VIII. On the Use of Mathematics in Economics - Koopmans, Tjalling C. (The Review of Economics and Statistics, 1954)
Dr. Novick has added his voice to the swelling chorus of exasperated voices protesting in a variety of ways, in rhyme 12 or with reason,13 the increasing use of formal language, constructs, and derivations in economic and statistical analysis, and the specialization that springs from this development. It is probable that the protest is not directed at the use of tools of long standing in economics, such as diagrammatic representation or simple calculus. Most of us, including the present discussant, have little knowledge of the battles over the usefulness of these tools, fought and decided long ago. It is rather the tools and concepts that have been brought into economics more recently, such as matrix algebra, set theory, difference equations, stochastic processes, statistical inference, and the axiomatic method, which are now the issue. There is a remarkable similarity between the present stage of economics and the situation which arose in physics in the I930's. intensive use of matrix algebra and group theory by the developers of quantum mechanics gave rise to strongly felt protests on the part of experimental, general, and even some theoretical, physicists. Alarm was expressed at the increasing tendencies to a formalism of which the function was neither generally apparent, nor even as yet fully visible to the developers of the new theories. However, since that time the clamor has abated and quantum mechanics has become an acceDted and fruitful Dart of physical theory. In fact, the headstart of physics over the political and social arts and has since become the major threat to contemporary civilization. There is nothing in the existing difficulties of communication between mathematical and non-mathematical economists-illustrated by the substantive contents as well as by the emotional overtones of Dr. Novick's comments that time and effort will not cure. Most of what needed saying on these matters was said eloquently, effectively, and wittily in the bargain, in two recent publications, by Professor Samuelson 14 and Professor Stigler.15 An equally thoughtful discussion in a more sober tone by Professor Allais 16 perhaps appeared too late to be taken into account by Dr. Novick. It is unfortunate, however, that Dr. Novick has not clarified his attitude, for instance, to Samuelson's arguments, on those points where his views appear to differ from Samuelson's. Without this clarification, mathematics as a quantitative method as used in the applied natural and physical sciences still looks to me like just one case a rather successful one of mathematics as a language form. In some of his statements Dr. Novick appears indeed to attribute to reasoning (as applied to the physical world) a power to establish the validity of propositions without reference to a set of premises. In any case, he fears that the innocent reader of economics studies may impute such powers to argument in eco* I am indebted to several colleagues at the Cowles Commission for Research in and in particular to Professor Kenneth Arrow of Stanford University, for valuable comments on a draft of this discussion. responsibility for errors is mine. 'See, for instance, F. Waugh, Applicability of Recent Developments in Methodology to Agricultural Economics, Journal of Farm 35 (December I953). 's See J. M. Clark, Economists and Others: A Plea for Communicability, Econometrica, I5 (April I947). ' Paul A. Samuelson, Theory and Mathematics -An Appraisal, Papers and Proceedings, American Economic Review, 42 (May I952), 56-66. 16 George J. Stigler, The Mathematical Method in Economics, the fourth of Five Lectures on Economic Problems, delivered at the London School of Economics (New York, I950). 16 M. Allais, L'Utilisation de l'Outil Mathematique en Economique, Econometrica, 22 (January I954), 58-7I.