How Economics Became a Mathematical Science

Published by EH.NET and H-Business (August 2002)

E. Roy Weintraub, How Economics Became a Mathematical Science. Durham,

NC: Duke University Press, 2002. xiii+ 313pp. ISBN 0-8223-2871-2.

Reviewed for EH.Net by Robert D. Tollison, Department of Economics, University

of Mississippi.

Professor Weintraub has given us two books in one. First, he traces from

Marshall forward how high-level mathematics came into and changed the

presentation of modern economics. What he means by mathematics is set theory

and the type of axiomatic proof offered by Arrow and Debreu in their famous

“existence” paper. Second, he tells the story of his father, Sydney Weintraub,

who was a pioneer in mathematical economics, his father’s influence over his

son’s work in economics, and the shaping of his son’s intellectual agenda as a

renowned historian of economics. Math happily runs in the Weintraub family.

His methodology is biographical. He relates many interesting stories of

mathematicians and economists and their interactions and intersections, at

times (as with Debreu) reporting discussions that he had with them. Stigler to

the contrary not withstanding, Professor Weintraub amply demonstrates the value

of a biographical approach to the history of science. Along the way, we meet

names that are familiar (Patinkin) and some new names (Volterra) from the

little known world between economics and mathematics. Particularly fascinating

is the tale of the editorial review of the Arrow-Debreu paper by Econometrica,

in which the mathematician referee (Phipps) argued strenuously against

publication while the economist referee (Baumol) easily acquiesced.

To me the central message of the book is that axiomatic economic theory was a

by-product of the intellectual curiosity of certain economists and

mathematicians working on the boundaries of their disciplines. But for these

personalities, there may not have been an axiomatic economics. In this respect

the biographical method is the core contribution of the book. And, of course,

Weintraub is correct. Great economists are highly specialized resources, and

the history of economics is undoubtedly greatly influenced by their preferences

and constraints. In this way we are the product of our own science. Axiomatic

proofs became a part of modern economics because scientific entrepreneurs

discovered their usefulness and import. The only oddity in this case is that we

know who the economists were but not for the most part the mathematicians.

Weintraub brings these scholars to light.

I have nothing critical to say about the book. It is an important contribution

to the history of economics, it is interesting in all respects, and I recommend

it to economists and historians of science. Ah, but I do have a few quibbles.

The first concerns the focus on the axiomatic mathematics that has enabled

general equilibrium theory to play such a prominent role in modern economics.

While quite important in its own right, this focus deflects us from such issues

as the role of R.G.D. Allen’s textbook, as well as Samuelson’s Foundations.

These works did not lead to existence theorems, but they were very important in

changing the presentation of modern economics. Perhaps some distinction between

higher and lower mathematics would be helpful here. It could be argued, for

example, that calculus has had more useful effects in economics than set

theory.

The second quibble is empirical in nature. The latter part of the last century

saw the rise and fall of mathematical economics. If one examines the leading

journals of economics, for example, the American Economic Review, from

1950-2000, the number of equations per page has been in decline since the early

1980s. For whatever reason, mathematical presentation is on the wane. What are

the causes of this development? Is there a turn away from math? Is the level of

math the same, with the result being driven by a more concise presentation?

There is work to be done here.

A related and final quibble is that the data may be suggesting that math has

diminishing returns in economics. Indeed, economics has diminishing returns in

that any science is finite; there is only so much that we can learn. And the

data indicate that citations to economic research have fallen dramatically over

recent years. Are we at the end of economics? Have all the crucial

relationships been discovered? Is mathematical economics the capstone to the

history of our discipline? Does mathematical economics mean that once we are

able to express ourselves in so precise and general a way, there is little of

value left to say?

These, of course, are all subjects for another time. For now, let us salute

Professor Weintraub for his excellent and stimulating book. One can only be

heartened at Duke’s continuing preeminence in and emphasis on the history of

economics.

Robert D. Tollison is the author of numerous books and papers. He is a Senior

Editor of Public Choice, and a past President of the Southern Economic

Association and the Public Choice Society.