### Alfred Marshall on the Use of Mathematics in Economics

I retired at the end of last year and have finally finished clearing out my office. In the process I found a number of interesting things, including a printout of a book on transport economics that I wrote thirty some years ago when teaching a course on the subject but never published and thought was lost.

I also found the original of a passage in a letter by Marshall that I have been misquoting for many years. Here it is.

Balliol Croft, CambridgeWhich leaves me wondering how much of the economics of the next century went into Marshall's fireplace.

27. ii. 06

My dear Bowley,

I have not been able to lay my hands on any notes as to Mathematico-economics that would be of any use to you: and I have very indistinct memories of what I used to think on the subject. I never read mathematics now: in fact I have forgotten even how to integrate a good many things.But I know I had a growing feeling in the later years of my work at the subject that a good mathematical theorem dealing with economic hypotheses was very unlikely to be good economics: and I went more and more on the rules—(1) Use mathematics as a shorthand language, rather than as an engine of inquiry. (2) Keep to them till you have done. (3) Translate into English. (4) Then illustrate by examples that are important in real life. (5) Burn the mathematics. (6) If you can't succeed in 4, burn 3. This last I did often.I believe in Newton's Principia Methods, because they carry so much of the ordinary mind with them. Mathematics used in a Fellowship thesis by a man who is not a mathematician by nature—and I have come across a good deal of that—seems to be an unmixed evil. And I think you should do all you can to prevent people from using Mathematics in cases in which the English Language is as short as the Mathematical.....

## 8 Comments:

How out of date is the transport economics draft book? Is it possible to scan it and put it online?

I've seen fine economic thinking without the overt use of mathematics, mathematics used to disguise absurd assumptions in support of conclusions that seem to have originated in normative commitments, and models of worlds not much related to ours.

But I'm inclined to believe that a better approach would insist that what seems to make sense in words yet cannot be expressed in some form of mathematics is every bit as dubious as what seems to make sense in mathematical expression but cannot be expressed in words.

A very real problem, as I see it, is that most economists effectively regard mathematics as conterminous with arithmetic. And, when Marshall was writing, it was still more common to make such a spurious equation.

I assume you haven't stopped lecturing or writing, so if you have more time post retirement, care to make a project of that transport economics book? Lots of people are mad in Europe over railroad privatization so an accessible transport economics book like Hidden Order or Law's Order could be a good call.

I should reread the transport econ book. I wrote it a long time ago and I assume there has been some progress in theory, or at least evidence, since. I have considered scanning it--what I have is a dot matrix printout and I'm not sure how well it would OCR. I doubt it's worth publishing.

Marshall clearly doesn't regard mathematics as merely arithmetic--note his comment about integrating. I don't think Ricardo uses anything beyond arithmetic, but his mathematical intuition was good enough to produce a general equilibrium theory, I think the first serious one, despite that limitation.

I agree that it is worth checking verbal arguments by putting them in mathematical form to make sure you haven't made a mistake obscured by a less precise language.

While integration and the calculus more generally are not

elementaryarithmetic, they are still mathematics of quantity. Note that the foundations of analysis consist of the construction of number systems and of arithmetic operations. (My favorite book on that subject is Landau'sFoundations of Analysis; your mileage may vary.) But, in the 19th Century, efforts were begun to find rigorous support for the mathematics of quantity, and the discernible scope of mathematics exploded in consequence, so that it came to be seen as more generally concerned withorder(rather broadly understood).We see some application of that more general conception in decision theory, but economists tend to move as quickly as possible to quantitative proxies, making strong and largely untested assumptions in order to apply familiar mathematics.

My understanding of Ricardo is very much informed by what I have read of your discussion of his work, which discussion seems excellent to me.

I agree, of course, that there is mathematics that isn't arithmetic or calculus and does not deal with quantity. The question is whether there is useful economics that got done using such mathematics and could not be done without it.

Let me give one example to support my skepticism. Akerlof, Yellin and Katz have a piece explaining why improved contraception and legal abortion were followed by an increase, rather than a decrease, in the number of children born to unmarried mothers [George A. Akerlof, Janet L. Yellin, J. L., & Michael L. Katz, "An Analysis of Out-of-Wedlock Childbearing in the United States," 111 Q. J. Econ. 277 (1996)]. It's a nice argument and might well be correct. They make it using game theory.

The same analysis can be done using nothing Marshall didn't know--arguably nothing Smith didn't know. It's just an application of joint product theory–Smith's wool and mutton. If sex and children are joint products and rearing children is expensive, women can insist on support as a condition for sleeping with men. Once the link is broken, competition from women who don't want children drives down the market price for women's sexual services, so women who don't want children are less able to find a man willing to support them. For details see Chapter 13 of _Law's Order_ (webbed on my site).

It may be true that the mathematics sometimes provides more rigorous support for things we already know, but where does it point us at interesting things we didn't know?

Although I certainly won't insist that there is no useful economics that can be done with non-quantitative mathematics that could not be done with quantitative mathematics, that is not the only issue here. Do we affix nails to everything so that we may use a hammer?

It is, for example, quite possible to use intervals as proxies for incomplete preörderings. But these proxies increase the underlying complexity of the system. And people are apt to confuse the map with the territory, treating characteristics of the proxy as characteristics of the thing proxied.

And, when I wrote of strong and untested assumptions, I had in mind such things as presumptions of

completenessof preörderings, which would allow those preörderings to be proxied by point-values. Economists usually don't even know how to look for failure of those assumptions, because they don't have a more general system with which to make comparisons.De gustibus non est disputandumwhen it comes to that which isinteresting, but I think that weknowsomething to the extent that it is either a direct datum of experience or we canrigorouslyderive it as a result. I think that when you or I are persuaded by a non-rigorous argument, we either merelybelieveit, or we have learned enough from the argument to produce a more rigorous argument.And, because economics has policy implications, it is prone to disputation and to outright corruption. If we want to make the world a better place, then we cannot be content with knowing things and not demonstrating them with as much rigor as is practicable.

I would argue mathematics was essential for the development of "macroeconomics", i.e. the theory of business cycles. In the 20s and 30s, people were having very confused debates about how interest rates, output, price level, money supply, employment, etc. were determined, until Hicks showed that it's all quite straightforward once you are willing to write down a simple system of equations. Of course, the particular system Hicks wrote down (the ISLM model) was flawed, but it had the effect of bringing some clear thinking to the debate. Later, in the 60s and 70s, dynamic programming helped correcting many of the flaws which beset the old Keynesian models. It's not that important ideas in macro like, say, the Euler equation could not be derived without math, it's just that, as a matter of fact, they were not derived before people started using math.

Similarly, while I don't believe it would have been impossible to develop some of the key ideas in modern finance without stochastic calculus, it would have been awfully hard. For instance, I don't know how anybody could have derived the Black-Scholes equation just by verbal reasoning.

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