**The Mathematical Aspect**

On its formal side then, all "neo-classical" economics represented an early stage of the long, slow development, which still is going on today, of "mathematical economics" or what may be called a gradual "mathematicization" of economic theory. Yet in a sense the real origins or "roots" of this development are very old;

only the "fruits" have been slow to appear and "ripen." To a great extent many of the central problems of all economic theory always have been, in one aspect, mathematical in kind or nature, i.e., problems needing to be studied with the aid of mathematics. Thus there were already gropings and anticipations in this direction in quite early times and by economists with no command of any mathematics beyond arithmetic. To go no farther back, the Ricardian-classical "law of diminishing returns" of additional produce from more intensive cultivation of agricultural land was an insight which really needed, for its proper statement, a simple use of differential calculus; though Ricardo and the Ricardians could only explain it clumsily, in words and with the aid of arithmetical illustrations. And in fact that "law of diminishing returns" was only a special case of a much more general principle, the elaboration of which in all its bearings came to pervade, and make up a great part of, all "neo-classical" economic theory.

Moreover, another indication of the real shortness of the distance separating "classical" economic theory from the later growth of "mathematical" economic theory, is the fact that Malthus, of all men, foresaw the latter and expressed his belief that "fluxions" (as the branch of mathematics that is now called "calculus" was called then) would someday come into use in economics, as it already was in physics and other sciences. Nor were the first steps toward the fulfillment of that prediction long in coming, though they came in other circles unknown to Malthus and Ricardo. In all a good many scattered mathematicians and economists produced significant pieces of work in mathematical economics within the first half of the nineteenth century. The most important of them all, probably, was the French mathematician Cqurnot, whose Mathematical Theory of Wealth, published in 1830, was later to influence Marshall and others. But the modern growth and general spread, among all economists, of the vogue of this kind work, to the point of causing most young economists to feel that they must become equipped and able to take part in it, was long delayed and has been slow.

Thus there were great differences among the various leading late-nineteenth-century theorists already mentioned here as regards the senses and degrees in which their works were "mathematical." Jevons strongly urged his view that the science was inherently mathematical in nature and must become so overtly, or be developed with full use of all the relevant mathematical aids to both research and exposition; and tried in his own work to make full use of his rather limited knowl-edge or command of mathematics. Although he was a brilliant econo-mist and theorist, his limitations on the mathematical side were such that—so a competent judge,4 with a high opinion of his work, has said - a skilled reader of his writings can see that he evidently had to this economic-theoretical thinking "in English" (not in mathematics), firs and then "translate" it, or a few parts of it, rather painfully and awkwardly, into mathematical formulations. On the other hand, the same authority has said that the converse of that procedure was the practtice of Alfred Marshall. A first-rate mathematician and a great economist, Marshall thought out his system mathematically, but then expressed or presented it mainly and as far as possible—wholly in his main text—in ordinary verbal language, "hiding away" all his supple-mentary mathematical formulations in footnotes and appendixes, in order to avoid "scaring away" otherwise qualified but not mathematically

equipped readers, and to make his work attractive and intelligible to awide public of businessmen and other "lay" citizens as well as economits and scholars of all kinds.

Nor does this contrast between Jevons and Marshall indicate by any means the full range of the variety of attitudes, abilities, and practices among the "neo-classical" economists. The members of the Austrian school completely avoided any overt use of mathematics, and performed extraordinary feats of complete, thorough, lucid exposition of their analyses in ordinary verbal language; although in essence their ideas and reasonings were mathematical, and could have been presented in mathematical "language" with far greater economy as compared with the prolix or repetitive verbal expositions needed to fully clarify and drive home" their points. Again in complete contrast, the work or system of Walras was uniquely and fully mathematical in both conception and development, and presentation—using sets of simultaneous equations to display the general forms of the relations of interdependence among all economic variables. The Walrasian system long stood almost alone as representing "mathematical economics" par excellence, and in consequence the system long had, among economists in general, a quite limited receptive "public." Apart from it, some works by Irving Fisher, a very few other economists of this era, and the "real" Mar-shallian system as fully understood only by the few fully, mathematically equipped students of it—most of the literature of economic theory produced in this era was not, or hardly, on its face, "mathematical" at all. Simple graphs—demand, supply, and cost curves, and the like—. appeared in all or most theoretical treatises and textbooks, but that as a rule was the only "mathematics" used in exposition or presentation. However, the common, prevalent body of "conceptual tools" and reasonings was, to repeat this point, in real essence mathematical all the same—a set of applications of those used in differential calculus—even though they were generally expressed only in verbal language.

Since that era, down to the present day, the continuing development of mathematical economics in countless newer forms has gone far beyond all that was known or conceived in the time of the "neo-classical" systems. But because the latter did represent the virtual beginnings— in the bulk of the literature—of this kind of work or economic thinking, and do need appraisal in this aspect or respect, among others, I venture to present here a few words of comment on the value and limitations of mathematical economics, as I see it. I am sure that this kind of work does have an important place and value as one of the kinds of investigative work to be carried on by economists. It is the best way of doing the useful things it does do—with intellectual economy and full precision, rigor, and thoroughness; and no doubt in many of its more advanced developments, it is the only way of dealing at all with the problems therein considered. Some of the deficiencies that are often present in such work only reflect the still transitional state of the profession in this matter, or the fact that not all economists who try to work in this way or field are adequately equipped as both economists and mathematicians. Too often, those who are really good mathematicians are not very good economists, and vice versa. Able mathematicians generally, it has often been noticed, "mature" as such or reach the "peak" of their powers as mathematicians while still quite young—since this is purely a matter of developing and exercising intellectual ability, and not at all a matter of acquiring experience and knowledge of "the real world." But mastery of economics—of its real subject matter—requires accumulation of experience and knowledge, and growth of personal maturity and wisdom through the greater part of one's lifetime. The good, mature economist, who did not become a good mathematician in his youth, generally finds it extremely difficult or impossible to do so by starting late in life. On the other hand, the brilliant young mathematician and mathematical economist may in some cases never become in all respects a really good economist, because for him it may be easy to attain high prestige without ever learning much about the real economic subject matter. It can be easy and fascinating to play all kinds of intellectual games with arbitrarily devised and chosen sets of postulates and reasonings having little relevance to any social-economic realities and real problems; and within a profession in which many of the members are easily overawed by such mathematical-intellectual performances, they may bring a false prestige resulting from the human tendency to confuse exact knowledge of the implications of ideas or propositions with exact knowledge of substantial truths about the real, empirical world. The developments, which are going on today, of scientific (mainly statistical) empirical research in economics, and fruitful unions and reciprocal interactions of these with mathematical-theoretical research hold greater promise. But in my opinion even this kind of work can never become more than one small part of all the investigative work and thinking which should be carried on by economists. Aside from he transitional difficulty already mentioned, there are permanent or inherent limitations of or around the important value of all mathematical economics, even when it is combined with or includes statistical research and is thus in its own way or sense empirical as well as theoretical. All good work of this kind helps to meet important real needs and I do not mean to belittle it. But I think there are and will always be other equally great needs for studies and reflections of quite other kinds, concerned with the nonmeasurable and not even conceptually Quantitative, intangible, elusive "human elements" of the economic object matter and the complete human-social contexts within which always occurs; and with all the values, of all kinds (aesthetic and ethical as well as economic) which are at stake in all human, private and public, action or conduct. These very extensive and supremely Important parts of the subject matter that should properly concern all economists are of such a nature that they necessarily elude the grip of all mathematical and indeed all strictly scientific, intellectual machinery; but the tendency to exclude them, on that ground, from the field of the economist's interests is I think a radical and tragic error.

Achieving in the study of economics and political economy a well-balanced union of all of the appropriate mathematical and statistical work, with all the appropriate work of other kinds—in economic psychology and sociology, economic and general history, and general and social, moral, and political philosophy—achieving this is not easy. It never has been done ideally well, and the modern trends in intellectual life lead mainly away from it, but I think it remains the supreme need and proper ideal of the inquiry.