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1 VII. A Catechism: Mathematics in Social Science Author(s): Robert Dorfman Source: The Review of Economics and Statistics, Vol. 36, No. 4 (Nov., 1954), pp Published by: The MIT Press Stable URL: Accessed: 26/05/ :37 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. The MIT Press is collaborating with JSTOR to digitize, preserve and extend access to The Review of Economics and Statistics.

2 374 THE REVIEW OF ECONOMICS AND STATISTICS duce good economics then, still as a Darwinian, I predict that long before the appendix has disappeared from the human digestive tract most people interested in economic theory will as a matter of course learn some mathematics.11 (By the way, is it theory that Novick really deplores?) Is there anything to the charge that much mathematical economics is merely trivial? Certainly. Every economist who learns his calculus must come to the realization that this very powerful tool when applied to easy problems gives remarkably easy answers. I have long been considering the purchase of an electric drill, and I am sure that when I finally buy one I will drill dozens of holes I do not really need. But triviality is a universal failing; you can be trivial in any language, as a look at the journals will show. The remedy against it is strength of character, not technological backwardness. Problems must dictate methods, not vice versa. Incidentally, as knowledge of ele- mentary mathematics spreads, the payoff to this kind of thing will disappear. There is also a certain amount of purely mathematical muscle-flexing that goes on. This is the inevitable reaction of the ex-ninetypound-weakling. But there's really not much. It is the exception for the train of thought to go: My, here's a nice piece of mathematics - how can I make it look like economics. Much more often it goes: economic intuition tells me that something like this ought to be true - how can I find out exactly what is true, and prove it. In any case it doesn't take a very expert reader to tell one kind of mathematical economics from the other. One often sees the queston: "Have economists gone too far in the use of mathematics?" as if mathematics were the gin in the martini of economics. The only test is the flavor and potency of the result, its economic content, not the weighing of ingredients. On this the profession (or Natural Selection or Supply and Demand) will judge. VII. A CATECHISM: MATHEMATICS IN SOCIAL SCIENCE Robert Dorfman The best way I can comment on Novick's paper is to subject myself to a brief catechism and to compare my responses with those implicit or explicit in Novick's treatment. What is mathematics? It is presumptuous for an economist to answer this question, but to get on with the argument I propose that mathematics is the technique of expressing relationships, usually symbolically, in such a way as to bring out their formal structure and then of taking advantage of accumulated knowledge about the properties of such formal structures to reveal further relationships which are not immediately evident. The advantage of this procedure is its enormous economy. Consider, for example, the distinction between marginal productivity and net marginal productivity, "I have considered and rejected an alternative answer to the above questions, namely that there is a Gresham's Law according to which bad (mathematical) economics drives out good (literary) economics. Gresham's Law depends on the fact that people are motivated to hoard the undervalued currency. There is no evidence that untold fascinating literary-economic ideas are piling up in the profession's mattresses. which worried J. B. Clark and J. Robinson among others. It is possible to prove ab initio that these two concepts are identical (J. Robinson did so); it saves a lot of brain-wracking to notice that the problem concerns the relationships among certain total and partial derivatives, relationships which were worked out generations ago. The best clue to Novick's answer is contained in the sentence, "It is high time therefore that we recognize the difference between mathematics as a language form and mathematics as a quantitative method...." Novick seems to feel that there are two kinds of mathematics, one purely linguistic and the other perhaps corresponding to the concept I have proposed. The old aphorism that "mathematics is a language" underlies much of his discussion. This maxim is at best half true, the true half being that mathematics has a language and that the first step in "doing mathematics" is to translate non-mathematical concepts into that language. Even as a linguistic device mathe-

3 matics has a service to perform. Every bilinguist knows that there are things which can be said in French but not in English, and vice versa. Every economist knows that there are things which can be said in literary economics but which elude mathematical formulation. But many economists forget the converse: there are some things which can be said in mathematics but which cannot be paraphrased with tolerable accuracy in ordinary English. Consider, for example, that old stumbling block: is marginal cost the cost of the last unit produced or the first one not produced. Mathematics, I am contending, is neither a language form nor a quantitative method but a branch of logic. I don't know exactly how to distinguish it from the rest of logic, and my definition, apart from the remark about symbols, would apply about as well to logic as to mathematics. The distinguishing mark of mathematics probably lies in the volume of knowledge which has been amassed about certain types of formal relationships; when we make use of these time- and thought-savers we are engaged in mathematics. Are scientific theories proved by mathematics or statistics or neither? The first question encroached on the mathematician's field of expertness, now we are trespassing in the philosopher's domain. We must proceed, but with diffidence. The situation seems to be that scientific theories are not proved ever, but that both mathematics and statistics (which in this context means merely observation) provide criteria for the provisional acceptance of theories for the purposes and at the times that they are accepted. The mathematical criteria are that if two theories are logically contradictory they cannot both be accepted or, what is the same thing, if any theory is accepted so must be all its logical implications. The statistical criterion is that in choosing among several internally consistent theories that one is to be accepted which would most successfully predict observed phenomena. Both criteria are complicated and they work together in complicated ways. Two of Novick's passages bear on this question. "The richness of social science today arises chiefly from the stimulus to thinking which follows the fact that little or nothing in the theorems now used is susceptible to MATHEMATICS IN ECONOMICS 375 absolute proof and therefore cannot be accepted without challenge." This source of wealth, alas, is open to all sciences; none is paralyzed by the misfortune of absolute proof. The second relevant passage is, "Mathematics is now used in the social sciences chiefly as it has been used in theoretic physics or chemistry and not as the mathematical results of theory proved by statistics in physics or chemistry are applied in everyday engineering or mechanics." This makes Novick's stand clear. Theories can be proved and, moreover, by statistics. I can barely hint at the reasons for rejecting these positions. In the first place the agreement between observation and theory is never perfect. The possibility always remains, therefore, that a new theory will be discovered which entails fewer discrepancies than any accepted theory. In the second place, agreement with statistics is not the only criterion a theory must meet, and internal consistency and consistency with other accepted theories cannot be verified by statistics. A theory may be supplanted by an alternative whose predictive power is no greater but whose logical implications comprehend a wider realm of phenomena. In the third place, observations do not uniquely determine theory. For example, both underconsumption and overconsumption theories of the business cycle are consistent with the same data. In evaluating a theory statistics and mathematics complement each other, the one testing its logical acceptability and comprehensiveness, the other its empirical relevance. Proof we cannot hope for. Is the role of mathematics different in social science, physical science, and engineering? It is not. In every case mathematics is used to bring to light relationships implicit in the postulates which are used as a point of departure. Judging from the last passage which I have quoted Novick would hold that the role of mathematics in engineering is different from its role in physical science, that its role in economics is similar to its role in physical science, but that its role in economics should be similar to its role in engineering. I am unable to see the distinction, especially since engineers use precisely the same mathematics and formulas as physicists and chemists. Is mathematics necessary in social science?

4 376 THE REVIEW OF ECONOMICS AND STATISTICS I suppose not. It is quite conceivable that all problems could be solved by verbal means, just as it is quite possible to find that the square root of CLXXXXVI is XIV. Such methods, though, would be not only painful but fearfully inefficient. They would mean, in effect, rediscovering or reproving mathematical theorems every time they arose in a new substantive context. They would eliminate from fruitful research many men who are competent to learn and use some mathematics but not competent to reinvent it when and where needed. I am saying, in effect, that mathematics is a practical necessity for the solution of many economic problems. Consider, for example, the elaborate apparatus which was eventually needed to solve the "identification problem" originally raised by E. J. Working in the I92O'S. Another example is the theory of general equilibrium and its recent highly technical development. As a matter of history, no one has succeeded in exploring the existence theorems and welfare implications of general equilibrium without a liberal application of fancy mathematics. Superficial treatments of this problem have led to fallacy, and fallacy, I feel, should be extirpated even at the cost of having to learn some mathematics. Physicists have had to learn tensor analysis willy-nilly, or to come closer to home, statisticians have become involved in the higher reaches of measure theory. This has been hard on those physicists and statisticians who lacked the tools which their developing sciences demanded, but no one seriously contends that this made the tools inappropriate. We economists are in the same fix. If mathematics solves otherwise intractable problems, and it does, there is little point in attacking these solutions because the bulk of the profession cannot understand them. Mathematics is an aid to reasoning. If we could dispense with it, I'm sure everyone would be glad to and thereby widen his audience and economize on printing bills. But some impor- tant problems - I have already mentioned a few - are too hard to solve any other way. This being so, we economists will just have to learn some mathematics. Novick could hardly deny this, and yet he wrote, "It is not impossible that some day mathematical methods can be applied to economic and social activity. However, that day seems remote...." I should agree that the day is remote and should hazard, just as a guess, I838 (Cournot). Are mathematical results often unintelligible and subject to misunderstanding? Alas, they are. This, of course, is Novick's chief charge against mathematical economics, and it is a remarkable charge. I have many times had to argue that the church cannot be attacked because of the sins of the clergy, but this is one of the few times that I have seen the church roundly condemned for the shortcomings of the laity. The specific misunderstanding which concerns Novick most relates to the significance of mathematical results. All that mathematics purports to establish is statements of the form, "If A is true then B is." What mathematics does is enable use to see more clearly and more deeply into the consequences of assertions which may or may not be true. Novick feels that some economists regard a mathematical theorem as a statement which has been established absolutely, independently of assumptions. Such economists, if such there be, should take warning from his paper and regard it as an important step in their mathematical education. Should mathematical economists restate their theories in literary form? Novick urges them to do so and Marshall gives the same advice. I must dissent. Practitioners of mathematical economics already have their hands full coping with some of the toughest problems which the science offers and it is unfair to impose on them the special problems of literary lucidity. Mathematical and literary talents do not always dwell in the same man. An amelioration of the communications difficulty lies with translators or popularizers who undertake the ungrateful task of trying to grasp the meaning of the mathematics and restate it with tolerable accuracy in English. The ultimate answer lies with the professional reader, who must equip himself to read what he wishes to understand. My very strong feeling is that any discussion, like this one, of the legitimacy of mathematical methods in economics is doomed to be fruitless. Time and tide will not be rolled

5 MATHEMATICS IN ECONOMICS 377 back. The profession will not relinquish powerful instruments of reasoning. The only real question is a question of fact: do mathematical methods have any power. That they do has been demonstrated repeatedly by the problems they have solved and by the muddled impasses which literary reasoning has not been able to break through. VIII. ON THE USE OF MATHEMATICS IN ECONOMICS Tjalling C. Koopmans * 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 mathematical 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 mathematical 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. The 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 * I am indebted to several colleagues at the Cowles Commission for Research in Economics, and in particular to Professor Kenneth Arrow of Stanford University, for valuable comments on a draft of this discussion. The responsibility for errors is mine. 'See, for instance, F. Waugh, "Applicability of Recent Developments in Methodology to Agricultural Economics," Journal of Farm Economics, 35 (December I953). 's See J. M. Clark, "Mathematical Economists and Others: A Plea for Communicability," Econometrica, I5 (April I947). physical theory. In fact, the headstart of physics over the political and social arts and sciences 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 Sam- uelson 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 mathematical 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 mathematical economics studies may impute such powers to mathematical argument in eco- '" Paul A. Samuelson, "Economic Theory and Mathematics -An Appraisal," Papers and Proceedings, American Economic Review, 42 (May I952), 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.