Inflection Points in Mathematical Discourse

American Economic Association

The When, the How and the Why of Mathematical Expression in the History of Economics
Analysis
Author(s): Philip Mirowski
Reviewed work(s):
Source: The Journal of Economic Perspectives, Vol. 5, No. 1 (Winter, 1991), pp. 145-157
Published by: American Economic Association
Stable URL: http://www.jstor.org/stable/1942707 .
Accessed: 30/06/2012 20:21
Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .
http://www.jstor.org/page/info/about/policies/terms.jsp

.
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.

.

American Economic Association is collaborating with JSTOR to digitize, preserve and extend access to The

Journal of Economic Perspectives.

http://www.jstor.org

Journal of EconomicPerspectives- Volume5, Number1- Winter1991-Pages 145-157

the

When,

The

Mathematical
of

History

How

and

Expression

Economic

the
in

Why

of

the

Analysis

Philip Mirowski

W^

then

one reads about the history of the use of mathematicalexpres-

sion in economics, one gets the impression of a few brave but lonely
innovators-like
Daniel Bernoulli (1738), Achille Isnard (1781),
N.-F. Canard (1801), Johann von Thiunen (1826) and Antoine-Augustin Cournot
at the creation of economic thought; and later a feisty band of
(1838)-present
visionaries associated with the "Marginalist Revolution" at the turn of the
century who had the right tools in their rucksacks; and then, finally, the full
flowering of activity in the postwar period. Through it all there is the unmistakable conviction that this movement was cumulative, inevitable, and indeed,
natural.
I do not think this characterization extreme. Consider, for instance, the
opinion of Gerard Debreu (1986, p. 1261):
[T]he development of mathematical economics [was] a powerful, irresistable current of thought. Deductive reasoning about social phenomena
invited the use of mathematics from the first... economics was in a
privileged position to respond to this invitation, for two of its central
concepts, commodity and price, are quantified in a unique manner, as

soon as units of measurement are chosen ... The differential calculus and
linear algebra were applied to that [commodity-price] space as a matter of
course.
While such notions are undoubtedly widely held among practicing economists,
I want to argue that they are not historically valid. The purpose of this paper is
* Philip Mirowskiis Carl Koch Professorof Economicsand the Historyand Philosophy
of Science, Universityof Notre Dame, Notre Dame, Indiana.

146

journal of EconomicPerspectives

to suggest that the deployment of mathematicalexpression in economic discourse enjoyed neither an inexorable nor unhindered progress,but rather was
characterizedby two primary ruptures in the history of economic thought,
episodes marking the inflection points in the rise of mathematicaldiscourse.
The main reason for such a disjointed narrativeis that, in the context of the
evolution of economic thought, most of the participantswere not convinced
that the subject matter intrinsicallydemanded mathematicalexpression, while
those so enamored experienced great difficultyin creating a communitywhich
could agree upon a formalismwhich was thought to be well-suitedto economic

issues.

Inflection Points in MathematicalDiscourse
Whilst spotty and isolated instances of mathematicalreasoning exist prior
to the work of Cournot, they had absolutely no impact upon the quotidian
discourse of political economy; indeed, as Baumol and Goldfeld (1968, p. xi)
put it, "the theorist of today can manage quite well without most of them."
Bernoulli's"solution"to the PetersburgParadoxwas a local suggestion regarding problems in the classicaltheory of probability,and was neither intended
nor entertained as a theory of relative price (Daston, 1988, p. 76). Canard's
attempt to assimilate market exchange to D'Alembert'sPrinciple and the
equilibrium of the lever was widely judged a failure (Mirowski, 1989b, pp.
203-205). WilliamWhewell himself admitted that his formalizationof supply
and demand only bore a "faint and distant resemblanceof the state of things
produced by the perpetualstruggle and conflictof such principleswith variable
circumstances"(Whewell, 1971 [1831] p. 12; Mirowski,1990a). Even Cournot's
work on his "law of sales" might be regarded as a "freak,"in the words of
Baumol and Goldfeld.
Such observationsraise a number of provocativequestions. For example,
how does one judge mathematicalreasoning to be a "failure"in economics?
Was the flawed character of mathematical discourse prior to Cournot (or

Walrasor Samuelson)responsiblefor the great indifferencewhich greeted it?
How can the historicalindifference of economists be squared with the above
assertionsof Debreu that price and quantity are naturallyquantitativeterms,
and that the applicationof the differentialcalculusshould have been naturally
applied as a matter of course?'
The earliest proponents of mathematicalexpression in economics from
Ceva and Beccaria(Theocharis, 1983) to Canardand Whewellall noticed that
prices were expressed as ratios, but had great difficultyin conceptualizingthe
IThe idea that failure of mathematical discourse may be located in a failure of analogy has been
discussed by Polya (in Tymoczko, 1986), Mary Tiles (1984) John Maynard Keynes (1973, chs.
19-21) and the present author (1988, ch. 8). The narrative which follows should be regarded as
implicitly based upon these philosophical texts.

PhilipMirowski 147

underlying primitives of the analysis:in other words, what principle rendered
those ratios determinate, and how could one maintain that any of those
conditions themselves exhibited mathematicalintegrity?Without exception, all
of these so-called "precursorsin mathematicaleconomics"looked to the physics
of motion, referred to as "rational mechanics"in the 18th century, to provide

them with the analogies needed to guide them in their conceptualization of
value; indeed, many from Bernoulli to Whewell and Cournot explicitly discussed the role of this analogy in their reasoning. The one thing which links
together all the mathematicalwriters prior to the neoclassicalsis the admission,
grudging or no, that the failure of analogy between rational mechanics and the
price system was so pervasive and that their own precursors'versions were so
flawed that this research program had yet to attain a state of cumulative
self-assured internal development. This suggests that the mere fact of the
numerical characterof prices was not sufficient to justify applying mathematics
to economic discourse.
This situation changed dramatically after the middle of the nineteenth
century, which provides the first major discontinuity in the history of mathematicaleconomics. It was never quite enough merely to borrow some particular
mathematicaltechnique from elsewhere: after all, the calculus was invented in
the seventeenth century, but only managed to diffuse into economic discourse
in the late nineteenth century. What happened after roughly 1870 was that the
analogicalbarrier to a social mechanicswas breached decisivelyby the influx of
a cohort of scientistsand engineers trained specificallyin physics who conceived
their project to be nothing less than becoming the guarantors of the scientific
character of political economy: among others, this cohort included William
StanleyJevons, Leon Walras,FrancisYsidro Edgeworth, Irving Fisher, Vilfredo
Pareto, and a whole host of others.2 They succeeded where others had failed

because they had uniformly become impressed with a single mathematical
metaphor that they were all familiarwith, that of equilibriumin a field of force.
They were all so very taken with this metaphor which equated potential energy
with "utility" (or rarete or "ophelimity"or "wantability"
-the specific name
was irrelevant)that they-in some cases even unaware of each others' activities
-copied the physical mathematicsliterally termfor termand dubbed the result
mathematicaleconomics.
Hence the key to the rise of neoclassical economics, which is coextensive
with the institution of the first ongoing program of mathematicaleconomics, is
not the fact that an analogy was drawn from physical theory-all precursors of
mathematicaleconomics engaged in that endeavor-but rather that a critical
mass of theorists each (independently or not) adopted the same mathematical
metaphor. Since there finally was a shared language and a shared metaphor,
2Both the specific physics background of each of these neoclassical economists and the term-for-term
translation of energy physics into the jargon of neoclassical economics is discussed in Mirowski
(1989b, chap. 5).

148

Journal of EconomicPerspectives

serious discourse concerning the implications of this construct could begin in
earnest. Indeed, since mathematical research in any applied discipline consists
of orderly criticism of analogy, only at the point where common analogic
ground was jointly acknowledged to exist could sustained mathematical reasoning be said to commence. Because there were no economics journals nor other
continuous indicators of the nature of economic discourse which cover the
entire nineteenth century, this first great discontinuity of mathematical economics in the period 1870-90 must be demonstrated by means of indirect and
narrative evidence.3
Prior to the 1930s, the physical metaphor of the vector potential field
adopted by neoclassical economics was the only coherent mathematical
metaphor readily available to economists. For instance, there was absolutely no
agreement on how one might mathematically express or formalize Marxian
economic theory; or again, there was much uncertainty over how to accommodate neoclassicism with the novel probabalistic mathematics, as evidenced by
the work of Edgeworth (Mirowski, 1989a; forthcoming). However, mathematical expertise was as yet nowhere a hallmark of the professional economist, and
the level of mathematical acquaintance of many neoclassical theorists of the
next generation such as John Bates Clark, Jacob Viner, Langford Price, Eugen
von Bohm-Bawerk, Paul Leroy-Beaulieu, Edwin Seligman and Frank Knight
was negligible. This gave rise to the curious situation that all sorts of claims and
counter claims were made about neoclassical theory in this period, such as, say,

the requirements of a theory of utility, without any framework or criteria to
evaluate their validity.
By 1920, the neoclassical "marginalist revolution" had achieved little more
than a tenuous beachhead. There was widespread resistance if not outright
hostility towards the core neoclassical tenet of a social mechanics, which was
often associated with the phenomenon of mathematical economics. One observes this stance among the American Institutionalist movement and the
Historicist schools of Germany and England in this period.4
Moreover, the original marginalist cadre had not been able to require the
same training in physics and mathematics for their own students which they
themselves had enjoyed, largely because economics was only just becoming
professionalized in this period. As a consequence, the next generation of
neoclassical economists were not equally unalloyed enthusiasts of a rigidly
mathematical discourse, and their publications reveal this ambivalence. Finally,

3The reader is directed to (Mirowski, 1989b; 1990a; forthcoming) for discussion of the specific

motivations, both internal to and external to the economics profession, which caused this cohort to
find the particular metaphor of a force field so compelling.
4The attitude of American Institutionalists such as Thorstein Veblen, John R. Commons and
Wesley Clair Mitchell towards the imitation of physics is discussed in Mirowski (1988, ch. 7). The

best source on the English Historicist movement is Koot (1987). There is not yet any synoptic
survey of the German Historicist school in English; but Werner Sombart's Der ModerneKapitalismus
is slated to be published in English translation by Princeton University Press.

MathematicalExpressionin the Historyof EconomicAnalysis

149

the early marginalists had come under some heavy fire from physicists and
mathematicians such as Vito Volterra, Joseph Bertrand, Paul Painleve and
Henri Laurent for their incorrect and infelicitous uses of the physical metaphor,
especially with regard to questions of integrability and the integrity of a
conservative preference field (Mirowski, 1989b, pp. 241-250). Consequently,
earlier claims to have attained definitive scientific status simply by means of
mathematical expression had grown vulnerable and hard to justify. Thus,
mathematical discourse occupied only a tenuous position within economics in
the half-century or so after the rise of neoclassical economics.
The second quantum leap in the application of mathematical discourse to
economic theory may be dated with somewhat more precision in the decade
1925-35. One index of this discontinuity is presented in Figure 1, which
reports the results of an extensive survey and review of the journal literature in
economics for the period 1887-1955. In this tabulation, an independent
determination has been made whether mathematical discourse was central to
each individual article: if the answer was yes, then the entire page count was
tabulated; if the answer was no, a second determination was made as to
whether some subset of article pages legitimately used mathematical reasoning,
and that page count was tabulated. The tabulation was not based upon a
sample; every volume of the journals in the sample was examined. This
procedure was carried out for representative generalist journals of the fledgling
economics profession in the three countries of France, Great Britain and the
United States. These criteria of long-term publication and representative appeal for professional economists resulted in the choice of the Revue D'Economie
Politique [RDP], the EconomicJournal [EJ], the QuarterlyJournal of Economics
[QJEI, and the Journal of Political Economy[JPE].5
5The present tabulation diverges dramatically from other such exercises to quantify the extent of

mathematics in economics (such as Debreu, 1986; Grubel & Boland, 1986; Anderson, Goff and
Tollison, 1986; Stigler, 1965, pp. 48-49). In those papers, some mechanical decision procedure is
used to quantify the appearance of mathematical discourse, such as tabulation of numbered
equations per page, percentage of pages containing one or more equations, or number of articles
using algebra or geometry. However, the bare format of symbolic expression should not be
confused with mathematical reasoning. A more satisfactory approach needs to take into account the
subtleties of mathematical discourse: an article couched entirely in prose but concerned with an
elaborate discussion of the use of mathematics in economics may make many profound mathematical points, whereas another article which includes an equation or two in a footnote as irrelevant
window-dressing effectively has little or no mathematical content at all. Further, examination of
content allows assessment of less tangible attributes such as hostility to mathematical expression, or
to neoclassical theory, or to particular classes of formalism.
Also, AEA members may wonder why the American Economic Review is not included in our
tabulation. The primary reason is that it only began publication in 1911, much later than the JPE
and Q JE. But there is also the consideration that due to the influence of Richard Ely on the early
AEA, the AER was not considered to be as sympathetic to "theory" as the JPE and Q JE in their
early years.
Our tabulations are of course biased by the omission of journals such as Econometricaand
Review of EconomicStudies which were entirely devoted to mathematical economics. But since these
journals were started up in the 1930s and follow the pattern described above, our index understates the magnitude of the rupture in the period 1925-35.

Journalof EconomicPerspectives

150

Figure1
MathematicalDiscourse in Four Economics Journals
40
Revue D'Economie Politique
4 30 a 20 110
0
1880

1900

1920

1940

1960

40
Quarterly journal of Economics

rA

c 30
bo 20
10
0
1880

1900

1940

1960

1920

1940

1960

1920

1940

1960

1920

40
Journal of Political Economy
c 30
bu

20 10

A~

0

1880

1900

40
Economic journal
30
20
10
0
1880

1900

As Figure 1 shows, most economnicsjournals look very much alike when it
comes to mathematical discourse from roughly 1887 to 1924. Journals rarely
devote more than 5 percent of their pages to mathematical discourse, and in no
journal does the proportion of mathematical pages venture beyond one standard deviation of zero. Interestingly enough, the mean proportion for the
Revue is actually a little higher than the JPE, which suggests that, at least in this

PhilipMirowski 151

respect, France was not substantially different from the United States. The
EconomicJournal may appear somewhat aberrant with respect to the 5 percent
ceiling, until one notices that in this period 39 percent of its total mathematical
pages are accounted for by a single author: Francis Ysidro Edgeworth, one of
the editors of that journal from 1891 until his death in 1926. When it came to
mathematical expression, even a crusading editor could not make all that much
difference.
Nevertheless, a pronounced change of regime with regard to mathematical
discourse happened within the decade 1925-35, with the QJE leading the way
and the JPE tracking it to 1933, resisting intensification from 1934 to 1944,
and then joining the QJE at the new plateau of roughly 25 percent in the early
1950s. A less marked increase in mathematical discourse can be observed in the
EJ in the same period (partly because it began from a higher mean proportion);
and yet, for both the EJ and the RDP the new plateau clearly dates from just
after World War II, with roughly 20 percent of journal pages devoted to
mathematical discourse. From this evidence, it seems that the critical second
"rupture" in the economics profession took place in the decade of the Depression.
Examination of the character of the papers before and after the second
"rupture" indicates clearly that the newly-achieved level of mathematical discourse was fairly narrowly associated with the neoclassical research program. It
is a mistake to associate the new format of discourse and technique as some
general improvement in mathematical techniques or sophistication in economic
concepts; instead, it was more nearly a reflection of a change in the structure
and character of the neoclassical research program. This watershed in the
program was multi-faceted, including not only more self-consciousness in the
formalization of discrete models, but also the reconciliation of the neoclassical
program with some of its empirical shortcomings, its troublesome infelicities left
over from the 19th century field-of-force metaphor, and the cautious accommodation with stochastic mathematics (Mirowski, 1989a; 1989b). After this rupture, neoclassical articles tended to displace those of other rival research
programs in page volume, and each neoclassical article was more likely to have
a primarily mathematical orientation.
What accounts for this second rupture? Full exploration of this question
would demand extensive historical illustration; for present purposes we shall
simply state a few stark theses. By the 1920s, the neoclassical research program
was in trouble in most academic contexts. Few economists placed much credence in the concept of utility; many mocked it openly, like the partisans of the
American Institutionalist school. It had never gained much of a foothold in
France. Marshall's Cambridge was the primary stronghold; but the denizens of
that citadel mostly maintained an ironic distance with regard to the physics
formalism and its attendant mathematical devices. Moreover, since most of the
second generation of neoclassical theorists were not as well-versed in mathematics or physics as their predecessors, their defenses of the program grew more

152

Journalof EconomicPerspectives

and more inconsistentand idiosyncratic,primarilybecausethey had no conception of what neoclassicaltheory ruled out and what it permitted. The internal
consistencyof the Marshallianprogram of supply and demand schedules then
came under attack by J. H. Clapham, Piero Sraffa and others in the 1920s,
particularlyin the EJ, impugning the only textbook literature neoclassicals
could point towards.After 1917, Marxismloomed on the horizon as something
more than an irrelevantfringe doctrine.
Into this unstable situation,propelled largely by contractingopportunities
in the physical sciences, but sometimes also by a fervent desire to apply the
scientificmethod to the social bettermentof mankind,came an unprecedented
wave of trained scientistsand engineers into economics. The roll call is stunning, and included RagnarFrisch,TjallingKoopmans,Jan Tinbergen, Maurice
Allais, Kenneth Arrow and a host of others. For the first time, noted mathematicianssuch as John von Neumann, Griffith Evans, Harold Thayer Davis,
Edwin Bidwell Wilson and others were induced to turn their attention, however briefly, to economics, and to participatein its elaborationrather than to
jeer from the sidelines. When this influx of talent became acquaintedwith the
corpus of neoclassicaltheory, they discovered that it consisted largely of the
formal models which they had already mastered in their earlier training in
physics,with the only differencebeing that the vintage of the model was clearly
that of the later 19th century. Thus, with only the most passing acquaintance
with the long traditionof economic theorizing, these tyros couldjump right in
and apply more up-to-date mathematicaltechniques and metaphors to the
neoclassicalprogram and come up with far-reachingresults.
I will providejust one illustrationfor those who find such claims inflammatory. In a talk delivered to the AmericanPhysicalAssociationin New York
on January 29, 1979, Tjalling Koopmans (Koopmans Collection, Sterling
ManuscriptRoom, Yale University,Box 18, folder 333) explained:
Why did I leave physicsat the end of 1933? In the depth of the worldwide
economic depression, I felt that the physicalscienceswere far ahead of the
social and economic sciences. What had held me back was the completely
different, mostly verbal, and to me almost indigestablestyle of writing in
the social sciences. Then I learned from a friend that there was a field
called mathematicaleconomics, and that Jan Tinbergen, a former student of Paul Ehrenfest, had left physics to devote himself to economics.
Tinbergen received me cordiallyand guided me into the field in his own
inimitableway. I moved to Amsterdam,which had a facultyof economics.
The transitionwas not easy. I found that I benefited more from sitting in
and listening to discussions of problems of economic policy than from
reading the tomes. Also, because of my reading block, I chose problems
that, by their nature, or because of the mathematicaltools required, have
similaritywith physics.

Mathematical
in theHistoryof Economic
Expression
Analysis

153

Further, new formalisms developed during the intervening evolution of
physics-the increased stress on stochastic formalisms, the improved mathematics of vector fields and phase spaces, the increased familiaritywith linear
algebra and constrained optimization-which could be readily superimposed
upon the older structure of discourse.6 The net result was a new discourse
self-consciouslypatterned upon the rhetoric of the scientific research report,
shifting the intellectual center of gravity from the book or essay to the journal
article constructed around a mathematical "model," eschewing the earlier
discursive mode of expression accessibleto economist and non-economist alike.
The calling card of the new, improved neoclassicalprogram was the mathematics of someone trained in physics circa 1935; if mathematics were indeed a
language, neoclassicaleconomics at this time was a local dialect.

Taking the Measure of Man and His Commodities
Mathematicsis not singular, but rather plural. Because of this fact, there
was no single unique way that mathematicscould or should enter economics,
contrary to those who see it as some generic species of rigorous rational
discourse. As we have seen, neither could it be a simple process of diffusion
across disciplinaryboundaries, since that would precisely beg the question of
the appropriate discipline to imitate. The mere fact that price and quantity are
expressed as numbers in the quotidian operation of markets was not sufficient
to persuade generations of political economists that mathematical discourse
would be regarded as uniquely correct or appropriate in discussions of the
economy.
In fact, the notion that commodities exhibit a natural isomorphism to a
real Euclidean vector space is the most deeply rooted unobtrusive postulate in
modern economic theory. The modern view is captured by Debreu, as in the
quotation at the start of this article. Or as he has put it elsewhere (Debreu,
1984, pp. 267-268), "The fact that commodity space has the structure of a real
vector space is a basic reason for the success of the mathematicization of
economic theory." To gain some perspective on this issue, we must have
recourse to one of the other human sciences which has not been quite so
interested in the imitation of physics. Let us begin with a quote from the
anthropological literature, which was intended to illustrate the cultural bias of
many discussions of mathematics(Ascher and Ascher, 1986, p. 128):
The conventional wisdom is well encapsulated in an anecdote that is
repeated wholly or in part with such frequency that it must have special
6The one exception to this transfer of mathematics from physics to economics has to do with
techniques developed to handle physical dynamics, such as Hamiltonian methods and Liapunov
techniques. On the reasons for this curious lacuna, see Weintraub (forthcoming) and Mirowski
(1989b, ch. 7).

154

Journalof EconomicPerspectives

appeal. The anecdote tells of an exchange between a native African
Demara sheep herder and someone else variously described as an explorer, trader, scientist,anthropometrist,or ethnologist. It is intended to
show that the herder cannot comprehend the simple arithmeticfact that
2 + 2 (or 2 x 2) = 4. It describes how the herder agrees to accept two
sticks of tobacco for one sheep but becomes confused and upset when
given four sticksof tobaccoafter a second sheep is selected. Of course, the
problem is not that the shepherd does not understand arithmetic, it is
rather that the scientist/trader does not understandsheep. Sheep are not
standardizedunits. Since the Demara herder finally agreed to the trade,
his confusion could be attributed to the trader's willingness to pay an
equal amount for the second, different animal.7
If the algebraicstructureof commoditymeasurementsis fixed by nature, then
such analyticalfreedom should not be allowed. Does nature impose the mathematical structure, or is it the analyst?The answer can be tendered on two
levels, one abstract,and the other historical.
On the first level, it should be clear that commodityidentity(and hence the
group characterof commodityaddition) is not at all given by nature.Take that
favorite of textbooks, the example of apples and oranges. To every individual
qua individual (and especially for a committed utilitarian), each apple is
different: some bigger, some stunted, some mottled, some McIntosh, some
coated with stuff that looks prettybut will kill you slowly,and so forth. The date
of its existence may be relevant, as may the situation within which it is
regarded. Pushing this argument to its extreme, from a personal point of view
one can alwaysfind something about each apple which renders it unique. More
disturbingly, what is a legitimate metric in apple/orange space? I am not
claiming here that modern actors do not in fact treat apples and oranges as
indifferentlysubstitutablein certain situations;only that from the vantage of
methodologicalindividualism,the analogybetween commoditiesand directions
in space is certainlynot dictatedby nature.
Such abstractnotions can themselvesbe rendered more concreteby historical research. For instance, the eminent historian Witold Kula (1986) has
7A neoclassical economist might immediately think that he or she has an explanation for this
incident-that increased demand should raise the price-but the economists should reflect upon the
anomaly that the herder ultimately accepted the first price, as he should if the commoditieswere
correctlydefined as identical and the law of one price actually is a "law." Of course, the neoclassical
explanation can be rescued by a judicious redefinition of commodity identity, as it has been on
numerous other occasions (witness Debreu's own redefinition of the dated commodity), but this is
precisely the gist of the present objection.
It should also be pointed out that the intruder upon the sheep herder is not a neoclassical
economist, which raises the issue of whether this blind spot is not a broader cultural phenomenon.
I would venture that it is a broad characteristic of social scientists bent upon quantification as part
of a program of rendering their discipline a science.

PhilipMirowski 155

reminded us that standardized commodity measurements are a relatively recent phenomenon, becoming instituted long after market relations were prevalent. In earlier times many commodity metrics were gauged by arbitrary
anthropometric units, such as butter by the "round," wool by the "fleece,"
honey by the "hand," land by the amount that could be plowed in a single day,
and so on. Of course, one man's plowday was not identical to another's; one
man's ell could not be divided by his own "foot" nor by another's; the spice
merchant's pound was not the same as the butcher's; the bushel of one serf
need not be the same as that of a second. By and large, measures did not
exhibit the invariance required to constitute algebra as we know it. Moreover,
in such a milieu price could not perform the functions which are commonly
attributed to it in neoclassical models: indeed, in many situations the money
price was held invariant in order to carry out calculations, while the various
physical measures of the commodities were altered in reaction to alterations in
supply (Kula, 1986, pp. 72-78).
The lesson to draw from such work is that quantification is itself not an
invariant in human history, even within the more limited subset of marketorganized structures. Prices in modern markets obviously conform to specific
algebraic structures, but they are not the a priori products of nature or of the
individual mind (through projection of completeness, reflexivity, transitivity,
and so on upon preference structures); rather, they are provisional invariances
imposed upon the motley variety of human perception by various conventions
and social structures (Mirowski, 1990b).
If this be the case, then the argument becomes stronger that the mathematicization of economic discourse should not be traced to the natural quantification of commodities, but rather should be explained empirically by changing
social perceptions of the symmetries and invariances read into market activities
through the instrumentality of social institutions. For instance, the standardization of commodity identities is a market process of progressive abstraction from
artisanal idiosyncracies and enforced by auxiliary social structures, like trademarks and other such "advertising." In this view, the reason that modern
economic actors express prices as ratios is that the following regularities are
being projected onto their quotidian exchange activities:
(1) The commodity preserves its identity through the exchange process;
(2) Buying nothing should cost nothing;
(3) The order in which items are presented for purchase should not
influence the amount paid in the aggregate;
(4) Dividing the aggregate into subsets and paying for each separately
should not influence the sum paid;
(5) The net result of buying an item and then returning it should be zero;
(6) Everyone should pay the same price for the same item.
Hence the reason that algebra (and probably abstract algebra) are necessary to describe modern market activity is that market structures have histori-

156

Journal of EconomicPerspectives

cally evolved to the point where these six principles reify the impersonal
character of appropriation. This has absolutely nothing to do with "equilibrium" or any other metaphor borrowed from physics.
Such considerations lead to a very different conception of the role of
mathematics in the history of economic discourse than that found in previous
discussions. Here, history and mathematics, so often regarded as polar opposites in economic discourse, are united in a single narrative, albeit one far
removed from the neoclassical penchant to root social relations in some purely
natural determinants. Here, mathematics both expresses and enforces the way
our culture has decided to organize exchange.
* The author would like to gratefully acknowledgethe support of a grant from the
National Endowmentfor the Humanities,whichgreatlyassistedin thepreparationof this
paper. Joshua Rosenfeldand John Pepper went above and beyondthe call of research
assistantship,which made the survey of economicsjournals possible.TimothyTaylorand
Gavin Wrightmade extremelyperceptiveand helpful commentson an earlier draft, but
shouldnot be held responsible
for any awkwardnessor errorswhichremain.Thisarticleis
based upon my books Against Mechanism (1988) and More Heat Than Light
(1989).

MathematicalExpressionin the Historyof EconomicAnalysis

157

References
Anderson, G., B. Goff, and R. Tollison,
"The Rise and (Recent) Decline of Mathematical Economics," Bulletin of the History of Economics Society, 1986, 8, 44-47.
Ascher, M., and R. Ascher, "Ethnomathematics," History of Science, 1986, 24, 125-144.
Baumol, W., and S. Goldfeld, eds., Precursors in Mathematical Economics: An Anthology.
London: London School of Economics Series
of Reprints of Scarce Works on Political Economy, no. 19, 1968.
Bernoulli, Daniel, "Exposition of a New
Theory." In Baumol, W., and S. Goldfeld,
eds., Precursors in MathematicalEconomics: An
Anthology. London: London School of Economics Series of Reprints of Scarce Works on
Political Economy, no. 19, 1968, pp. 15-26.
[1738.]
Canard, N.-F., Principes D'Economie Politique. Rome: Edizioni Bizzarri, 1969. [1801.]
Cournot, Antoine Augustin, Researchesinto
the MathematicalPrinciplesof the Theoryof Wealth.
New York: Kelley, 1971. [1838.]
Daston, Lorraine, Classical Probabilityin the
Enlightenment. Princeton: Princeton University
Press, 1988.
Davis, P., and R. Hersh, The Mathematical
Experience.Boston: Houghton Mifflin, 1981.
Debreu, Gerard, "Economic Theory in the
Mathematical Mode," American Economic Review, 1984, 74, 267-278.
Debreu, Gerard, "Theoretical Models:
Mathematical Form and Economic Content,"
Econometrica,1986, 54, 1259-1270.
Douglas, Mary, How InstitutionsThink. Syracuse: Syracuse University Press, 1986.
Grubel, H., and L. Boland, "On the Efficient Use of Mathematics in Economics," Kyklos, 1986, 39, 419-442.
Isnard, Achille, Traite' des Richesses. Lausanne: Grasset, 1781.
Keynes, J. M., A Treatiseon Probability,Vol.
VIII of CollectedWorks.New York: St. Martins,
1973. [1921.]
Koot, Gerard, English Historical Economics,
1870-1926. New York: Cambridge University
Press, 1987.
Kula, Witold, Measuresand Men. Princeton:

Princeton University Press, 1986.
Mirowski, Philip, "Mathematical Formalism and Economic Explanation." In Mirowski,
Philip, ed., The Reconstructionof Economic Theory. Boston: Kluwer-Nijhoff, 1986.
Mirowski, Philip, Against Mechanism. Totowa, NJ.: Rowman & Littlefield, 1988.
Mirowski, Philip,
"The Probabalistic
Counter-Revolution," Oxford Economic Papers,
1989a, 41, 217-235.
Mirowski, Philip, More Heat Than Light.
New York: Cambridge University Press,
1989b.
Mirowski, Philip, "Smooth Operator: How
Marshall's Demand and Supply Curves Made
Neoclassicism Safe for Public Consumption but
Unfit for Science." In Tullberg, Rita, ed., Alfred Marshall in Retrospect. Cheltenham: Edward Elgar, 1990a.
Mirowski, Philip, "Learning the Value of a
Dollar: Conservation Principles and the Social
Theory of Value," Social Research, 1990b.
Mirowski, Philip, Edgeworthon Chance, Economic Hazard and Statistics. Totowa, NJ: Rowman & Littlefield, forthcoming.
Stigler, George, Essays in the History of Economics. Chicago: University of Chicago Press,
1965.
Theocharis, Reghinos, Early Developmentsin
Mathematical Economics, 2nd ed. London:
Macmillan, 1983.
Thunen, Johan Heinrich von, Der Isolierte
Staat in Beziehung auf Landwirtschaftund National Okonomie. Dusseldorf: Wirtschaft & Finan, 1986. [1826.]
Tiles, Mary, Bachelard: Science and Objectivity. Cambridge: Cambridge University Press,
1984.
Tymoczko, Thomas, ed., New Directions in
the Philosophy of Mathematics. Boston:
Birkhauser, 1986.
Weintraub, E. Roy, StabilizingDynamics.New
York: Cambridge University Press, forthcoming.
Whewell, William, Mathematical Exposition
of SomeDoctrinesof Political Economy.New York:
Kelley, 1971.