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ALFRED MARSHALL AND MATHEMATICS
Marco Dardi (*)
1. Marshall the mathematical economist and Marshall the methodologist on the use of mathematics
in economics should be kept separate. The former was active approximately until the early editions
of the Principles of Economics, after which mathematics as a heuristic tool was renounced and
remained solely as a forma mentis, in the background. The latter became more and more prominent
as the former faded away, but the methodological criteria he elaborated — skilfully condensed in
frequently quoted passages of the correspondence — do not seem to convey the gist of his previous
work as a mathematical economist. Well-known texts such as the letters to Hewins of 12 October
1899 and to Bowley of 27 February 1906 (Whitaker 1996, II and III) were written as a reaction to
what he saw as biases on the part of his interlocutors, whom he tried to direct towards a midway
path between rarefied abstraction and a down-to-earth observation of facts. However valuable they
may be, it is not in these pedagogical disquisitions that we can find an explanation for his early
enthusiasm that later turned into a coldness towards mathematical economics. In this paper we shall,
instead, look into (i) the peculiarities of Marshall's mathematical formation, which bears the mark
of the Cambridge Mathematical Tripos in the early 1860s; (ii) the concept of economics that he
gradually built up, with its stress on mechanical agencies embedded in evolutionary processes
which continuously transform the conditions under which the former operate; (iii) and lastly, the
practices he developed during the course of his professional life, particularly his reliance on
personal enquiries and direct acquaintance with selected samples of persons and places. The picture
that we draw from these combined sources will be seen to have some relevance also for
understanding Marshall's attitude towards statistics.
2. The parabola of Marshall the mathematical economist started with the long and intensive
preparation for the Cambridge Mathematical Tripos between 1861 and 1865 (Groenewegen 1995,
80 ff.). Much has been written on the characteristics of this examination, which in mid-century
Cambridge still constituted the backbone of a so-called "liberal education" (from Whewell),
intended as the teaching of a mental discipline of rigour and clarity in the pursuit of truth, whatever
the field — science, profession, theological or philosophical speculation — in which the intellectual
powers of the learner were to be engaged (Becher 1980; Richards 1988, 1992; Fisch 1991, chap. 2,
1994; Warwick 2003). Historians of mathematics generally agree on a certain relative
(*)
[email protected], Dipartimento di Scienze Economiche, University of Florence. This draft is provisional, please
do not quote. The research is part of a project on "The role of mathematics in the history of economics" funded by
MIUR, 2002-04.
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backwardness of Cambridge with respect to continental, mainly French, advances in mathematical
scholarship. Since the Cambridge didactical approach was meant to reinforce the nexus between
intuition and abstract ideas, it relied mainly on geometrical ways of presentation and on the
derivation of fundamental concepts from classical mechanics, in the wake of the 18th-century
Newtonian tradition of mathematical physics. This certainly acted as a constraint on the possibility
of analytic developments, but was also the outcome of a deliberate philosophical and pedagogical
choice that did not ignore what was going on contemporaneously on the continent. Attempts at
reforming the syllabus in the early part of the 19th century, and the heavy (regressive, some say)
influence exerted by Whewell resulted in the mid-century compromise of splitting the examination
into two parts: a first, elementary one, intended for all, and a second, advanced part, for those who
aimed at academic distinction or at pursuing a scientific career. While the former was mainly based
on geometrical treatment and physical applications, the latter was more abstract and closer to
continental algebra and analysis than the former. It remained true, however, that mathematics was
generally seen as "intellectual training", rather than "an autonomously valuable specialized study"
(Richards 1988, 7).
The Cambridge pedagogical formula has also been criticised for insisting too much on the
mechanical aspects of calculus and on rapidity of execution to the detriment of mathematical
imagination and inventiveness. But ease and confidence in finding one's own way in complex
problems have their merits too, and it should not be forgotten that highly creative mathematicians
and physicists like Cayley, Sylvester, De Morgan, Maxwell (just to name a few) were educated
there.
Marshall sat the advanced examination and came out second to that J.W. Strutt who was to
become the eminent physicist, Lord Rayleigh. He did not go on to attempt the Smith Prize, which
was a further and most challenging mathematical test usually taken by top wranglers. This fact may
bear out J.K. Whitaker's remark that Marshall did not feel completely at home in the realm of
abstract mathematics (Whitaker 1975, I, 5). If this were so, however, it also confirms that he was
perfectly attuned to the Cambridge view of the aim and essence of mathematical studies. In any
case, he was an excellent product of a school which, for all its peculiarities, was at the top of
European mathematical teaching. In all likelihood his training was superior to that of most
mathematical economists of the previous and his own generation. The reason why economists like
Edgeworth and Pareto are considered to be, and probably were, his superiors in mathematical skills
(see Weintraub 2002; on Pareto’s mathematical training at the Torinese school of Angelo Genocchi
see Mornati 2005), is that Marshall did not care to keep up with the latest advances in the field; thus,
by the turn of the century, his mathematics were outdated even by Cambridge standards. Through
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his acquaintance with Clifford he certainly caught a glimpse of the non-Euclidean geometry
revolution, but all we know about his reactions concerns the philosophical significance of it, not the
mathematical implications (Raffaelli 1994, 76).
Even after giving up mathematical economics, Marshall always expressed satisfaction with
the mathematical education he had received. He attributed to it the credit for the "scientific
instincts", the "sound instinctive habits", which in his view made the difference between "scientific"
and "literary" economics (see Whitaker 1996, II, 112, 140, 306-7; III, 227-8 etc.). At the time of
planning the new Cambridge Economic Tripos (1902), he insisted on students receiving "scientific
training of the same character […] as that given to physicists, or psychologists or engineers"
(Marshall 1961, II, 171): for physicists and engineers this training certainly included a familiarity
with basic mathematics. And in 1906, when his mathematical economics was already dissolving
into "very indistinct memories", he still could write, in what looks like a Whewellian vein, "I
believe in Newton's Principia Methods, because they carry so much of the ordinary mind with
them" (Whitaker 1996, III, 130).
That mathematics provided the young scholar with a bridge to pass from his undergraduate
studies to economics is well known (Groenewegen 1995, chap. 6). By "translating" Mill into
mathematics he opened a major breakthrough in two areas which would mark his theoretical
horizon for a long time: the pure theory of trade, both foreign and domestic, concerned with the
relation between price-quantity equilibria and the cost structure of the countries or industries
involved; and the problem of the mutual determination of prices and distributive variables through
the interaction of connected supply and demand functions for products and productive services.
However, it was not long before the very ease with which he progressed in these areas brought
about the inevitable crisis. On paper, the results of the analysis were indubitable but, Marshall
wondered, had they any relation with problems people care about in real life? In the pure theory of
foreign trade he hesitated in front of the case of decreasing costs and, before going into it, raised the
question: does it bear on situations which have occurred or may occur in history, or is it merely an
aesthetically attractive case? The "Cournot problem" was already there to admonish the young
scholar that assembling hypotheses and working them out mathematically may lead into a trap. And
Jevons and Walras, rather than luminous forerunners, seemed to him to have taken tangential paths
going in wrong directions. Purely mathematical probing of economic doctrines then came to a halt,
and Wander-jahre among the factories, the journey to America, and other real-life experiences took
its place, like rites of passage that the bookish young man saw as inescapable if he wanted to
become a fully-fledged economist.
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3. The post-graduate period between the tripos and the opening of a life career as an economist was
also the one during which Marshall became involved in philosophy and psychology, a parallel field
of interest on which he brought to bear his familiarity with the Cambridge epistemological debates
of the age. The most remarkable outcome of this activity was a short manuscript essay containing a
model of a learning machine, "Ye machine". This is an elaborate mechanical analogy meant to
represent the modus operandi of the mind as a system of routines organized into hierarchical levels
and capable of self-expansion in response to external stimuli (Raffaelli 1994, 116-32).
Two aspects of this paper may be noted in connection with Marshall's concept of
mathematics. First, the close similarity between the functioning of the machine, its very physical
structure consisting of a two-layered system of wheels and bands, and the mechanical images that
Marshall often associated with mathematical reasoning — the "cog-wheels of […] mathematical
machinery" (Whitaker 1996, II, 307), just to quote one. These conveyed an idea of mental processes
turned into unconscious and automatic routines through repetitive exercise "like the practising of
scales on the piano" (ibid.). It is difficult not to see a link between this representation of
mathematics and the rigorous exercise required in preparation for the Tripos (see Warwick, 2003).
The efficiency of mathematics in intellectual work, similar to that of industrial machinery in
manufacturing, was seen to lie in the "stored up force" (Pigou 1925, 172) that descends from the
freezing of complex operations in compact algorithms which allow the performing of them in a
short time and with little conscious effort.
The second point is the clear-cut separation between routine operations and activities
involving invention and creativity. The latter utilise the available routines as far as they can go, but
necessarily transcend them and rely on controlling agencies and faculties other than those embodied
in the mechanical body of the machine. Indeed, the main object-lesson of the essay lies in showing
how the development of routines liberates the superior faculties of the mind from ordinary tasks, so
that they can bring their powers to bear on new problems posed by unprecedented external
circumstances. Here again, Marshall's concept of the role of mathematics in the intellectual process
comes out by association: in all the situations which cannot be dealt with satisfactorily by means of
ready-made routines, mathematics plays the essential but subservient role of paving the way to
those mental faculties, instincts and imagination, that take on the task of establishing new
connections and finding new procedures, as the case requires. With time and repetition, the solution
to a new problem may get set in a new routine and be added to the existing body. But however
closely connected they may be, the mechanical body and the creative mind make up a dual entity.
A recent interpretation (Raffaelli 2003) of Marshallian economics has pointed to the many
similarities between the capacity for self-development of the mental machine and Marshall's view
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of economic progress as gradual absorption of novelties in an increasingly complex structure
through successive phases of standardization and specialization. Marshall himself gave away his
deep source of inspiration in revealing passages of Book IV of the Principles and of Industry and
Trade. The theme of mechanism as the precipitate of invention and the instrument for further
invention resurfaced repeatedly and in various forms around the turn of the century: at the
methodological level, in the theorising of two distinct and connected ways of reasoning, itemised as
mechanical and biological analogies (Marshall 1898); and at a more metaphysical level, in the notes
for the 1901 lecture on "Machinery and life". In this, mechanisms and life are said to interact at all
levels, the former as agencies of the latter but also, thanks to their orderly and disciplined character,
as constraints and burdens on the latter (Raffaelli 1994a).
This dialectics between order and creativity, antithetical yet in need of each other, lyes deep
at the heart of Marshall's ideas regarding both social and economic development and the intellectual
enterprise of understanding and interpreting it. The latter does indeed, in Marshall's view, partake of
the former, as economics is concerned with the same problems as economic agents in their everyday
business, and employs the same mental processes, but at a more refined level. The position of
mathematics in it, like the position of machinery in industrial development and that of any kind of
order in human development, is uncomfortably antithetical, actually on a razor edge: "Order is an
evil […] A necessary evil in its place. Find its place" (Raffaelli 1994a, 18).
4. Against the background of Marshall's fundamental dialectics, the youthful quest for "real-ising
abstract reasoning" (Whitaker 1996, I, 110) which made him raise his head from his papers and
look around him, takes on particular prominence. The train of mathematical deduction as Marshall
conceived it has an inertia of its own which allows the researcher to carry on indefinitely without
any need of taking in new information from the outside. Social life, too, involves inertial processes
that imply no novelties, as if they were subjected to the discipline of predictable rules. In Marshall's
evolutionary view, the principle of natural selection through which evolution operates "involves
only purely mechanical agencies" (Raffaelli 1994, 119); and in the economic field, these are
identified with the principle of substitution (Marshall 1961, I, 596-7). Purely inertial processes,
however, are comparatively few; they are more often immersed in an environment characterized by
frequent casual variations which impart unexpected twists in their regular development. Thus,
deduction on the one side, and social life on the other, may take paths that, although starting from
the same point and keeping close to each other for a while, are bound to diverge sooner or later. It is,
therefore, the researcher's duty to stop treading the deductive path from time to time and to check
his position with respect to social reality. Marshall's exhortation to keep to "short deductive chains"
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is well-known but, taken literally, is of little help: when is it that a deductive chain is short or long?
If we look at his practice as a mathematical economist, we get a clearer idea: "short" simply means
"interrupted before completion", that is, before all the possible consequences are drawn from the
given premises.
Marshall's applications of mathematics to economics, especially those that found their way
to publication, from the "Pure Theory of Foreign Trade/Domestic values" of the 1870s to the
Mathematical Appendix of the Principles, share a common pattern. The set up is usually, and not
surprisingly, mechanical in type, with the mathematical model of the planetary system lying more
or less explicitly in the background, to provide the ideal prototype for the study of economic
interdependencies. The forces that drive economic agents to action are analyzed at the margin,
where they vie with each other and determine whether observable state variables, such as prices and
produced or traded quantities, will change or maintain their current values. Propositions about
equilibria and stability are argued either by means of differential calculus or, more frequently,
geometrical illustrations ("Newton's method" again). Whichever the chosen way of presentation
may be, the object is always to derive a simplified force field from a few intuitively plausible and
distinctly stated economic hypotheses.
So far, everything is consistent with Marshall's own idea of economics as being the closest
to physics and, therefore, also the most "scientific" of all the moral sciences — a privileged position
which is due to the fact that the forces that ultimately interact through the market system can be
measured by means of the transfers of money between individuals that they bring about. When
human motivations are set against money, it is possible to trace a visible record of their relative
strength. For all that, however, in Marshall's view economics remains unquestionably a moral
science. As he says in a passage of Appendix C of the Principles, the forces it deals with act on a
living material, one that has the property of reacting to forces by modifying its "inner nature and
constitution" (Marshall 1961, I, 772). In the mechanical terminology that Marshall liked to utilize, it
is as if each movement in a state space regulated by a dynamic law changed the position of the point
in the space and the law itself at one and the same moment: the model of the oscillation of a
pendulum would not be of much use if each oscillation changed the nature of the pendulum
(Whitaker 1975, II, 163). Thus, evolution — intended here as emergence of novelty through
variation — steps in and dictates a halt to mechanical analysis. In the writings of the 1870s, the
difficulty was raised in terms of the "rigidity" of the curves of the pure theory of domestic/foreign
values. It resurfaced as a problem of the "irreversibility" of movements along supply curves right in
the middle of Book V of the first edition of the Principles; and although in the later editions the
issue was relegated to an appendix, it remained as a sort of absolute limit to the mechanical
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treatment of economic phenomena. Here also lay the nec plus ultra of the application of
mathematics to economics.
This limit is not a technical one. Marshall admitted that more complex dynamic models that
allowed for switches between different overlapping force fields could deal with certain types of
irreversibility (the "catastrophes of mechanics", Marshall 1898, 42). He might also have observed,
of course, that a limit to mechanical treatment is not in itself a limit to mathematics, but the
Cambridge mathematical imprinting evidently mattered and, as far as we know, there is no hint of
an idea of mathematics as disjoint from mechanics in his writings. His point was a different one and
may be put this way. Mathematics helps to organize reasoning from exact premises to conclusions.
A change in a force field is always due to non-routine responses to unusual circumstances, that is, to
the occurrence of something the possibility for which was not implicit in the premises. Submitting
that change to mathematical reasoning would amount to turning some of the initial premises into
conclusions which, in turn, can be drawn from deeper and wider premises concerning creative
responses to novelties. This is what Marshall was not willing to accept. There is no routine, no
predictable regularity nor repetitiveness, in the way in which novelties come about. Assuming the
opposite would mean opening a breach in his "machinery and life" philosophy and removing
exactness and order to where they do not belong.
The sketch of general economic equilibrium contained in the notes XIV-XXI of the
Mathematical Appendix of the Principles is perhaps the place where we can see Marshall's
inhibition at trespassing on the limits of mathematical treatment in the clearest manner. Along the
lines of his youthful attempts at integrating Mill's defective analysis of competitive equilibrium, the
object of the notes was to demonstrate that every new variable that is added to the system of
equilibrium equations is accompanied by an independent condition, so that, however extended the
system may be, it will always contain "enough, and only enough, premisses for conclusions"
(Marshall 1961, I, x). At this point, the idea of constructing an all-inclusive system would seem to
follow as the next natural step, but Marshall explicitly rejected it as inappropriate. The reason he
put forward was that, if we did so, we would sooner or later allow into the system conditions that
are not homogeneous with the others; and he cited the supply of human skills as an example.
"Could anyone believe that this is ruled by the same laws as the supply of industrial machinery?"
seems to be the underlying question, and the suggestion is that while a function, i.e. a set rule, is
appropriate for describing the supply of an industrial product, it would not be so for the outcome of
decisions concerning children and their education.
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5. The Marshallian exhortation to interrupt analytical work frequently and "[look] at the world with
wide open eyes for a few minutes" (Whitaker 1996, III, 291) appears, therefore, to be a
consequence of the application to economics of the general principle according to which analysis
and instincts, routine-like order and creative liberty, complement each other in all intellectual tasks
that cannot be reduced to the mere performing of pre-established patterns. In this light, Marshall's
commitment to "real-ising" his equations represents a more ambitious undertaking than simply a
call to generic empiricism and to the possibilities of combining induction and deduction. In addition
to testing analytical results, the non-analytical part of the economist's work should also be the phase
during which the direct experience of life comes in and provides a boost to the generation of new
ideas. Evidently, Marshall did not believe that heuristics could be generated from inside purely
analytical work.
The totally idiosyncratic and unsystematic way in which he combined sources of
information and methods of the most varied sorts seems to bear out this interpretation. Visiting
places and meeting people figured high on his list, higher than consulting statistics and elaborating
data sets. In private correspondence, he was outspoken as to the relative importance of statistics in
his empirical work: for example, "I rely more on my 'field work' […] and on my conversations […]
than I do on statistics"; and, more disparagingly, "that which offers the best guidance to me, is too
subjective for external use: so I have to waste time on analysing statistics for other peoples benefit"
(Whitaker 1996, III, 146, 202). Marshall did indeed use statistical time series as instruments for
investigating causal connections through correlations — a method that he derived from the
"concomitant variations" of Mill's System of Logic — but shunned the mathematical methods of
statistics, the mechanical character of which he did not like, although he admitted that they can
"grind out results wh. are officially pure & above reproach" (ibid., II, 305-6). Characteristically,
given his fondness for geometrical representations, he relied on the intuitive appeal of graphs, on
their capacity to attract our attention "through the eye" (Pigou 1925, 174 ff.). He also made it a
point to always keep quantitative and qualitative information together, as he considered the former
too dubious and limited on its own.
The essence of such an eclectic approach to empirical work is well represented by Marshall's
well-known "Red Book": an assemblage of time series chronologically aligned with lists of
historical facts and episodes, through which he would roam randomly and repeatedly until he came
across some unexpected figure or correlation which set his mind off on some new trend of ideas.
His personal heuristic formula revolved around this practice of free, almost instinctive association.
Intuition and imagination rank higher than quantitative methods also in the problem of
selecting representative cases or agents for the study of social and economic tendencies. Marshall
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recommended intensive study of few "carefully chosen" cases, rather than "the extensive method of
collecting […] very numerous observations […] obtaining broad averages in which inaccuracies
and idiosyncrasies may be trusted to counteract one another to some extent" (Marshall 1961, I, 116
n.). In this remark, which incidentally (as noted by Groenewegen 1995, 168) shows that Marshall's
"representative" agents bear no kinship to Quetelet's "average man", built out of normal
distributions of characters, we can see a reflection of the generally negative attitude of 19th-century
social science towards the adoption of statistical methods which had been proven valid for
astronomy and geodetics (Stigler 1986, chap. 5). Marshall repeatedly expressed his objections to the
law of error through the simile of the marksman who, when shooting from inside a shelter, does not
realize that the wind is deviating his bullets always in the same direction (Whitaker 1996, II, 301;
III, 265). Edgeworth, too, was aware of the problem, and went perhaps more directly to the root of
it when he pointed out the difference between observations ("different copies of one original") and
statistics ("different originals affording one 'generic portrait'"): averages are real in the former case
and fictional in the latter (quoted in Stigler 1986, 309). But for Edgeworth, this was the starting
point for an attempt at refining statistical techniques, while for Marshall no rules could be given and
all ended up with a generic mention of the need for "a rare combination of judgement […] and of
insight and sympathy" (Marshall 1961, I, 116 n.).
When he is not engaged in analytical work, then, the ideal economist as Marshall conceived
him is not the sort of empirical economist that we would expect nowadays, equipped with all the
statistical and econometric implements of the trade. He could be described, instead, as one who can
"know roughly, without calculation" (Whitaker 1996, II, 301), or, in the same words with which
Marshall described the businessman, one who "works generally by trained instinct rather than
formal calculation" (Marshall 1961, I, 406). According to today's dominant taste, such a reliance on
personal wisdom to the detriment of mathematical and statistical technique would be considered
excessive. Professional standards have changed, so that a contemporary economist like H. Brems
could condemn Marshall outright for not making his sources and tools fully visible, thus eschewing
public check and criticism (Brems 1975, 585). To this, we can imagine that Marshall's reply would
be that not all the resources which are mobilised in a scientific enterprise — at least in the field of
the social sciences — have such an externally observable and controllable nature.
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REFERENCES
Becher, H.W. (1980) "William Whewell and Cambridge Mathematics", Historical Studies in the
Physical Sciences, 11, 1-48.
Brems, H. (1975) “Marshall on Mathematics”, Journal of Law and Economics, 18, 583-5.
Fisch, M. (1991) William Whewell Philosopher of Science, Oxford: Clarendon Press.
Fisch, M. (1994) “’The emergency which has arrived’. The problematic history of 19th Century
British algebra — A programmatic outline”, British Journal for History of Science, 27, 24776.
Groenewegen, P.D. (1995) A Soaring Eagle: Alfred Marshall 1842-1924. Aldershot: Elgar.
Marshall, A. (1898) “Distribution and exchange”, Economic Journal, 8, 37-59.
Marshall, A. (1961) Principles of Economics, Ninth (variorum) edition with annotations by C.W.
Guillebaud. Two vols. London etc.: Macmillan.
Mornati, F. (2005) “Gli studi matematici di Pareto all’Università di Torino: le coordinate
istituzionali ed intellettuali generali, gli indirizzi epistemologici, i contenuti formali e
l’influenza esercitata sull’opera paretiana”, Università di Torino, Department of Economics
‘S. Cognetti de Martis’, Working paper N° 05/2005
Pigou, A.C., editor (1925) Memorials of Alfred Marshall. London: Macmillan.
Raffaelli, T., editor (1994) The Early Philosophical Writings of Alfred Marshall. Research in the
History of Economic Thought and Methodology, Archival Supplement 4.
Raffaelli, T. (1994a) “Marshall on ‘Machinery and Life’”, Marshall Studies Bulletin, 4, 9-22.
Raffaelli, T. (2003) Marshall’s Evolutionary Economics. London and New York: Routledge.
Richards, J.L. (1988) Mathematical Visions: The Pursuit of Geometry in Victorian England. New
York: Academic Press.
Richards, J.L. (1992) "God, truth, and Mathematics in Nineteenth Century England", in: M.J. Nye
et al. (editors), The Invention of Physical Science, Kluwer, 51-78.
Stigler, S.M. (1986) The History of Statistics: The Measurement of Uncertainty Before 1900.
Cambridge (Mass.) and London: Belknap Press.
Warwick, A. (2003) Masters of Theory. Cambridge and the Rise of Mathematical Physics. Chicago
and London: The University of Chicago Press.
Weintraub, E.R. (2002) How Economics Became a Mathematical Science. Durham and London:
Duke University Press.
Whitaker, J.K., editor (1975) The Early Economic Writings of Alfred Marshall, 1867-1890. Two
vols. London and Basingstoke: Macmillan.
Whitaker, J.K., editor (1996) The Correspondence of Alfred Marshall, Economist. Three vols.
Cambridge (UK), New York and Melbourne: Cambridge University Press.