Philosophy of mathematics and economics image, context and perspective

Philosophy of Mathematics and Economics With the failure of economics to predict the recent economic crisis, the image of economics as a rigorous mathematical science has been subjected to increasing interrogation One explanation for this failure is that the subject took a wrong turn in its historical trajectory, becoming too mathematical Using the philosophy of mathematics, this unique book re-examines this trajectory Philosophy of Mathematics and Economics re-analyses the divergent rationales for mathematical economics by some of its principal architects Yet, it is not limited to simply enhancing our understanding of how economics became an applied mathematical science The authors also critically evaluate developments in the philosophy of mathematics to expose the inadequacy of aspects of mainstream mathematical economics, as well as exploiting the same philosophy to suggest alternative ways of rigorously formulating economic theory for our digital age This book represents an innovative attempt to more fully understand the complexity of the interaction between developments in the philosophy of mathematics and the process of formalisation in economics Assuming no expert knowledge in the philosophy of mathematics, this work is relevant to historians of economic thought and professional philosophers of economics In addition, it will be of great interest to those who wish to deepen their appreciation of the economic contours of contemporary society It is also hoped that mathematical economists will find this work informative and engaging Thomas A Boylan is Professor Emeritus of Economics of the National University of Ireland, Galway His main research and teaching interests have been in Economic Growth and Development Theory; Applied Econometrics; Philosophy/Methodology of Economics; Post-Keynesian Economics; and the History of Irish Economic Thought Paschal F O’Gorman is Professor Emeritus of Philosophy of the National University of Ireland, Galway His main research and teaching areas have been in the Philosophy of Science; Logic; Philosophy of Mind; and, since the 1980s, the Philosophy and Methodology of Economics Routledge INEM Advances in Economic Methodology Series Edited by Esther-Mirjam Sent, the University of Nijmegen, the Netherlands The field of economic methodology has expanded rapidly during the last few decades This expansion has occurred in part because of changes within the discipline of economics, in part because of changes in the prevailing philosophical conception of scientific knowledge, and also because of various transformations within the wider society Research in economic methodology now reflects not only developments in contemporary economic theory, the history of economic thought, and the philosophy of science; but it also reflects developments in science studies, historical epistemology, and social theorizing more generally The field of economic methodology still includes the search for rules for the proper conduct of economic science, but it also covers a vast array of other subjects and accommodates a variety of different approaches to those subjects The objective of this series is to provide a forum for the publication of significant works in the growing field of economic methodology Since the series defines methodology quite broadly, it will publish books on a wide range of different methodological subjects The series is also open to a variety of different types of works: original research monographs, edited collections, as well as republication of significant earlier contributions to the methodological literature The International Network for Economic Methodology (INEM) is proud to sponsor this important series of contributions to the methodological literature For a list of titles please visit: www.routledge.com/Routledge-INEM-Advancesin-EconomicMethodology/book-series/SE0630 13 The End of Value-Free Economics Hilary Putnam and Vivian Walsh 14 Economics for Real Aki Lehtinen, Jaako Kuorikoski and Petri Ylikoski 15 Philosophical Problems of Behavioural Economics Stefan Heidl 16 Philosophy of Mathematics and Economics Image, Context and Perspective Thomas A Boylan and Paschal F O’Gorman Philosophy of Mathematics and Economics Image, Context and Perspective Thomas A Boylan and Paschal F O’Gorman First published 2018 by Routledge Park Square, Milton Park, Abingdon, Oxon, OX14 4RN and by Routledge 711 Third Avenue, New York, NY 10017 Routledge is an imprint of the Taylor & Francis Group, an informa business © 2018 Thomas A Boylan and Paschal F O’Gorman The right of Thomas A Boylan and Paschal F O’Gorman to be identified as authors of this work has been asserted by them in accordance with sections 77 and 78 of the Copyright, Designs and Patents Act 1988 All rights reserved No part of this book may be reprinted or reproduced or utilised in any form or by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying and recording, or in any information storage or retrieval system, without permission in writing from the publishers Trademark notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloging-in-Publication Data Names: Boylan, Thomas A., author | O’Gorman, Paschal F (Paschal Francis), 1943- author : Philosophy of mathematics and economics : image, context and perspective / Thomas A Boylan and Paschal F O’Gorman cription: Edition | New York : Routledge, 2018 | Series: Routledge INEM advances in economic methodology | Includes bibliographical references and index tifiers: LCCN 2017051813 (print) | LCCN 2017054526 (ebook) | ISBN 9781351124584 (Ebook) | ISBN 9780415161886 (hardback : alk paper) | ISBN 9781351124584 (ebk) ects: LCSH: Economics, Mathematical Classification: LCC HB135 (ebook) | LCC HB135 B69 2018 (print) | DDC 330.01/51 dc23 LC record available at https://lccn.loc.gov/2017051813 ISBN: 978-0-415-16188-6 (hbk) ISBN: 978-1-351-12458-4 (ebk) We would like to dedicate this book to our respective grandchildren, Aifric Donald, Lara Boylan, Alison O’Sullivan and Symone O’Gorman Contents Preface Introduction Economics and mathematics: Image, context and development Walras’ programme: The Walras–Poincaré correspondence reassessed The formalisation of economics and Debreu’s philosophy of mathematics The axiomatic method in the foundations of mathematics: Implications for economics Hahn and Kaldor on the neo-Walrasian formalisation of economics Rationality and conventions in economics and in mathematics The emergence of constructive and computable mathematics: New directions for the formalisation of economics? Economics, mathematics and science: Philosophical reflections Appendix Bibliography Index Preface This book has had arguably a uniquely long gestation period The original proposal was first drafted while we attended the 10th International Congress of Logic, Methodology and Philosophy of Science in the magnificent setting of the Palazzo dei Congressi in Florence in 1995 In its initial formulation the central figure was Henri Poincaré and his philosophy of science, and more particularly what would have been the consequences for economics, as a discipline, if it had followed the path contained in Poincaré’s philosophy of science as we interpreted it at that time, based on Paschal O’Gorman’s earlier work on Poincaré’s conventionalism A central informing thesis for us at that time was what we regarded as the high epistemic cost that economics had inflicted on itself arising from the application of a particular philosophy of mathematics from the late nineteenth century, which was further intensified in the formalistic philosophy of mathematics that dominated for most of the twentieth century in economics In contrast to the formalism reflected in the neo-Walrasian programme and in particular in the work of Debreu for instance, our initial proposal was informed and heavily influenced by our attraction to the intuitionism of Brouwer, Heyting and later in the twentieth century in the work of Michael Dummett, which we felt provided a more adequate and epistemically satisfying philosophy for our discipline On reflection our work in economic methodology over the last quarter of a century has been, in a variety of ways, influenced by the broad philosophical framework of Brouwerian intuitionism Since 1995 and the intrusions of new and varied work demands at university and national levels, along with the completion of a number of book projects and edited collections, conspired to delay sustained engagement with this book project As a result the current book proposal was persistently relegated to the ‘for completion’ file! But we were determined to honour this project, which could only have been achieved with the extraordinary patience and forbearance of all at Routledge, for which we are deeply grateful But in this extended intervening period our thinking also underwent change which generated a considerable amount of reconfiguration with respect to the dimensions and topics to be addressed in the central domain of interest to us, namely the relation and influence of major developments in the philosophy of mathematics and their influence on economics It became clear to us arising from our teaching at both undergraduate and postgraduate levels, that the need for an extended reflection on the major developments within the philosophy of mathematics and their impact on economics was an emergent and pressing intellectual challenge which required inclusion as an integral component in the philosophical and methodological pedagogy of economics, particularly at the postgraduate level While this book is not a textbook in any conventional understanding of the term, it does explore in some considerable depth a range of topics that we deem paramount in any extended intellectual understanding of the complex interaction between economics and mathematics (incorporating the processes of quantification, measurement and formalisation) We can concur with Lawson that in ‘the history of the modern mainstream, the rise to dominance of formalistic modelling practices and the manner of their “survival” in this role, constitutes a central chapter in the history of academic economics that remains largely unwritten’ (Lawson 2003: 256) It awaits the comprehensive scholarly treatment exemplified for instance in Ingrao and Israel’s (1990) outstanding work on the emergence and development of general equilibrium theory In the remainder of this Preface we provide, albeit briefly, an outline of the rationale that motivated our initial formulation, but more particularly as we extended our consideration to embrace a more extended array of topics related to the focus of our central concern, namely the relation between the major developments in the philosophy of mathematics and their influence on economic theorising and modelling This book is motivated by the conviction that both philosophers of economics/economic methodologists and theoretical economists have much to gain by addressing the philosophy of mathematics The indispensable skill set of mainstream theoretical economists includes a competent, preferably expert command of particular areas of advanced mathematics to facilitate the construction of sophisticated economic models required for the rigorous analytical exploration of complex economic systems The challenges theoretical economists face presuppose the acquisition of increasing competence Consequently our recourse to the philosophy of mathematics is predicated on two objectives Firstly, it is used to critically interrogate the intricate and complex process of what is called the formalisation/mathematisation of economics Secondly, the philosophy of mathematics opens up ranges of novel logico-mathematical techniques for the theoretical modelling of rationality on the one hand and economic systems on the other, which result in outcomes at variance with orthodox/mainstream economic theorising Vis-à-vis the philosophy of economics/economic methodology this work may be read as a contribution to the research agenda identified by Weintraub namely ‘to study how economics has been shaped by economists’ ideas about the nature and purpose and function and meaning of mathematics’ (Weintraub 2002: 2) In this connection we distinguish the philosophy of pure mathematics from the philosophy of applied mathematics and analyse how these evolved over the course of the twentieth century (with particular reference to the twentieth century) Thus we examine how major figures, including Walras and Debreu among others exploited divergent philosophical perspectives on applied mathematics along with their relationship to pure mathematics in their methodological defences of their mathematical economics This book, however, is not limited to enhancing our understanding of the intriguing, often divergent, defences of the formalisation of economics by some of its major architects It is also engaged in the critical evaluation of these defences, thereby complementing challenging critiques by, among others, Bridel and Mornati (2009), Mirowski and Cook (1990), Ingrao and Israel (1990), Lawson (1997, 2003a) and his critical realist colleagues Our critique is distinct in its extensive exploitation of both the philosophy of pure mathematics and the philosophy of applied mathematics For instance, in connection with the neo-Walrasian programme, our critique is based on Brouwer’s novel philosophy of pure mathematics and his alternative mathematics which is fundamentally grounded in that philosophy This Brouwerian mathematics is not another chapter in the vast book of advanced mathematics exploited by mainstream mathematical economists: it is a different mathematics grounded in a different logic A central contention of this book is that this Brouwerian mathematics is crucially significant for the epistemic critique of the neo-Walrasian programme If a Brouwerian analysis of a rigorous mathematical proof stands up to critical scrutiny the very core of the neoWalrasian programme, i.e the existence proof of general equilibrium is mathematically undermined This book is not, however, limited solely to the enrichment of our understanding of the dynamics of some of the major critical junctures in the long and intricate process of the formalisation of economics and to the epistemic critique of the often intriguing views of some of its major contributors It is also concerned with the current state of academic economics Vis-à-vis mainstream mathematical economics some commentators such as Dow (2002) perceive a process of increasing fragmentation Others, for instance Lawson (1997, 2003a) maintain that current academic economics is experiencing a deep malaise The extent of either increasing fragmentation or malaise within academic economics is a question of empirical research which is not a central focus of this book Rather, assuming an increasing fragmentation whether slow or fast, we pose the question: what is the relevance of the philosophy of mathematics in the contemporary economic climate? We argue that the relevance of the philosophy of mathematics is both positive and negative Firstly, developments in twentieth-century logic expose the logical limitations of the advanced mathematics exploited in academic economics even when fragmented Simultaneously, these developments offer theoretical economists a novel range of rigorous logical tools to be used in their mathematical modelling of rationality – a range not exploited in mainstream academic economics Secondly, we explore the thesis that Brouwerian mathematics – used in our critique of the neo-Walrasian programme – is a more appropriate mathematics than the mathematics exploited in mainstream economics By recourse to the philosophy of mathematics we expose the descriptive inadequacy of various results of orthodox theorising In particular we show how these results exploit non-algorithmetic theorems of advanced mathematics and argue that non-algorithmetic models of economic decision making hold only for God-like beings whose decisions are not in any way constrained by temporal considerations Nonalgorithmetic models cannot be applied to actual economic decision makers who, even with the aid of the most sophisticated computers, take real time to make decisions On the other hand if economic modelling was limited to computable mathematics then its models of rational decision making, by virtue of the algorithmetic nature of the mathematics used, would in principle be compatible with the time constraints of actual economic decision making In this vein our sympathies totally lie with those economists, such as Velupillai (2000), who argues for the limitation of mathematical modelling to a judicious synthesis of computable mathematics with developments in Brouwerian mathematics A few caveats are in order We are not claiming that the philosophy of mathematics should colonise the philosophy of economics/economic methodology Rather the philosophy of mathematics makes an intriguing and unique contribution to our reflections on how economics became a mathematical science and on the contemporary status of mathematical economics Neither is there any suggestion that mathematical economics exploiting the resources of Brouwerian-computable mathematics should colonise the discipline of economics In so far as mathematical economics influences the construction of specific economic models, these models must be empirically interrogated While theoretical economics cannot emulate the experimental sophistication of theoretical physics, economics is an empirical endeavour and thus its models must come before the bar of experience Whether or not its models will or will not successfully pass this indispensable constraint, is a question for sophisticated economic testing which cannot be answered by any philosophy of economics As with the production of any book there are many debts incurred, of which we would like to mention a small number We would like to convey our thanks to our colleagues in the Departments of Economics and Philosophy respectively at the National University of Ireland Galway (NUIG) for their support and co-operation during the writing of this book In particular we would like to thank Professors John McHale and Alan Ahearne and Dr Aidan Kane for their support, encouragement and assistance during their respective terms as Head of the Department of Economics at NUIG We also recall with fond memory the friendship, generosity and intellectual insights of Professor Vela Velupillai during his sojourn as the J.E Cairnes Professor of Economics at NUIG Vela’s work in computable or algorithmic economics stands, in our estimation, as one of the truly pioneering contributions of the late twentieth century in the area of mathematical economics which is informed by , J.B (1971 [1826]) ‘Discours Préliminaire’, in Traité d’Economie Politique, 5th edn, with a preface by G Tapinos, Paris: CalmannLevy muelson, P.A (1947) Foundations of Economic Analysis, Cambridge, MA: Harvard University Press muelson, P.A (1952) ‘Economic theory and mathematics – an appraisal’, American Economic Review, 42(2): Papers and Proceedings of the Sixty-Fourth Annual Meeting of the American Economic Association (May): 56–66 rf, H with the collaboration of Terje Hansen (1973) The Computation of Economic Equilibria, New Haven: Yale University Press elling, T (1960) The Strategy of Conflict, London: Oxford University Press mid, A.M (1978) Une Philosophie de Savant Henri Poincaré et le logique mathématique, Paris: Francois Muspero umpeter, J.A (1954) History of Economic Analysis, London: George Allen & Unwin epanti, E and Zamagni, S (1995) An Outline of the History of Economic Thought, Oxford: Clarendon Press ckle, G.L.S (1955) Uncertainty in Economics and Other Reflections, Cambridge: Cambridge University Press ckle, G.L.S (1972) Epistemics and Economics, Cambridge: Cambridge University Press nker, S.G (ed.) 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(2010) Computable Constructive and Behavioural Economic Dynamics Essays in honour of Kumaraswamy (Vela) Velupillai, London: Routledge Index henwall, Gottfried 43–44n5 nar, S 45n20 gorithme of Algebra’ (Petty, W.) 22 orithms, economic theorising and: Brouwer, L.E.J 190; Church, Alonzo 187–189; Debreu, Gerard 189, 190; Gödel, Kurt 187–188; Herbrand, Jacques 188; Kaldor, Nicholas 190; Kleene, S.C 187; Mirowski, Philip 191; Simon, Herbert 190–191; Turing, Alan Mathison 187, 188–189, 190, 191; Velulillai Vela, Kumaraswamy 189–191; von Neumann, John 187, 191; Zambelli, Stefano 189, 190 orithms, mathematisation of economics and: Hilbert, David 183, 185, 186; Turing, Alan Mathison 183–187 ais, Maurice 41 ati, F and Aspromourgos, T 19 derson, G., Goff, B and Tollison, R 12 na, R 35, 45n23 umentation, grammar of 135–136 stotle 81, 82, 169; mathematics, philosophy of 205–206 hmetic, Foundations of (Frege, G.) 205 ow, Kenneth J 80–81, 98, 102n10, 130; constructive and computable mathematics, emergence of 175, 176; neo-Walrasian formalisation of economics 135, 139, 142; philosophical reflections 197, 202 hur, Brian 70 promourgos, T 19, 20, 22, 44n13 brey, John 44n9 mann, R.J 162n2 xier, R.E and Hahn, L.E 193n18 omatic method, economics and 105–128; applied mathematical sciences, axiomatisation and 111–114; commodity and price, concepts of 115; Debreu, Gerard 105–106, 109–110; distillation of axiomatic method 107–108; economic analysis, assets of Debreu’s formalist philosophy of 123–128; economic domain, axioms in 117; economic theory, Debreu’s formalist mathematisation of 114–118; Élements d’economie politique pure (Walras, L.) 105, 114; Euclidean axiomatisation 105–106, 107–108, 112–114, 117, 123–124, 126; formalist philosophy of applied mathematics, indispensable steps in 113; General Equilibrium Theory, axiomatisation of 105–106; Hahn, F.H 114, 123, 127; Hilbert, David 106, 107, 110, 111–112, 113, 123, 129n14; Hilbertian principle of uniformity of axiomatic method 111–112; ideology-free economics, Debreu’s ideal of 124; image of economic theory, body of knowledge and 117–118; logic, basic tool of any axiomatic system 108; logico-mathematical structures 112–113; mathematical form, semanticist conception of 119; mathematical model of Debreu, formalist or semanticist? 118–123; metamathematics, Hilbert’s notion of 110; pathology of the paradoxes in, formalist remedy for 109–110; Poincaré, Henri 109, 110, 120, 121, 122, 123; Poincaré malaise, concerns raised by 120–122; pure formalism, domain of 112; pure mathematics, mathematical content and 119–120; rigour, axiomatisation and 107–110; semantical domination, semanticism and 108–109, 110, 112, 113, 118, 119–120, 121–122, 123, 126, 129n13; Theory of Value (Debreu, G.) 105–106, 110, 114–115, 116, 117, 118, 119, 121, 123, 124, 128n2, 128n11; Walras, Leon 105, 114, 116, 117, 126; Weintraub, Steven H 110, 117, 128n1–3 omatisation 3, 6, 87, 132, 136, 143n2, 161, 176, 193n20, 208; aesthetic code of 125; of arithmetic and of logic 108–109; axioms, assumptions and grammar of argumentation 136–140; of economic theory 2; formalist conception of 7; non-Euclidean geometries and 107–108; philosophical admiration for 105, 125–126 chelier, Louis 50, 51 ckhouse, R.E 12 con, Francis 16, 17, 19–20, 21, 23, 44n6, 44n8, 44n12 ker, K 30 nard, T.C 44n10 tiat, Frédéric 33 hurst, Ralph 44n8 umol, W.J and Goldfeld, S.M 37 nacerraf, P and Putnam, H 212n1 nnett, Kate 44n9 keley, George 148 nays, P 106, 111–112, 114–115, 128–129n12, 128n10 hop, Errett 42 alence, doctrine of 176, 177, 193n20, 206, 209 nché, R 105 ug, M 12, 102n8 olos, G 212n2 tock, D 212n1 urbaki, Nicholas 175, 192n12; critique of Brouwerian intuitionism 172–174 wen, H.R 12 ylan, T.A and O’Gorman, P.F 45n30, 72n23, 104n24, 143n1, 162, 162n6, 164n25, 164n33 yle, Robert 17, 44n8 wer, A 45n16, 45n22 del, P 45n29 del, P and Momati, F 71n14 uwer, L.E.J 2–3, 41, 42, 162n3, 163n7; algorithms, economic theorising and 190; Brouwer-Hilbert controversy 167; constructive and computable mathematics, emergence of 165, 182, 183, 187, 192–193n15, 192n5, 192n7, 192n11, 193n16; formalisation of economics and Debreu’s philosophy of mathematics 77, 86, 87, 93, 98, 101, 102n7, 102n9, 103n15, 103n20–21, 104n27; formalism and economic theorising, internal critique of 181; intuitionism of, Bourbaki’s critique of 172–174; intuitionistic mathematics of 168, 193n18, 193n23, 194n27; mathematics, philosophy of 204, 208–209, 212, 212n4; neo-Walrasian formalisation of economics 110, 116, 119–120, 121, 122, 130n25; philosophical reflections 196, 197, 198, 199; pragmatic intuitionism, economic theorising and 178, 180; strict intuitionism, Dummett’s philosophical reconstruction of 174–175, 177; strict intuitionism, neo-Walrasian programme and 170, 171 wn, J.R 212n1 wn, Robert 11 , B 50, 71n2 nard, Nicholas-Francois 37, 39 ntillon, Richard 25, 45n16 ntor, Georg 169, 172, 185, 187, 192n6; formalisation of economics and Debreu’s philosophy of mathematics 74–75, 88, 89, 90, 92, 93, 98, 103n20; mathematics, philosophy of 205, 206–207, 209, 211, 212; neo-Walrasian formalisation of economics 110, 121, 122, 129n23 nap, Rudolf 5, 71n7, 72n17, 105, 111, 159, 164n26 tesian deductive method 26, 58 tesian geometry 62, 160 tesian tradition 107, 147, 168 sirer, Ernst 27 urch, Alonzo 3, 4, 42, 103n18, 129n14, 166, 183, 184; algorithms, economic theorising and 187–189; Church-Turing thesis 194n26–27, 197, 201; formalism and economic theorising, internal critique of 181; mathematics, philosophy of 208, 209; post Gödel-Church-Turing view of classical mathematics 3–4 sical mathematics 1, 2, 3–4, 82, 110, 165–167, 171–172, 175; constructive mathematics and, choice between 178–179; economic modelling and 199–200; excluded middle and 170; logical basis of 176; post Gödel-Church-Turing view of 3–4; pure calculus of 129n13; strict intuitionism and 168–169, 173–174; transfinite domain of 202–203 hen, B.I 27 ander, D., Follmer, H., Hans, A et al 13 yvan, M 212n1 mmerce en Géneral (Cantillon, R.) 25 Elements of Commerce (Tucker, J.) 25 mte, Auguste 28, 48, 50 ndorcet, Marie Jean, marquis de 30–32; social sciences, ambitions for 30–31 nring, Hermann 43–44n5 structive and computable mathematics, emergence of 165–191; algorithmetic revolution in economic theorising? 187–191; Arrow, Kenneth J 175, 176; Bourbaki critique of Brouwerian intuitionism 172–174; Brouwer, L.E.J 165, 182, 183, 187, 192–193n15, 192n5, 192n7, 192n11, 193n16; constructive or intuitionistic mathematics 165–166; Debreu, Gerard 165, 166–167, 169, 170–172, 176, 180, 184–185, 186, 192n9; Dummett, Sir Michael 168, 178, 187, 193n18, 193n21, 194n27; economic content and computable mathematics, fit between 166–167; excluded middle, Aristotlan principle of 169–170, 192n8, 193n20, 194n27; extensionality 175, 176, 206; external and internal critiques of Hilbert programme 165–166; formalisation of economics, choice in 166; formalism and economic theorising, internal critique of 181; Frege, Gottlob 169, 175, 192n10, 193n18, 193n23; Gödel, Kurt 166, 176–177, 184, 186; Hahn, F.H 165, 185; Hilbert, David 165, 166, 168, 171, 174, 175, 180, 192n11–12; Kaldor, Nicholas 165, 169, 172; Kant, Immanuel 168, 192n5, 193n23; mathematical discourse, infinite nature of 168; neo-Walrasian programme, poverty of prognosis for 166–167; Poincaré, Henri 168, 178, 185–186, 192–193n15, 192n5, 192n12; pragmatic intuitionism and economic theorising 178–181; Russell, Bertrand 167, 169, 175; strict intuitionism, classical mathematics and 168–169; strict intuitionism, Dummett’s philosophical reconstruction of 174–177; strict intuitionism, neo-Walrasian programme and 167–172; theorems of Gödel, Debreu’s philosophy of economic theorising and 182–183; Turing, algorithms and mathematisation of economics 183–187; Velulillai Vela, Kumaraswamy 185, 193–194n25, 193n19, 194n27–28; Weyl, Hermann 171, 172, 192n11 vention: custom and convention, relationship between 147; Keynes’ notion of, rationality in context of radical uncertainty and 5; Lewis’ six conditions for 153–154; in mathematical physics, indispensable role of 159; neo-classical rationality and 152–155; post-Keynesians on uncertainty and 155–158; promises, convention and 151; see also rationality, convention and economic decision-making nvention (Lewis, D.) peland, J 183 ry, L 106, 111, 128n2, 129n16 urcelle-Seneuil, Jean Gustave 33 urnot, Augustin 37–39, 41, 45n26, 76; mathematician-economist in tradition in France 38–39 urtault, J.M., Kabunov, Y., Bru et al 50, 51 lo, R 39 mwell, Oliver 17–18 nningham Wood, J and McLure, M 39 Alembert, Jean le Rond 26 sgupta, P 128n2 vidson, Paul 157–158, 159, 161, 196 wson, J.W Jr 182 breu, Gerard 2, 4, 6–7, 12, 40–41; algorithms, economic theorising and 189, 190; axiomatic method in foundations of mathematics 105–106, 109–110; constructive and computable mathematics, emergence of 165, 166–167, 169, 170–172, 176, 180, 184–185, 186, 192n9; on contemporary period of economic formalism 76–78; equilibrium, proof of existence of, Poincaré malaise and 97–99, 102; equilibrium, Walras and proof of existence of 99–102; formalism and economic theorising, internal critique of 181; formalist mathematisation of economic theory 114–118; formalist philosophy of economic analysis, assets of 123–128; generality, achievement of 84–89; Kaldor on achievement and legacy of 140–143; mathematical economics, ‘scientific accidents’ theory of the growth of 40–41; mathematical model, formalist or semanticist? 118–123; mathematical model of a private ownership 4; mathematisation of economics, global view on 94–97; neo-Walrasian formalisation of economics 131–132, 133, 134, 135–136, 139; philosophical reflections 195–196, 198–199; philosophy of economic theorising, Gödel’s theorems and 182–183; philosophy of mathematics, formalisation of economics and 73–74, 75–76, 102n6–8, 104n30–31, 104n35; rigour, achievement of 79–84; simplicity and existence proofs 90–94 la Moneta (Galiani, F.) 25 marcation 102n4 nnehy, C.A 44n10 scartes, Rene 16, 26, 62, 66, 107, 168 stutt de Tracy, Antoine 33 iplinary boundaries, Petty’s disregard for 16–17 bbs, B.J.T and Jacob, M 45n18 yfus, Captain Alfred 48, 71n1 mmett, Sir Michael 2, 88, 198; constructive and computable mathematics, emergence of 168, 178, 187, 193n18, 193n21, 194n27; intuitionism of 193n23–24; philosophical reconstruction of strict intuitionism 174–177 ppe, T and Weintraub, E.R 45n27 pré, L 45n14 puit, Jules 36–37, 39 lauf, S.N and Blume, L.E 45n28 namic Stochastic General Equilibrium (DSGE) model well, T., Milgate, M and Newman, P 45n29 undation of Economic Analysis (Sameulson, P.) 41 onomic Journal 12 nomic theorising: algorithmetic revolution in? 187–191; dynamic process of general equilibrium theorising 135; Gödel’s theorems, Debreu’s philosophy of economic theorising and 182–183; image of economic theory, body of knowledge and 117–118; philosophy of, Gödel’s theorems and 182–183; pragmatic intuitionism and 178, 180; simplification of assumptions in 138–139; theoretical economics as science, Hahn’s opposition to 134; Walras’ contribution to history of 46 nomics: axiomatisation of economic theory 2, 6–7; balance in curriculum, issue of seeking 13; Classical Situation in 15; commitment to use of mathematics and quantification in 20; curriculum structure and design 13; economic domain, axioms in 117; economic policy, debates about 133; economic theorising, developments in philosophy of mathematics and 11; engineer-economist tradition in France 36–37; formalisation of 1–2; formalism in, issue of 10, 35–43; general equilibrium formalisation of, radical questioning of 132; general equilibrium theory, neo-classical economics and 142; historical development of mathematisation of 15–16; ideology-free economics, Debreu’s ideal of 124; intellectual contextualisation for study of 13; laws of society, existence of 11; mathematical economics, Debreu’s ‘scientific accidents’ theory of the growth of 40–41; mathematical modelling in 2; mathematics and, relationship between 9–11; mathematics in, contentious nature of 10–11; mathematics in, intensification of use of 12; mathematisation of economics 2–3, 5–6, 11–14; methodological basis for Petty’s political arithmetic 22–23; methodology in, implications of Gödel’s theorems for 4; momentum for mathematisation of 14; neo-classical economics, scientific theory of value in 133; normative economics, distinction between positive economics and 133; numerical analysis, commitment to ‘spirit of’ 15–16; Petty’s economic writings 18–19; political arithmetic in 14–24; quantification of socio-economic phenomena 13; scientific investigations of actual economies 131–132; social laws of development, existence of 11; socioeconomic domain, Pareto’s contribution 39; socio-economic phenomena, emergence of quantification of 14–24; socio-economic phenomena, Petty’s quantification and analysis of 20–21; specificity as social science of 11; theoretical economics, applied mathematical science and 2, 5–6, 11–14; theoretical economics, mathematical science and 2; Walras as central figure in evolution of 39–40 w Economics became a Mathematical Science (Weintraub, S.H.) 42 geworth, Francis Ysidro 40 stein, Albert 72n21, 111, 164n31 lund, R.B and Hébert, R.F 36, 45n28 ments de Statiques (Poinsot, L.) 52, 71n6 ments d’economie politique pure (Walras, L.) 4, 105, 114; equilibrium, proof of existence of 99–100, 101, 102n2; formalisation of economics, Debreu’s philosophy of mathematics and 73, 74–75, 77; Walras-Poincaré correspondence, reassessment of 40, 46, 48–51, 54–59, 68, 70 s, W 45n21 pirical objects, status and quantification of 21 pirical sciences as ‘mixt mathematical arts’ 22 ghtenment 25–26, 30 tein, R.L and Carnielli, W.A 182 ilibrium: Debreu and proof of existence of 99–102; mathematical modelling of 2; near equilibrium state as causal factor in real world 140–141; Poincaré malaise and proof of existence of 97–99, 102; proof of existence of 99–100, 101, 102n2; Walras and proof of existence of 99–102 rit des Lois (Montesquieu) 24, 28 lid 6–7, 75, 85–86; applied mathematical sciences, axiomatisation and 112, 113, 114; axiomatic method in foundations of mathematics 105, 106, 123, 124, 126; Debreu’s formalist mathematisation of economic theory and 114, 117; Euclidean geometry 63, 75, 85, 107, 160, 204; neo-Walrasian formalisation of economics and 132, 136; ontological-epistemic indeterminacy and 159–160; rigour, axiomatisation and 107–108 luded middle, Aristotlan principle of 93, 108, 128n6, 169–170, 192n8, 193n20, 194n27, 209–210, 211 lanation, Kaldor on Debreu’s conception of 141 ensionality: constructive and computable mathematics, emergence of 175, 176, 206; formalisation of economics, Debreu’s philosophy of mathematics and 80, 81, 82, 83, 98, 102n10, 104n30 carello, G 45n23 ning, B 44n10 mat, Pierre de 16 rone, V 45n14 her, Irving 40 etwood, S 45n30 get, E.L 45n23 malisation of economics, Debreu’s philosophy of mathematics and 73–102; Brouwer, L.E.J 77, 86, 87, 93, 98, 101, 102n7, 102n9, 103n15, 103n20–21, 104n27; Cantor, Georg 74–75, 88, 89, 90, 92, 93, 98, 103n20; core theses 74–76; Cours d’Economie Politique (Pareto, V.) 77; Critique of Pure Reason (Kant, I.) 85; economic formalism, Debreu on contemporary period of 76–78; economy, Debreu’s perspective on an 96; Élements d’economie politique pure (Walras, L.) 73, 74–75, 77; equilibrium, Debreu, Walras and proof of existence of 99–102; equilibrium, Debreu’s proof of existence of 97–99; extensionality 80, 81, 82, 83, 98, 102n10, 104n30; fixed point theorem (Brouwer) 77, 79–80, 86, 93, 94, 97, 98, 100, 101, 102n7, 104n27; formalism, rejection of semantical philosophy of mathematics variables 86–87; The Foundations of Arithmetic (Frege, G.) 81; Frege, Gottlob 75, 79, 81, 86, 87, 90, 91–92, 103n12, 104n23; generality, Debreu and achievement of 84–89; Hilbert, David 77, 87, 89, 90, 103n18, 103n20–21, 104n35; Marginalist Revolution, Walras as key figure in 73; mathematisation of economics, Debreu’s global view of 94–97; Poincaré, Henri 79, 86, 87, 89, 90, 93, 102n1–2, 102n9, 104n33–4; Poincaré malaise, Debreu’s proof of existence of equilibrium and 97–99; Recherches sur les Principes Mathématiques de la Théorie des Richesse (Cournot, A.) 76–77; rigour, Debreu and achievement of 79–84; simplicity and existence proofs, Debreu and 90–94; Theory of Games and Economic Behaviour (von Neumann, J and Morgenstern, O.) 77; Theory of Value (Debreu, G.) 73–74, 76, 77, 79, 89, 90, 93, 94–95, 96, 99–100, 101–102, 102n7, 104n30; truth-tables 83–84, 118; Walras, Leon 74–76, 76–77, 98, 99, 102n1–2; Weintraub, Steven H 74, 76, 77, 78 malism 1; applied mathematics in context of 75–76; Debreu on contemporary period of economic formalism 76–78; economic theorising and, internal critique of 181; economic theorising and internal critique of 181; formalist mathematisation of economic theory, Debreu and 114–118; formalist philosophy of applied mathematics, indispensable steps in 113; pathology of the paradoxes in axiomatic method, formalist remedy for 109–110; philosophy of economic analysis, assets of 123–128; pure formalism, domain of 112; pure mathematics and metamathematics in 129n13 , A 44n10 ge, Gottlob 152, 196, 197; constructive and computable mathematics, emergence of 169, 175, 192n10, 193n18, 193n23; formalisation of economics and Debreu’s philosophy of mathematics 75, 79, 81, 86, 87, 90, 91–92, 103n12, 104n23; mathematics, philosophy of 204, 205, 206–207, 208, 209, 211; neo-Walrasian formalisation of economics 108–109, 110, 113, 128n9 nch Liberal School, emergence of 36 dman, M 133, 134, 137, 139, 141 brook, E 45n30 iani, Ferdinando 25 me theory 5, 9, 77, 145, 153–155, 162, 163n10, 164n24 nier, Josselin 33 sendi, Pierre 16 eral equilibrium theory 71n6, 86, 105, 106, 127, 131–132, 141, 187, 188–189; abstraction and 143; dynamic process of general equilibrium theorising 135; mathematics, economics and 14, 25, 36, 37, 39, 41, 45n25, 45n29; methodological approach to 133–134; neo-classical economics and 142 erality, achievement of 84–89 e, R 102–103n11 ria-Palermo, S 102n7 ddard, Jonathan 17, 44n8 del, Kurt 3–4, 104n33, 129n13–14, 129n24, 201; algorithms, economic theorising and 187–188; constructive and computable mathematics, emergence of 166, 176–177, 184, 186; formalism and economic theorising, internal critique of 181; incompleteness theorem (second) 176; mathematics, philosophy of 208, 209; post Gödel-Church-Turing view of classical mathematics 3–4; theorems of, Debreu’s philosophy of economic theorising and 182–183 d and silver as measures of exchange 151–152 sen, Hermann Heinrich 40 unt, John 19, 44n7 ffe, J-L., Heinzmann, C and Lorentz, K 72n21 enewegen, P 45n22 bel, H and Boland, L 12 nbaum, A 164n30 erlac, H 45n18 hn, F.H 4, 45n27, 101, 104n24; axiomatic method in foundations of mathematics 114, 123, 127; constructive and computable mathematics, emergence of 165, 185; grammar of argumentation, first step 135–136; neo-Walrasian formalisation of economics 131–132, 133–134, 136–140, 141, 142, 143; philosophical reflections 196–197, 198–199 nds, Wade 133 tlib, Samuel 17 usman, D.M 11, 133 ck, R 212n2 nzman, G and Nabonnard, P 192–193n15 brand, Jacques: algorithms, economic theorising and 188; Herbrand-Gödel general recursiveness 188 yting, Arend 168, 170, 172, 179–180, 192n7; mathematics, philosophy of 204, 211, 212n4 ks, John 41 archy of reservations, notion of 54–58 bert, David 2–3, 6, 7, 51, 71n5, 74, 75; algorithms and mathematisation of economics 183, 185, 186; axiomatic method in foundations of mathematics 106, 107, 110, 111–112, 113, 123, 129n14; Brouwer-Hilbert controversy 167; constructive and computable mathematics, emergence of 165, 166, 168, 171, 174, 175, 180, 192n11–12; economic theorising, Gödel’s theorems and Debreu’s philosophy of 182; formalisation of economics and Debreu’s philosophy of mathematics 77, 87, 89, 90, 103n18, 103n20–21, 104n35; formalism, economic theorising and internal critique of 181; formalism and economic theorising, internal critique of 181; mathematics, philosophy of 204, 207–208; neo-Walrasian formalisation of economics 131, 132, 136; philosophical reflections 196, 197 bbes, Thomas 16, 19, 20, 21, 22, 44n6, 44n12 chstrasser, T.J 45n22 dgson, Geoffrey M 9–10, 43n1 lander, S 45n23 ppit, J 44n13 ghes, G.E and Cresswell, M.J 102–103n11 l, C.H 18, 20 me, David 5, 25, 145, 155, 161–162, 163n11–17, 196; convention and foundations of justice 146–152; Human Nature, Treatise on (1739) 146–147 sserl, Edmund 193n16 chison, T 16, 18, 19, 24, 25, 45n20 ompleteness theorem (second) 176 rao, B and Israel, G 25, 26, 27, 28, 29, 30–31, 33, 37, 38, 45n29, 58, 71n3, 140 itionism 2, 3, 204, 208–212, 211–212, 212n1; Bourbaki’s critique of 172–174; Brouwer’s intuitionistic mathematics 168, 193n18, 193n23, 194n27; critique of Brouwerian intuitionism 172–174; Dummett and 193n23–24; economic theorising, pragmatic intuitionism and 178–181; formalism and intuitionism, conflict between 2–3; philosophical reconstruction of strict intuitionism 174–177; pragmatic intuitionism 178–181, 198–199; see also strict intuitionism ard Achylle-Nicolas 36, 39, 45n25 el, Jonathan 26, 45n17 é, William 45n25, 45n27; Walras-Poincaré correspondence 47, 49, 50, 52–53, 54–55, 56, 58, 62, 67, 71n6, 72n18 mes I of England 19 on, William Stanley: Memorial Fund Lecture 135 ons, William Stanley 40, 49 nk, A 45n27 dan, T.E 44n10 rnal of Political Economy 12 kutani, S 93, 97, 101, 102n7, 104n32 dor, Nicholas 96, 125; on achievement and legacy of Debreu 140–143; algorithms, economic theorising and 190; constructive and computable mathematics, emergence of 165, 169, 172; neo-Walrasian formalisation of economics 131, 132, 133, 136 nt, Immanuel 79, 85–86, 160; constructive and computable mathematics, emergence of 168, 192n5, 193n23; mathematics, philosophy of 204–205 es, S 45n23 nny, Anthony 81; mathematics, philosophy of 207 ynes, John Maynard 2, 5, 44n11, 139, 196; convention notion, rationality in context of radical uncertainty 5; post-Keynesian decision-making 5; and post-Keynesians on uncertainty and conventions 155–158; rationality, convention and economic decision-making 144–145, 146–147, 154–155, 162, 162n5 khoff, Gustav 62 man, A 45n29, 140 ene, S.C 182, 183; algorithms, economic theorising and 187 ght, Frank H 156–157 mogorov, A 179–180 nai, János 136 onecker, Leopold 89 czynski, M and Meek, R 45n21 hn, Thomas 127 grange, Joseph-Louis 31, 62 guage: ideal language, construction of 206; logico-philosophical and socio-historical views of 152–153; natural language, logical reasoning and 82–83, 84, 86, 202 nsdowne, H.W.E Petty-Fitzmaurice, marquis of 23 place, Pierre-Simon 31, 33, 36 son, B.D 37 sis, J 146 urent, Hermann 49, 51, 52–53 wson, Tony 42, 45n30, 144, 145, 155, 161, 162n5 bniz, Gottfried Wilhelm 164n28 ontief, W 29 win, W 17, 18 vasseur, Pierre Émile 40 wis, C.I 82 wis, C.I and Langford, C.H 102–103n11 wis, David 5, 196; convention and neo-classical rationality 152–155; rationality, convention and economic decision-making 145, 146, 158–159, 161–162, 163–164n21–22, 163n9, 163n19, 164n24 denfeld, D.F 43–44n5 nebo, Ø 212n1 bachevsky, Nikolai 85, 107 cBride, F 212n2 Cormick, T 44n10 Guinness, B and Oliveri, G 193n18 Lure, M 39 lcolm, N and Stedall, J 16 lthus, Thomas R 34 ncosu, P 192n11, 212n1 rginalist Revolution 1, 4, rsenne, Father Marin 16 rshall, Alfred 40 rtinich, A.P 44n6 rx, Karl 16, 22 s-Collel, A 45n27 hematics: classical logic, calculus of 179–180; convention in mathematical physics, indispensable role of 159; economics, Debreu’s global view on mathematisation of 94–97; in economics, quantification of extent of 12–13; economics and, Newton and 14, 25–28, 30, 31, 32, 33, 37, 39; general equilibrium theory, economics and 14, 25, 36, 37, 39, 41, 45n25, 45n29; intuitionistic mathematics of Brouwer 168, 193n18, 193n23, 194n27; logico-mathematical structures 112–113; mathematical discourse, infinite nature of 168; mathematical economics, formulation of 14, 24, 35, 36, 37, 39–40, 41, 45n25; mathematical economics, ‘scientific accidents’ theory of the growth of 40–41; mathematical model, formalist or semanticist? 118–123; mathematical physics as deductive a priori science, response to Poincaré critique 62–68; ‘mathematical reasoning’; methodological framework for 23–24; metamathematics, Hilbert’s notion of 110; philosophy of, formalisation of economics and 73–74, 75–76, 102n6–8, 104n30–31, 104n35; physico-mathematical approach 25–28, 32–33; Poincaré’s critique of mathematical physics as deductive a priori science, Walras’ response to 62–68; post Gödel-Church-Turing view of classical mathematics 3–4; precision in 9–10; private ownership, mathematical model of 4; quantification of socio-economic phenomena 13; rational inquiry and 22; social mathematics, Concordet’s notion of 30, 31–32; social sciences, mathematisation of 36; in socio-economic inquiry, employment of 11 hematics, philosophy of 2, 204–212; algorithm application 210; algorithmatically decidable and undecidable theorems 3–4; Aristotle 205–206; arithmatic 204, 205; arithmetic, foundations of 205–207; axiomatisation of mathematics 207–208; bivalence, philosophical thesis of 176, 177, 193n20, 206, 209; Brouwer, L.E.J 204, 208–209, 212, 212n4; Cantor, Georg 205, 206–207, 209, 211, 212; Church, Alonzo 208, 209; classical logic 204–205, 209–210, 211; classical mathematics 2, 3–4; classical mathematics, post Gödel-Church-Turing view of 3–4; completeness 208; computable mathematics 3; computable mathematics, orthodox modelling of rationality and 4–5; constructive existence proof 210; decidability 208; economic methodologies, creative destruction of 7–8; empiricism 204; Euclidean geometry 204; Euclid’s parallel postulate 210; excluded middle, principle of 93, 108, 128n6, 169–170, 192n8, 193n20, 194n27, 209–210, 211; finite systems, meta-empirical law of 210; formal axiomatic system, development of 207–208; formalism 204, 207–208, 211–212, 212n1; formalism and intuitionism, conflict between 2–3; formulae, symbols and 208; Frege, Gottlob 204, 205, 206–207, 208, 209, 211; geometry 205; Gödel, Kurt 208, 209; Heyting, Arend 204, 211, 212n4; Hilbert, David 204, 207–208; ideal language, construction of 206; infinity, actual or potential 209–210, 211; intuitionism 2, 3, 204, 208–212, 211–212, 212n1; Kant, Immanuel 204–205; Kenny, Anthony 207; logic, reasoning and 205–206, 209; logic, set theory and 206–207; logicism 204–207, 207–208, 211–212, 212n1; logico-mathematical analysis 3; logico-philosophical analysis 2, 8, 75, 152; logico-semantics 206; mathematical infinity 209; mathematical proofs 210–211; natural number system 205; natural numbers 209; a posteriori truth 205; a priori truth 204–205; pure mathematics 207–208; rational number system 205; rationalism 204; rationality, mathematical modelling of 4–5; real number system 205; reasoning, logic and 205–206, 209; reductio ad absurdum method 210; Russell, Bertrand 204, 206–208, 209, 211; sense and reference, distinction between 205; set theory 205, 206, 207–208, 209; set theory, logic and 206–207; symbolism, construction of 206; theorems 208; totality 209; Turing, Alan Mathison 208, 209; Walras, Leon 206 thematics and Modern Economics (Hodgson, G.) xwell, James Clerk 71n8 ek, R.L 45n21–22 nger, K 49 , John Stuart 192–193n15, 192n7 er, A 164n31 owski, P Cook, P 46, 47, 58, 60, 62 owski, Philip 6, 12, 42, 72n23, 166, 194n28; algorithms, economic theorising and 191 dus ponens rule of inference 193n22 ntesquieu 24, 27–28, 32 oduction to Moral Philosophy (Smith, A.) 24–25 al sciences, Walrasian programme in context of 47–49 ality, Hume’s philosophical foundations of 148–150 ravia, S 33 rgenstern, Oskar 41, 77 rishima, M 45n27 Essay Concerning the Multiplication of Mankind (Petty, W.) 19 rphy, A.E 16, 45n16 gel, E Newman, J.R 122 poleonic political order 31, 32 sh equilibrium 162n2–3, 163n19 ural and Political Observations mentioned in a following Index, and made upon the Bills of Mortality (Graunt, J.) 19 Nature of Social Laws (Brown, R.) 11 -Walrasian formalisation of economics 131–143; Arrow, Kenneth J 135, 139, 142; axioms, assumptions and grammar of argumentation 136–140; Brouwer, L.E.J 110, 116, 119–120, 121, 122, 130n25; Cantor, Georg 110, 121, 122, 129n23; Debreu, Gerard 131–132, 133, 134, 135–136, 139; Debreu’s formalisation, economists’ reception of 131–132; economic policy, debates about 133; economic theorising, simplification of assumptions in 138–139; economic theory, role of 132–134; equilibrium, near equilibrium state as causal factor in real world 140–141; explanation, Kaldor on Debreu’s conception of 141; Frege, Gottlob 108–109, 110, 113, 128n9; general equilibrium theory, methodological approach to 133–134; general equilibrium theory, neo-classical economics and 142; Hahn, F.H 131–132, 133–134, 136–140, 141, 142, 143; Hahn’s grammar of argumentation 135–136; Hilbert, David 131, 132, 136; Invisible Hand thesis, Smith and 135–136; Kaldor, Nicholas 131, 132, 133, 136; Kaldor on Debreu’s achievement and legacy 140–143; neo-classical economics, scientific theory of value in 133; normative economics, distinction between positive economics and 133; rational agent axiom 137–138; theoretical economics as science, Hahn’s opposition to 134; Theory of Value (Debreu, G.) 131–132, 133, 136, 140, 141, 142, 143 -Walrasian programme 3–4, 6, 7, 73–74, 123, 129n19, 131–143, 200–201, 202–203; constructive and computable mathematics, emergence of 165, 166, 167–172, 175–176, 177, 178, 179–181, 183, 185, 186; formalisation of mathematics 131, 140, 142; formalism and economic theorising, internal critique of 181; poverty of prognosis for 166–167; prices,xplanation of 4; strict intuitionism and neo-Walrasian programme 167–172; trajectory and development of 41–42 w Mathematics and Natural Computation 45n31 wton, Elements de la Philosophie de (Voltaire) 26, 27 wton, Isaac 62, 65–66, 72n21, 75, 85; mathematics, economics and 14, 25–28, 30, 31, 32, 33, 37, 39; mechanics of 160, 164n31; physicomathematical approach 25–28 wtonian analysis, infinitesimal calculus of 30 wtonian natural science 26–28 wtonian rational mechanics 39 -ergodicity 158, 159, 162, 196 -Euclidean geometries 74, 107, 160, 205 kham’s razor 69 Donnell, R 161, 164n33 Gorman, P.F 163n8, 164n31 ological-epistemic indeterminacy: conventions and philosophy of mathematics and 158–161; Euclid and 159–160; rationality, convention and economic decision-making 145–146 ological lock-in, Walras and 68–70 gden, A 45n14 mer, R.R 45n22–23 eto, Vilfredo 39, 41, 49, 77, 185; Pareto optimality 140, 141, 143 cal, Blaise 16 inetti, L.L 34 , John 16 ri, F and Hahn, F 45n29 tit, P 154–155, 161, 163–164n21, 164n23 ty, William 14, 15–24, 25, 30, 43–44n5, 44n7–13; ‘Algorithme of Algebra’ 22; disciplinary boundaries, disregard for 16–17; economic writings 18–19; economics, commitment to use and application of mathematics and quantification in 20; empirical objects, status and quantification of 21; empirical sciences as ‘mixt mathematical arts’ 22; ‘mathematical reasoning’; methodological framework for 23–24; mathematics, rational inquiry and 22; methodological basis for political arithmetic of 22–23; methodological commitments 19–20; Political Arithmetic (1690) 23; Sir William Petty’s Quantulumcunque concerning Money, 1682 (1695) 19; socio-econimic phenomena, quantification and analysis of 20–21; Treatise of Taxes and Contributions (1662), production models in 21–22 osophical reflections 195–203; algorithmetic mathematics, economic modelling and 201–202; Arrow, Kenneth J 197, 202; axiomatised mathematics 201; Brouwer, L.E.J 196, 197, 198, 199; Church-Turing thesis 194n26–7, 197, 201; classical mathematics, economic decision-making and 197–198; classical mathematics, economic modelling and 199–200; classical mathematics, transfinite domain of 202–203; computer/digital age, emergence of 201; conventions, reflections on 196; Debreu, Gerard 195–196, 198–199; Élements d’economie politique pure (Walras, L.) 195; geometrical conventionalism, Poincaré’s thesis of 196; Hahn, F.H 196–197, 198–199; hermenuetics 195; Hilbert, David 196, 197; Hilbertian mathematico-philosophical climate 196; intuitionistic mathematics, economic modelling and 200–201, 202–203; intuitionistic philosophy of mathematics, formalisation of economics and 200–201; logico-mathematical concepts, disputed nature of 196–197; mathematico-economic commitments of Debreu 196; non-constructive mathematics, economic modelling and 199–200, 202–203; non-ergodic economic world 196; orthodox formalisation of economics 198; philosophy of mathematics, new formalisation of economic theorising? 197–203; Poincaré, Henri 195, 196; Poincaré’s philosophy of applied mathematics 195–196; post-Keynesian economic decision-making 196; pragmatic intuitionism 198–199; pragmatic intuitionism, mathematical modelling and 199; praise for philosophy of mathematics 195–197; problems, methods and 202; Russell, Bertrand 196, 197; strict intuitionism 198; theoretical economics, experimental resources of 199; theoretical economics, predictive success of 198–199; Theory of Value (Debreu, G.) 195–196; Walras, Leon 195–196; Walras’s scientific realist defence of theoretical economics 195–196 sico-mathematical approach of Newton 25–28 card, Antoine Paul 71n3 o 192n10 onic philosophy 58–59, 61, 62, 70 onic realism 58–59, 61, 62, 192n10; Platonic-scientific realism 58–62 asure, Hume’s analysis of abstract idea of 148 ncaré, H Darboux, G and Appell, P 48 ncaré, Henri 2, 4, 5, 6, 40, 71n1–2, 71n5, 71n8–9, 72n16, 72n21; axiomatic method i foundations of mathematics 109, 110, 120, 121, 122, 123; constructive and computable mathematics, emergence of 168, 178, 185–186, 192–193n15, 192n5, 192n12; critique of mathematical physics as deductive a priori science, Walras’ response to 62–68; Debreu’s proof of existence of equilibrium, Poincaré malaise and 97–99, 102; formalisation of economics and Debreu’s philosophy of mathematics 79, 86, 87, 89, 90, 93, 102n1–2, 102n9, 104n33–4; hierarchy of reservations, notion of 54–58; philosophical reflections 195, 196; rationality, convention and economic decision-making 145–146, 158–159, 160–161, 161–162, 163n8, 164n31–2; Walras and 46–47, 60–61, 68–70, 73; Walras-Poincaré correspondence, mathematisation of economics 49–51; Walras-Poincaré correspondence, measurement of utility 52–54; Walras-Poincaré correspondence, Poincaré’s hierarchy of reservations 54–58 nsot, Louis 52, 71n6 itical Anatomy of Ireland (Petty, W.) 19, 23 itical Arithmetic (Petty, W.) 23 other Essay in Political Arithmetic (Petty, W.) 19 itical Discourses (Hume, D.) 25 tical economy 24–35; construction of, Say’s perspective on 33; Down Survey, quantitative surveying and 18; foundations of 16–20; historical development of 15–16; philosophical framework for 24–25 ovey, Mary 20 pper, Karl 59, 105 ter, T.M 48 t-Keynesian analysis 5, 145, 146, 154, 159, 161, 162, 165 t-Keynesian economics 196 t-Keynesian uncertainty and conventions 155–158 ter, M 129n14 ts, J 13 gmatic intuitionism, economic theorising and 178, 180 bam, K 45n20 est, G 212n4 or, A.N 83, 103n16 mises, convention and 151 endorf, Samuel von 28 nam, Hilary 111 hagoras 204 arterly Journal of Economics (QJE) 12, 156, 157 esnay, Francois 28–29, 34, 45n21; methodological approach 28–29; Tableau Économique 29 ne, W.V.O 82, 163–164n21 onality, convention and economic decision-making 144–162; appetites, emotions, passions and sentiments, interplay of 147–148; convention, Lewis’ six conditions for 153–154; custom and convention, relationship between 147; deductive system of pure mathematics 144; equilibrium, commitments to 144; game theory 153–155; General Theory (Keynes, J.M.) 144–145, 156, 157; geometrical conventionalism, Poincaré and 145–146; gold and silver as measures of exchange 151–152; goods, species of 150–151; Hume, convention and foundations of justice 146–152; Keynes, John Maynard 144–145, 146–147, 154–155, 162, 162n5; language, logicophilosophical and socio-historical views of 152–153; Lewis, convention and neo-classical rationality 152–155; Lewis, David 145, 146, 158–159, 161–162, 163–164n21–2, 163n9, 163n19, 164n24; mathematical modelling of rationality 2; morality, Hume’s philosophical foundations of 148–150; non-ergodicity 158, 159, 162; ontological-epistemic indeterminacy 145–146; ontological-epistemic indeterminacy, conventions and philosophy of mathematics and 158–161, 162; pleasure, Hume’s analysis of abstract idea of 148; Poincaré, Henri 145–146, 158–159, 160–161, 161–162, 163n8, 164n31–2; Poincaré’s conventionalist reading of principles of mechanics 158–160, 162; probability theory 144; promises, convention and 151; rational agent axiom 137–138; rational conduct in face of radical uncertainty, possibility of 145–146; rationality, commitments to 144; Risk, Uncertainty and Profits (Knight, F.H.) 156–157; society, Hume’s thesis on development of 150; Treatise on Human Nature (Hume, D.) 146–147; uncertainty and conventions, Keynes and post-Keynsians on 155–158; Walras, Leon 144, 158, 159 tansi, P.M 26 a recherche d’une discipline économique (Allais, M.) 41 ervations, notion of hierarchy of 54–58 ue D’Economie Politique 12 ardo, David 34–35; deductivist argumentation of 34–35 mann, Bernhard 85, 107, 160 ur, achievement of 79–84 k, Uncertainty and Profits (Knight, F.H.) 156–157 bbins, L 45n20 let, L 48, 71n1 ncaglia, A 19, 45n20 ser, Barkley J Jr 194n28 so, Giovanni 33 hschild, E 30, 45n24 sell, Bertrand 51, 71n5, 75, 90, 152, 164n27; constructive and computable mathematics, emergence of 167, 169, 175; mathematics, philosophy of 204, 206–208, 209, 211; philosophical reflections 196, 197 muelson, Paul A 12–13, 41, 43n2 , Jean-Baptiste 32–33, 34–35, 36, 37, 45n23 rf, H 100–101; algorithms, economic theorising and 189 elling, T 155, 163–164n21 mid, A.M 71n5 umpeter, J.A 15, 43–44n5, 43n4, 45n20 epanti, E and Zamagni, S 45n20 ckle, George L.S 157, 158 nk, J.B 26 on, Herbert 2, 4–5, 158; algorithms, economic theorising and 190–191 plicity, existence proofs and 90–94 nner, Quentin 20 rms, B 162n2, 163–164n21, 164n24 all, A 45n15 th, Adam 24, 32, 33–34, 96, 99, 135, 140, 182 iety, Hume’s thesis on development of 150 io-economic phenomena, Petty’s quantification and analysis of 20–21 ratic-Platonic tradition 147 ow, R 202 nenschein, Hugo 126, 138 ffa, Piero 146–147 atwissenschaft (von Justi, J.) 25 rk, W 14–15 ner, P 45n23 wart, L 45n19 ler, G.T 12, 37 ler, G.T., Stigler, S.M and Friedland, C 12 well, J 164n29 auss, E 18, 44n9 ct intuitionism: classical mathematics and 168–169; Dummett and philosophical reconstruction of 174–175, 177; neo-Walrasian programme and 170, 171; philosophical reconstruction of 174–177 den, Robert 146 herland, I 44n7 bleau Économique (Quesnay, F.) 29 , Peter Guthrie 62 qqu, M.S 50, 71n2 ascio, V 39 ski, A 103n17, 201 reatise of Taxes and Contributions (Petty, W.) 19; production models in 21–22 ocharis, R.D 45n25 orie de la Spéculation (Bachelier, L.) 50 ory of Games and Economic Behaviour (von Neumann, J and Morgenstern, O.) 41 ory of Value (Debrue, G.) 6, omson, Sir William 62, 64 poi 163n20 uffut, J.P 38 ité des Richesses (Isnard, A.-N.) 36 be, K 15 elstra, A.S 170, 178 elstra, A.S and van Dalen, D 191n1, 192–193n15, 192n8 ker, Josiah 25 got, Anne Robert Jacques 29–30, 45n22 ing, Alan Mathison 3, 4, 42, 129n14, 166, 184, 186–187; algorithms, economic theorising and 187, 188–189, 190, 191; algorithms and mathematisation of economics 183–187; Church-Turing thesis 194n26–7, 197, 201; formalism and economic theorising, internal critique of 181; mathematics, philosophy of 208, 209; post Gödel-Church-Turing view of classical mathematics 3–4 mer, J.H 19–20 ggi, G 45n21 ue and Capital (Hicks, J.) 41 n Daal, J and Jolink, A 45n27, 71n6 n Dalen, D 170, 178, 191n3 n der Berg, R 45n25 n Heijenoort, J 212n1 n Stigt, W.P 167–168, 173, 192n4, 192n7, 192n14 nderschraaf, P 146, 163n10, 163n17 ulillai Vela, Kumaraswamy 4–5, 13, 42, 45n31; algorithms, economic theorising and 189–191; constructive and computable mathematics, emergence of 185, 193–194n25, 193n19, 194n27–28 bum Sapienti (Petty, W.) 19 ks, A 192n12 taire 26, 27 Justi, Johann 25 Neumann, J and Morgenstern, O 163n10 Neumann, John 41, 42, 77, 102n7, 182, 212n2; algorithms, economic theorising and 187, 191 Thünen, Johann Heinrich 40 kefield, A 45n15 ld, A 41 lker, Donald A 45n27 llis, John 44n8 lras, Leon 2, 4, 5–7, 71n2–3, 71n15, 72n18–20, 185; axiomatic method in foundations of mathematics 105, 114, 116, 117, 126; economic thought, contribution to history of 46; equilibrium, Debreu and proof of existence of 99–102; formalisation of economics and Debreu’s philosophy of mathematics 74–76, 76–77, 98, 99, 102n1–2; mathematical economics, formulation of 14, 24, 35, 36, 37, 39–40, 41, 45n25; mathematical physics as deductive a priori science, response to Poincaré critique 62–68; mathematics, philosophy of 206; moral sciences, Walrasian programme in context of 47–49; neo-Walrasian formalisation of mathematics 131, 140, 142; ontological lock-in 68–70; philosophical reflections 195–196; platonic-scientific realism of, ‘Economics and Mechanics’ in context of 58–62; Poincaré and 46–47, 73; rationality, convention and economic decision-making 144, 158, 159; Walras-Poincaré correspondence, mathematisation of economics 49–51; Walras-Poincaré correspondence, measurement of utility 52–54; Walras-Poincaré correspondence, Poincaré’s hierarchy of reservations 54–58; see also neo-Walrasian theory lras-Poincaré correspondence 40; cardinal magnitude concepts 53; causal scientific realism 58, 62, 63, 67; consistency, pure mathematics and concern for 54; deductive method (Cartesian) 58, 62, 68; ‘Economique et Mechanique’ (Walras, L.) 46, 58–62; empiricalconventional account of principles of mathematical physics 58, 63–64; essentialist scientific realism 58, 59, 62, 69; hierarchy of reservations, Poincaré’s notion of 54–58; human action, dimensions of 47; infinite clairvoyance, Walras’ hypothesis of 56, 57–58; interpersonal comparison of utility, issue of 55, 57; intuition, Poincaré’s perspective on guidance by 51; Jaffé, William 47, 49, 50, 52–53, 54–55, 56, 58, 62, 67, 71n6, 72n18; linguistic-conceptual scheme of science 64–65, 66–67, 69, 158–159; logical validity, pure mathematics and concern for 54; mathematical physics as deductive a priori science, Walras’ response to Poincaré critique 62–68; mathematisation of economics 49–51; moral sciences, Walrasian programme in context of 47–49; ontological lock-in to Walras’ principles of mathematical economics 68–70; platonic-scientific realism, ‘Economique et Mechanique’ in context of 58–62; Poincaré’s hierarchy of reservations 54–58; principles and laws, Poincaré’s view of relationship between 65–66; probability theory, applicability of 50; probability theory, exposure of errors in use of 48; reassessment of 46–70; reassessment of, Élements d’economie politique pure (Walras, L.) and 40, 46, 48–51, 54–59, 68, 70; rebustness of principles 58, 60–61, 65–66, 67, 69; rigour, Poincaré’s quest for 51; Science and Hypothesis (Poincaré, H.) 55–56, 62, 67; scientific causality 64, 65; scientific conceptual schemes 64, 66–67; utility, measurement of 52–54 lrasian-Poincaré correspondence: measurement of utility 52–54 ng, H 129n13 rd, Seth 44n8 alth of Nations (Smith, A.) 32, 33–34 lections on the Production and Distribution of Wealth (Turgot, A.R.J.) 29–30 intraub, E.R and Mirowski, P 73, 96, 192n13 intraub, Steven H 12, 41, 42, 131, 157, 192n13; axiomatic method in foundations of mathematics 110, 117, 128n1–3; formalisation of economics and Debreu’s philosophy of mathematics 74, 76, 77, 78 ir, A 212n2 yl, Hermann 75, 79, 90, 93, 103n21, 104n27–8, 104n35; constructive and computable mathematics, emergence of 171, 172, 192n11 atmore, R 45n23 cksell, Knut 40 kin, John 44n8 lis, Thomas 44n8 tgenstein, Lugwig J.J 152, 193n18 rsley, Benjamin 18 en, Christopher 44n8 en, Matthew 44n8 ght, C 193n21 ung, H Peyton 146 mbelli, Stefano 45n31, 193–194n25, 194n27–28; algorithms, economic theorising and 189, 190 melo, Ernst 185 ... Philosophy of Mathematics and Economics Image, Context and Perspective Thomas A Boylan and Paschal F O’Gorman Philosophy of Mathematics and Economics Image, Context and Perspective Thomas A Boylan and. .. that the philosophy of mathematics should colonise the philosophy of economics The philosophy of economics has many mansions, one of which is constructed on the site of the philosophy of mathematics. .. Rationality and conventions in economics and in mathematics The emergence of constructive and computable mathematics: New directions for the formalisation of economics? Economics, mathematics and science: