2. Some Homework on Mechanics

I must emphasize at once that I am not concerned here with rhetorical remarks using images from mechanics, such as someone saying that he is just a “cog in the works” of his organization; nor do I treat uses of “mechanics,” or cognates like “mechanism,” which refer in some general way to the “workings” of an institution or social system. Such language often occurs in economics discourse and, indeed, in many walks of life, and I make no objection to it. However, unless it is explicitly indicated otherwise, in this article I am concerned with contentual allusions to, and deployments of, elaborated theories in classical mechanics, including exercises in modeling that relate to mechanical devices and machines, in the states that the subject had reached by around 1800 and later.

By then mechanics had developed in three different traditions, somewhat in competition with each other, especially in dynamics.Footnote ² They may be summarized as follows.

Firstly, Newtonian mechanics was most prominent in celestial and planetary mechanics, and in parts of terrestrial mechanics. His laws were at once both mathematical and mechanical. The second one came to be used in the form

(1)

including by Newton himself (Maltese Reference Maltese1992); but he actually formulated in terms of a relationship between (micro) increments of impulse and increments of momentum. The first law, that an undisturbed body stays stationary or moving with uniform velocity along a straight line, was well understood to apply both to static and to dynamic equilibrium. However, within dynamics the proof of some results was dubious until Leonhard Euler and others realized that for true generality the principle of angular momentum had to be adopted as a fourth law. Widely adopted was Newton’s theory of central forces acting along the line joining between bodies, obeying an inverse square law of attraction in this part of the universe, and their balance by reaction according to his third law.

Secondly comes the relationship in mechanics between kinetic and potential energies, to use the modern terms. This tradition gained its best credentials in engineering and technology; by the 1780s it was elevated into a general approach, especially in France, with special strength in cases of impact and percussion where disequilibrium occurred and energy was lost. The main advocate was Lazare Carnot in the 1780s, who was followed especially from the 1820s onwards by graduates of the Ecole Polytechnique (founded 1794), such as Claude Navier, Gaspard Gustave de Coriolis, Jean Victor Poncelet, and Charles Dupin (Grattan-Guinness Reference Grattan-Guinness1984; Reference Grattan-Guinness1990a, chs. 8, 16). Its practitioners explicitly followed positivistic principles; indeed, they may well have influenced their younger colleague, Auguste Comte, to formulate in the 1820s the philosophy to which he gave that name. For example, Poncelet distinguished between forces (obscure) and the “effect” of forces, such as water hitting the blade of a turbine (experientable); he also preferred work to “live forces” as the primitive notion, because ([the effect of] force x distance) was closer to experience than (mass x velocity squared). This tradition came to have some influence on topics such as the efficiency of machines, the management of water, and ergonomics, where both mechanics and economics were involved.Footnote ³

The third tradition in mechanics grew part in reaction against Newton’s. Jean le Rond d’Alembert suggested that (1) should be taken as a definition of force, and to replace it he offered a rather incoherent statement of what is now known as “d’Alembert’s principle,” which described the motion of a mechanical system under constraints when it was disturbed from a position of equilibrium. This tradition also adopted “the principle of least action,” an optimizing law formulated during the 1740s with the help of the calculus of variations: “action” was a technical term, denoting (force x velocity x distance) in a variety of contexts; it was utilized comprehensively first by the Italian Joseph Louis Lagrange. The final main principle was that of “virtual velocities” (not “work,” a word that hinted at the mysterious notion of force): a refinement of d’Alembert’s principle, it gave a sufficient condition for a system of masses acting under forces to be in static equilibrium. This kind of mechanics was formulated and developed in algebraic terms; indeed, it is often called “analytical,” and Lagrange’s treatise Méchanique analitique (first edition 1788) was the definitive statement of it for a long time. There are no diagrams in the book, the author tells us early on, and he means it, seriously.

Finally, we note three features of mechanics irrespective of tradition. Firstly, in 1803 French mathematician Louis Poinsot notably extended understanding of static equilibrium with his theory of the “couple” (his name), two coplanar and non-collinear forces of equal magnitude and opposite sense (Poinsot Reference Poinsot1842, ch. 1); economists did not normally note this aspect of statics, but we shall see a trace of it in Fisher. Secondly, when appealing to instruments to represent static equilibrium, economists favored the mechanical balance, which was in such a state when the moments of two weights about the pivot were equal; Jevons, Walras, and Fisher are the most prominent enthusiasts. Thirdly, students of fluid mechanics long worried about the motion of various kinds of it while engineers struggled with hydraulics (Rouse and Ince Reference Rouse and Ince1957, chs. 9–14); we shall see that some economists were wont to appeal to analogies with hydrodynamics from time to time, and note an irony in subsection 19.4.

3. Mechanics ≠ Physics

All three traditions in mechanics continued to develop in the nineteenth century, and interacted with the inauguration of classical mathematical physics, especially heat diffusion, physical optics, electrostatics, magnetism, electromagnetism, and thermodynamics. These new mathematical sciences were initiated between the 1800s and the 1820s largely by the French, especially Pierre Simon Laplace, Joseph Fourier, Etienne Louis Malus, Jean Fresnel, Siméon Dénis Poisson, André Marie Ampère, and Carnot’s son Sadi (Grattan-Guinness Reference Grattan-Guinness1990a, chs. 9–12). They were all teachers, former teachers, and/or graduates of the Ecole Polytechnique. Mathematical physics came to international attention from the mid-1820s onwards. For example, analytical mechanics was notably extended from the 1830s by William Rowan Hamilton, being applicable to physics as well as mechanics; indeed, his own initial motivation came from optics. In addition, from the 1850s, William Thomson (later Lord Kelvin) raised the status of potential theory across both mechanics and physics with his advocacy of the work of George Green and his own contributions.

Kelvin was also an important figure in the development of thermodynamics, and especially of “energetics” physics, where the energy–work tradition in mechanics mentioned above, including the loss of energy in cases of mechanical impact, was transformed into a conservation of energy now taken across all physical phenomena, not just mechanical ones. The difference between energy–work mechanics and energetics physics is not well recognized even by its historians; the best account to bring it out is Haas (Reference Haas1909).

An important context for this difference concerned the work term ∑_r F_r dx_r for forces F_r displaced from static equilibrium by infinitesimal distances dx_r. Lagrange and his followers in analytical mechanics assumed that this term was always an exact differential, so they equated it to the differential of a potential P,

(2)

and integrated (2) to obtain P, the potential energy of the system, which was called “conservative.” When the magnitude of the work term depended only on the initial and final configurations of the system, it was said to be “path independent.” This equation has epistemological import, firstly in that potential was the primitive concept and force derivative from it, and secondly that any situation involving dynamic equilibrium could be reduced to one of static equilibrium, so that dynamics reduced to statics. Adherents of energy–work mechanics objected to both claims; for when mechanical impact occurred between bodies, then some or all of the force functions would be discontinuous, so that the mixed second-order partial derivatives of P were no longer equal (Scott Reference Scott1970). By contrast, under energetics physics (2) was acceptable, because the summation there was taken over all ‘forces’ involved, whether mechanical or physical.

This disagreement over (2) was significant, since it involved claims over the (lack of) generality of theories, and also the ubiquity of the notion of continuity. It also affected the influences of mechanics and of physics on the development of neoclassical economics, for the distinction between energy–work mechanics and energetics physics suggests three lines of influence:

Two points of historiography arise. Firstly, I find that the first line was more influential than the other two for several of the economists discussed below. This view runs counter to that adopted in the important contributions made to the history of economics in Mirowski (Reference Mirowski1988, Reference Mirowski1989), and Ingrao and Israel (Reference Ingrao and Israel1985, Reference Ingrao and Israel1990). They rightly emphasize the importance of the third line, and do not restrict it to energetics; as they note, economists invoked thermodynamics from time to time. But these historians express everything pertinent in terms of the second and third lines, and Mirowski underrates, and Ingrao and Israel largely ignore, the energy–work tradition; so they underestimate the importance of the first line.Footnote ⁴

Secondly, it is widely thought among historians in general that during the eighteenth century (and later), the Newtonian tradition dominated the physical sciences; among historians of economics, see, for example, Husseini (Reference Husseini1990) or Keita (Reference Keita1992, ch. 4). But this situation held only for Britain: as even the short summary above suggests, in the far more important venue of the Continent, especially France, Newton had serious competition, especially in dynamics. This feature applies to the use of mechanics in economics. The analytical tradition enjoyed most popularity because equilibrium took prime place there, while little use was made of Newtonian mechanics (Cohen Reference Cohen and Mirowski1994); in particular, the talk of “force” in economic contexts was rarely accompanied by appeals to Newton’s three laws, the inverse square law, or central forces.

4. A Guide for the Perplexed on the Calculus

As was mentioned in section 1, the neoclassical economists brought the differential and integral calculus into economics. Some features need to be explained, for they are not obvious: different versions had been proposed, and notations from one version were (mis)used in another one.Footnote ⁵

The first version of the calculus, due to Newton from the 1660s, was obsolete by the time treated here. The second one was due to G.W. Leibniz from the 1670s. The “differential calculus” (his phrase) was based on the infinitesimally small variable forward increment dx on x, a kind of extrapolation of the finite forward difference Δx, and like Δx of the same dimension as x. “dy/dx,” the ratio of differentials on variables x and y, was normally finite in value. An important modification was made by Euler in the 1750s when he stipulated that if dx := 1 then y was a function of x, and dy/dx became known as its “differential coefficient.” This is the forerunner of the derivative “f′(x)” of a function f(x): that word and notation belonged to a different tradition of the calculus due to Lagrange in the 1770s and popularized in the late 1790s by his teaching at the Ecole Polytechnique, in which he assumed, mistakenly, that every function f(x) took a convergent power-series expansion for all values of its variable x.

From the 1820s onwards, another tradition was launched at the Ecole Polytechnique by professor Augustin Louis Cauchy, a graduate of the school (Grattan-Guinness Reference Grattan-Guinness1990a, ch. 11); although it was badly received there, it gradually gained favor internationally. He started out from a theory of limits, in which infinite sequences of values of variables might (not) converge to a limiting value, and (not) do so monotonically. He defined the “derived function” f′(x) as the limiting value, if there was one, of the forward difference quotient Δy/Δx as Δx approached 0 (and dimension changed). He used both Lagrange’s word and notation for the derivative, but his theory was quite different: indeed, in 1822 he was the first to refute Lagrange’s theory by means of counter-examples (and Hamilton was the second ten years later). Such transfers of notation from one version to another was and is a significant source of perplexity; in particular, a mongrel version of the differential calculus developed and is still widely taught, in which limits form the ground but both the symbol “dy/dx” (and maybe “f′(x)” also!) and the name “differential coefficient” are used.

This summary deals with the calculus of one variable. From the mid-eighteenth century onwards, the calculus of several variables had developed, initially by d’Alembert and Euler. The range of applications of the calculus increased enormously, with partial differential equations supplementing and even supplanting ordinary ones (Caramalho Domingues Reference Caramalho Domingues2008). A wide range of notations for partial derivatives was deployed; the use of “∂” gradually became common from the 1830s onwards.

In all these traditions the “integral” (Leibniz’s word) was construed as the inverse of the differential coefficient or derivative, and/or as an area, and/or as a sum, and/or as the limiting value of a sequence of partition sums. Only with Cauchy did the inverse relationship with differentiation become a genuine theorem, with sufficient conditions of continuity imposed upon f(x) to guarantee its truth.

Finally comes the calculus of variations, pioneered by Euler and others but given its full generality by Lagrange, with the symbol “δ” marking the variation δx of x (Goldstine Reference Goldstine1980), where x may be a parameter now taken to be a variable. (The distinction between variables and parameters was not always carefully observed, a sloppiness over which economists are not alone!) The theory was especially powerful as a means of expressing problems involving optimization (shortest distance, least time, and so on).

For Lagrange the use of “δ” exemplified his desire to “algebraise” mathematical theories as much as possible. Other examples include his foundation of the calculus, and especially the no-diagrams analytical mechanics noted in section 2, where he favored especially the principle of least action, since it invoked optimization and so could be expressed in terms of the algebraic calculus of variations. Indeed, the use of that calculus caused analytical mechanics sometimes to be called “variational mechanics.”

Cauchy used his theory of limits not only to re-found the calculus but also to express the theory of functions and the convergence of infinite series; the whole doctrine became known as “mathematical analysis” (Jahnke Reference Jahnke2003, ch. 6). One of its features is the prominent place given to inequalities, of which the place in mathematics in general is underrated (Tanner Reference Tanner1961); thermodynamics, parts of mechanics, and mathematical economics are other branches where they play an important role.

Among our mathematical economists, Cournot brought in both the univariate and the multivariate differential and integral calculus. Jevons worked mainly with the univariate calculus, and in the mongrel version; a consequence is noted in section 8. Edgeworth also favored the multivariate calculus, rather unhelpfully using his own notations; he also innovated the calculus of variations, which was used less widely than one might have expected in economics where optimization came to be a marked feature. The other figures normally used differentials and/or mongrel. No economist tried to algebraize his theory to a Lagrangian extent; on the contrary, diagrams abound in the writings of most of our authors.

One objection made to the use of the differential and integral calculus is that many economic procedures take place from time to time and that some commodities are only finitely divisible, so that the difference calculus is more appropriate (and indeed was deployed: subsection 20.2). This is a fair point, but it can be made about parts of mechanics itself, for example; so I shall not refer to it again.

5. A Methodological Trio: Analogizing, Modeling, and (De)simplification

5.1 Analogies and emulations. Various kinds of relationship may hold between theories in applied mathematics (Grattan-Guinness Reference Grattan-Guinness2008a). One is reduction, where some theory B is reduced to a special kind or case of theory A; two examples were mentioned above in connection with Lagrange.

More important here is analogy, where B resembles A in using similar principles and/or the same mathematics, without any claim of reduction. The “analogy content” is the level of detail with which the analogy between the two theories holds. For example, if they both use integrals, then the level is modest; but if they also both use the same definite or indefinite integrals, say, or both define an integral in terms of an area or a sum rather than an anti-derivative, then the level increases.

If the creator of the new theory B consciously imitated features of A in his construction, and included this feature in his advocacy of B, then he saw B as emulating A. As with analogy, he may even hope to find more similarities when developing B further, thereby increasing the emulation content of B relative to A. Alternatively, he may wish to decrease it by reducing the similarity to the status of an analogy. When our economists mentioned similarities, normally they used the only the word “analogy” and probably did not consciously distinguish it from “emulation”; but in my jargon they were emulating in many cases. Particularly in the context of this article, we need to distinguish the intention of the theorist from his achievement: a small measure of emulation content does not imply that the desire to achieve it was also small.

Analogy and emulation content between theories are found not only in their technical aspects: they can be present also, or instead, in philosophical aspects such as generality, (lack of) rigour, use of the same theory of knowledge, reliance on algebra, common historical routes or backgrounds, and so on. For example, Adam Smith admired Newton’s methods (as he understood them), but his invisible hand does not necessarily bring equilibrium to a competitive market by mechanically Newtonian means.Footnote ⁶

An important philosophical case is the balance between induction and deduction in a theory, of any kind. For example, the creation of a theory is likely to be helped by the accumulation of similar phenomena or relationships, which may suggest properties, laws, or analogies (“quite a lot of swans are white,” and so on). When a theory is systematized enough to have in place some seemingly basic principles or laws, consequences are deduced from them, in order to elaborate or test it (the hypothetical-deductive method). In general, deduction has been seen as a good sign of development of a theory, and so has enjoyed a higher status than induction, especially among searchers for certainty in knowledge. Further, theories in mechanics contain a more substantial deductive component than those in economics; hence, there is a motive for neoclassical economists to emulate mechanics. Some of their considerations and quotations below fit this sentiment, and section 9 contains an example of the difference in balance in Walras.Footnote ⁷

Analogy and emulation are relations consistent with Thomas Kuhn’s theory (1970) of paradigms, especially the status of normal science as the target of such imitation. They are also especially important in the building and appraising of theories (Kaushal Reference Kaushal2003). Finally, as Schumacher sagely warns above, influence is not always positive: there are the opposites such as disanalogies, disemulations (a word that, curiously, does not seem to exist in English), and negative influences in general.

5.2 Some examples of analogizing and emulating in mechanics and physics. The rise of mathematical physics noted in section 3 provides several important and successful examples of all these considerations (Grattan-Guinness Reference Grattan-Guinness1990a, chs. 7, 13, and 14 passim). For example, already in the 1770s and 1780s, Charles Coulomb had argued on experimental grounds that a Newton-like inverse square law obtained in both electrostatics and magnetism; Poisson’s mathematical elaboration of this theory in the 1810s and 1820s was partly based on adapting results from planetary mechanics, and also helped to develop potential theory. Again, Ampère repeated Coulomb’s emulation of Newtonianism in the 1820s for the new subject of electrodynamics. Meanwhile, in his second and definitive wave theory of light, Ampère’s lodger Fresnel assumed that when the particles comprising his punctiform aether were disturbed from equilibrium, they performed simple harmonic motion, after the manner of the vibrating string; he also saw his cosine law of intensity of the ray as akin to the parallelogram of forces in statics. In addition, he came to see that his analysis of double refraction was consistent with the conservation of kinetic energy in his wave system, although conservation principles had not helped him to form his theory in the first place; this case is an example of a fourth methodological category, which I call corroboration.

Among later figures,Footnote ⁸ from the start of his career in the early 1840s, Kelvin made an explicit methodology out of analogy by transferring results from mathematical analysis and potential theory between heat diffusion, electricity, magnetism, and hydrodynamics without advocating the emulation of any topic over the others (Grattan-Guinness Reference Grattan-Guinness, Flood, McCartney and Whitaker2008c; Wise Reference Wise1981). This policy in turn influenced the mechanical models made by James Clerk Maxwell of his electromagnetic theory from the 1860s onwards (Siegel Reference Siegel and Harman1985). For him, they aided the development of theory in areas where knowledge was scrappy, but no claim either of comprehensiveness or of truthhood was made of them. This latter caveat was especially welcome concerning his famous model of electromagnetic induction in terms of a mechanism in which idler wheels rotated between tiny hexagonal vortices, for, in fact, it is inapplicable.Footnote ⁹ In addition to models of and in mechanics and physics, Maxwell deployed mathematical ones, which were often based on structural similarity between mathematical theories.

In turn, Maxwell’s advocacy of models, together with the stress laid on the mental formation of images made from the 1860s onwards by Hermann von Helmholtz, led Heinrich Hertz in the 1880s to emphasize the role of “images” of mechanics; he required that the chosen theories (for example, parts or all of any of the three traditions outlined here) are logically consistent and faithfully represent the principal relationships of the pertinent phenomena (Lützen Reference Lützen2005, chs. 7–10). Other figures around 1900 who attended to models included physicist Ludwig Boltzmann; mathematician and historian and philosopher of science, Pierre Duhem, who devoted a chapter of his widely read book on “the object and structure of physical theory” to “abstract theories and mechanical models,” critical of some supposed “English” predilection for modeling (Duhem Reference Duhem1906, pt. 1, ch. 4); and the English mathematician and historian of mathematics, Philip Jourdain, who stressed models in his popular account of dynamics (Jourdain Reference Jourdain1912, ch. 4).Footnote ¹⁰

5.3 Modeling ≠ applied mathematics. These last cases show modeling being applied to mechanics, but the Maxwell case exemplifies the converse process of applying mechanics to modeling phenomena that are not mechanical. Both procedures subtly affected the development of mechanics and physics (d’Agostino and Petruccioli 1987).

Normally in applied mathematics, one takes a theory already formed and uses it in the physical application as a ready-made package, unlikely to undergo any major changes in the process; the traditions of mechanics and of the calculus described above are usually deployed this way. But modeling involves much more interaction between the mathematical theory and the application at hand, usually due to the great complications of the latter, for a principal aim is to pick out the seemingly most important features of the phenomena under consideration. Other aims include helping to develop a theory in the first place, representing reasonably well as many of the pertinent phenomena or data as possible, suggesting lines of investigation, furnishing testable predictions, and switching readily between induction and deduction. Simplification and desimplification are prominent, and oversimplification is not unknown; analogies and emulations may well be involved.

It is a moot point as to whether the difference between modeling and application is one of degree rather than of kind, but it is still well worth noting, for this emphasis on models in mechanics and physics came to have some effect on the development of neoclassical economics, especially Fisher from the 1910s (section 13); both lines ME and PE of influence of section 3 are evident. Apart from his, most of the uses of mechanics were applications in the above sense up to the 1930s, when modeling came into prominence in economics with figures such as Jan Tinbergen.Footnote ¹¹

5.4 Desimplification. This is another powerful method of building up a theory. The world is a very complicated place, as a student of any of its phenomena soon realizes. So he will deliberately omit factors pertaining to his objective study, and simplify others, and then he will attempt to develop a theory of this simplified situation. Having done so (let us assume), he may put back in those factors that he omitted or simplified (and using analogies and emulations might play a role), producing a desimplified theory. One consequence, maybe intentional, is that the analogy or emulation content with some other theory is increased. These procedures are consistent with both applications and modeling.

In mechanics, for example, the theorist may at first ignore the rotation of the Earth about the Sun, assume that the density of a solid body is constant, and so on; then he will try to allow for these factors and thereby desimplify the theory that he has already produced. Similarly, in economics he may at first leave out real-life factors such as panic buying, hoarding, gender (male utility functions can differ markedly from female ones), and non-purchase because the buyer forgot to bring along enough money to the market or found that the queue was too long to wait. He may also assume that, say, an economic agent has perfect knowledge of his situation. Then, after developing a theory, he will try to take note of some of these factors. An especially important desimplification in economics is to attempt to move from static to some form of dynamic equilibrium.

6. Preamble

Normally, classical economists took notions such as production, supply, and demand to be basic categories, or at least they worked with them without specifying them in terms of other notions. Further, their mathematical equipment rarely strayed from arithmetic and common algebra, and curves and graphs of various kinds.Footnote ¹² A striking case is William Whewell, who deliberately eschewed using the calculus or mechanics in his economics, despite his expertise in both areas and his interest in the efficiency of machinery; in the case of mechanics, the avoidance might well have been deliberate.Footnote ¹³

One of the main motives of neoclassical economistsFootnote ¹⁴ was to start out from different primitives. In addition, the mathematically inclined of this persuasion brought in the differential and integral calculus, and some also were influenced by mechanics. This last feature is my dominating concern; the (very limited) role of mechanics among their predecessor classical economists is not discussed, the physics story is restricted to energetics and a touch of thermodynamics,Footnote ¹⁵ the statistics and calculus stories are abbreviated, and the place in economics of mathematics of any kind is noted only briefly, at the very end of the article. All these topics deserve general treatments of the kind offered here for mechanics.

The eight chosen figures are considered in chronological order of appearance. They count among the most influential voices in the period covered, and collectively they seem to encompass the scope of attitudes to mechanics and the uses to be made of it (or not). The original literature by them is far too large to receive even a summary here: I hope that I have made a reasonably representative selection of their pertinent contributions. An interesting exercise would be to examine all the book reviews that they wrote on each other.

More detailed historical sources for each economist will be cited in situ. Among the pertinent general histories of economics, I have found particularly useful the appropriate parts of Howey (Reference Howey1960), Blaug (Reference Blaug1968), Ingrao and Israel (Reference Ingrao and Israel1985, Reference Ingrao and Israel1990), Mirowski (Reference Mirowski1989), Screpanti and Zamagni (Reference Screpanti and Zamagni1993), Theocharis (Reference Theocharis1993), and Perlman and McCann (Reference Perlman and McCann1998), and the source books Baumol and Goldfeld (Reference Baumol and Goldfeld1968) and Marchionatti (Reference Marchionatti2004); they consider both further aspects of the work of my octet and other contemporary economists. In contrast, the mathematics abstracting journals, Jahrbuch über die Fortschritte der Mathematik (1868–1942) and the Zentralblatt für Mathematik und ihre Grenzgebiete (1931–), were not much use; despite the name of the latter, the coverage of mathematical economics in both was slight,Footnote ¹⁶ exemplifying the very limited interest that the professional mathematical community has taken in mathematical economics.

7. The Place of Cournot

Cournot was not a neoclassical economist because he did not take utility as a primitive; further, his influence grew so slowly that it was still progressing in the 1920s (Sigot Reference Sigot and Martin2005). But several important features of his work are consistent with neoclassicism, and he influenced all the other figures discussed below. So let us give him honorary membership of the club.

Antoine-Augustin Cournot (1801–1877) was not a graduate of the Ecole Polytechnique but studied at the Ecole Normale in Paris; however, he was there for only the academic year 1821–1822 because it was then closed down for several years as part of the repressions in the politically volatile situation that was to be resolved only with the revolution of 1830. Living in the leading mathematical town of his time (Grattan-Guinness Reference Grattan-Guinness1990a, chs. 16–17), his personal contacts with senior figures such as Fourier (presumably) and Poisson (certainly) would have stood him in good stead. He took some courses in the Paris Faculté des Sciences of the national so-called Université education system,Footnote ¹⁷ in order to take their doctorate (Cournot Reference Cournot and Bottinelli1913, pp. 76–80;154–155). He chose to study mechanics, and between 1829 (with his doctoral dissertations) and 1834, he produced ten articles and a dozen reviews of books or papers, around 200 pages in all (Grattan-Guinness Reference Grattan-Guinness2006a); he treated, or reviewed writings by others on, several specific questions in celestial and planetary mechanics, the theory of machines, impact between hard bodies, and the motion of bodies under friction.

Of especial interest are the studies that were motivated by the innovation by Fourier of linear programming (as we now call it) in the 1820s (Grattan-Guinness Reference Grattan-Guinness1994b, pp. 48–52). Under this inspiration Cournot analyzed the paradox of statics (that the basic laws of static equilibrium are not sufficient to determine the pressures on the legs of a table if there are more than three legs), and the principle of virtual velocities when expressed as an inequality. However, Fourier’s initiative died out fairly quickly, seemingly for lack of effective methods for finding solutions: matrix theory still lay in the future.

Cournot’s work on economics began to appear in the late 1830s, by when he was holding senior posts within the Université (Hulin-Jung Reference Hulin-Jung1989passim). The chief sources are two books on wealth. In the first one, Recherches sur les principes mathématiques sur la théorie des richesses, Cournot (Reference Cournot1838) pioneered the use in economics of the differential and integral calculus, already used in his studies of mechanics and on which he would soon publish a substantial textbook covering also the theory of functions (Cournot Reference Cournot1841); but the very presence of such mathematics in a book on economics at that time may well have guaranteed him a tiny readership. Much later he produced Principes de la théorie des richesses (1863), spare in its use of mathematical symbols but nevertheless also spare in readers. In his economics he assumed that a mathematical function could always be developed in a power series, although in his textbook he did mention one of Cauchy’s counter-examples (Cournot Reference Cournot1841, p. 179).

The main contributions to economics using the calculus due to Cournot include the notion of the elasticity of price (1838, esp. ch. 4), to use the quasi-mechanical name that Marshall will give it; and marginal cost and revenue, especially the conditions for maximizing profit under monopoly and oligopoly. However, he did not draw on his earlier use in linear programming of convexity, even though it was important in his curves that represented turnover and competitive supply (chs. 4, 7–9). The quotation at the head of this section exemplifies the place of mechanics in his economics: as a source of analogies but not of emulations. His restraint may be due to his considerable knowledge of mechanics, in which case he is to be complimented. The topics “mechanics,” “physics,” and “equilibrium” rarely feature in his books. The word “work” appears more frequently, but normally used in a general way concerning the activities of “workers” in industry and elsewhere. Similarly, “force,” often qualified with adjectives such as “productive,” did not invoke theories in mechanics.Footnote ¹⁸

Concerning analogies, Cournot’s main source seems to have been Leibniz on the conservation of forces vives in mechanics, a pre-Kelvin version of energetics physics (Vatin Reference Vatin and Mosini2007b), and so an example of line ME of influence in section 3. Vatin (Reference Vatin1998) provides a rich picture of the various contexts of Cournot’s interests, including machine theory, ergonomics, thermodynamics, evolution, and forestry; but his account of the books on economics does not stress mechanics in general—correctly, in my view. There are occasional exceptions: in one, Cournot feared that his theory of wealth was only an “idle speculation” if it was too far distant from actual wealth accumulation, just as theory of a perfect fluid in hydrostatics may be far removed from “the constitution of the most widespread fluids in nature” (1838, art. 6). In another, he made an analogy between the unequal distribution of wealth in a community and the balance of several connected vertical tubes filled with fluids of varying densities (1863, pp. 55–56).

Cournot’s attitude to data in economics was not aloof, but he seems to have been preparing for them rather than actually handling them (Le Gall Reference Le Gall2007, ch. 2). Further, his allegiance to the physical sciences, while modest, was robust enough to exclude economics in any significant way from his analysis of the philosophy of probability theory.Footnote ¹⁹

8. Jevons

Stanley Jevons (1835–1882) may have been partly self-taught, both in mechanics and physics, initially during his early years in Australia. In the late 1850s he was a mature student at University College London, where his teacher in the calculus was Augustus De Morgan (Grattan-Guinness Reference Grattan-Guinness2002, pp. 692–696). He might have taken mechanics courses there also, though we shall see that his facility in the subject was limited. After spells at the colleges in Liverpool and Manchester, he came back to the college in 1876 as professor, retiring four years later. The features discussed below largely refer to his main book Theory of Political Economy (1871 and later editions, including 1970).

Jevons was an important pioneer of neoclassical economics from the early 1860s onwards (White Reference White, Aspromourgos and Lodewijks2004, Reference White2005), and his allegiance to mechanics (line ME of influence of section 3) was quite substantial although not often explicitly stated (Maas Reference Maas2005). To replace supply and demand with new primitives, he chose the “utility” of the economic agent, which maximized his pleasure and minimized his pain. Of special importance was the derivative function, now often called “marginal utility,” measuring the rate of change of utility. Like some of his classical predecessors in other contexts, he assumed that utility was a decreasing function of the commodity involved, and that its second difference (or derivative) was positive, so that it was represented by a convex or a concave curve (1970, pp. 104–107). Well familiar with logic, he was aware of roles for both induction and deduction in the “logical method of economics” (1970, pp. 87–91), as was Whewell before him (Hollander Reference Hollander1983).

Already a published author on the mechanical balance in 1863, Jevons made some strong appeals to mechanics, especially that trading took place in a market or auction when the utilities of each agent were in balance. He took this to be static equilibrium: while “The real condition of industry is one of perpetual motion and change,” it was too difficult to study; hence “It is only as a purely statical problem that I can venture to treat the action of exchange,” he declared in his Theory of Political Economy (1970, p. 138). In its preface he stated (p. 44) that

In the first sentence he states the general analogy based on the moments of forces, followed by a stress on the place of the calculus; the last sentence looks forward to further use of mechanics-like economics. The aim for emulation content is high, but it rarely manifests in the book. The most pertinent part is a passage on an “analogy,” dependent on static equilibrium, between “the theory of exchange” and “the theory of the lever”; drawing upon conditions for equilibrium, he found the general law dy/dx = y/x, which has no obvious parallel in mechanics itself (pp. 144–147).Footnote ²⁰

Again, in the lengthy preface of 1879 to the second edition, Jevons discussed all sorts of matters, but none pertaining to mechanics. However, in a new passage he formed an “analogy to the theory of the lever” (1879, pp. 110–115), in which he fulfilled, rather tardily, a promise made in the first edition to deploy the principle of virtual velocities. The purpose was to justify his conditions for achieving the state of equilibrium for two agents to exchange two commodities; however, he did not use the principle properly, for he set the displacements from equilibrium in such a way that the lever principle was already assumed (Grattan-Guinness Reference Grattan-Guinness2009). Moreover, while his “proof” attracted some attention among followers, and some criticized it from economical points of view, none of them noticed the illegitimacy of the basic argument. Indeed, one of them reproduced the pertaining diagram with admiration.

9. Walras

This culprit was none other than Léon Walras (1834–1910), in a set of notes seemingly compiled in the 1900s (1995, p. 639). He was an external student (that is, not a graduate of the Ecole Polytechnique) at the Ecole des Mines in Paris from 1854 to 1858, when he might have taken the mechanics course given by Charles Combes, a very distinguished mining engineer. As a young man, he acquired a copy of the eighth edition of the Eléments de statique by Poinsot, which contained a lovely discussion of the circumstances under which (2) held (Poinsot Reference Poinsot1842, ch. 2) that seems to have influenced him considerably. He was appointed a professor at the University of Lausanne in 1870.

Walras studied in detail “pure” economics, which resembled rational mechanics in concentrating on simplified situations (Walker Reference Walker2006, ch. 5); applied economics was concerned with industry, for example. He cited Jevons and Cournot and, indeed, corresponded with both of them in the mid-1870s. Like them (and also some classical economists), he assumed that a law of diminishing returns would apply. A major notion was a version of marginal utility, called the “scarcity” of a commodity to a buyer or seller. Allowing for any number of economic agents, he construed the equilibrium of exchange as the state in which total scarcities perceived by its agents was zero, or when prices became steady. He also studied similarly the maximization of utility in the contexts of production (for example, the equality of cost and sale prices allowing for profit), capitalization and credit (for example, investment and expenditure), and circulation of money; here I have followed the titlings and order of the parts of his main textbook on pure theory (Walras Reference Walras1998). He popularized the theory of “groping” (“tâtonnement,” or, as Anglo-Saxon authors usually write, “tatonnement”), which asserted that if a state of equilibrium was reached, it was independent of the particular sequence of states that occurred en route. This property was redolent of Cauchy’s theorem that the value of an integral is independent of the sequence of partition sums used to define it; it also smacks of conservative fields of forces in mechanics. He used the calculus, but most of his mathematics came from the theory of linear equations, drawing on linear combination as in (2); however, he was working during a period before matrix theory became widely known,Footnote ²¹ so that some of his analyses were limited in rigor and maybe in detail. He argued for the existence and uniqueness of these various equilibria on the weak grounds that the number of unknown constants equalled that of the equations, and he seemed unable to exclude negative solutions for inappropriate variables such as prices.

Walras claimed that his theory of equilibrium was general, and it was so admired by many successors. The generality was achieved by accumulation: after forming the fairly small theoretical core of maximizing scarcities, he applied versions of it to this topic in economics and then to that one and then … This version of generality differed in degree from that found in many general mathematical theories, which contain a substantial core and are developed in a more implicational way; if (some parts of) the core are assumed, then further parts of the theory are deduced, and from them still further parts follow, with the core frequently in action (Grattan-Guinness Reference Grattan-Guinness2010). Six important examples are the three traditions in mechanics and in the calculus rehearsed in sections 2 and 4; to note two of them, the core of mathematical analysis is the theory of limits, which is frequently deployed in the further exegesis; similarly, the core of analytical mechanics includes the principle of least action, which is often used in its development. This difference in the ways of achieving generality is an example of the difference in balance between induction and deduction in economics and mechanics that was noted in section 5.1; it is like that between having the parts of a theory laid out in a horizontal row and having them stacked up in a structure.

Like Jevons, Walras exemplified strongly the line ME of influence in section 3 and did not appeal to physics outside mechanics. In an appendix on the “geometrical theory of prices” added to the 1890 edition of his main textbook, he noted that geometry, algebra, the calculus, and mechanics had led successfully to “the marvel of modem industry,” and he made the strong claim that following “the same procedure in economics” will lead to similar success “in the economic and social order” (1998, p. 699). Again, in his book of 1898 on applied economics, he rather suddenly cited Lagrange’s Mécanique analytique (second edition 1811–1815) as the source of the view that the equation Pdp + Qdq = 0, akin to the principle of virtual velocities, established “the equilibrium of two forces.” He then made the striking statement that economics “is itself the mechanics of the equilibrium and motion of social riches” in the way that “hydraulics” is applied to liquids, and “is thus incontestably” an “abstract and deductive theory but mathematical also” (1992, pp. 405–406). It is nice to see him emulate the analysis of liquids via hydraulics, the engineers’ study of the behavior of bodies of water in various natural and man-made circumstances, rather than hydrodynamics (subsection 5.3), as Cournot had done: he did not discuss the distinction (on which see subsection 19.4).

How much did mechanics emerge into the foreground of his writings, and how often? A clue is provided by the excellent volume of tables and indices that completes the edition of his and his father’s works: the entry for him on “Mécanique” runs for several lines but is not extensive, especially when one notes that it covers nine substantial volumes in the edition (Walras Reference Walras2005, p. 505). Similar remarks apply also to the entry for “Physique” (p. 525). Moreover, several of the passages cited, for both entries, are of some very general kind, making no special contentual appeal to their parent disciplines; for example, rarities or intensities are just like mechanical forces. Some of the most significant cases involve consideration of static and dynamic equilibrium, and Walras never successfully got his theory out of the static into the dynamic case, which is where most economic transactions take place.

Walras retained his adhesion to mechanics up to his last paper (Walras Reference Walras1909), entitled “Economics and Mechanics,” from which the quotation at the head of this section comes. The most striking invocation of mathematics there was for vector algebra (‘I want to buy your apples,’ not vice versa), which, while valuable in mechanics, was not confined to it. Overall, his expressions of enthusiasm for using mechanics in economics resemble the occurrences of isolated waves in the motion of the seas: occasional but strong.

10. Edgeworth

A major economist who dipped into mechanics and physics was Francis Edgeworth (1845–1926). He spent much of the 1860s being educated at Trinity College Dublin and then at the University of Oxford. Mathematics does not seem to have been much on his agenda, but later he taught himself enough of it to express himself in economics. He took a chair at Oxford in 1891.

Edgeworth’s most influential book was Mathematical Psychics (1881). The title already suggests that analogies with mathematical physics are in the offing; maybe a motivation was John Herapath’s book Mathematical Physics, published in 1847. His choice of topics from mechanics and physics was quite eclectic; as well as contexts for optimization (for example, of energy and in analytical mechanics in general), it included using coordinates as in analytical mechanics and features of hydrodynamics such as irrotational flow. He exhibits the line ME of influence of section 3, and also PE to some extent; but in many cases the content is modest. For example, he stated that “The Newtonian astronomy is the model for our Science” (1909, p. 80), but he was surely wishing economists to emulate its precision and certainty rather than its content.

This remark was made in a paper discussing the use of the differential calculus in optimization problems in economics; unusually for economics but quite legitimately, Edgeworth included the calculus of variations. Elsewhere he discussed at some length the role of mathematics in economics, and promoted the calculus and the associated theory of mathematical functions and planar curves, but little mechanics (2003, pp. 211–239; 278–339; 369–376 passim).

One important change from Jevons was that Edgeworth took an economic agent to work with one utility function for all commodities involved instead of with a function for each one, thereby bringing in the calculus of several variables and so following Cournot. He stressed the “indifference-curve” of an agent, the locus of values (x, y) along which his utility function of commodity variables x and y was constant; as he surely knew, it emulated that of equipotential curves (and surfaces) in statics and physics. Assuming laws of diminishing returns, he argued that these curves were (continuous and) either concave or convex; in a seemingly novel move, he proposed that equilibrium of exchange occurred when the curves of two agents touched, for then their degrees of utility functions (in Jevons’ terms) would be equal in value and opposite in sign (1881, pp. 33–36). But he admitted that in a given state of perfect or imperfect competition, several possible situations for trading could obtain, that each one depended upon its initial launch, and that the process of achieving any one of them, whether by groping or some other means, could not be analyzed (1881, pp. 44–50; 1891, pp. 177–181).

Edgeworth also advocated the use of probability theory and mathematical statistics in various contexts, such as bimetallism, banking, and index numbers (Edgeworth Reference Edgeworth and McCann1996, pt. 1); indeed, he stands out among all his contemporary economists in the fervor of his advocacy (S. Stigler Reference Stigler1986, pp. 305–325). Unfortunately, he found little response; mathematical statistics had still to wait (section 16).

As the quotation at the head of this article shows, Edgeworth was very realistic about the quality of mathematical economics as compared with mathematics elsewhere. In that widely published address (1889) to the British Association for the Advancement of Science, he stuck largely to arithmetic and algebra, and geometrical diagrams, especially to illustrate supply and demand curves. With mathematical physics he made only some general analogies; on mechanics he said nothing at all.

11. Marshall

Mechanics did not capture the enthusiasm of all leading mathematical economists of the time. For example, Scottish engineer H.C.F. Jenkin (1833–1885) advocated energetics physics in science and engineering, and in this spirit he produced a graphical means of assessing the economic efficiency of machinery. But when he wrote on economics in general, he was not neoclassical in his approach; in particular, there was hardly a trace of influence from mechanics there.Footnote ²² He is best remembered for another graphical theory: intersecting supply and demand curves (respectively, convex and S-shaped ones).Footnote ²³

Even more striking a case of abstention is Alfred Marshall (1842–1924). Second Wrangler in the Cambridge Mathematical Tripos in 1865, he possessed a more solid mathematical training than the other economists discussed here, and, moreover, one in which mechanics and mathematical physics played large roles (Warwick Reference Warwick2003, esp. ch. 5). After three years at Bristol College, he returned to take a chair at Cambridge in 1888, and taught there for his whole career.

Marshall’s main work, Principles of Economics, first published in 1890 and in several further editions, shows him subscribing to many features of neoclassical economics, but it is also notable for the very modest place of mathematics in general and mechanics in particular. He was skeptical of all three lines of influence in section 3, especially ME. He confined his symbolic treatments to an appendix (1949, pp. 690–706), and in his prosodic text he stressed the limitations of mechanics more than its possible scope; he even consigned his diagrams to footnotes. However, he likened the first stages of the analysis of equilibrium in supply and demand with value to simple cases of static equilibrium such as “the mechanical equilibrium of a stone hanging by an elastic string”; again, the law of composition of (two) mechanical forces involves a “method of combination” suitably analogous to the action of “two economic forces” (pp. 269, 637). He also advocated partial equilibrium (although he did not call it that), when only some of the variables in an economic situation are considered and the others are set aside, which has some analogy with the simplifications that are effected in dynamics when one ignores air resistance, for example. But “even in mechanics long chains of deductive reasoning are directly applicable only to the occurrences of the laboratory,” and “the uses of the statical method in [economic] problems relating to very long periods are dangerous” (pp. 637, 315). The warning at the head of this section comes in a passage to which he adjoined an appendix on the “Limitations of the use of statical assumptions in regard to increasing return” (pp. 664–669); he made it also in other contexts, such as ones involving increasing returns (Bharadwaj Reference Bharadwaj1972).

12. Pareto

Vilfredo Pareto (1848–1923) graduated as an engineer at the Istituto Politecnico in Turin in 1869, and for twenty years he worked as a railways and mining engineer before moving across to economics and sociology. In 1893 he succeeded Walras at the University of Lausanne, whom he followed, and also Jevons, on various aspects of economics (Walker Reference Walker2006, pp. 259–278). At first he worked with “ophelimity,” which he saw as the desire or need of an economic agent, differing from Jevons’ utility in that it did not necessarily refer to pleasure or have to be numerically measurable (1896, pp. 3–6; 35). Like Edgeworth, he took it to be a function of all the variables in a trading situation and so worked with multivariate mathematical analysis.

Pareto emphasized the process of a sequence of theories successively approximating to the phenomena under study (Bruni Reference Bruni2002, ch. 1); it is broadly similar to my notion of desimplification. It featured in mechanics, as in the quotation above; he seems not to have known that Cournot had studied the paradox of statics (section 9). In economics, the pure part was the first approximation, akin to rational mechanics.

Analogies also played an important role in his economics (and also in his sociology, which I do not discuss here), and some of them drew (fragilely?) on mechanics or physics: for example, kinetic energy/budget expenditure, potential energy/total utility, and force/marginal utility (McLure Reference McLure2000, esp. pp. 20–47). Thus, he exhibited both lines ME and PE of influence of section 3.

However, despite his experience as an engineer, Pareto’s competence in mechanics is cast in doubt by an analogy that he once made between the contrast of “a man sliding down a slope on a sledge, while another man descends the same slope on foot, stopping at every step” (Pareto Reference Pareto1896, art. 587). He allied the slider with the smooth operations of the mechanician and the stepper with the economists’ concern with situations where equilibrium may be broken or change in discrete amounts (when, for example, a trader has suddenly changed his prices). However, the analogy fails, since the slider is not in any state of equilibrium in the first place. Pareto seems to have confused dynamical equilibrium with a mechanical situation lacking impact (and so no broken legs), and thereby involving the conservation of energy and the assumption of potentials.

In the early 1900s Pareto tried a more straightforward approach to dynamic equilibrium by allowing all terms, price included, to become variable over time and indeed differentiable (in Leibniz’s sense, using differentials). Assuming some kind of condition for the equality of incomings and outgoings allowed him to leading to form differential equations. Although this procedure gave him some freedom from mechanics, typically he pressed an analogy: “Mathematically the prices double as auxiliary unknowns, adopted to resolve the equations of economic equilibrium; and a very notable analogy lets one see that they are introduced in those equations like the tensions of links in the equations of mechanical equilibrium” (1901, p. 247).

In the mid-1900s Pareto moved back towards the worlds of mechanics when he replaced ophelimity by the “index functions” of the economic agents, their (non-unique) curves of indifference; he saw their role as similar to that of equipotential surfaces. In this connection he significantly extended the use of mechanics in economics with his emulation of d’Alembert’s principle, especially its claim that a state dynamic equilibrium could be reduced to one of static equilibrium. He assumed that his fields were conservative; that is, that the work done by a system of forces (or utilities) when displaced from equilibrium was path-independent. Defining the price of a good as the negative of the partial derivative of the money variable with respect to the corresponding commodity variable, he converted the normal linear combination of the utility function into first-order partial differential equations, after the manner of the mechanics in which d’Alembert’s principle (not named) played a role; so he could claim that dynamic economic equilibrium was reduced to the static situation, and so overcome a major limitation in Walras’ theory of economic trading (see, for example, (Pareto Reference Pareto1911, arts. 2–22, esp. arts. 3 and 15)).

Although the presentation is somewhat odd, Pareto’s use of mechanical principles (and the mongrel calculus) is correct in itself. However, its interpretation as economics is extremely debatable; one might wonder if he has even reached a first approximation. In later life he seems to have reduced the role of mechanics in his work (Bruni Reference Bruni2002, pp. 96–97). His name is attached to an important type of economic equilibrium that has no analogues in mechanics: when the utility of every agent involved in that situation cannot be improved without causing the decline of the utility of least one other agent.

13. Fisher and Then Fisher

Irving Fisher (1867–1947) studied at Yale University from 1885 to 1891, when he wrote his doctoral dissertation in economics. As we see him testifying below, he was profoundly influenced by the mathematical physicist, J. Willard Gibbs, and also by the mathematician, Edwin B. Wilson, himself a student of Gibbs. He started his career as a professional mathematician before switching to economics in 1895; as he put it charmingly in a letter of that time, “My one regret about a mathematical life has been its lack of direct contact with the living age” (Allen Reference Allen1993, p. 69). He passed his entire career at the university.

Fisher launched his career as an economist with a remarkable doctoral dissertation containing “Mathematical investigations in the theory of value and prices” (Fisher Reference Fisher1892). The essay fell into two main parts: when the “utility” of each commodity depended respectively 1) only on its own values, and 2) on the values of all the commodities in the trading context. This sounds like a transition from Jevons to Edgeworth, and, indeed, he praised Jevons highly in the first part; but in the preface he stated he had developed the second part before reading Edgeworth (p. vi).

Wishing to eschew psychology from his theory, Fisher followed principally Jevons in his analysis of utility and developed a model of trading of “m commodities—n consumers” (ch. 4) using the calculus of differentials. Fond of geometrical thinking and talented at making gadgets and lashups, he illustrated several results in his thesis with sophisticated mechanisms of levers and pumps, including hydrostatic ones. These formed striking analogies; but since “The student of economics thinks in terms of mechanics far more than geometry” (p. 24), he went further along this line ME of influence. In a chapter on “mechanical analogies,” he compiled a table of comparable notions in economics and in mechanics, noting their status as scalar or vector (another Gibbs speciality) (pp. 86–87): for example, space/commodity; force/marginal utility, both addable vectors; work/disutility, and (total) energy/(total) utility, both scalars; and equilibrium of imposed and resistant forces/equilibrium of marginal utility and marginal disutility. The analogy content here is substantial, though corrigible, as Mirowski nicely observes (1989, pp. 224–231): particle/individual is pretty dehumanizing for a start, while “work or energy”/utility is already confused in the mechanics. His later discussion of the issue of statics versus dynamics was more acute: “The ideal statical condition assumed in our analysis is never satisfied in fact,” while “the dynamical side of economics has never yet received systematic treatment” (pp. 103, 104), including not here.

As part of his transition from mathematician to economist, Fisher published a short elementary textbook on the calculus (1897) expressly for use by students of mathematical economics and statistics. His version was a healthy mongrel, bred on limits, differential as zeros, and the “differential quotient” symbolized using the primes notation. The integral was the anti-derivative, but also noticed as an area. The multivariate calculus was briefly summarized, and a few economics contexts noted in passing.

In Fisher’s research work after his thesis, and in his teaching, neither mechanics nor physics seemed to play significant roles, nor were they needed; emulation had softened to analogizing. In his textbook of 1912, he highlighted the numerically measurable financial factors in economics, such as wealth, income, capital, and prices; utility was relegated to an early footnote. No calculus was used, and no topic from mechanics or physics gained an entry in his excellent index. A whiff of statics came into his discussion of accounting, where he contrasted the “method of balances,” where the books of each economic agent were balanced, against the “method of couples,” where “we cancel each liability against an equal and opposite asset, which will be in some other individual’s account” (Fisher Reference Fisher1912, pp. 51–54).

In his work on the quantity theory of money, Fisher introduced the notion of its “velocity,” the number of times that it circulated (in all its various forms) around an economic community in a given period of time. The dynamic background of the name reflected that of its definition, which resembled (2) in its use of linear combination: if quantity Q_r of a commmodity g_r is sold at price p_r over the period, and quantity M of money is available, then its velocity V is defined from

(3)

where P is the weighted average price. His attachment to modeling led him to describe his mathematical treatments as “illustrations” even if they were as elementary as the algebraic example given in (3). He also drew upon mechanical and physical modeling, including once again from hydrodynamics (Morgan Reference Morgan, Morgan and Morrison1999). His liking for diagrams taken from mechanics led him, for example, to show mechanical balances to picture the “equation of exchange mechanically expressed between money and commodities, the velocity of circulation of money, and especially the yearly balances in the US economy between 1896 and 1911 (1912, pp. 157–160; 177–180; 254). Again, he illustrated Gresham’s law on bad money driving good money, and the circulation of money in a bimetallic market, by a hydrodynamic model of three linkable reservoirs of water (pp. 225–230; 235).

Especially from the 1920s onwards, Fisher was a prominent advocate of index numbers in economics. From mathematics he required only arithmetic, common algebra, some basic statistics, and lots of diagrams (Boumans Reference Boumans2001).

Fisher discussed “The application of mathematics to the social sciences” in detail in the seventh Josiah Willard Gibbs Lecture, which he read in 1929 to a joint session of the American Mathematical Society and the American Association for the Advancement of Science (1930). He praised Gibbs hugely, especially for three characteristics he had himself followed: the aim for generality in theorizing, the use of geometric tools (including the promotion of vector algebra), and the usefulness of inequalities (as in thermodynamics). Regarding economics, he again praised the contributions of Cournot, for him still not appreciated enough. Contrary to Marshall’s doubts long before, he welcomed lengthy lines of reasoning, and noted various recent developments in which statistics were playing an important role; he also affirmed the importance of the calculus. However, in contrast to the early enthusiasm of his doctorate, about the influence of mechanics (or of thermodynamics, Gibbs’ speciality) he uttered—not a single word.

14. The Place of Evans

Our final major economist is the American Griffith C. Evans (1897–1973), who wrote a doctorate in the early 1910s on integral equations under Italian mathematician Vito Volterra (Weintraub Reference Weintraub2002, ch. 2). Upon appointment as a founder staff member in 1912 at the Rice Institute in Texas (Rice University from 1916), he pursued a notable career not only in mathematical economics but also in potential theory (Morrey Reference Morrey1983). We consider a textbook in English and a short monograph in French (Evans Reference Evans1930, Reference Evans1932), published shortly before he moved to Berkeley and building on papers published from the early 1920s onwards.

Evans followed Fisher in preferring to emphasize numerically measurable quantities in economics, especially labor, production, and financial factors such as money, taxation, and price, and related theories such as index numbers. His discussion of utility came only two-thirds of the way through his textbook, and was critical of the concept as handled by his predecessors, including the claim to establish indifference curves and surfaces (1930, ch. 11). Working out from marginality, his account of economic equilibrium was dominated by a consideration of demand and selling prices. He transited from static to dynamic equilibrium (ch. 14) by including not only prices but also their rates of change over time, an approach for which he cited Pareto (Reference Pareto1901). Assuming linearity, his functions of price took the form

(4)

where the constants were given positive, zero, or negative values according to economic circumstances. Assuming that the quantity produced of a commodity equaled that of its demand gave him a first-order differential equation, which was easily solved. He could then proceed to find an expression for profit at any given time, with its integral representing total profit over some specified period.

Apart from his attitude to utility and choice of topics, so far Evans has more or less followed the normal mathematical concerns of his predecessors. His use of the calculus of variations to maximize the total profit function was still unusual in mathematical economics, and he devoted several pages to explaining the subject (1930, pp. 167–173; 1932, pp. 35–41). But his place at a leading edge of mathematical economics shows itself in the final chapters of his book, where he considered the time lag between production and sale (which Walras had set to zero!), discontinuous production functions, and the possibility that the demand price offered at any given time depended on all previous prices over some specified period (1930, pp. 154–163). These considerations allowed him to deploy the skills of modern mathematical analysis that he had learnt from Volterra: integral equations, functional analysis, and measure theory, the last being the theory of the integral that had been proposed in the 1900s as an extension of Cauchy’s version described in section 4 (Jahnke Reference Jahnke2003, chs. 9, 13). This was the mathematical way forward, to replace the simultaneous equations of Walras and his kin: as he queried to his French audience, “how can one resolve 5209 [linear] equations concerned with 5209 unknowns?” (1932, p. 5). But even then he was skeptical of the progress made in mathematical economics: “one must confess that these general economic theories are as yet useful only for the indication of particular instructive cases” (1932, p. 54). The quotation at the head of this section expresses a similar forward-looking sentiment. To this we now turn.

15. On the Rise of “Dynamic Economics”

Evans was averse to using mechanics; in particular, while he named his method of transiting from static to dynamic equilibrium “dynamic economics,” he felt that the name “is not altogether happy on account of the special reference to the subject of mechanics and on account of the artificiality of method which a loose analysis suggests” (Evans Reference Evans1930, 143). He was alluding to his former student C.F. Roos (1901–1958), who had already taken up using derivatives of the price function and the calculus of variations to form what he called “dynamical economics” (Roos Reference Roos1927). Despite the legitimacy of Evans’ complaint about the name, Roos continued to use it for his theory, though it had seemingly little or no analogy content with mechanics. An important new context arose in 1934, when he was in his early thirties and based at the University of Chicago, for he was appointed the founder director of the Cowles Commission for Research in Economics. It published as its first monograph his book (1934) on dynamic economics, where he deployed both the calculus of variations and integral equations.Footnote ²⁴

Thanks to this book and the efforts of others, dynamic economics has become a part of economics in its own right (Weintraub Reference Weintraub1991, ch. 2). Yet, the analogy content with mechanics could increase at times. For example, in an essay of 1933 on “Propagation problems and impulse problems in dynamic economics,” Norwegian economist Ragnar Frisch explicitly likened economic cycles to free oscillations of the simple pendulum. His insight was to distinguish the “propagation” of the oscillation itself, which belonged to the pendulum and might well be regular and dampen down over time, from the “impulse” of the external disturbing force, which could be occasional (1933, pp. 171–173). Assuming linearity of relationships, he formed and solved first-order differential equations to represent the rate of change over time of economic variables such as total capital stock and the production of goods, with solutions representing economic propagation by their damped oscillations (pp. 177–181). He also considered the pendulum moving under constraints of friction and subject to “erratic shocks,” and interpreted the solutions of these differential equations in terms of fluid motion (pp. 197–205). His paper was significant for its generality and for the solutions as examples of time series analysis;Footnote ²⁵ but its mechanical analogies are rather weak, and the usual reservations about relying upon mechanics apply. A similar judgment applies to those students of economic dynamics of that time who relied on a mechanical background for their work; for example, Italian followers of Pareto (Pomini and Tusset Reference Pomini and Tusset2009).

16. On the Rise of Statistics

Another principal concern of the Cowles Commission was the links between economics and probability theory and especially mathematical statistics. While those topics had received some attention, largely in contexts that we would regard as macroeconomic (Krüger and others 1987, chs. 7–8 more than ch. 6), Edgeworth had been unique in the strength of his advocacy. More typical was the sincere but patchy support of Jevons (1874) and elsewhere (S. Stigler Reference Stigler1982). Again, Section F (“Economic Science and Statistics”) of the British Association for the Advancement of Science had been founded in Britain in 1833 (by Whewell and others) to raise the scientific status of two fledgling disciplines, with the emulation of astronomy as a target; however, as the quotations at the head of this article suggest, for many decades the quality of the work in each discipline was often poor, and the intersection between them remained very modest. In particular, in 1877 Galton tried to have Section F shut down for want of scientific quality (Henderson Reference Henderson and Mirowski1994). A story of similar modesty attends the Statistical Society of London, which had been founded in 1834 (Whewell involved again) and included economics in its remit (Henderson Reference Henderson and Rima1995).

However, as Fisher noted, the status of mathematical statistics and probability theory in economics had eventually improved, especially from the 1910s onwards. An early interest was detecting and analyzing economic cycles of various kinds; they have a special place in economics, for they do not exhibit equilibrium as normally understood, though they may show recurrence. A major student of cycles was the American, Henry L. Moore (1869–1958), who passed his career at Columbia University in New York. Anxious to determine their causes as well as their character, his initial cases came from the economics of agriculture, where he sought to link cycles of rainfall with cycles of crop production. To this end he represented their periodicity in terms of Fourier series, and analyzed the pertaining data by copying the methods used in celestial mechanics to fit such series to the data on sunspot cycles.Footnote ²⁶

Moore saw his work as an example of the need for reform of economics in general. Here are a few sentences from a fine passage:

Moore was advocating the statistical part of econometrics, the branch of applied economics that analyzes quantitative data pertaining to some context in order to determine properties and relationships, such as cause, that lie behind the data, not only the handling and especially the classification of data to which the Statistical Society had committed itself.Footnote ²⁷ Frisch was to introduce the word “econometrics” in a paper (1926) to denote the activity “intermediate between mathematics, statistics and economics” in which “abstract laws were to be subject to “experimental and numerical verification” of all kinds; thereby he hoped to fulfil the aspirations of Jevons (Reference Jevons1879, ch. 4) to determine the “numerical determination of the laws of utility.” In the next part of his paper, Frisch presented a rather axiomatic treatment of an individual’s choices within an opportunity space, although he related his theory to data on parameters such as sales figures. The word was used again around 1930 when Frisch, Fisher, and Roos founded the Econometric Society in the USA, with Fisher as founder president; their journal Econometrica started in 1933 under Frisch’s editorship, and the first number included a survey of Cournot (Roy Reference Roy1933), not an obvious figure for a journal in this field!

Thereafter econometrics grew in importance, though not at rapidly as may be thought (Rima Reference Rima and Rima1995b). Statistical methods were very prominent, including in modeling (Morgan Reference Morgan1990, chs. 2, 4).Footnote ²⁸ For me it is a great mystery that probability theory and especially mathematical statistics were such latecomers to the mathematical sciences (S. Stigler Reference Stigler1986; Porter Reference Porter1986).

17. From the Equilibrium of Mechanics to the Stability of Dynamical Systems

Apart from Frisch (Reference Frisch1933), these new initiatives in dynamic economics and in mathematical statistics took economics away from Walrasian principles (Jolink Reference Jolink, Backhaus and Maks2006) and taints of mechanics. However, certain advances in dynamics itself from the late nineteenth century onwards led to another range of possibilities for emulating mechanics, or at least analogizing with it; they were taken up, especially in the USA and from the 1920s. I sketch briefly some principal features.Footnote ²⁹

17.1 Poincaré and Lyapunov. In a study of 1889 of the three-body problem in celestial mechanics, and then in a lecture course of 1892 on celestial mechanics, Henri Poincaré brought in a new approach to the qualitative solutions of differential equations, including recurrence of dynamical systems to previous nearby states of equilibrium. Partly under this influence, Russian mathematician A.N. Lyapunov published as Ob′Shchaya Zadacha Ob′Ustoichivosti Dvizheniya [The General Problem of Stability of Motion] (1892) his doctoral dissertation, a lengthy essay on types of dynamic stability and their bearing upon equilibrium. Stability concerned situations where small oscillations from a state of equilibrium damp down and eventually return to it. Among physical examples were the near-recurrence of the orbit of a mass-point moving around a source of central force (Poincaré’s exercise of 1889) and the small disturbance of rotating ellipsoids from an equilibriate shape (Poincaré again, in planetary mechanics). Mathematically the main issue was solutions to ordinary differential equations or systems of them, for which Lyapunov proposed two “methods.” One of them found power-series solutions of the equations that ensured that the corresponding unperturbed motions were stable; the other extended known stability theory to non-conservative systems of forces by finding particular kinds of differentiable function, now named after him, that satisfied specific mathematical conditions and thereby guaranteed some kind of stability.

The lack of knowledge of Russian prevented rapid circulation of Lyapunov’s work abroad. In particular, when Emile Picard reported at length on the topic in the first edition of his treatise on differential equations, he drew on only Poincaré’s contributions (1895–1896, ch. 8); so Lyapunov promptly sent Picard a long letter in French describing his own work (Mawhin Reference Mawhin1994, pp. 34–43). Picard mentioned Lyapunov in a trio of popular lectures (1899) delivered in the USA;Footnote ³⁰ eventually he added a passage in the second edition of his treatise (1908, pp. 199–205), mainly related to the second method, where he cited a later paper that Lyapunov had written in French and published in France (Lyapunov Reference Lyapunov1897). In addition, around that time, a French translation of the original essay appeared, checked by the author and containing an additional note (Lyapunov Reference Lyapunov1907). Thereafter, both Lyapunov’s and Poincaré’s work inspired several leading mathematicians to an analysis of dynamical systems, which became a significant topic in mathematics (Kolmogorov and Yushkevich Reference Kolmogorov and Yushkevich1998, pp. 173–196). The impact in the USA was especially marked, at first due to G.D. Birkhoff.

A further mathematical source for economists was a dispute over methods of the modeling of predators and preys in mathematical biology that took place in the late 1920s between Volterra and American statistician Alfred J. Lotka (Israel and Millán Gasca Reference Israel and Millán Gasca2002): while the context was not mechanics, Lotka was much influenced by energetics physics (line PE in section 3). The third edition of Picard’s volume also appeared at that time, as Traité d’Analyse (1928).

17.2 Economic stability and equilibrium. The kinds of dynamical situation envisioned by these new theories were more similar to the volatility and unpredictability of real-life economic situations than, for example, Pareto’s vision of mass-point shoppers working their ways pseudo-dynamically around a market under the constraints of d’Alembert’s principle. A special attraction was the sufficient conditions established by Lyapunov for a small disturbance from some kind of equilibrium to return, stably, to it. So some economists gradually began to use or adapt these various theories, especially from the 1930s onwards (Weintraub Reference Weintraub1991, chs. 2–3).

An important later figure was Paul Samuelson, whose work began to appear in the late 1930s; we note here his monograph Foundations of Economic Analysis (1947), in which he was especially concerned with appropriate forms of equilibrium. An eclectic collector of theories from across and outside economics as sources of analogizing, through Birkhoff he learnt of Lyapunov and dynamical systems; through Wilson, his teacher at Harvard, he came across Lotka, thermodynamics, and the Le Chatelier principle in chemical equilibrium. He also emulated the correspondence principle in quantum mechanics. He even opined that analogizing from statics and dynamics “is a fruitful and suggestive approach [that] cannot be doubted” (1947, p. 311), which was not a common view by then, at least as far as invocation of theories in mechanics is concerned! Indeed, he added here that “it is too much to suppose that very many economists have the technical knowledge necessary to handle the formal properties of analytical mechanics. Consequently they became bogged down in the search for economic concepts corresponding to mass, energy, inertia, momentum, force and space,” perhaps with Pareto and Fisher in mind. Unsurprisingly, he did not revive mechanics; surprisingly, he did not cite Evans.Footnote ³¹

Samuelson’s enthusiasm for analogies was not just a personal inclination. On the title page of his monograph, he quoted from Gibbs the view, popular among mathematicians, that mathematics is a language, implying that it transcends any analogies that may be invoked. But this position is rather naïve: mathematics has languages, quite a lot of them, and always has been so; thus analogies cannot be taken as purely syntactical.

In addition, Samuelson had found a grounding for the use of analogies. Initially inspired by the mathematical logic of Giuseppe Peano, American mathematician E.H. Moore had sought from the 1900s onwards to develop “general analysis,” an indeed general theory that would incorporate as many linear theories in mathematical analysis as possible (Siegmund-Schultze Reference Siegmund-Schultze1998). He was motivated by the metaphysical assumption that if two theories exhibit analogies, then there exists a more general theory of which they are special cases. A statement of Moore’s assumption formed the opening paragraph of Samuelson’s monograph (1947, p. 1).

During the 1950s, the Russian-born mathematician, Solomon Lefschetz, who had taken courses with Picard in Paris in the early 1900s before moving to the USA, promoted Lyapunov’s work in connection with solving non-linear differential equations representing dynamical systems (Dalmedico Reference Dalmedico1994); economics followed suit once again to some extent.Footnote ³² Further study of dynamical systems has led on to catastrophe theory, chaos theory, and the analysis of attractors (that is, stable systems such as points of equilibrium or periodic orbits) (Aubin and Dahan Dalmedico Reference Aubin and Dalmedico2002), and again some economists have taken note while mindful of the limited analogy content (Weintraub Reference Weintraub1991, chs. 4–6).

18. The Persistence of Equilibrium in the Growing Federation of Economics

In addition to the rise of econometrics and dynamic economics, and the use of dynamical systems, other concurrent major developments in mathematical economics took place from the 1920s onwards; mechanics was not prominent, but equilibrium was still a staple.Footnote ³³ For example, macroeconomics gradually developed as a branch in its own right, taking over existing concerns such as business cycles unemployment, recession, and monetary policy, and developing new ones. Major figures include Fisher, Frisch, and J.M. Keynes: indeed, in the paper on cycles discussed in section 15, Frisch clearly innovated the general distinction “between two types of analyses: the micro-dynamic and the macro-dynamic types” (1933, p. 172). This is the difference of category between “microeconomics” and “macroeconomics,” to use the names that were introduced soon after the Second World War.

In addition, game theory arrived in the late 1920s, especially with John von Neumann,Footnote ³⁴ after quite a variety of partial anticipations and special cases.Footnote ³⁵ One aim was to provide a new basis to several notions and parts of microeconomics, including equilibrium; in their book The Theory of Games and Economic Behavior, von Neumann and Oskar Morgenstern deplored “too much calculus” and doubted the efficacy of notions such as utility and indifference curves, and instead extolled imputations and preferences among game players, though they admitted that their theory was “thoroughly static” (1944, esp. ch. 1). A further boost to game theory came after the Second World War when von Neumann and George Dantzig connected it up with linear programming;Footnote ³⁶ both theories have been applied to various parts of mechanics, seemingly the only notable case of converse influence, from economics.

In his game theory von Neumann drew upon fixed-point theorems from topology, which were used later to specify general types of economic equilibrium. He also participated in the late 1920s in the spread of David Hilbert’s program of “metamathematics” for the foundations of mathematics, which was to have an impact on economics during the 1950s.Footnote ³⁷ The rise of these and other branches of economics eliminated the dominance of traditional microeconomics, and so reduced the roles in it for mechanics; for some, game theory was an especially attractive alternative. Hence, those economists who wanted to pursue Walras’ vision of general equilibrium (for example, Samuelson) realized that “general” had to carry a somewhat more modest connotation than that adopted in the political economy of Walras’ time.

However, equilibrium remains as a major category in economics (Walker Reference Walker2000), and textbooks are still devoted to it. Usually specified in terms either of static-like states, or of asymptotically dynamic ones to which a disturbed situation will stably return (Weintraub Reference Weintraub2002, ch. 7), conditions for its existence and uniqueness have continued to receive much attention, often larded with axiomatization and topology (Ingrao and Israel Reference Ingrao and Israel1990, chs. 11–13). But its status among economists varies between habitual assumption to explicit rejection (Mosini Reference Mosini2007, pt. 3). In particular, somewhat in echo (though presumably not influence) of the worry long ago of the French engineers about the status of impact in mechanics (section 2), some economists such as Samuelson tried to ground equilibrium economics in disequilibria, where variables such as excess demand are on a par with the rest and arbitraging is an important process. In order to establish conditions for local and global stability of equilibria, they have used Lyapunov functions and related theories rather than traditional Walrasian methods of groping (Fisher Reference Fisher1983, esp. ch. 2).

19. Appraising the Influence of Mechanics

As the later Fisher and Evans exemplify, the emulation of classical mechanics and physics died down after Pareto; at least, I have not noticed any other notable emulator or analogizer, not even Samuelson, although some modeling was practiced. So a few summary observations about the role of mechanics are in order.

19.1 Tableau. As a Paretian first approximation, the upper half of Table 1 indicates the three possible positions normally taken by our eight main economists on the role of mechanics. Apart from the difference between Fisher/early and Fisher/later, their positions did not seem to change substantially during their careers. When a position was held with some fervor, the customary “+” is replaced by “++.” The lower half of the table indicates their attitudes to involving physics (not necessarily energetics), encouraging mathematical statistics (not just using data), and deploying the multivariate calculus and the calculus of variations; surprisingly, only Edgeworth and Evans (and also Roos) advocated the latter topic, despite the importance of optimization in economics.

Table 1. Profile of interests of the economists

19.2 The lure of mechanics.

It was very understandable that, during the period studied, economists such as Jevons, Walras, Pareto, and the early Fisher saw the contrast between the massive and long-running stumbling of their subject and the massive and long-running success of mechanics (and also of physics), and were seduced into emulation. They hoped to apply mechanics to much of the economics of their time, which, in modern terms, were largely microeconomic contexts in which equilibria of some sorts played major roles. Their intentions to use mechanics were quite sincere and ambitious even if they were not often stated explicitly.

The efforts of the economists exhibit nicely the tension between the desire to develop at least some theory and the quality of its representation of the phenomena to which it refers, even after desimplification—an issue of which the history of mechanics is also rich in cases! A fine example of this tension is a serious disagreement around 1890 on the theory of exchange between Edgeworth and Walras, who was supported by the young Polish mathematician, Ladislaus von Bortkievicz (Marchionatti Reference Marchionatti2007). Some of the issues lay within economics itself; for example, the representation of capital, the efficacy of the method of groping, and the legitimacy of the notion of ideal entrepreneur who neither gains nor loses. But Edgeworth queried the usefulness of the mechanics behind Walras’ pure theory rational à la Poinsot, surely too simplified to capture the complications of economic activity that was his own main concern; as usual, he was also skeptical about proving the uniqueness of a equilibriate state, or that groping methods would determine it (them?). Differences arose over a new theorem by Walras on the change in (static?) equilibrium, and conditions for maximizing utility, in a trading situation following the influx of new capital and consequent production (Walras Reference Walras1998, lectures 26–27); understandably, Edgeworth (Reference Edgeworth1891) was surprised that Walras had not formulated the theory in terms of the calculus of variations.

Analytical mechanics still has some economist fans; for example from several, in the use of Hamiltonians.Footnote ³⁸ But appeals to the details of mechanics or to classical physics as such seem to have become rare, and in many contexts non-existent: reduction is out, and analogizing takes mild forms, of limited content, maybe linked to modeling. Similarly, economists studying dynamical systems do not commit themselves to the dynamics (and so resemble those mathematicians in the field who just enjoy playing with some difficult differential equations!). The continuing common talk in economics of “equilibrium,” “force,” “mechanism,” “elasticity,” “friction,” “statics,” “dynamics,” and “work” is rhetorical, not tied to appropriate parts of mechanics or physics.Footnote ³⁹ Again, much anxiety is currently being expressed about the “whirlwinds” circling the world of banking and finance, but nobody is turning to meteorology for enlightenment.

19.3 The trap of mechanics.

My skeptical position about the place, or even the need, for mechanics in economics was not formed from wisdom garnered long after the events; for it was adopted by Cournot, Edgeworth, the later Fisher (and also Jenkin), and especially disemulaters Marshall and Evans, who regarded the efforts to emulate mechanics as mistaken: the outcomes were at most analogies, mere metaphors, of modest content. No enthusiast made a Kelvin-like success of analogizing; on the contrary, when Pareto and the early Fisher drew up their lists of correspondences between mechanical and economic notions, some of their assignments were dubious, as Samuelson noted. On the important business of bridging the gap between static and dynamic equilibria, Edgeworth and Walras did not sort it out in their dispute: Pareto tried by implicitly imitating analysis of the motion of mass-points under constraints as described by d’Alembert’s principle, but out of sight goes most real economic activity. Any small nuggets of mechanical/economic wisdom were gained by ignoring most of the personal and social events and processes adhering to economic transactions, for which there are no strong analogies either in mechanics or physics. On the contrary, two major disanalogies come to mind, both for classical mechanics and dynamical systems.

Firstly, what corresponds in economics, or in any social theory, to the steady and uniform action of gravity? Gravity dictates many aspects of the activities of economic agents as they move and work on (or above or under) the surface of the Earth, but it rarely rules over their economic choices or decisions. Prices may go “up” or “down,” but not in a gravitational way.Footnote ⁴⁰ The planets move around the Sun, or rather around the center of gravity of the planetary system; but where is the center of gravity of an economic situation?

The disanalogy is exposed in the following scenario. I make a table upon which to place my oranges in the market. The table remains in static equilibrium under gravity when loaded with oranges, although the paradox of statics (sections 7, 12) shows that determining the loadings on its four legs is not straightforward. By contrast, when I sell you six oranges, the state of economic equilibrium in the market could change at once; for now I have six less oranges to sell and may be tempted to raise the price, and you decide that you have finished shopping for the day and leave the market. Meanwhile, the table continues to be in static equilibrium under gravity, even though the distribution of oranges upon it has changed again. When I go home, the masses of my unsold oranges remain the same, but their utility is quite different. As economist W.W. Carlile put it more generally in an excellent extensive review of the disanalogies, “The things of physics, we see thus, take their names from their outward qualities, their colour, texture, atomic weight, and so on. The things of economics, on the other hand, take their names from our more or less transient mental attitudes with regard to them” (1904, pp. 41–42).

Secondly, one aim of developing an economic theory, felt by Walras, among others, is that it should be learnt by people who then behave better as economic agents; this feedback aspect may give an economic theory a normative cast, or at least tie it to ethical principles such as prices at equilibrium being “fair,” even though the mathematical treatment itself is ethically neutral (Walker Reference Walker1984). By contrast, mechanicians were not trying to teach solid bodies how to fall better. The closest analogue to feedback is servomechanisms, such as governors on steam engines, that “teach” the apparatus to which they are attached not to operate too quickly; more sophisticated examples can be built in to electrical and electronic equipment by suitable circuit design. But none of these arrangements begins to compare with the fast-developing interactions evident in human discourse in social situations such as market trading.

The modesty of the link between mechanics and economics is evident also in the intermittent developments of topics where both subjects played a role. One topic is locational equilibrium, which had a bright start in 1829 (Franksen and Grattan-Guinness Reference Franksen and Grattan-Guinness1989) but then slumbered until late in the century (Smith Reference Smith1981, ch. 4). Another is linear programming, of which we noted one episode in section 7 in connection with Cournot; it started and stopped several times over 150 years before its rapid rise in the 1940s, and mechanics had featured in only a few of the motivations, such as with Cournot (Grattan-Guinness Reference Grattan-Guinness, Knobloch and Rowe1994b). A third topic is the economic efficiency of machines (section 2); apart from some possible influence on Walras, it did not feature prominently in the concern of our economists. But maybe as a result, the following possible analogy went largely begging.

19.4 An irony in mechanics.

We have seen that several economists made analogies with hydrodynamics; this practice continued up to the Philips machine of the late 1940s, which at least surpassed its predecessors in being three-dimensional, one more than usual (Morgan and Boumans Reference Morgan, Boumans, De Chadarevian and Hopwood2004). These analogies assumed that the water would move in a well-behaved way: Fisher’s many diagrams of connecting reservoirs, resembling flows of money in a bimetallic economy, show it quite explicitly. But the parallels with hydraulics, which Walras made once (section 9), are much more appealing: change a few nouns appropriately in Forchheimer’s lament above, and one has the thinking economist’s lament about his subject!

This analogy relates to the use made in economics of modeling rather than of applied mathematics (subsection 5.3); namely, the long tradition of modeling in technology, where engineers have long been aware of the extreme complications that they treat. While they did not speak of modeling, their procedures, such as (de)simplification, strongly resemble it. One reason for their liking for energy–work mechanics mentioned in section 4 was to avoid many of these complications and just determine energy levels instead. The motions of large bodies of water form excellent instances of their difficulties. For example, the passage of water out of a lock into a canal is greatly dependent on the size, shape, and location of the sluice gate, and the behavior around especially a small gate can exhibit complicated effects such as evection and cavitation.

Marshall once used a similar kind of analogizing in 1895 when he claimed in the third edition of his Principles that “The science of the tides presents many close analogies to economics,” on the grounds that complications and uncertainties such as the weather can beset them both. But again disanalogies come to mind: both kinds of phenomena may indeed be complicated, but the reasons thereof are quite different, for gravity is involved only in the action of tides. Indeed, in the fifth edition of 1907, he watered down his claim (as it were) to an assertion that analogies for economics with tidal theory were better than those with celestial mechanics.Footnote ⁴¹

A similar case is the analysis of the laws governing the different velocities taken by different parts of a large body of a viscous fluid, such as a river or an oil pipeline: under reasonable simplifications, fluid dynamicists formed the Navier–Stokes equations (Darrigol Reference Darrigol2002), of which both hydraulic engineers and economists later became aware. In an excellent review of the use made in economics of this theory, and also of dynamical systems, Bausor (Reference Bausor and Mirowski1994, p. 121) notes that viscous flow is well represented by these equations, and why this is so; but “Nothing even remotely analogous graces economic dynamics.” In similar vein he (1995) nicely stresses the differences between the use of Lyapunov functions in economics and in thermodynamics.

19.5 An irony in philosophy. Mechanics exhibits a rich and varied panoply of philosophical problems (Grattan-Guinness Reference Grattan-Guinness, Loewe, Peckhaus and Räsch2006b); so it is not surprising that some of the connections and influences between mechanics and economics are of a philosophical character. One concerns rigor, especially the prominence often given to deduction from premises in mechanics, and also in some other parts of mathematics (subsection 5.1); theories in economics were frequently less thoroughly deductive than their practitioners claimed (Hutchinson Reference Hutchinson1998). Another common factor is generality: contested between the energy workers and the analyticals in mechanics over phenomena involving impact, very important for Walras and von Neumann, for example, and a matter of skepticism for Evans.

A particularly important issue is determinism versus indeterminism; that is, whether the future states of a process are predestined in the specification of the present state. This formulation of the difference is objective; there are subjective versions of these positions, concerning the uncertainty or incompleteness of an individual’s knowledge of a situation that, however, might be determined. Especially during the second half of the nineteenth century, some figures held that at least parts of physics, including mechanics, were deterministic, and only an individual of sufficient mental capacity could know it all (Laplace’s demon). In particular, the irreversibility of heat processes, especially when construed within energetics physics, was felt to be conducive to predestination (Brush Reference Brush1967; Reference Brush1976, chs. 13–14). Within mechanics itself, the analytic tradition was regarded as teleological, mainly for its heavy use of the calculus of variations. But by the 1920s, for various reasons, of which some are internal to the development of physics itself (Krüger and others Reference Krüger1989, pt. 6), determinism was much less popular (Wallace Reference Wallace1974, chs. 3–5).

Modeling was and is generally sympathetic to indeterminism, in that it allowed its practitioners a range of possible models. Indeed, another analogy comes to mind. The difference between applied mathematics and modeling (subsection 5.3) is similar to that in (the history of) management science between deterministic strategies developed by F.W. Taylor from around 1900, in which rather impersonal hierarchical structures were applied to the workplace, and the indeterministic practices followed forty years later in fields such as operational research and linear programming, in which far more democratic working conditions obtained and the tasks set could be questioned and changed.Footnote ⁴²

The philosophical issues concerning determinism versus indeterminism in economics are nicely laid out in Katsinelinboigen (Reference Katsinelinboigen1992). Among our octet of economists, one of the most philosophically sensitive was Cournot, who, however, seems not to have explored this issue in his economics. In his philosophy of probability, he admitted both objective and subjective interpretations and allowed both that an effect had to have causes and that causes could be mutually independent (Cournot Reference Cournot1843, esp. ch. 4), which leaves ajar the indeterminists’ door that Laplace had closed. Another economist philosopher was Jevons, who explicitly allowed for an uncertain future (1970, pp. 99–100); however, he seems to have had in mind only the subjective perception of uncertainty, akin to personal risk, rather than the objective construal. Edgeworth emphasized the freedom of the economic agent to change his strategies, albeit under “gross force” (1881, pp. 16–24). Of the later generation, Fisher comes over as indeterministic, especially when he loosened the bonds of mechanics in his later work. The rise of statistics encouraged indeterminism in that it raised the status of uncertainty; on the other hand, the supporters of statistical inference, perhaps including Moore, may have done so with a deterministic ring. What is a statistical regularity in this context, as Lionel Robbins was to query in the 1930s (Sutton Reference Sutton2000, pp. 13–23)?

The development of dynamical systems is also double edged. It advocated recurrence rather than repetition and, indeed, led on eventually to chaos theory, but its stability theorems on the return to equilibrium after disturbance could be taken as inevitabilities, with implications for the interpretation of business cycles.

My impression is that the issue of indeterminism and determinism was not deeply considered either by our economists or so far by their historians. (The same can be said of mechanicians and their historians, incidentally.) The irony is that, assuming that the physical world is open to influence by human decisions and their consequences, the deterministic interpretation of physical phenomena is flawed. In particular, in any context, including economic ones, the outcome of a competition cannot be determined in advance, as each competitor needs to know the first moves of all the others. Again, the predictions of future states require that the initial state be set up exactly, so that it can be repeated; but this is humanly impossible (Popper Reference Popper1951; Reference Popper1982, chs. 1–2). The history of the place of (in)determinism in neoclassical economics needs more attention; whether they will be tackled, who knows?

20. Which Mathematics for Economics?

20.1 The lure and trap of measurement.

In mechanics (and physics) measurement is often extolled; one is accustomed to work with precise numerical values (of velocity, density, and so on). Although it can be difficult to determine values to the accuracy required, especially in dynamics (Roche Reference Roche1998), this feature must have formed part of the attraction of mechanics for neoclassical economists (G.J. Stigler Reference Stigler1950). For economics also works with mounds of numbers, and values have to be assigned to many variables and parameters such as price, rent, and tax; so maybe we should favor mathematicized theories involving economic variables and parameters that are numerically precisible and/or statistically assessable. It is very striking that when Frisch introduced the word “econometrics” in 1926, he immediately cited Jevons (section 16), given the special difficulties concerning measurement that attended the notion of utility in both classical and neoclassical economics (Peart Reference Peart and Rima1995). Indeed, measurement, including the graphical representation of data, enjoyed a high status in many contexts in economics long before econometrics was individuated (Rima Reference Rima1995a). Among other contexts, Marshall raised the status of money and pricing, as a means of linking utility with demand.

But in various cases the reliance on numerical values can be queried; for example, what is the value of my utility function right now in the market? Pigou (Reference Pigou1951) nicely pointed out the difficulties in trying numerically to specify satisfaction in welfare economics, quoting in support a passage from Russell (Reference Russell1903, pp. 182–183) on the distinction between extensive and intensive magnitudes. Another example is indifference curves and surfaces, the analogues in economics of mechanical equipotentiality. A critic such as Evans could ask: how can you be confident that two points on such a surface have the same utility when that notion cannot be precisely measured in the first place? The reply from Edgeworth might be: no, I cannot know utility as an extensive magnitude, but I can know intensively equiutility, that the two corresponding economic situations attract me equally.

So we might recognize with Pareto that it may be sufficient to work with intensive magnitudes and simple-order some economic variables by choosing and comparing preferences without being forced into numerical calculation; one can work with a law of diminishing returns, for example, without having to calibrate it. The theory of revealed preferences for demand, promoted by Samuelson and others from the late 1930s, assumes only ordinal utility and tries to do away with indifference curves or surfaces.Footnote ⁴³ We may even dispense with continuity; in some circumstances nature can make jumps, to contradict the motto that Marshall placed at the head of his Principles.

20.2 The large question.

This skeptical opinion, uttered by the founder director of the London School of Economics, followed several lines of enthusiasm, maybe inspired by reading Marshall, about influences from biology and jurisprudence. Fifty years later Morgenstern was of like mind, in a lecture on the limitations of mathematics in economics delivered to a symposium on the “utility and inutility” of mathematics in the social sciences in general: “As far as the use of mathematics in economics is concerned, there is an abundance of formulas where such are not needed. They are frequently introduced, one fears, in order to show off” (1963, p. 18). Similarly, Mirowski (Reference Mirowski1988) argues for “protecting economics from science” tout court.

My position, not so extreme, is that we should protect economics from attractive but weak analogies with more established sciences. One notes with Velupillai (Reference Velupillai2005) that much mathematics was devised for use in the physical sciences, and so may not be effective elsewhere, such as in the social sciences. This point raises our large question: do we need different kinds of mathematics for the different sciences? It raises fundamental issues about applied mathematics: the various contexts for which it is created and in which it is used, and its (in)effectiveness there (Grattan-Guinness Reference Grattan-Guinness2008a). The question was beautifully posed by Lesk (Reference Lesk2000) in connection with the gap between the physical and the life sciences in contexts such as developing complexity theory. For him the answer is that the various non-physical sciences may well need some of their own mathematics, different for each kind of science as well as from that used the physical sciences. Morgenstern advocated such a position in (1963), and had hinted at game theory as an example in von Neumann and Morgenstern (Reference Von Neumann and Morgenstern1944, pp. 3–4).

Economists have made a habit of going into the mathematicians’ supermarket and choosing products off the shelf; they could be more radical and visit the design laboratories to examine the vast available array of notions and theories to create new products, or even form entirely new notions and theories. Regarding mathematical and physical notions already on the shelf, I predict a good future in economics for several ubiquitous ones, whether they are allied to numbers or not: in particular, time, (dis)continuity, convexity, optimization, linearity and especially non-linearity, (in)equality, (dis-)equilibrium, and causality. Among available theories, it seems rational to preserve at least some uses of arithmetic, common algebra, vector and linear algebra, the calculus and mathematical analysis, game theory and (non-)linear programming, probability theory and mathematical statistics. Add in solutions of difference equations, an established topic that mathematical economics encouraged: Samuelson’s monograph is noteworthy for an extensive appendix on them (1947, pp. 380–439). Further in this direction lie finite element methods, a branch of numerical analysis in which approximate solutions of (especially) partial differential or integral equations are obtained (Zienkiewicz, Taylor, and Zhu Reference Zienkiewicz, Taylor and Zhu2005); their current use in economics, as part of economic dynamics, may well be a worthwhile imitation (but not emulation) of mechanics and mathematical physics.

This last topic makes much use of computer power, which is another feature of mathematical economics that should be profitably durable. For example, thanks to numerical linear algebra, we can answer, positively, Evans’ rhetorical question about handling 5209 equations in 5209 unknowns (section 14). Such uses of computer power are preferable to the current fashion of using the computer to hunt mindlessly for correlations among vast hoards of economic parameters, a good example of the abuse of numbers in economics if statistical significance is automatically linked to economic significance.Footnote ⁴⁴

Regarding non-numerical theories, in some contexts it might be more realistic, although more challenging mathematically, to replace calculus-style maximizing over the real numbers by bounded rationality and satisficing, which was launched in the 1950s in connection with economics by Herbert Simon (Husseini Reference Husseini1990) and enjoys some currency in various disciplines (Byron Reference Byron2004, esp. ch. 11). Again, topology related to fixed-point theorems is quite influential at present.

Regarding testability, theories in economics, whether mathematical or not, do not emulate mechanics or mathematical physics enough in delivering falsifiable predictions involving at least intensive magnitudes and, if possible, numerical ones (Goldfarb Reference Goldfarb and Rima1995). At times economics seems to exhibit Maier’s law in psychology: if the data do not fit the theory, then the data must be discarded. Testing models of a theory is necessary but not sufficient.

Apart from this aspect, appealing to mechanics did and does not have much to offer to economics, so its revival is not to be encouraged. But confident prediction is hazardous; investments in analogies and emulations can go up as well as down!