ways to strengthen the rather simple, quasi-constructive

R AMSEY AND THE VIENNA C IRCLE ON LOGICISM

111

approach used by resorting to what may be called “hyper-processes” of perception and inference.28 For example, instead of somehow sequentially running through an infinite set, we might imagine an infinitely large array of receptors (sensory organs) with denumerably many or even continuum receptors wired together in parallel yielding an output in finite time and viewing infinitely much during one observational cycle. Not only that, we could also imagine what may be called “hyper-feedback”, such that the global state of a system is input back into one of the local registers of the system with no wait-cycles (action at a distance). In such a way, we could “amplify” the perceptual and cognitive powers of an ideal mind, obtaining perhaps what Laplace had in mind with his demon. Notice that Maxwell’s demon seems to be like Laplace’s in the first instance, but according to Szilárd’s (1929) explication of Maxwell’s demon, it obeys the laws of thermodynamics actually thought to hold in our universe; whereas Laplace’s clearly cannot; cf. Frank (1932, Ch. II, 1). This is because Maxwell’s demon, although very quick and accurate, is not assumed by Szilárd to execute “supertasks”, whereas Laplace’s demon clearly must to be able to predict the entire future of the universe from a single time-slice: Laplace’s demon must have a continuum of infinitely fast receptors, and he must solve infinitely many differential equations infinitely quickly to make his infinitely complex prediction in finite time. By the way, Parsons (1977) makes an interesting point that the ideal intuition used to iterate sets seems to be incompatible with a widely held view of theology that God cannot be active but is static, because he must be eternal, beyond time and space. This may be solvable by considering cosmologies with non-Archimedean time orderings: a “hyper-universe” could occasionally gain access to a sub-universe through worm-holes, such that, from the point of view of an observer in the hyper-universe, an infinitely long activity in the sub-universe appears to occur instantaneously in the hyper-universe. Of course, succession remains normal. But there are also cosmologies such as Gödel’s where there are closed time loops here and there. These could also be considered in configuring mental processes for ideal intuition. (Our imagination need only be limited by the demand for consistency.) Such hyper-physical processes (super-tasks) would become mental merely by being used in brain functions of an infinite mind in a hyper-world. Intelligence is just process, as Whitehead was wont to say. My guess is that, by proper configuration of suitable processes such as “hyper-feedback” and closed time-loops, an ideal intuition could be established with the capacity to “run through” any impredicative concept, so long as it’s consistent, thereby “verifying” the principle it’s based on. Constructivists won’t be pleased, because their restrictions will be violated; but their restrictions arbitrarily restrict concept formation to what they think is humanly possible – even thought they all go way beyond human capacities anyway. As Ramsey said, human ability to perceive (and reason with) only finitely many objects at once is a merely empirical accident which logic should not be bound by.

112

ECKEHART KÖHLER

N OTES 1.

2.

3.

Thus the authoritative Barwise (1977, p. vii): “Mathematical logic is traditionally divided into four parts: model theory, set theory, recursion theory and proof theory”. It should also be noted, however, that computer science departments have requisitioned logic as well: most courses taught and textbooks written in the areas of model theory and recursive functions are doubtless taught and written there. And lest anyone surmise that these courses and textbooks are restricted to finitary theories such as Turing Machines, he will be disabused of his error by glancing at IEEE Transactions periodicals in engineering libraries brimming with articles on transfinite machines and continuum game-playing automaton scenarios! All too few know that, in addition to his famous definition of cardinal number, in his Grundgesetze II, Frege (1903) also made considerable progress towards a very beautiful theory of real quantities (Positivklassen). The relation-theoretical notion of the ancestral of a relation (independently discovered by Russell 1901) was already published in Frege (1879). The explication of cardinal number was published in Frege (1884), who first stated all the Peano (1889) axioms, also independently anticipated by Dedekind (1888). Dedekind (1888, p. iii) is clearly a Logicist: “In science, nothing capable of proof ought to be accepted without proof. As reasonable as this demand seems, yet has it by no means been held to in recent expositions [a footnote refers to Schröder, Kronecker and Helmholtz], even in the foundations of the simplest science, namely that part of logic dealing with number theory. In speaking of arithmetic (algebra, analysis) as a part of logic, I frankly mean to say that I hold the number concept to be entirely independent of our notions or intuitions of space and time, but rather that it proceeds immediately from the laws of thought.” [Translation E.K.] So much for Kant! But Dedekind, on p. x of the 2nd edition of his Was sind und was sollen die Zahlen?, also selflessly gives Frege priority to his own discovery of the number axioms. Despite Dedekind’s demand for a reduction of arithmetic to logic, it needs to be emphasized that Dedekind did not actually prove his number axioms from (some version of) set theory, as Frege (1893) did! In the 3rd edition of 1911, well after the appearance of Russell’s Paradox, Dedekind movingly upholds his conviction in Logicism: “When I was requested about eight years ago to replenish the 2nd edition, already out of print, by a 3rd, I had reservations in complying because meanwhile doubt had been thrown on the reliability of important foundations of my viewpoint. Even now I make no mistake about the importance and legitimacy, in part, of this doubt. But my trust in the inner harmony of our logic has not been shaken thereby; I believe that an exact investigation of the creative power of our mind to create out of specific elements a specifically new object, their system [set], which necessarily differs from every one of its elements, will certainly lead to the design of a flawless foundation of my work [Schrift].” This important statement, so different from Frege’s disappointed withdrawal of his Logicist program, was unfortunately not included in the 1963 Dover reprint of Dedekind’s famous brochure. Dedekind seems to indicate some kind of Vicious Circle Principle in his concluding sentence to eliminate membership loops like that of Russell’s Paradox, but Dedekind’s formulation, as it stands, would exclude e.g. even his own Dedekind cuts from analysis. Ramsey (1926, § III) provided a proposal using “predicative functions” to solve this famous problem (see also Ramsey’s letter to Fraenkel in the Appendix of the present article); but ZF going back to Zermelo (1908), unfortunately not referred to by Dedekind, is already thought to have dealt adequately with the problem. Finally, Dedekind’s frank Idealism is notable in his reference to the “mind’s creation of systems”, which clearly distinguishes his Logicism from the strictly anti-psychologist Logicism of Bolzano and Frege. However, in Köhler (2000), I show that neither Bolzano nor Frege can escape psychologism, as the reference to “pure thought” in the subtitle of Frege (1879) veritably concedes. A treatment expressing rather standard negative views was Black (1933), who had studied in Göttingen and who argued that Whitehead & Russell’s Principia Mathematica made essential use of non-logical axioms in its “reduction” of mathematics to logic, violating Logicism’s original intention. In particular, the Axiom of Infinity remains even after Ramsey’s attempt to elimi-

R AMSEY AND THE VIENNA C IRCLE ON LOGICISM

113

nate the Axiom of Reducibility. Gödel basically agreed with similar lines of criticism in discussion remarks with Carnap in 1929. 4. Davis (1965) is the standard source book on the original recursion-theoretical (computability) literature, containing classic pioneering papers by Gödel, Church, Turing, Rosser, Kleene and Post, which now form a central part of the computer science curriculum. 5. Ramified Type Theory didn’t even contain the principal semantic relations of naming or satisfaction, to say nothing of logical consequence, so it was very paltry in semantics, and we may really ask: why bother with it at all? In Russell’s Foreword to the English version of Wittgenstein’s Tractatus, he famously called for the creation of a metatheory to solve Wittgenstein’s “puzzles” about allegedly not being able to say what could at best be shown about logical truths. Wittgenstein disavowed Russell’s Foreword; and metatheory had to wait for Hilbert, Carnap and Tarski. 6. The literature on set theory is extremely rich and variegated, ranging e.g. from “topoi” to “fuzzy sets”! For an authoritative treatment of much of set theory, mainly from Zermelo’s point of view, see Fraenkel, Bar-Hillel & Levi (1973); the system of von Neumann–Bernays (NB) is also discussed here, but the main source is Bernays (1958). The standard source for NBG is Gödel (1940). The early history of Z’s reception is dealt with in Moore (1977). For Quine’s NF and ML see Quine (1937) and (1940); but these systems are widely considered non-standard. 7. Originally, Quine (1937, p. 70 f.) put set theory in logic, but later, e.g. Quine (1963a, § II) and Quine (1970, pp. 64–74), he pulled it out again, for methodological reasons: first-order predicate logic is decidable and accepted as a universal standard since Hilbert & Ackermann (1928), whereas set theory is a quasi conventionalist, experimental science; cf. Orenstein (1977, p. 94) and Romanos (1983, § 2.4). This is like punishing logic for bad behavior by cutting off one of its main branches. To be sure, mathematics has misbehaved often, worst of all before Bolzano and Weierstrass cleaned up analysis. If we let the puritan Quine have his way, much of mathematics should, by parity of reasoning, be demoted to the status of accounting rules! 8. The “iterative concept of set” developed by Boolos (1971), Wang (1974, Ch. 6) and Parsons (1977) makes sets seem independent of concepts, but in fact the set-iteration procedures themselves constitute concepts! Remember that concepts are any procedures usable to classify objects, and are identical with Frege’s intensions (Arten des Gegebenseins von Gegenständen). 9. Oddly enough, Frege’s concepts (Begriffe) are in fact not any more intensional than Frege’s objects, as Pavel Materna has shown (they are “functions-in-extension”, in Russell’s terminology), whereas Bolzano’s concepts are truly intensional. 10. Leibniz’s fixation on monadic predicates is of course the main story in the pioneering book by Russell (1900): much of Leibniz’s metaphysics of monadology was dominated by his logic of monadic predicates. Couturat (1901, 1905) followed Russell. Körner (1979) thinks Russell went awry, but authoritative Leibniz-scholars such as Mates (1986), Rescher (1967) and Wilson (1989) largely side with Russell. BocheĔski (1956) and Kneale & Kneale (1962) provide fragments of relation-theoretical logic scattered throughout the history of logical texts; but (of course aside from Frege 1879), the first systematic treatment of relations was Schröder (1890– 1905), based mainly on the incomplete researches of Peirce; and, independently of Frege, Russell (1901). Structural features of relations such as symmetry, transitivity or reflectivity were naturally always felt to belong to logic, despite the tenacious fixation on syllogistics. Of course Poincaré is right in feeling that the same structures are dealt with by mathematical functions (which Frege, Peirce and Russell easily defined in terms of relations) and the algorithms they represent. One may perhaps say logic and mathematics have “equal claim” to relations. 11. The most detailed and authoritative history of logic for this period is Grattan-Guinness (2000); Coffa (1991) also provides very solid philosophical guidance, especially on semantic aspects; Hylton (1990) provides a very illuminating account of Russell’s transition from Idealism to the logic of the Principia and discusses in particular Russell’s approach to comprehension and perception of classes (i.e., sets). NB: Only after von Neumann–Bernays set theory was introduced in the thirties were classes distinguished from sets in the way now widely accepted by mathematicians. Earlier on, Bolzano had a place for them in his ontological panoply of logical objects, identifying sets with manifolds, i.e. Mannigfaltigkeiten. 12. This is close to the classic view of Cantor (1895, p. 481): “By a set ( Menge) we are to understand any collection into a whole (Zusammenfassung zu einem Ganzen) M of definite and sepa-

114

13.

14. 15. 16.

17. 18.

19.

20.

ECKEHART KÖHLER

rate objects m (which are called the ‘elements’ of M) of our intuition or our thought.” Dedekind was also a quasi-Idealist. The “collection” follows a “law” according to Cantor (1883, fn. 1): “Theory of Aggregates. With this word I denote a theoretical concept encompassing very much, which I have previously tried to develop only in the special guise of an arithmetic or geometric set theory. By an ‘aggregate’ or ‘set’ I understand in general every multiplicity which can be thought of as a unity, i.e. every quintessence of specific elements which can be combined by a law to a whole; and I believe I am thereby defining something which is related to the Platonic İȚįȠȢ or ȚįȑĮ , as well as with that which Plato calls μȚțIJȩȞ in his dialogue Philebus, or the Highest Good. He contrasts this with the ȐʌİȚȡȠȞ, i.e. the unbounded, indefinite, which I call the improper infinite (Uneigentlich-unendliche), as well as with ʌȑȡĮȢ , i.e. the bounded, and declares it to be an ordered ‘mixture’ of the latter two. That these concepts are of Pythagorean origin is indicated by Plato himself … .” This was not so for Cantor and Dedekind, however, insofar as both let sets be “legislated” by a mind. In Köhler (2001), I argue that the subject–object (or mental–physical) dichotomy is relative, and that both mental and physical interpretations can be found for both intensional and extensional entities in logic. The standard treatment of this famous episode in the history of logic and mathematics is Moore (1982). Immediately concerned is the famous “Vicious-Circle Principle”, first promulgated by Poincaré; see the authoritative treatment by Hallet (1984). This had of course been developed most fully in Whitehead & Russell (1910–13), but Russell originated its main ideas already in 1903 and developed them in detail in 1908 – the same year as Zermelo’s main publication on axiomatic set theory. Linsky (1997) thought the notorious Axiom of Reducibility could indeed be considered logical. The case is not closed. Ramsey’s ideas were discussed inside and outside the Schlick Circle meetings on Thursday evenings, but unfortunately none of the protocols by Rose Rand mention this. Of course, when Ramsey (1931) appeared, it was immediately read and discussed, in particular by Carnap and Gödel, but also by Schlick and Waismann. In correspondence, we have a better idea of what was discussed, and I reproduce a letter by Ramsey to Fraenkel of which Carnap copied a part in an appendix to this paper. Or infinitely wide dimensions. But once we consider “adjusting” dimensions, we will quickly arrive at Carnap’s clever solution of letting the individuals of type level 0 be – not physical bodies but – positions on scales of measurement. See below. At this Point in my original paper, I tendered the idea of proving the Axiom of Infinity – rather than Russell’s assuming it – by adding to the positive types negative ones descending infinitely deep. I thought this plausible on the ground that pure logic should not prejudice the empirical question of how deep the foundation of real objects goes, including whether the foundation is finitely deep. With infinitely many descending types, every type level automatically obtains infinitely many infinitely large propositional functions (or classes, or sets). In the discussion after my paper, Wolfgang Degen objected that no known proof rule could show this. I now conjecture that the situation is far worse: the cardinality of every type probably is immeasurably large, presumably exploding any and all limitations of size which are characteristically employed in the known axiom systems to avoid the paradoxes. So much for my idea of negative types. Carnap obviously came up with this idea through his familiarity with coordinate systems assigning numbers to objects through scales of measurement such as Relativity Theory presupposes; cf. Carnap (1922, 1926). Nowadays we would justify the “logical necessity” that infinitely many indexical positions exist with Carnap’s much later idea of “Meaning Postulates” (1952), which essentially state measurement norms as logic postulates. For example “nothing blue is red” or “nobody is taller than himself ”. Carnap (1963, p. 11 ff. ) was one of the very few students Frege had at the University of Jena; according to Gabriel (1996), only Carnap provided complete sets of lecture notes. Flitner (1986, p. 126f.) describes how Carnap even kept Frege’s lectures alive by motivating other students to attend. Carnap (1929, § 21b), (1950, §6), (1953, §34c) presented Frege’s famous explication of cardinal number as a model for all philosophy and mathematics of how to explicate a concept by characterizing it in terms of more elementary ones. Carnap emphasized that, in contrast to Peano’s

R AMSEY AND THE VIENNA C IRCLE ON LOGICISM

21.

22. 23.

24. 25.

26.

27.

28.

115

treatment of number, only Frege’s definition clarified exactly how cardinal numbers are used to count things, thereby proving that the cardinal number concept has to be defined the way he did. One wonders why Carnap didn’t prominently say this in his Syntax (1934) as well. Perhaps he skirted the issue because of the general disaffection with Logicism. Knowing that both Wittgenstein and Gödel rejected it, it was difficult to present a view that did not represent important members of his circle of colleagues, as Carnap strove to present his circle’s position. Geometry is a bad example, since it should properly be reckoned to physics; but it is the example foremost in the minds of Platonists, including Plato himself. Arithmetic, on the other hand, clearly is mathematical; unfortunately it’s not at all a plausible example, as Frege discovered. In Köhler (2002), I provide an explication of Platonism partly based on Gödel’s own definition, although I criticize Gödel for repeating the ancient error of assuming a Verdopplung of object domains and failing to realize that Platonic Dualism is instead based on the fact that mathematical truth has a different modality, namely normative validity. Using this simple insight, I could also solve the famous dispute between Carnap and von Neumann on information theory and entropy in Köhler (2001). My apologies to Gödel; even Frege sometimes says this. Bohnert adds the significant counter that “On the other hand, he had great hope for man’s ability to use formalism to construct an instrument whereby he could double-check intuitions by making them explicit … Indeed, he saw in formalism an instrument capable of leading man over intellectual chasms where his intuition seemed to fail altogether.” These comments are well-motivated but misleading, as “formalism” is also based on intuition, as Gödel insisted and as Carnap should have admitted. After all, the leading formalist, Hilbert, gave concrete intuition the central rôle in his epistemology, a fact consistently avoided in Carnap’s Syntax (1934). Here Ramsey was much closer to the ultra-Platonist Gödel, the “Mozart of mathematics” (Karl Menger), with whom he also shared a great and multifaceted mathematical creativity. A really “robust” Realist would claim that our intuition is powerful enough to see and identify the theoretical entities specifically ; like Russell’s “the king of France”, the variables used for theoretical entities aren’t assumed to denote anything specifically identifiable anymore. Still, by Quine’s famous existence criterion, the mere use of the variables at least implies ontological commitment to something not directly observable. But meaning is also normative, embedded in norms of rationality constitutive of mind, a fact which even Gödel missed, as a consequence of which he and Quine and many others failed to capture the true difference between logic / mathematics and natural science. Davidson’s philosophy makes essential use of just this modal distinction without realizing it. E.g. in Wang (1996), 8.1.9, 8.5.6. In Köhler (2001) and (2006), I treat the relation between the mental and the physical in much detail. On the one hand, I argue that all proofs or any other mental acts can somehow be “physicalized”; but conversely that all sets – as well as all physical objects – can be “mentalized”. The key to understanding this is simply to attend to whether one is on an object-level or a meta-level. The reason why semantics always involves mind is that semantics always uses a metatheory – whose purpose is to describe entire conceptual frameworks in order to reflect reality on some “higher level”, i.e. in some mind. Any attempt to separate theory from mind, as Bolzano, Frege and Husserl tried to do, is impossible, because theories constitute (part of) the knowledge of a mind. What Bolzano, Frege and Husserl really wanted was to distinguish empirical from normative psychology, but they didn’t realize this because they didn’t realize that logic is normative. Only in the title of Frege (1879) may the embarrassing implication be made that “pure thought” is the object of logic; and only later did Kantians widely realize that “pure” means normative. Benacerraf (1963) calls these intellectual processes “super-tasks” in honor of Zeno and the Eleatics. It is clear that the foundations of analysis and the infinitesimal calculus presuppose their existence in some sense or other; and that is a central problem for the epistemology of mathematics. To help understand super-tasks, I propose in Köhler (2002b) that they are to be situated in what I call “hyperworlds”, i.e. quasi-physical universes whose laws permit instantaneous communications which allow infinitely large receptors to observe infinite data, and infinitely fast processors to allow infinite proofs, etc.

116

ECKEHART KÖHLER

R EFERENCES Jon Barwise (ed.) (1977): Handbook of Mathematical Logic, North-Holland Publ. Co., Amsterdam. Yehoshua Bar-Hillel, E.I.J. Poznanski, Michael O. Rabin and Abraham Robinson (eds.) (1966): Essays on the Foundations of Mathematics, dedicated to A.A. Fraenkel on his seventieth birthday, Manes Press, Jerusalem. Paul Benacerraf (1963): “Tasks, Super-Tasks, and the Modern Eleatics”, The Journal of Philosophy LIX, 765–784. Paul Benacerraf (1965): “What Numbers Could Not Be”, Philosophical Review 74; reprinted in Benacerraf & Putnam (1983). Paul Benacerraf and Hilary Putnam (eds.) (1983): Philosophy of Mathematics: Selected Readings, 2nd edition, Cambridge University Press, Cambridge. Gerhard Benetka (2000): “Der ‘Fall’ Stegmüller”, in Stadler (2000). Paul Bernays (1958): Axiomatic Set Theory, with a Historical Introduction by Abraham A. Fraenkel, North-Holland Publishing Co., Amsterdam. Max Black (1933): The Nature of Mathematics, Harcourt, Brace & Co., New York. Herbert Bohnert (1963): “Carnap’s Theory of Definition and Analyticity”, in Schilpp (1963). Herbert Bohnert (1975): “Carnap’s Logicism”, in Hintikka (1975). George Boolos (1971): “The Iterative Conception of Set”, Journal of Philosophy LXVIII, 215–231; reprinted in Benacerraf & Putnam (1983) and in Boolos (1998). George Boolos (1998): Logic, Logic, and Logic, edited by Richard Jeffrey, with Introductions and Afterword by John P. Burgess, Harvard University Press, Cambridge MA. Bernd Buldt, Eckehart Köhler, Michael Stöltzner, Peter Weibel, Carsten Klein, Werner DePauliSchimanovich-Göttig (eds.) (2002): Kurt Gödel: Wahrheit und Beweisbarkeit 2. Kompendium zum Werk, öbv&hpt, Vienna. Georg Cantor (1883): Grundlagen einer allgemeinen Mannigfaltigkeitslehre, Teubner, Leipzig. Georg Cantor (1895, 1897): “Beiträge zur Begründung der transfiniten Mengenlehre”, Mathematische Annalen 46, 481–512; 49, 207–246; translated by P.E.B. Jourdin as Contributions to the Founding of the Theory of Transfinite Numbers, Open Court Publ. Co., Chicago 1915; reprinted Dover Publ. Co., New York 1952. Rudolf Carnap (1922): “Der Raum. Ein Beitrag zur Wissenschaftslehre” (Dissertation), Kantstudien, Ergänzungsheft Nr. 56, Reuther & Reichard, Berlin. Rudolf Carnap (1926): Physikalische Begriffsbildung, Verlag G. Braun, Karlsruhe; reprinted by the Wissenschaftliche Buchgesellschaft, Darmstadt 1966. Rudolf Carnap (1928): Scheinprobleme der Philosophie: Das Fremdpsychische und der Realismusstreit, Weltkreis-Verlag, Berlin; reprinted together with Der logischer Aufbau der Welt by Felix Meiner Verlag, Hamburg 1961. Rudolf Carnap (1929): Abriss der Logistik, mit besonderer Berücksichtigung der Relationstheorie und ihrer Anwendungen, Schriften zur wissenschaftlichen Weltauffassung 2 (ed. by Philip Frank & Moritz Schlick), Springer-Verlag, Vienna; this later evolved into Carnap (1954). Rudolf Carnap (1934): Logische Syntax der Sprache, Schriften zur wissenschaftlichen Weltauffassung 8 (ed. by Philip Frank & Moritz Schlick), Springer-Verlag, Vienna; transl. as Carnap (1937). Rudolf Carnap (1935): “Ein Gültigkeitskriterium für die Sätze der klassischen Mathematik”, Monatshefte für Mathematik und Physik, 42, 163–190; translated as §§34a–34i of Carnap (1937). Rudolf Carnap (1937): The Logical Syntax of Language, translation of Carnap (1934) by Amethe Smeaton (Countess von Zeppelin) with supplements, Routledge & Kegan Paul, London. Rudolf Carnap (1947): Meaning and Necessity, a Study in Semantics and Modal Logic, University of Chicago Press, Chicago. Rudolf Carnap (1950): Logical Foundations of Probability, University of Chicago Press, Chicago.

R AMSEY AND THE VIENNA C IRCLE ON LOGICISM

117

Rudolf Carnap (1952): “Meaning Postultates”, Philosophical Studies 3, 65–73. Rudolf Carnap (1954): Einführung in die symbolische Logik mit besonderer Berücksichtigung ihrer Anwendungen, Springer-Verlag, Vienna; translated by Wm. H. Meyer & John Wilkinson as Introduction to Symbolic Logic and Its Applications, Dover Publications, New York 1958. Rudolf Carnap (1956): “The Methodological Character of Theoretical Concepts”, in Herbert Feigl and Michael Scriven (eds.): Minnesota Studies in the Philosophy of Science I, The Foundations of Science and the Concepts of Psychology and Psychoanalysis, University of Minnesota Press, Minneapolis 1956. Rudolf Carnap (1958): “Beobachtungssprache und theoretische Sprache”, Dialectica 12, 236–248; this double issue was a Festschrift for Paul Bernays and was published separately as Logica Studia Paul Bernays dedicata, by Editions Grifon, Neuchâtel. Rudolf Carnap (1962): “The Aim of Inductive Logic”, in Ernest Nagel, Patrick Suppes & Alfred Tarski: Logic, Methodology and Philosophy of Science, Stanford University Press, Stanford. Rudolf Carnap (1963): “Intellectual Autobiography”, in Schilpp (1963). Rudolf Carnap (1966): “On the Use of Hilbert’s İ-Operator in Scientific Theories”, in Bar-Hillel et al. (1966). Rudolf Carnap (1968): “Inductive Logic and Inductive Intuition” (paper given in London 1965), in Lakatos (1968). Alonzo Church (1940): “A Formulation of the Simple Theory of Types”, Journal of Symbolic Logic 5, 56–68. Alonzo Church (1951): “A Formulation of the Logic of Sense and Denotation”, in Henle, Kallen & Langer (1951). Alonzo Church (1956): Introduction to Mathematical Logic I, Princeton University Press, Princeton. Peter J. Clark (2004): “Frege, Neo-Logicism and Applied Mathematics”, in Galavotti & Stadler (2004). J. Alberto Coffa (1991): The Semantic Tradition from Kant to Carnap, Cambridge University Press, Cambridge. Louis Couturat (1901): La Logique de Leibniz d’après des documents inédits, Alcan, Paris; reprinted Olms, Hildesheim 1961. Louis Couturat (1902): “Sur la métaphysique de Leibniz”, Revue de métaphysique et de morale 10, 1–25; reprinted in Frankfurt (1972). Louis Couturat (1905): Les principes des mathématiques, avec un appendice sur la philosophie des mathématiques de Kant, Alcan, Paris; reprinted Olms, Hildesheim 1965. Johannes Czermak (ed.) (1993): Philosophy of Mathematics, Proceedings of the 15th International Wittgenstein Symposium, Hölder-Pichler-Tempsky, Vienna. Martin Davis (ed.) (1965): The Undecidable. Basic Papers on Undecidable Propositions, Unsolvable Problems and Computable Functions, Raven Press, Hewlett NY. Richard Dedekind (1888): Was sind und was sollen die Zahlen?, Vieweg, Brunswick; 2nd edition 1893; 3rd edition 1911; reprinted 1965; 2nd edition translated by Woodrow Woodruff Beman in Essays on the Theory of Numbers, Open Court Publishing Co., Chicago 1901; reprinted by Dover Publ. Co., New York 1963. Wolfgang Degen (1993): “Two Formal Vindications of Logicism”, in Czermak (1993). Wolfgang Degen (1999): “Complete Infinitary Type Logics”, Studia Logica 63, 85–119. Wolfgang Degen and Jan Johannsen (2000): “Cumulative Higher-Order Logic as a Foundation for Set Theory”, Mathematical Logic Quarterly 46, 147–170. Wilhelm Flitner (1986): Gesammelte Schriften 11, Paderborn. Abraham Adolf Fraenkel, Yehoshua Bar-Hillel and Azriel Levi (1973): Foundations of Set Theory, 2nd revised edition with the collaboration of Dirk van Dalen, Elsevier, Amsterdam. Philipp Frank (1932): Das Kausalgesetz und seine Grenzen, Schriftenreihe zur wissenschaftlichen Weltauffassung 6 ed. by Ph. Frank & M. Schlick, Springer-Verlag, Vienna; reprinted by Suhrkamp stw734, Frankfurt am Main 1988. Harry Frankfurt (ed.) (1972): Leibniz. A Collection of Critical Essays, Anchor Books AP16, Doubleday, New York.

118

ECKEHART KÖHLER

Gottlob Frege (1879): Begriffschrift. Eine der arithmetischen nachgebildete Formelsprache des reinen Denkens, Louis Nebert, Halle; reprinted Georg Olms, Hildesheim 1964. Gottlob Frege (1884): Die Grundlagen der Arithmetik. Eine logisch-mathematische Untersuchung über den Begriff der Zahl, Breslau; reprinted by Georg Olms, Hildesheim 1961; translated by J.L. Austin, Oxford 1960. Gottlob Frege (1893, 1903): Grundgesetze der Arithmetik, begriffschriftlich abgeleitet I, II, Jena; reprinted Georg Olms, Hildesheim 1962; translated by Montgomery Furth as The Basic Laws of Arithmetic, University of California Press, Berkeley 1967. Gottfried Gabriel (1996): “Gottlob Frege. Vorlesungen über Begriffschrift. Nach der Mitschrift von Rudolf Carnap”, History and Philosophy of Logic 17. Maria Carla Galavotti and Friedrich Stadler (eds.) (2004): Induction and Deduction in the Sciences, Vienna Circle Institute Yearbook 11, Kluwer, Dordrecht. Kurt Gödel (1931): “Über formal unentscheidbare Sätze der Principia mathematica und verwandter Systeme I ”, Monatshefte für Mathematik und Physik 38, 35–72; translated in van Heijenoort (1967); reprinted with commentary by S.C. Kleene in Gödel (1986). Kurt Gödel (1940): The Consistency of the Axiom of Choice and of the Generalized ConinuumHypothesis with the Axioms of Set Theory, Annals of Mathematics Studies 3, Princeton University Press, Princeton; 7th reprinting 1966; reprinted with commentary by R.S. Solovay in Gödel (1990). Kurt Gödel (1944): “Russell’s Mathematical Logic”, in Schilpp (1944); reprinted with commentary by Charles Parsons in Gödel (1990). Kurt Gödel (1951): “Some Basic Theorems on the Foundations of Mathematics and Their Implications”, in Gödel (1995); this is the so-called “Gibbs-Lecture” held at the AMS meeting at Brown University. Kurt Gödel (1953): “Is Mathematics Syntax of Language?”, in Gödel (1995); this was originally intended for the Schilpp volume (1963) on Carnap, but withdrawn. Kurt Gödel (1986, 1990): Collected Works I, Publications 1929–1936; II Publications 1938–1974, edited by Solomon Feferman (editor-in-chief), John W. Dawson, Jr., Stephen C. Kleene, Gregory H. Moore, Robert Solovay and Jean van Heijenoort, Oxford University Press, New York. Kurt Gödel (1995): Collected Works III, Unpublished Essays and Lectures, edited by Solomon Feferman (editor-in-chief), John W. Dawson, Jr., Warren Goldfarb, Charles Parsons and Robert Solovay Oxford University Press, New York. Ivor Grattan-Guinness (2000): The Search for Mathematical Roots, 1870–1940: Logics, Set Theories and the Foundations of Mathematics from Cantor through Russell to Gödel, Princeton University Press, Princeton. Michael Hallet (1984): Cantorean Set Theory and Limitation of Size, Oxford Univ. Press, Oxford. Jean van Heijenoort (ed.) (1967): From Frege to Gödel. A Source Book in Mathematical Logic, 1879–1931, Harvard University Press, Cambridge MA. Paul Henle, Horace Kallen and Susanne K. Langer (eds.) (1951): Structure, Method and Meaning: Essays in Honor of Henry M. Sheffer, with a forward by Felix Frankfurter, Liberal Arts Press, New York. David Hilbert (1926): “Über das Unendliche”, Mathematische Annalen 95, 161–190; partially translated in Benacerraf & Putnam (1983). David Hilbert and Wilhelm Ackermann (1928): Grundzüge der theoretischen Logik, Springer-Verlag, Berlin; 4th edition 1959 ; translated as Principles of Mathematical Logic, Chelsea Publishing Co., New York 1950. Jaakko Hintikka (ed.): Rudolf Carnap, Logical Empiricist, Reidel Publ. Co., Dordrecht 1975. Peter Hylton (1990): Russell, Idealism and the Emergence of Analytic Philosophy, Oxford University Press, Oxford. Felix Kaufmann (1930): Das Unendliche in der Mathematik und seine Ausschaltung, Franz Deuticke, Vienna; reprinted by the Wissenschaftliche Buchgesellschaft, Darmstadt 1968; translated by Paul Foulkes as The Infinite in Mathematics, edited by Brian McGuinness with an Introduction by Ernest Nagel, Vienna Circle Collection 9, Reidel, Dordrecht 1978. Hubert C. Kennedy (1980): Peano. Life and Works of Giuseppe Peano, Reidel, Dordrecht.

R AMSEY AND THE VIENNA C IRCLE ON LOGICISM

119

William and Martha Kneale (1962): The Development of Logic, Oxford Univ. Press, Oxford. Eckehart Köhler (2000): “Logic Is Objective and Subjective”, in Timothy Childers & Jari Palomäki (eds.): Between Words and Worlds. A Festschrift for Pavel Materna, Filosofia, Prague. Eckehart Köhler (2001): “Why von Neumann Rejected Carnap’s Dualism of Information Concepts”, in Rédei & Stöltzner (2001). Eckehart Köhler (2002): “Gödel und der Wiener Kreis”, in Köhler, Weibel et al. (2002). Eckehart Köhler (2002a): “Gödels Jahre in Princeton”, in Köhler, Weibel et al. (2002). Eckehart Köhler (2002b): “Gödels Platonismus”, in Buldt, Köhler et al. (2002). Eckehart Köhler (2006): “Intuition Regained. Gödel’s Views on Intuition, and How Carnap Abandoned Empiricism by Accepting Intuition as Evidence”, forthcoming, Synthese. Eckehart Köhler, Peter Weibel, Michael Stöltzner, Bernd Buldt, Carsten Klein und Werner DePauliSchimanovich-Göttig (eds.) (2002): Kurt Gödel: Wahrheit und Beweisbarkeit 1. Dokumente und historische Analysen, öbv&hpt, Vienna. Stephan Körner (1979): “On Russell’s Critique of Leibniz’s Philosophy”, in Roberts (1979). Imre Lakatos (ed.) (1968): The Problem of Inductive Logic. Proceedings of the International Colloquium in the Philosophy of Science, London, 1965, Vol. 2, North-Holland Publ. Co., Amsterdam. Leonard Linsky (1997): “Was the Axiom of Reducibility a Principle of Logic?”, in Tait (1997). Benson Mates (1986): The Philosophy of Leibniz, Oxford Univerity Press, Oxford. Gregory H. Moore (1982): Zermelo’s Axiom of Choice, Its Origins, Development, and Influence, Springer-Verlag, Heidelberg. Bryan Norton: Linguistic Frameworks and Ontology. A Re-Examination of Carnap’s Meta-Philosophy, Janua Linguarum, Mouton Publishers, The Hague 1977. Alex Orenstein (1977): Willard Van Orman Quine, Twayne’s World Leaders Series 65, H.K. Hall, Boston. Jeff Paris and Leo Harrington (1977): “A Mathematical Incompleteness in Peano Arithmetic”, in Barwise (1977). Charles Parsons (1977): “What Is the Iterative Conception of Set?”, in Logic, Foundations of Mathematics, and Computability Theory, Proceedings of the 5th International Congress of Logic, Methodology and the Philosophy of Science (London ON 1975), edited by Robert Butts & Jaakko Hintikka, Reidel, Dordrecht 1977; reprinted in Benacerraf & Putnam (1983). Giuseppe Peano (1889): Arithmetices principia, nova methodo exposita, Bocca, Turin; reprinted in Opere scelte 2, Edizione cremonese, Rome; transl. with a biographical sketch by Hubert C. Kennedy: Selected Works of Giuseppe Peano, University of Toronto Press, Toronto 1973. Willard Van Orman Quine (1937): “New Foundations for Mathematical Logic”, American Mathematics Monthly 44, 70–80; reprinted in Quine (1953). Willard Van Orman Quine (1940): Mathematical Logic, Harvard University Press, Cambridge MA; rev. 1951. Willard Van Orman Quine (1951): “Two Dogmas of Empiricism”, in Quine (1953). Willard Van Orman Quine (1953): From a Logical Point of View: Logico-Philosophical Essays, Harvard University Press, Cambridge MA. Willard Van Orman Quine (1963): Set Theory and Its Logic, Harvard University Press, Cambridge MA. Willard Van Orman Quine (1963a): “Carnap and Logical Truth”, in Schilpp (1963); reprinted in Quine (1966). Willard Van Orman Quine (1966): The Ways of Paradox and Other Essays, Random House, New York. Willard Van Orman Quine (1970): Philosophy of Logic, Prentice-Hall, Englewood Cliffs NJ. Frank Plumpton Ramsey (1925): “Foundations of Mathematics”, Proceedings of the London Mathematical Society 25, 338–384; reprinted in Ramsey (1931, 1978). Frank Plumpton Ramsey (1926): “Truth and Probability”; reprinted in Ramsey (1931, 1978). Frank Plumpton Ramsey (1928): “On a Problem of Formal Logic”, Proceedings of the London Mathematical Society 30, 338–384.

120

ECKEHART KÖHLER

Frank Plumpton Ramsey (1929): “Theories”, in Ramsey (1931, 1978). Frank Plumpton Ramsey (1929a): “Philosophy”, in Ramsey (1931). Frank Plumpton Ramsey (1931): The Foundations of Mathematics and Other Logical Essays, ed. by R.B. Braithwaite with an Introduction by G.E. Moore, Routledge & Kegan Paul, London. Frank Plumpton Ramsey (1978): Foundations. Essays in Philosophy, Logic, Mathematics and Economics, re-edited version of Ramsey (1931) with a different selection of texts, including especially the two papers on taxation and savings, by D.H. Mellor with introductions by D.H. Mellor, L. Mirsky, T.J. Smiley and Richard Stone, Routledge & Kegan Paul, London 1978. Miklós Rédei and Michael Stöltzner (eds.) (2001): John von Neumann and the Foundations of Quantum Mechanics, Vienna Circle Institute Yearbook 8, Kluwer, Dordrecht. Nicholas Rescher (1967): The Philosophy of Leibniz, Prentice-Hall, New York. George W. Roberts (ed.) (1979): Bertrand Russell Memorial Volume, Allen & Unwin, London. George Romanos (1983): Quine and Analytic Philosophy, MIT Press, Cambridge MA. Bertrand Russell (1900): A Critical Exposition of the Philosophy of Leibniz, Cambridge University Press, Cambridge; 2nd ed. Allen & Unwin, London 1937. Bertrand Russell (1901): “Sur la logique des relations avec des applications à la théorie des séries”, Revue de Mathématique (Rivista di Matematica) VII, 115–148; translated in Russell (1956). Bertrand Russell (1903): The Principles of Mathematics, Allen & Unwin, London; 2nd ed. 1937. Bertrand Russell (1908): “Mathematical Logic as Based on the Theory of Types”, American Journal of Mathematics 28, 222–262; reprinted in Russell (1956). Bertrand Russell (1914): “On the Nature of Acquaintance”, The Monist XXIV, 1–16, 161–187, 435– 453; reprinted in Russell (1956). Bertrand Russell (1956): Logic and Knowledge: Essays 1901–1950, edited by Robert Charles Marsh, Allen & Unwin, London. Paul Arthur Schilpp (ed.) (1944): The Philosophy of Bertrand Russell, Library of Living Philosophers V, Northwestern University Press, Evanston IL. Paul Arthur Schilpp (ed.) (1963): The Philosophy of Rudolf Carnap, Library of Living Philosophers XI, Open Court Publishing Co., La Salle IL. Ernst Schröder (1890–1905): Vorlesungen über die Algebra der Logik I–III, Leipzig. Abner Shimony (1953): A Theory of Confirmation, Ph.D. dissertation at Yale University. Abner Shimony (1955): “Coherence and the Axioms of Confirmation”, Journ. o. Sym. Log. 20, 1–28. Friedrich Stadler (ed.) (2000): Elemente moderner Wissenschaftstheorie, Veröffentlichungen des Instituts Wiener Kreis 8, Springer-Verlag, Vienna. Leo Szilárd (1929): “Über die Entropieverminderung in einem thermodynamischen System bei Eingriffen intelligenter Wesen”, Zeitschrift für Physik, 53, 840–856; translated as “On the Decrease of Entropy in a Thermodynamic System by the Intervention of Intelligent Beings” by Anatol Rapoport and Mechthilde Knoller in Behavioral Science, 9, 301–310; the latter reprinted in Szilard (1972). Leo Szilárd (1972): The Collected Works of Leo Szilard: Scientific Papers, ed. by B.T. Field and G. Weiss, MIT Press, Cambridge MA. William W. Tait (ed.) (1997): Early Analytic Philosophy: Frege, Russell, Wittgenstein, Open Court Publishing Co., La Salle IL. Hao Wang (1974): From Mathematics to Philosophy, Humanities Press, New York; Ch. VI “The Concept of Set” is reprinted in Benacerraf & Putnam (1983). Hao Wang (1987): Reflections on Kurt Gödel, MIT Press, Cambridge MA. Hao Wang (1996): A Logical Journey: From Gödel to Philosophy, MIT Press, Cambridge MA. Alfred North Whitehead and Bertrand Russell (1914): Principia Mathematica, Cambridge University Press, Cambridge; the 2nd edition of 1925 included a long new Introduction largely influenced by Ramsey. Catherine Wilson (1989): Leibniz’s Metaphysics: A Historical and Comparative Study, Princeton University Press, Princeton. Ernst Zermelo (1904): “Beweis, dass jede Menge wohlgeordnet werden kann”, Mathematische Annalen 59, 139–141; translated in van Heijenoort (1967).

R AMSEY AND THE VIENNA C IRCLE ON LOGICISM

121

Ernst Zermelo (1908): “Neuer Beweis für die Möglichkeit einer Wohlordnung”, Mathematische Annalen 65, 107–128; translated in van Heijenoort (1967). Ernst Zermelo (1908a): “Untersuchungen über die Grundlagen der Mengenlehre I”, Mathematische Annalen 65, 261–281; translated in van Heijenoort (1967).

Dept. of Business Administration University of Vienna – BWZ Brünnerstraße 72 1210 Vienna Austria [email protected]

J.W. Degen

Logical Problems Suggested by Logicism

The mathematics of logic is difficult, the logic of mathematics is even more difficult.

1. Introduction Let us call the Logicist Thesis, or the Thesis of Logicism, or simply Logicism the thesis that [LT] Pure Mathematics is part of Logic. The founding fathers of Logicism are Frege and Russell. Roughly, Frege maintained that (at least) higher-order arithmetic is part of logic, but definitively not geometry, whereas for Russell even all of pure mathematics was to be part of logic. Unfortunately, Frege’s system GGA (Grundgesetze der Arithmetik, 1893, 1903) [6] by means of which he wanted to prove his version of [LT] was shown to be inconsistent by Russell. However, had GGA been consistent it would have proved, provided it is logic, a much stronger version of [LT] than Frege envisaged since GGA contains all of set theory despite the more modest title Grundgesetze der Arithmetik.1 Russell’s manifesto of his Logicism is to be found in his Principles of Mathematics of 1903; it is firmly repeated in the second edition of 1938 [11]2. The formal implementation followed 1908 in Russell’s Mathematical logic as based on the theory of types [12], and then in the Principia Mathematica written with Whitehead, published 1910–13. The second edition of PM of 1927 [14] seems to be still in print. The sentence [LT] as stated above contains three undefined phrases: (1) Pure Mathematics (2) is part of (3) Logic Furthermore, even if these three notions are defined in some way or other, there remains the following ambiguity in [LT]: nonuniform[LT]: For every (sharply delineated) piece M of Pure Mathematics there exists a logic LM such that M is part of LM .

123 M. C. Galavotti (ed.), Cambridge and Vienna: Frank P. Ramsey and the Vienna Circle, 123–138. © 2006 Springer. Printed in the Netherlands.

124

J.W. Degen

uniform[LT]: There is one universal logic Luniv such that all of Pure Mathematics is part of Luniv . Finally, we have the following weak version of the logicist thesis. weak[LT]: The (or some) main part of Pure Mathematics is part of some Logic. The cautious distinctions just given are necessitated by the mathematical and logical experience between 1903 and 2003. It is possible to refute even the version weak[LT]. For instance, let us say that Logic is just first-order logic, and that each main part of pure mathematics must contain some nontrivial arithmetic. Then weak[LT] is false. Also, even if we admit pure classical type theory CT , i.e. P M \ inf inity as a logic, weak[LT] will become false. On the other hand, if we admit ZF C as Logic, and are not squeamish about the vast incompleteness of ZF C, then we may even argue for uniform[LT]. Considering as the main part of Pure Mathematics ordinary mathematics as known by Russell, namely classical analysis, algebra and certain parts of Cantor’s set theory (below ℵω ) and admitting P M as logic, then at least weak[LT] can be vindicated.

2. Some Preparatory Clarifications In my talk I do not want to refute any of these logicist theses, not even the strongest among them. Rather, I will prove (a version of) weak[LT]. Furthermore, I claim that my proof is non-trivial and will yield new information and logical (or mathematical) problems about the logical (or mathematical) status of CT , P M , extensions and variants thereof. I must admit that a philosophically satisfactory analysis of Logicism should dwell more carefully on big questions such as: What is mathematics? What is Logic? I decide these questions by fiat in order not to impair my message by difficulties extraneous to its (rather precisely statable) mathematicallogical content.3 2.1 Rich Model-theoretic Logics For our purpose, let us define a model-theoretic logic L to be a triple (M, |=, L). Here M is a class of models, L is a language4 conceived of as a class of sentences; and for M ∈ M and ϕ ∈ L we mean by M |= ϕ that M makes ϕ true. By V alid(L) we understand the valid sentences of L, i.e. those sentences which are true in all models M ∈ M. We call a (model-theoretic) logic rich if M is large, i.e. a big set or a proper class of pairwise non-equivalent5 models. If L is rich then V alid(L) captures the intuitive notion of a universally (or logically) true sentence, i.e. a sentence true under all possibilities or true in all possible worlds. Of course, if M consists of just one or two models, then V alid(L) is far from being a set of universally true sentences, in the intuitively required sense.6

Logical Problems Suggested by Logicism

125

Let now L be a rich logic, and P a part of pure mathematics. We will say that ∗ L (semantically) captures P if there is a syntactic interpretation (−) such that (I) For all results π of P : π ∗ ∈ V alid(L). Thus far, we have not said anything about the way P is presented; nor what it means that π is a result of P . Indeed, we have not said what P really is. It may be a structure or a theory, or an activity or whatnot. However, it must be something definite if we want to prove something about it. Our discussion hitherto is connected with Logicism by the stipulation that the interpretation π ∗ is to be a universally valid sentence of a rich logic. Nothing is stipulated about the genuine or intrinsic logicality of the notions representable by the language L of the model-theoretic logic L. In spite of this generality, several possibilities are already ruled out, e.g., the case that π ∗ is a set-theoretic sentence (of the first-order standard language {∈} of set theory) which is true just in the model (Vω+ω , ∈). But it does not rule out the related case where the model class considered is Mzermelo = {(Vα , ∈) : α a limit ordinal ≥ ω + ω}. Definition (I) gives a semantic version of weak[LT] with respect to the rich model-theoretic logic L, and the chosen part P of pure mathematics. Now, if P is ordinary pure mathematics formulated in set-theoretic terms, and we take the rich model-theoretic logic Lzermelo = (Mzermelo , |=, {∈}), then Lzermelo captures P (semantically). Neither Frege nor Russell had such a purely semantic version of Logicism in mind, although there exists a precise one, as just explained. Nevertheless, something like our semantic version of Logicism was surely implied by their logicisms. Moreover, Frege and Russell had (also) some system of proofs in mind, and – being of the highest importance for their project – several lists of so-called logical definitions of mathematical concepts. However, the semantic version of Logicism has, as a foundational standpoint, no great a priori plausibility, and the possible fruits of a realization of this version are rather dubious. For let us take as an example of pure mathematics the (or a) second-order theory T of the real numbers qua completely ordered field (this is really third-order arithmetic in so far as real numbers can be modelled as sets of natural numbers). Then, why should every theorem ϕ of T translate into a sentence ϕ∗ ∈ V alid(L) for some rich model-theoretic logic L? That is to say: why should all or at least many sentences which are true in some individual structure translate into sentences which are true in all structures of a certain kind? Neither Frege nor Russell had the conceptual tools even to formulate this question. But we can see already from these considerations that there is a big lack of motivation in the very idea of Logicism, at least when formulated semantically; for, why should a sentence which is true in few mathematical structures, or just in the field of real numbers, not belong to pure mathematics? Does truth in all structures endow a sentence with a dignity over and beyond those sentences which are true in just one structure?

126

J.W. Degen

2.2 Logical proofs Besides the rich logic L we are now going to introduce proofs into our picture since, as just mentioned, both Frege and Russell explicitly envisaged proofs as part of their logicist programme. Suppose that Σ is a system of proofs which is correct with respect to L, i.e. if Σ  ϕ, then ϕ ∈ V alid(L). Then we will say that Σ captures (syntactically) the piece P of mathematics if (II) For all results π of P : Σ  π ∗ . Of course, (II) implies (I), but not the other way around unless Σ is (semantically) complete with respect to (the model class of) L. Version (II) commands the most interest when the proofs of the proof system Σ are logical proofs. But what is a logical proof? An unobjectionable definition would run like this. Let a rich logic L be given. A proof is a sort of tree whose nodes are sentences from the language L connected by applications of inference rules. Such a proof is logical (with respect to L) if its leaves are members of V alid(L) and the inferences preserve membership in V alid(L). Note that nothing in our definitions presupposes that the sentences of the rich logic L are finite symbolic configurations, or that the proofs in Σ are finite trees. Moreover, we do not assume that the models in L are finite. Why should we? 3. The Argument Now I will present the promised proof; it will use a certain system Σℵ1 of logical proofs, and two associated rich logics. In order to prove (mathematically, or logically) my claim that Σℵ1 syntactically (and therefore semantically) captures a part P of mathematics, this P must be made precise. Although it may seem, prima facie, both logical and historical nonsense, we set P := P M , that is, unramified Principia Mathematica with a full comprehension schema and an axiom of infinity. The perfectly exact definition of P M will be given presently. I have promised a nontrivial proof of weak[LT]. If P M is a logic (and I do not deny this), then we have a proof of weak[LT] via P M  π =⇒ P M  π ∗ ∗ with (−) the identity function. Certainly, this proof is trivial. Regardless of whether P M is a logic or not, Σℵ1 will be logical in a higher degree than P M , and that in a precise sense of logical; moreover, Σℵ1 will turn out to be a natural and systematic strengthening of P M . That P M , in turn, captures a large part of mathematics is well known; it captures more mathematics than anyone of us will ever learn. Disregarding incompletenesses of G¨odelian 1931-type and purely set-theoretic questions like hypotheses about the continuum, P M is “practically complete" for ordinary mathematics (perhaps when enlarged by forms of the axiom of choice). Thus, if P M as it stands is a logic, Russell and Whitehead have proved Logicism at least in the version weak[LT].

127

Logical Problems Suggested by Logicism

The announced system Σℵ1 is just one in an infinite hierarchy of type logics all of which have a very simple definition. This hierarchy, along with two related hierarchies concerning forms of AC, are fully presented and investigated in my paper [3]. First, the types. Let κ be ℵ0 , ℵ1 , ℵ2 , . . .7 Then the κ-types are defined as follows (1) 0 is a κ-type. (2) if α < κ, and if (τξ )ξ