i A Structuralist Theory of Economics Economists have long grappled with the problem of how economic theories relate to empirical evidence: how can abstract mathematized theories be used to produce empirical claims? How are such theories applied to economic phenomena? What does it mean to “test” economic theories? This book introduces, explains, and develops a structural philosophy of economics which addresses these questions and provides a unifying philosophical/logical basis for a general methodology of economics The book begins by introducing a rigorous view of the logical foundations and structure of scientific theories based upon the work of Alfred Tarski, Patrick Suppes, Karl Marx, and others Using and combining their methods, the book then goes on to reconstruct important economic theories – including utility theory, game theory, Marxian economics, Sraffian economic theory, and econometrics – proving all the main theorems and discussing the key claims and the empirical applicability of each theory Through these discussions, this book presents, in a systematic fashion, a general philosophy of economics grounded in the structural view Offering rigorous formulations of important economic theories, A Structuralist Theory of Economics will be invaluable to all readers interested in the logic, philosophy, and methodology of economics It will also appeal particularly to those interested in economic theory Adolfo García de la Sienra is research fellow at the Instituto de Filosofía, and professor of logic, philosophy, mathematics, and economic theory at the Universidad Veracruzana, Mexico Routledge INEM Advances in Economic Methodology Series Editor: Esther-Mirjam Sent 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 The Individual and the Other in Economic Thought An Introduction Edited by Ragip Ege and Herrade Igersheim Economics and Performativity Exploring Limits, Theories and Cases Nicolas Brisset A Structuralist Theory of Economics Adolfo García de la Sienra For more information about this series, please visit: www.routledge.com/ Routledge-INEM-Advances-in-Economic-Methodology/book-series/SE0630 iii A Structuralist Theory of Economics Adolfo García de la Sienra First published 2019 by Routledge Park Square, Milton Park, Abingdon, Oxon OX14 4RN and by Routledge 52 Vanderbilt Avenue, New York, NY 10017 Routledge is an imprint of the Taylor & Francis Group, an informa business © 2019 Adolfo García de la Sienra The right of Adolfo García de la Sienra to be identified as author of this work has been asserted by him 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: García de la Sienra, Adolfo, author Title: A structuralist theory of economics / Adolfo García de la Sienra Description: Edition | New York : Routledge, 2019 | Series: Routledge INEM advances in economic methodology | Includes bibliographical references and index Identifiers: LCCN 2018039630 (print) | LCCN 2018041087 (ebook) | ISBN 9781315100609 (Ebook) | ISBN 9781138295643 (hardback : alk paper) | ISBN 9781315100609 (ebk) Subjects: LCSH: Economics | Econometric models | Game theory Classification: LCC HB141 (ebook) | LCC HB141 G367 2018 (print) | DDC 330.1–dc23 LC record available at https://lccn.loc.gov/2018039630 ISBN: 978-1-138-29564-3 (hbk) ISBN: 978-1-315-10060-9 (ebk) Typeset in Times New Roman by Apex CoVantage, LLC v This book is dedicated to Luz María and to the memory of Elena Minerva and Adolfo vii Contents Preface Acknowledgments x xi 1 Introduction 1.1 A new view of scientific theories 1.2 The S-view in action: an example 1.3 Problems of the S-view 13 Models and structures 2.1 2.2 2.3 2.4 2.5 The rise of the concept of structure 13 Models for first-order logic 14 Elementary classes in economics 19 Theory of classes 25 The general concept of structure 30 35 The structuralist view of theories 3.1 3.2 3.3 3.4 3.5 Origins 35 The concept and relevance of a metatheory 35 The ways of theoretical practice 38 A dialogue between the L-view and the S-view 38 The structuralist view of theories 48 55 Idealization and concretization 4.1 4.2 4.3 4.4 Motivation 55 Empirical structures 57 Construction of idealized models Concretization 65 Measurement 5.1 The nature of measurement 69 5.2 Representational measurement 72 59 69 viii Contents viii 5.3 Measurement by means of theory 74 5.4 Measurement without theory 78 82 A general concept of an economy 6.1 Fixing the reference of economic theories 82 6.2 A general concept of an economic system 82 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 114 Introduction 114 Finding the (lost) beings, stories and narratives 115 Conceptual prolegomena 116 Concepts of strategy 117 Probability spaces induced by behavioral strategies 119 The equilibrium concept 123 The game concept 125 Empirical applicability of GT 129 132 Abstract labor and labor-value 9.1 9.2 9.3 9.4 9.5 10 Introduction 90 The conditions of Debreu and Mas-Colell 92 Motivation 93 The algebraic theory of difference 94 The geometric theory of difference 97 The theory of preference 106 The empirical meaning of differentiability 107 New foundations of preference theory 111 The logical structure of game theory 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 90 Preference and utility Introduction 132 Abstract labor 134 Market-determination of abstract labor 140 Labor-value 145 The meaning and relevance of abstract labor 153 155 Classical economics 10.1 10.2 10.3 10.4 10.5 Introduction 155 The fundamental Marxian theorem General equilibrium 160 Equilibrium 163 The empirical claim 172 156 ix Contents 11 The logical structure of Sraffian economic theory 11.1 11.2 11.3 11.4 11.5 11.6 12 174 Introduction 174 The conceptual framework 174 The basic theory-element 177 The standard commodity 179 The empirical claim 180 What can/ought be done? 181 The logical structure of econometrics 12.1 12.2 12.3 12.4 12.5 12.6 12.7 12.8 12.9 ix 182 Motivation 182 Real concrete DGPS 183 Random experiments 184 Probability spaces 185 Random variables 189 Conditional probability as fundamental 191 From conditional to absolute probability 194 Statistical inference 195 The logical structure of econometrics 196 References Author Index Subject Index 199 207 211 208 Author Index Hammond, P J., Hands, D W., 66, 67 Hausdorff, F., 24 Hegel, G W F., ix, 37 Helmer, O., Hendry, D., 182 Henkin, L., Hestenes, M R., 143 Hicks, N J., 93 Hilbert, D., 112 Houthakker, H S., 115, 130 Hölder, O., 11–12, 14, 18, 22, 69, 70, 92, 97, 99, 110, 112 Mirowski, P., Mirrles, J A., 148 Mitchell, W C., 79 Morishima, M., 157 Moulines, C U., 14, 30, 65 Muller, F A., 1, 4, 6, 9, 11, 13, 25, 55 Mäki, U., 64 Jammer, M., 79 Jeffrey, R., 34 Jerison, M., 178 Jónsson, B., 24 Okishio, N., 158 Oliu, J., 178 Kamke, E., 14 Karlin, S., Katzner, D W., 91 Kemp, M C., 138 Kepler, J., 50, 67, 197 Kimura, Y., 138 Kolmogorov, A., 184–185 Kolmogorov, A N., 114 Koopmans, T C., 79–80, 148 Krantz, D., Krantz, D H., 69 Kreps, D M., 128 Kuhn, H W., 118 Kuhn, T S., 174, 182 Kurz, H D., 178 Kusch, M., Lakatos, I., 182 Levahri, D., 148 Lindenbaum, A., 19 Lorenzano, P., 67 Lotka, A J., 59 Lowen, R., Luce, R D., 3, 69, 72–73, 128, 192 Löwenheim, L., 13, 19, 24 Maltsev, A., 19 Mandel, E., 133 Marx, K., 37–38, 55, 61–62, 65, 68, 84–85, 89, 132, 134, 140–142, 154–155, 161, 172, 180 Mas-Colell, A., 91–93 McKinsey, J C C., 1, 2, 23 Michell, J., 11, 73 Nagel, E., Nash, J F., 83, 124, 126 Newton, I., 2, 18, 37–38, 42, 46, 69 Nicholas, H., 180 Nowak, L., 68 Peano, G., 13 Pearson, K., 197, 198 Philosopher, the see Aristotle, 80 Ponce de León, J., 72 Popper, K., 182, 185 Portides, D., 64 Punzo, L F., 179 Raiffa, H., 128 Ramsey, F P., 50 Richter, M K., 23 Robinson, A., 24 Roemer, J E., 163 Salvadori, N., 178 Samuelson, P A., 75, 114–115, 130, 148 Schefold, B., 178, 179 Scott, D., 21, 23 Seton, F., 158 Skala, H J., 19, 21 Skolem, T., 13, 19, 24 Slomson, A B., 24 Slutsky, E., 75 Smith, A., 79 Sneed, J D., 14, 30, 35, 48–50 Spanos, A., 80, 82, 182–184, 193, 195–197 Spiegel, M R., 43 Sraffa, P., 174, 175–181 Stegmüller, W., 39 Stiemke, E., 138, 139 Stiglitz, J E., 148 Stigum, B., 68 Struik, D J., 79 Author Index Sugar, A C., Suppe, F., 182 Suppes, P., 1–5, 14, 20–21, 23, 35, 40, 42, 57, 59–60, 69–74, 79–81, 184–185 Sydenham, P H., 71–72 Symon, K R., 43 Takayama, A., 173 Tarski, A., 1–3, 13, 16, 19, 24 Truesdell, C., Tverski, A., 3, 69 van der Waerden, B L., 13 Van Heijenoort, J., 13 Varian, H R., 67, 75, 81, 130 Velupillai, V., 179 Volterra, W., 59 Wade, H., 11 Walras, L., 63, 65 Whinston, M D., 91–92 Zermelo, E., 30 Zoubek, G., 55, 68 209 Subject Index abstract labor, 132, 134–135, 141, 143, 145, 151, 155–156 structure, 136 as a weak ordering, 135, 137, 143 cancellation law, 135, 137, 144 continuity, 136–137, 144 homotheticity, 136–137, 144 monotonic, 78, 92, 144 monotonicity, 135, 137 reduction, 164, 172 representation theorem, 70, 137, 140 abstraction, 11 added value, 157 admissible approximation, 52 Afriat’s inequalitites, 78 agent behavior, 172 aggregated maximum profit, 162 algebra modern, 13, 17 σ-, 186, 187 universal, 13, 17 algebraic structure, 13–14 field, 18 group, 18 monoid, 18 ring, 18 semi-group, 18 algebraic topology, 38 algebraic-difference measurement, 11, 70, 79, 112 structure, 96, 103, 106, 107 geometric representation, 10, 13 numerical representation, 10 representation, 96 theory, 91, 94 allocation of resources by the market, 87 through distribution, 88 allocation rules, 84 alternative techniques, 145, 172, 175 analytical geometry as numerical representation, 70 Antonelli’s integration problem, 75 approximation, 65 intratheoretical, 51 approximation relation, 52 ARC, 25 ARCU, 25, 30, 40 arithmetic foundations, 13 Peano’s axioms, 13 arithmetic or elementary class, 23 ars inveniendi, 37 Atwood’s machine, 48 augmented reproduction, 160 axiom of choice, 83 of class existence, 26 of class separation, 26 of completeness, 25, 28 of extensionality, 26 of regularity, 29 of urelements, 25–26 schema of set existence, 28 axiomatic method, axiomatization kinds, 39 basic theory-element, 39, 41, 65 Berge’s maximum theorem, 166, 168–169 Berkeley University, Bertrand’s game, 125 equilibrium, 114–115, 125, 127, 154–155, 160, 164 pure strategy, 23, 117, 125–126 Book of the Dead, 79 Borel algebra, 187 set, 187 212 Subject Index Bourbaki, N., 14 budget restriction, 85 calculus, differential and integral, 22 capitalist economy, 61, 65 capitalist society, 85 cardinal number, 14 cardinal of the continuum, 22 cardinal preference theory, 113 Carnap’s approach, see scientific theory L-View, categorical theory, 18 category theory, 25, 30 central planner, 84 characterization, 8–9, 41, 49, 51 choice action, 83 circumstance, function, 88 structure unintended consequence of, 62 choice axiom, 25 City College, New York, class, 25 disordered pair, 28 empty, 27 intersection, 29 κ-class, 24 M-homogeneous, 23 M-universal, 23 of all sets, 25 of urelements, 25–26 subclass, 27 classical equilibrium, 114–115, 125, 127, 154–155, 160, 164 analysis, 2, 5, 23 dynamics, 7, 90 economics, 155 economy, 148, 160, 164, 172 equilibrium, 114–115, 125, 127, 154–155, 160, 164 mechanics, see classical particle mechanics particle mechanics, 38–39, 48–49 axiomatization, 40 basic theory-element, 8, 40, 114 specialization, 39 sun-earth system, 50 system, 39, 40 theoretical terms, 49 closed convex cone, 160 closed first-order theory, 17 closure under isomorphism, 18 Columbia University, compactness theorem, 19 comparative statics, 115 concept of magnitude, 70 concept of preference, 47 concept of truth, 11 concrete system, 47 concretization, 11, 55, 57, 61, 65, 67–68 as intertheoretical relation, 55 conditional probabilities, 194 conditional-probability structure, 185 Condorcet condition, consequences of a set of sentences, 17 conservative force, 39 consistent set of sentences, 17 constant capital, 157 constraint, 50–51, 53 in game theory, 126, 130 consumer behavior, 58, 90 choice, 58, 61, 65, 175 concept of, 62 ideal concept, 75 idealized concept of, 63 subsistence, 85 theory, 83, 90, 93, 126 neoclassical, 90 consumption, 68, 83 bundle, see consumption menu final, 84 menu, 83, 85, 88, 93, 160 minimum, 85 productive, 84 countable unions, 186 cutting-off function, 50 data generation process, 80, 82, 183, 193 and random experiments, 184 as a random mechanism, 183 as a real-concrete system, 183 as a source of observable data, 183 data-structure, 6, 9, 58 decidable set of formulas, 18 decision theory, 19 deductive system, demand theory, 74, 129, 130 as a specialization of game theory, 129 demand function, 75 empirical, 2, 75 Walrasian, 75–76, 81, 92, 173 C1, 92 Subject Index derivation relation, 17 derivation rules, dialectical materialism, 37 difference structure, 94, 96 numerical representation, 10 difference relation, 17–18, 94–95 geometric representation, 18, 112 numerical representation, 17 difference structure, 112 geometric representation, 18, 112 differentiable manifold, 93 disjointed set, 29 distribution, 68, 83–84 rule, 85, 86 system, 15, 47, 51–52, 64, 66, 82, 84, 157 Douglas Aircraft Company, dynamic theory, 155 dynamical representation, 46 dynamics, 7–8 econometric structure, 197 econometrics, 12, 57, 79, 182 T-theoretical terms, 178, 181, 196 and probability theory, 197 basic theory-element, 54, 114, 174, 184, 196 density functions, 197 domain of intended applications, 178, 196 empirical claim, 49–51, 53, 178, 198 explanation in, 197 fundamental law, 198 histogram, 196 as a step function, 196 cumulative, 196 dispersion, 196 higher central moments, 196 kurtosis, 196 location, 196 mean, 196 median, 196 mode, 196 shape, 196 skewness, 196 standard deviation, 196 variance, 196 logical foundations, 182 logical structure, 196 observable data, 197 observed data, 193, 195 partial potential models, 197 Pearson frequency curve, 197 213 probability model, 193–195, 197 random variables, 194 sampling model, 193, 195, 197 statistical model, 195, 197 confidence estimation, 195 hypothesis testing, 195 misspecification, 195 point estimation, 195 prediction, 195 specification, 193 structuralist characterization of, 184 true density function, 198 economic coordination, 87 by the market, 87 equilibrium, 114–115, 125, 127, 154–155, 160, 164 existence of, 115, 164, 172 methodology, 18, 37–38 and econometrics, 182 system, 15, 47, 51–52, 64, 66, 82, 84, 157 as data generation process, 89 as organic whole, 61, 83 as target system, 89 concept, 82, 123–124 cycle of, 83 disequilibrium, 87 equilibrium, 83, 128–129, 154 feasibility condition, 85 general concept, 66, 79 labor resources, 85 productive cycle, 85 reproduction, 87, 177 total resources, 85 theory, 19, 23 unit, 82 household, 82 production unit, 82 economy, 85, 88, 172 efficient process existence, 70, 146 elementary class, 18, 24 in economics, 19 event, 30, 185, 189 elementary or arithmetical class, 17–18, 21 empirical claim, 61, 64, 66 of classical economics, 172–173 of game theory, 131 of Sraffian economic theory, 180, 181 214 Subject Index data, 38, 48, 57, 64, 66, 68, 91, 183 in consumer theory, 130 meaning, 41 observation, 38 phenomenon as target of game theory, 126 structure, 6, 57–58, 66–67, 74–75 theory, ens rationis, 25 equal-arms balance, 57, 73, 79 measurement, 11, 70, 79, 112 equilibrium Marxian, see equilibrium, classical equilibrium price as a sink, 155 Euclid’s Archimedian axiom, 112 Eudoxo’s theory of proportions, 69 exact value of a magnitude, 42 excess demand, 163 exchange, 68, 83, 84 as a social process, 88 as reallocation of initial endowments, 87, 88 by means of a general equivalent, 87 by means of barter, 87–88 by means of familiar agreements, 87 by means of money, 88 exchange process, 88, 134 exchange-value, 175 expected utility maximization, 124, 160 of player in a game, 126 explanation in game theory, 126–130 exploitation rate, 157 extended completeness theorem, 19 factual predicate, 38 feasibility set for consumers, 162–163 aggregated, 53, 162–163 for firms, 161–162 aggregated, 53, 162–163 feasible choices, 83 final allocation, 88 financial capital market, 155, 160, 164 first-order logic, 1, 5, 13, 40 formalization in, 40 formulation of a theory in, language, 4, 6, 14, 40 cardinality of a, 16 connectives, 16 constants, 16 formulas, 16 individual variables, 16 model of a, 16 notion of satisfaction, 17 notion of truth, 17 parameter, 16 sentences of a, 16 terms of a, 16 theory, 17 formalized language, 35, 40 foundations of measurement, 3, 22 foundations of physics, Fountain of Eternal Youth, 72 Frobenius root, 173 fundamental measurement, 11, 70, 79, 112 measuring, 71 fundamental law, 67 of game theory, 127–128, 131 of Sraffian economic theory, 177, 179–180 Fundamental Marxian theorem, 156–158, 160 fuzzy set, 52 game behavioral strategy, 114–115, 117–119 profile, 117–118, 124 concept, 7, 90, 125 definition dynamic, 114, 125 equilibrium, 114–115, 125, 127, 154–155, 160, 164 concept, 82, 123–124 definition, 124 existence of, 115, 164, 172 Nash, 124 feasible action, 117 history, 116 feasible, 123 initial, 117 mixed strategy, 23, 117–118 profile, 117–118, 124 perfect recall, 118 pure strategy, 23, 117, 125–126 profile, 117, 118, 124 strategy profile, 115, 123 temporal horizon, 114, 116 theory, 11, 65, 114–115 T-theoretic term, 127 application to empirical phenomena, 130 basic theory-element, 8, 40, 114 constraint, 115 empirical applicability, 114 Subject Index empirical claim, 125 equilibrium, 83, 128–129, 154 fundamental law, 114–115, 125, 129 guiding principle, 125, 127 logical reconstruction of, 114 models, 114 problem of lost beings, 115 problem of lost stories, 115 two-person, 23, 65 with discrete time, 125 with finite horizon, 125 with multiple equilibria, 128 zero-sum, 65 Gaussian curvature, 93 general concept vs theoretical model, 68 general equilibrium, 160 general equilibrium theory, 67 intended applications, 66 general productive system, 175 Generality I, 36, 38 II, 36, 38 III, 36–37 Genesis, 79 geometric interval, 97, 112 congruence, 97 direction, 97 equivalence class, 97, 98 Abelian group, 99 order relation, 99 order relation, 99 sum, 97–98 geometric-difference structure, 111 theory, 97 Gödel’s completeness theorem, 18 Haavelmo’s methodological foundations of econometrics, 182 Hausdorff ηα-sets, 24 Hegel’s dialectics, 37–38 homogeneous labor, 172, 176 household consumption menu, 84 hypothetical deductive system, 38 Hölder axioms, 18, 97 cuts, 18 theory, 91, 112 ideal model, 47, 50–53, 57, 59, 172 system, 15, 47, 51–52, 64, 66, 82, 84, 157 type, 63 idealization, 11, 54–55, 57, 61–63, 65–68 as decomposition, 64 as deformation, 64 as intertheoretical relation, 55 as isolation, 64 as stabilization, 64 in economics, 55 in physics, 55 nullifying, 64 stabilizing, 64 idealized concept, 63 imbedding, 6, 58, 60 isomorphical of a data-structure, inclined plane, 42, 46–47, 50 independent producer, 84 indifference curve Gaussian curvature of, 92 sets, 92 indifference sets, 92 individual choice, 153 structure, 7–9 similarity type, 9, 15, 33 theory, basic theory-element, 8, 40, 114 inference rule, 18 modus ponens, 18 generalization, 18 informal mathematics, narrative, 41 initial distribution, 86 endowment, 86–88 of labor-power, 86 of material resources, 86 resources, 161 instrumentalism, 73 integration method, 67 intended application, 82 interest rate, 160, 171 intertheoretical relation, 67 investment return, 161 isomorphism, 14 joint production, 145, 172, 176, 179 Jónsson’s theorem, 24 Kepler’s laws, 50 kinematic term, 215 216 Subject Index kinematic vs dynamic terms, kinematical representation, 46 structure, 47 kinematics, Kolmogorov’s axioms for probability theory, 184–185 Kolmogorov-Bochner theorem, 114 Kuhn’s theorem, 118 logic, methodology and philosophy of science congresses, logical axioms, 18, 97 calculus, empirical interpretation of, primitive symbols, consequence, 17 Löwenheim-Skolem-Tarski theorem, 19, 24 labor -power, 85, 87, 134, 176 reproduction, 87, 177 -value, 132, 145, 151 expenditure, 135 allocation, 87, 133–134 as an indispensable input, 146 division, 132 equalization, 134 heterogeneous, 67, 141, 177 homogeneous, 65, 140–141, 177 input, 84, 134 normal, 134, 135, 146 process, 83–85, 87, 135, 156, 175, 184–185 efficient, 145–146, 148 productivity, 146 simple, 21, 141, 156, 187 theory of value, 65, 67, 156, 160 unskilled, 141, 156 labor-power allocation, 87, 133–134 normal, 134–135, 146 labor-value, 156 as representation of abstract labor, 151-153 market independent, 140 market-dependent, 141–142 law of excluded middle, 180 of utility maximization, 58, 63 of value, 67, 132–133, 174 in Leontief economies, 173 manifestation form, 132 least squares, 57 Leontief economy, 148, 160, 164, 172 model, 47, 50–53, 57, 59, 172 Leviticus, 79 lexicographic preferences, 22 Lindenbaum’s theorem, 19 linearly ordered sets, 22 logic formalized language of, 13, 18 mathematical, 13 manifold C2, 92 Ck, 92 differentiable, 70, 90, 92 marginalists, 22 market participant, 154 price, 142, 145, 151, 155–156, 158–160, 164, 171, 173 Marx’s dialectics, 37, 38 method of political economy, 11, 37, 89 Marx’s method of political economy, 55 Marxian economics, see classical economic theory, 12, 154–155 mathematical logic, 179 method, 65 vs experimental method, 63 structure, 74 mathematical structure, see set-theoretical structure maximum profit, 162–163 means of production, 134 measurement, 47 and weak orderings, 71 as an empirical operation, 71 as assignment of numbers, 69 as homomorphism, 72, 80 as isomorphic imbedding, 70–71 as numerical representation, 70 by geometric entities, 69 by real numbers, 69 conjoint, 79 econometric, 79–80 existence, 70, 146 fundamental, 79 in psychology, 72 instrument, 52 of a magnitude, 71 of idealized properties, 69 procedure, 74 representation theorem, 70, 137, 140 Subject Index representational, 69, 70, 72, 74 as a measuring procedure, 73 set-theoretic structures in, 73 representational theory, 70 uniqueness theorem, 70 utility, 74 without theory, 80, 81 measuring procedure, 74 structure, 71 fundamental, 72 mechanical law, 39 metaphysical theory, 68 metatheory, 38 method of production, 174, 175, 177 all-engaging, 178–179 of subsistence, 177 of surplus, 177 self-replacing, 177 methodological individualism, 62, 153 metric concept, 47 metrization, 72 of a property, 74 minimum wage, 157 mode of production, 84 model as a set-theoretical structure, 5, 11 as imaginary system, 11 as set of sentences, 59 characterization of extrinsic, 35 intrinsic, 35 complete theory, 18 econometric, 79–80 elementary equivalence, 18 elementary extension, 18, 24 extrinsic characterization of, for first-order logic, 13–15 in Tarski’s sense, 172 of a first-order language, 11, 15 of a first-order theory, 35 of a sentence, 16 of a theory, 5, of first-order logic, 30 system, 15, 47, 51–52, 64, 66, 82, 84, 157 theory, 18 model of data, see data-structure model system, 59–61, 63–65, 75 as idealization, 61 as target, 60 monetary value of net product, 157 Mose’s law, 79 motion composition, 95 217 Nash equilibrium, 126 National Research Council, National Science Foundation, neoclassical economics, 155 basic theory-element, 54, 114, 174, 184, 196 net output, 84 Newton’s laws, 46 Newton’s second law, 38, 42 Newton’s theory of proportions, 18 non-T-theoretical structure, 47 non-T-theoretical term, 50–51 of Sraffian economic theory, 177, 179–180 non-linear programming, 90, 112 non-substitution theorem, 148–149, 151 nonstandard model of real-number theory, 23 normal science of game theory, 127–128, 131 of Sraffian economic theory, 177, 179–180 nullifying assumption, 67 numerical conditional-probability representation, 192 numerical representation, 70 observable data, 82, 183 observation sentence, 4, observational vs theoretical terms, observationally adequate theory, observed behavior of an agent, 47 ontological theory, 68 opportunity set, 7, optimality set for consumers, 162–163 aggregated, 53, 162–163 for firms, 161, 162 aggregated, 53, 162–163 order -type, 14 dense strict, 24 isomorphism, 14 strict partial, 21 strict simple, 21–23 ordering, 19, 20 dense simple, 18 linear, 18 partial, 18, 21 quasi-, 20–21 simple, 21, 141, 156, 187 strict partial, 21 strict simple, 21–23 weak, 21 ordinal preference theory, 113 218 Subject Index p-representation, 61 Paleolithicum, 79 parametric method, 66, 75, 81 partial ordering, 39 partial potential model, 48, 50, 57 imbedding of a data-structure into, 58 imbedding of an empirical structure into, 58 of consumer theory, 76 of game theory, 127–128, 131 particle mechanics, see classical particle mechanics, perfect information, 63 knowledge, 115 memory, 115 recall, 63 philosophy of science, 36, 38 Polish school of logic, positive integers, 13 potential model, 49, 51 practice, 35 pre-theoretical thought, 183 preference as a game-theoretic term, 129 cardinal, 93–94, 97 concept, 7, 90, 125 strength, 91 structure, 8–9, 18, 22, 24 preference ordering, see preference relation preference relation, 1, 7, 17–19, 23, 47, 61, 63, 74, 85, 88, 90–92, 111–112 cardinal, 93–94, 97 connected, 20, 63 continuous, 25, 78, 90–93, 160, 189 convex, 91 differentiability condition, 17, 111 empirical meaning of, 18 differentiability condition, 91 empirical meaning, 17 locally nonsatiated, 63, 92, 160 maximization of a, monotonic, 78, 92, 144 nonsatiated, 25, 64, 78, 90 order properties, 91 rational, 21 rationalization of a choice structure by, regular, 8, 21, 63, 90, 160 smooth, see uniform differentiability strictly convex, 63, 90, 160 transitive, 20, 62–63 uniform differentiability empirical meaning of, 18 uniformly differentiable, 19 preference structure generation of a choice structure by a, in a game, 126 regular, 8, 21, 63, 90, 160 smooth, 18 preference theory, 18, 91, 111 new foundations, 111–112 price system, 157, 161, 179 induced by a reduction, 156 induced by abstract labor, 151 representation of, 142 price-distribution system, 176 prime model, 18 test, 19 private ownership economy, 84, 160 private property, 174 probability as propensity, 185, 191 as relative frequency, 184–185 probability distribution, 187 simple, 21, 141, 156, 187 probability measure, 185–186 of a countably additive probability space, 188 probability space, 30, 33, 185–186, 193 algebra of events, 185 as formalization of a random experiment, 191 continuous, 25, 78, 90–93, 160, 189 countably additive discrete, 189 elementary event, 185–186 elementary identities, 187 event, 30, 185, 189 generated by a qualitative conditional probability Archimedian structure, 194 sample point, 185 sample space, 185 continuous, 187 countably infinite, 187 finite, 187 standard, 117 probability theory, 34, 80 in econometrics, 184, 193 problem of lost beings, 9, 13, 38, 41, 68 of lost content, 9, 11, 13, 38, 41, 68 of theoretical terms, 50 of unavailable stories, 9, 10, 38, 41, 68 Subject Index producer, 86 production, 68, 83 means, 84 mode, 88 possibility set, 84, 88 price, 142, 145, 151, 155–156, 158–160, 164, 171, 173 equilibrium, 83, 128–129, 154 process, 83–85, 87, 135, 156, 175, 184–185 efficient, 145–146, 148 flow version, 84 global, 163 stock version, 84 structure, 146, 177 activation, 157 activation or state, 149 system, 15, 47, 51–52, 64, 66, 82, 84, 157 as a convex polyhedral cone, 142 profile of actions, 83 of choices, 83 profit maximization, 175 proletariat, 62, 85 proper class, 14 pure mathematics, 4–5 pure or safe predicate, 25–26 purely mathematical theory, 38 qualitative conditional probability, 192–193 Archimedian structure, 192–194 quantifier elimination, 18 theorem, 19 quantitative relationships, 70 Ramsey sentence, 11, 50–51 Rand, see Rand Corporation Rand Corporation, 1, random experiment, 184–185 process, 83–85, 87, 135, 156, 175, 184–185 variable, 187, 189, 190 density function, 193, 195 distribution function, 191 joint distribution function, 193, 195 probability density, 191 probability distribution, 193 probability function, 191 true density function, 195 vector, 193 219 rate of profit, 142, 156, 158, 161, 173, 177 average, 157 maximum, 157 uniform, 142, 145, 155, 159–160 rate of surplus value, 160 rational agent, 123 choice, 58, 61, 65, 175 in a game, 126 individual, 90 theory, 90–91 player, 125 rationalization of data, 58, 81 real analysis, 2, 5, 23 number system, 23 real-concrete economic system, 66 objects, 74 system, 15, 47, 51–52, 64, 66, 82, 84, 157 mechanical, 50 reduction, 140 of heterogeneous labor, 145, 151, 154 induced by a price system, 143 standard, 145 relation antisymmetric, 20 asymmetric, 20 connected, 20, 63 intransitive, 20 irreflexive, 20 reflexive, 20 strongly connected, 20 symmetric, 20, 75 transitive, 20, 62–63 relational structure, 13, 42 M-homogeneous, 24 M-universal, 24 relational system, 14, 18–19 chain of, 24 class of, 21 elementary extension, 18, 24 relative prices, see exchange-value, 175 representational measurement, 11, 70, 79, 112 theory of measurement, 73 reproducible economy, 148, 160, 164, 172 social choice, 132 reproduction, 174 220 Subject Index response theory, 74 revealed preference strong axiom, 75, 78, 130 as a constraint, 130 weak axiom, 130 ring of sets, 186 Robinson’s model completeness test, 19, 24 sample space, 30 Samuelson-Houthakker’s strong axiom of revealed preference, 115 Santa Monica, California, satisfaction difference, 94 interval, 94, 99 level, 93, 113 rate of change, 113 of a first-order formula, 17 of a set of first-order formulas, 17 scale transformations, 71 scientific theory, 3, 5, 38 application, 51, 57, 66, 75, 81 approximated application, 51 as a methodological tool, 74 as a sup-semilattice, 39 as a system of set-theoretical structures, 6, 74 as set of statements, 39 axiomatization of, 40 basic theory-element, 54, 114, 174, 184, 196 empirical claim, 49–51, 53, 178, 198 aggregated, 53, 162–163 empirical content, empirical testing, 38 guiding principle, 126 intended application of, 66 intended scope, see domain of intended applications, see domain of intended applications logical reconstruction of, 39, 41 L-View, 1, 3–6, 38 metamathematical concept of, 39 metatheory of, 37 model, 47, 50–53, 57, 59, 172 nature, nomological statement of a, 51 observational term, 48 partial potential model, 50, 52–53 positivist conception, potential model, 50 rational reconstruction, 36, 38, 54 regional theory, 53 specialization, 39, 53 standard formalization, S-View, 5–11, 13, 38 problems, testing, 75 T-theoretical term, 48–49 semantic of theories, 55 sentences formulated in a formal language, set theory, 5–6, 35 term, 25 set-theoretical analysis, 2, 5, 23 characterization, 33 methods, object, 33 characterization of, 33 predicate, 5–8, 14, 33, 35, 40–41, 49 structure, 11, 57–58, 59, 61, 63–64, 66, 70, 91 in the empirical sciences, 41 similarity type, 9, 15, 33 simple reproduction, 160, 175 Slutsky matrix, 75 negative semi-definite, 75 symmetric, 20, 75 social equilibrium, 114–115, 125, 127, 154–155, 160, 164 labor, 156 wealth, 163 social choice feasible, 86 reproducible, 87 social wealth, 174 socially necessary labor-time, 134, 157 society differentiated, 134 undifferentiated, 134 Spanos’ econometrics, 182 specialization, 65, 67 Sraffa’s mathematics, 179 constructivist, 179 intuitionist, 179 Sraffian economic theory, 12, 174, 177 T-theoretical terms, 178, 181, 196 basic theory-element, 54, 114, 174, 184, 196 domain of intended applications, 178, 196 Subject Index empirical structure, 181 of value, 67, 132–133, 174 standard commodity, 174, 176, 179 standard sequence, 96 Stanford University, 1–3 Department of Economics, Institute for Mathematical Studies in the Social Sciences, model revolution at, statement view of theories, see scientific theory L-View, see L-View statistical inference, 193, 195 Stiemke’s theorem, 138 stochastic process, 193 Stone Age, 79 strategic rational behavior, 115 strong axiom of revealed preference as a constraint, 114 structural claim, see scientific theory empirical claim, 60 structuralism, structuralist theory of economics, 12 structuralist theory of science, see structuralist view of theories structuralist view of theories, 11, 35, 38, 48, 67, 182 view of theoretical progress, 67 structure Bourbaki’s concept of, 14, 30 domain, 15 enlargement or extension of a, 15 for first-order logic, 13–15 fundamental relations of, 15 homomorphic image of, 15 homomorphism, 15 automorphism, 15 epimorphism, 15 isomorphism, 15 monomorphism, 15 imbedding, 15 M-homogeneous, 22 ðM; kÞ-universal, 22 M-universal, 22 numerical, 20 of data, 58, 81 of real numbers, 22 order of, 14 set-theoretical, 31–32 auxiliary sets, 30, 31 canonical transformation of, 30 first-order, 33 general concept of, 30 principal sets, 30, 31 similarity type, 9, 15, 33 221 type of n species, 31–32 similarity, 15 similarity type, 8, 15, 32 substructure of a, 15 theory of a, 17 subsistence basket, 85, see subsistence consumption menu, 161, 177 subsistence consumption menu, 157 Suppe’s semantic view of theories, 182 Suppes’ learning theory, 81 view of theories, 4, 35 surplus rate, 157, 158 synthetic geometry, 70 system reproduction, 87 t-representation, 61, 64 target system, 60–61, 64, 66, 68, 75 as a data generation process, 193 in econometrics, 184, 193 Tarski’s concept of model, 16 tautology, 18 technological, 146 possibility, 87 set, 145 technology, 146 theorem Afriat, 78 theoretical idealized model, 68 model, 47, 50–53, 57, 59, 172 practice, 35–36, 38 ideological, 36 pre-scientific, 36 scientific, 36 systematization, 57, 65 terms, 35, 50 T-theoretical, 35, 47, 48 T-theoretical function in economics, 80 theory confirmation of, core, 51–52 empirical, 2, 75 falsification of, finitely axiomatizable, 18 logical structure of a, net, 53 of classes, 25 of labor-value, 155 of measurement, 21, 47 of science, 37 of theoretical practices, 37 primitive terms, 222 Subject Index theory-dependence of observation, 183 theory-element, 52–53, 66 idealized, 67 specialization, 39, 53 total benefit, see total profit labor-power, 88 material resources, 88 profit, 157 resources, 161 trade, 88 union, 155 truth for formalized languages, 13 of a first-order formula, 17 unavailable stories, problem of, 13 universal numerical system, 23 unlimited computational powers, 115 urelement, 25 utility as a T-theoretic function, 81 as a game-theoretic term, 129 compared with labor-value, 132 maximization, 124, 160 utility function, 17, 20, 67, 75, 126 as a game-theoretic concept, 130 as a random variable, 190 C2, 92 cardinal, 93–94, 97 Ck, 92 concave, 78 continuous, 25, 78, 90–93, 160, 189 continuously differentiable, 19, 63, 90–91, 93, 111–112 C1, 92 differentiable, 70, 90, 92 integration of, 75 locally nonsatiated, 63, 92, 160 monotonic, 78, 92, 144 monotonically increasing, 90 nonsatiated, 25, 64, 78, 90 quasi-concave, 90 quasiconcave, 63 utility theory, 21, 23 with respect to Q, 22 with respect to a numerical system, 22 validity of a first-order formula, 17 value substance, 141 variable capital, 157 wage equilibrium, 114–115, 125, 127, 154–155, 160, 164 rate, 177 system, 15, 47, 51–52, 64, 66, 82, 84, 157 representation of, 142 Walrasian demand, 115, 126 weak order, 18, 21 Wissenschaftstheorie, 38 World War II, ZFC, 14 ... Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloging-in-Publication Data Names: Garc a de la Sienra, Adolfo, author Title: A structuralist. .. project Filosof a de la Econom a 127380 Section of Chapter is a modified translation of A Garc a de la Sienra “Teor a general de las clases”, originally published in A Garc a de la Sienra (ed.) Reflexiones... structure A and the Rξ are the fundamental relations of the same The elements of A are also called elements of A Two relational systems are similar if they are of the same similarity type The class of