Thomas E. Leslie on the use of mathematics in Economics

In a review of William Jevon’s The Theory of the Political Economy, economist Thomas E. Leslie has some interesting thoughts on the use of mathematics in economics:

Within that narrower field can we proceed, as Mr. Jevons contends, not only by simple deduction, but by mathematical process?  “There can be,” he says,  “but two classes of sciences those which are simply logical, and those which, besides being logical, are also mathematical. If there be any science which determines merely whethera thing be or not, whether an event will or will not happen, it must be a purely logical science ; but if the thing may be greater or less, or the event may happen sooner or later, nearer or farther, quantitative notions enter, and the science must be mathematical in nature, by whatever name we call it.” Nevertheless, it can hardly be contended that Adam Smith’ s reasoning respecting the nature and causes of the wealth of nations is in its essence, and ought to be in actual form, mathematical ; or that the process by which his main propositions are established is anything more than logical. We might add that they rest in good part on inductive, and not simply on deductive, logic; but the question before us is whether mathematical methods could properly be applied to their demonstration. That wealth consists, not of money only, but of all the necessaries and conveniences of life supplied by labour, land, and capital; that man’s natural wants are the strongest incentives to industry ; that the best assistance a government can give to the augmentation of national opulence is the maintenance of perfect liberty and security ; that the division of labour is the great natural organization for the multiplication of the products of industry ; that it is limited by the extent of the market; and that the number of persons employed in production depends in a great measure upon the amount of capital and the modes of its employment—these are the chief propositions worked out in the ‘ Wealth of Nations,’ and it can hardly be said that mathematical symbols or methods could fitly be used in their proof. We need not controvert Mr. Jevons’ proposition that ‘ pleasure, pain, labour, utility, value, wealth, money, capital, are all notions admitting of quantity ; nay, the whole of our actions in industry and trade depend upon comparing quantities of advantage or disadvantage.’ But the very reference which Mr. J evons proceeds to make to morals militates against the assumption that ‘ political economy must be mathematical, simply because it deals with quantities,’ and that ‘ wherever the things treated are capable of being greater or less, there the laws and relations must be mathematical.’ The author instances Bentham’s utilitarian theory, according to which we are to sum up the pleasures on one side and the pains on the other, in order to determine whether an action is good or bad. Comparing the good and evil, the pleasures and pains, consequent on two courses of con duct, we may form a rational judgment that the advantages of one of them preponderate, that its benefits are greater, its injurious results, if any, less; but it by no means follows that we can measure mathematically the greater or less, or that the application of the differential calculus would be appropriate or possible in the matter.

Leslie was a famous Irish historicist who in the late 19th century made significant theoretical contributions to the development of historical economics in England. Math was never really used in economics during this time (at least not on the level like what we see today), so it’s understandable that he would be tentative in embracing the use of mathematical modeling in economics. But aside from that, he has a point here. How can we use mathematical symbols to properly model economic phenomenon like the division of labor, the invisible hand (spontaneous order), market organization, and a bunch of other important things that shape the economic system we live in? When does economic modeling go beyond solving elaborate “chess problems” (to borrow from Deirdre McCloskey)? Based on my current knowledge of economics (which by most standards is very limited), most of the current models used rest on dubious assumptions that render the model almost useless when applied to any real market system. Because of the complex math being used, many of the major problems in these model are obscured by an elaborate “superstructure” of mathematical symbols. Leslie made a similar point:

We regret that so much of Mr. Jevons’ own reasoning is put into a mathematical form, because it is one unintelligible or unattractive to many students of considerable intellectual power and attainments. On the other hand, we not only concede that a mathematical shape might have been given to a great part of Ricardo’s system, but we regret that it ever received any other, because his theory of value, wages, profits, and taxation is misleading and mischievous. Assume that the products of equal quantities of labour and abstinence are necessarily of equal value and price, and that exertions and sacrifices of different kinds are commensurable, and a number of mathematical equations and calculations can be based on those assumptions. But since the basis is false, the more the superstructure is hidden the better; and we should be glad to see it obscured, in every treatise in which it is put forward, by a liberal use of the calculus.

That’s not to say I completely embrace Leslie’s position. I’m not against the use of mathematic in economics and it can be a very useful tool*. But mathematical modeling is no substitute for studying actually economic systems, i.e. history, institutions, social relations, evolutionary systems, division of labor, and many other phenomenon.

*See this post by Noah Smith on why we should use mathematic in economics. There is another fantastic post on the blog Magic, Maths, and Money. Also see this post by Paul Krugman. There has been extension discussion on the econblogsphere on this topic and if you search around a little, you’re bound to find some interesting commentary.

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