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Time in Economics, part I

Marshall on Time and Mathematical Analysis

Marshall sur le Temps et l'Analyse Mathématique
Vlassis Missos
p. 61-86

Résumés

L’essai offre un compte rendu de l’approche d’Alfred Marshall aux mathématiques et vise à approfondir notre compréhension de ses considérations méthodologiques et de ses vues sur le Temps. Contrairement à ce que l'on croit, les Mathématiques du 19e siècle se sont développées de manière similaire à l’évolution des idées politiques et économiques, c’est-à-dire par le biais de débats intellectuels passionnés entre des camps opposés. Le Temps et la recherche de son vrai sens, constitue un aspect particulier de cette controverse dont la Tradition Mathématique de Cambridge offre un espace vital pour la réflexion. Cela est dû à cette formation que Marshall sépare entre ‘t’ et ‘T’ et développe son scepticisme contre le raisonnement analytique. Ses réflexions sur les processus en temps réel et sur la traduction des déclarations mathématiques au langage économique, plaide en faveur de sa profondeur intellectuelle unique. La recherche est textuelle, basée sur des sources primaires aussi bien que secondaires.

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Texte intégral

1In mathematical analysis time is set forth as an independent entity, unaccompanied by all other variables. The pattern of numerical sequencing which it follows offers a dynamic flavor to the specified relations formed between these variables. Time advances like a rhythmical counter crying out the consecutive moments, while all variations are linked by an abstract timeline that extends from the past to the future. From this particular feature of counting, certainty and foresight effortlessly arise. A foreseen, yet unpaved, numerical ‘path’ cannot suffice as a close replication of an unforeseen future that will succeed the present state of affairs. As has been noted:

Some mathematicians may, indeed ... have thought of the path of the “particle” of classical physical dynamics as requiring to be traced out of the movement of the particle itself. Perhaps this was the view of some ... who thought of such a curve as “becoming” rather than “being”. But in thinking thus they were bringing in ideas quite alien to the texture of their mathematical analysis itself. The solution of the differential equation, if it can be found, is complete in an instantaneous and timeless sense” (Shackle, 1967, 23).

2Clearly, the idea of time or of change is “quite alien to the texture of ... mathematical analysis itself”. Arguments in favor of this view can be traced back to Bergson’s ([1907] 1912) Creative Evolution, in which the ontological state of modern science is thoroughly examined. Bergson argues that the dated events are detached from the whole, ‘which at every moment puts on a new form and which communicates to them something of its novelty’ (ibid., 361). The creativity of the human mind is not an arbitrary thought but demands a conscious action, through which our present status is associated with a potential future one. Hence, time steps in. “Thought-constructions which are meant to show us order in the general scene of human action are bound to involve the ideas which we group under the heading time” (Shackle, [1965] 2010, 1).

  • 1 For a view on the reasons why Marshall may have abandoned all efforts to have his promised second v (...)
  • 2 24th March 1908 (incomplete), letter n. 899. References from Marshall’s correspondence will hencefo (...)

3Raffaelli (2006) has underlined the important effect of William K. Clifford on Alfred Marshall’s appreciation of the role of human creativity and spontaneity in a model that interprets human action. In the Principles ([1920] 1961) he undertakes the challenging task of building upon the complexity of the human character, along with one of the most genuine attempts to designate a suitable role for time within the inherited rigid framework of political economy that was in circulation. For arcane reasons,1 the intended project of transforming static and dynamic mechanics into biological concepts, in a manner that would respect the role of time, was left incomplete. Over a period of almost twenty years, spanning from the publication of the first edition of the Principles, where“the element of Time ... is the center of the chief difficulty of almost every economic problem” (Marshall, [1920] 1961, vii), to a letter Marshall sent to John Bates Clark, in which he asserts that “there is a great deal to be done still” on the “elaboration of the influence of time”,2 Marshall remained skeptical of the way in which mathematical analysis had been blended with economic theory and policy. It is the relationship between mathematical analysis and Time that is the focus of this article.

4Recent research (Dardi, 2014) on Marshall’s mathematical thought has revealed a clear inclination towards an idiosyncratic, intuitive approach that allowed him to instill life into the rigid armor of mathematical symbols. The rationale behind this was intended to methodologically enrich the dominant economic theory while at the same time revealing the limits of economic interpretation. What is more, his examination of the issue of time should be approached as a special intuitive category that gives meaning to economic phenomena and it should not be confined within the technicalities generously offered by the “toolbox”. Marshall’s careful use of the analytical tools can also be found in Pigou (1925, 84), who points out that “though a skilled mathematician, he used mathematics sparingly”, only to suggest that they “might distort our sense of proportion by causing us to neglect factors that could not easily be worked up”. Marshall’s methodological argument progresses gradually from the first to the eighth edition of the Principles, without his initial theoretical core being impacted in any way.

5In the present article, a link between Alfred Marshall’s approach to mathematical analysis and time is drawn. Both concepts occupy a major part of his methodological ideas, reflecting the intellectual environment in Cambridge, where he studied. His approach to the subject was, however, unsystematic although the general stance he took in different areas of his writings is altogether consistent. In the majority of the notable works of dominant economic thought, over a period spanning at least from as early as Gossen ([1854] 1983) and the marginalists to Hicks ([1939] 1946) and Cassel ([1924] 1932), time was captured indirectly through the broad concept of causality. Consequently, the article begins with an interpretation of that abstract concept, serving as a subtle nexus to the importance of analytical reasoning. It then proceeds to a discussion of the influential Cambridge figure of William Whewell and his ideas on Mathematics, finally to conclude with the evidence found in Marshall’s correspondence, where his dissenting attitude to the use of mathematical analysis can be traced, and with a focus on the issue of time.

1. Causality, Time and Marshall

1.1 Causality and Time

6Recollecting events that may once have occurred in the past implies ordering events that happened “earlier” and “later” according to how recent or how remote we take them to be from the present moment or from any other date. Such practices constitute the very essence of logical time, which, although it finds no foothold in reality, is used as a flexible tool in order to describe any group of events, just like flipping backwards through the pages of a book, or even at random. When exercising this capacity of our mind, logical time allows for the reversibility of the previously ordered events and, thus, for any one of these to be arbitrarily considered as the starting point. Economists are interested in particular in the specific course of events and, more demandingly, in the causal links that may be drawn among them. Causality, however, is also closely related with the idea of time through the essential property of directionality, a concept which can be found from as early as Carl Menger:

The idea of causality, however, is inseparable from the idea of time. A process of change involves a beginning and a becoming, and these are only conceivable as processes in time. Hence it is certain that we can never fully understand the causal interconnections of the various occurrences in a process, or the process itself, unless we view it in time and apply the measure of time to it (Menger, [1871] 1950, 67).

7Time is conceived as change based upon some measurable causal processes whereas our essential knowledge and understanding is obtained only through the latter, i.e. causality. Before examining this further, we should reflect upon this metaphysical basis of the idea itself.

  • 3 Temporal sequence, in Hume’s Treatise of Human Nature (part 3, section 2), was considered an essent (...)

8The necessary connexion (Hume, [1748] 2007) that binds a cause to a single effect abides as an outward impression, contemplated upon the human mind, without revealing the actual powers and forces through which its occurrence is provoked: “experience only teaches us, how one event constantly follows another; without instructing us in the secret connexion, which binds them together, and renders them inseparable” (ibid., 13). Consequently, what binds them together remains difficult for us to grasp or, as Menger puts it, “can never be fully understood”. Can we therefore suggest that a particular effect is followed by an “infallible consequence” produced by a cause? Or is there some hidden quality from which we can safely conclude that two separate instances irrefutably follow one another in a causal link? Unless we tacitly assume that the immediate occurrence would be disclosed within the cause itself, the effect would be but arbitrary. Time allows for a separation between the two instances3 since:

every effect is a distinct event from its cause. It could not, therefore, be discovered in the cause; and the first invention or conception of it, a priori, must be entirely arbitrary (ibid., IV, 11)

An act of volition produces motion in our limbs, or raises a new idea in our imagination (ibid., VII, 9).

9Our knowledge is bound to the impressions obtained by our senses, which also impede us from realizing the actual forces that give birth to a subsequent instant or state of being. The process of causality, the very means with which it is unfolded, may remain a mystery to human understanding, since the analytic representations of a process which interconnects two distinct instances is not capable of providing answers to questions concerning the potency—or “reasons”—that blows in one direction (end) rather than the other. The human incapacity to reasonably draw specific functions that produce various and multiple effects from particular causes is also attributed to the very flow of time itself. Since, if the future is uncertain, then the effect should also be unknown. Accordingly, Hume’s critique targeted the heart of the general scientific view of social life as a natural phenomenon:

[Past experience] can be allowed to give direct and certain information of those precise objects only, and that precise period of time, which fell under its cognizance: But why this experience should be extended to future times, and to other objects, which for aught we know, may be only in appearance similar; this is the main question on which I would insist (ibid., IV, 16).

10Uncertainty, Hume seems to suggest, characterizes the arbitrary scenarios concerning future instances. Below, I suggest that Marshall was not generally opposed to Hume’s treatment of the issue, and that his restricting of his mathematical analysis to the appendices of his Principles was not an arbitrary choice.

1.2 Marshall

  • 4 Marshall was absorbed with Kant’s philosophical inquiries during his vacations on the Continent. Th (...)

11The concept of undirected causality has long permeated the writings of the major representatives of economic theory. It has also initiated a solemn debate, the fruits of which did not suffice to alter the essence of the dominant theory. According to Marshall, his general views were influenced by the “mathematical conceptions of continuity, as they were exposed in Cournot’s Principes. Cournot’s contribution to economic thought was that, when confronting the great difficulty of the innate complexity of economic problems, he urged us to distinguish between continuity and causality. What Hicks (1979, 18) described as “mutual causation” is in Marshall transformed into the “mutual determination” of two contemporaneously assumed elements4 that are interactively related. This is exactly the analytical operation between two abstract arguments x and y. The one determines the other in a static or dynamic context of mathematical analysis, while this method of tackling “the various elements of an economic problem” is conceived “not as determining one another in a chain of causation, A determining B, B determining C, and so on—but as all mutually determining one another” (Marshall, 1961, ix). Mutuality and causality are considered to be incongruous, although both express continuity in this context. Marshall labels this type of continuity “mathematical”. Shove ([1942] 1982, 717), moreover, maintains that Cournot’s influence is more apparent if someone studies Marshall’s endeavors as those of a Cambridge student attempting to translate the doctrines of Ricardo and Mill into differential equations. It is crucial to understand that both the technical tools with which Marshall was familiar and the assumption of continuity, as expressed by Cournot, are complementary elements that form an integral method of economic analysis.

12There are at least two reasons as to why economics, unlike physics, lacks the exactness of its laws. Firstly, it is uncertain whether the causes being investigated are the true ones as regards the event under consideration, and, secondly, their results cannot expect to enjoy the confidence of a physicist who measures the velocity of a body that moves linearly across a table in his or her laboratory. “[T]he actions of men are so various and uncertain” that economic laws resemble more to “statements of tendencies, which we can make in a science of human conduct”, and which are ultimately “inexact and faulty” (Marshall, 1961, 32). Indicative of the latter is the term “norma” which was used by Marshall as a substitute for the delusive certainty emanating from “law”, and was mostly used to express the particular conditions under which specific causes generate largely anticipated effects. Normality is detached from any moral argument and it “implies the predominance of certain tendencies which appear likely to be more or less steadfast and persistent in their action over those which are relatively exceptional” (ibid., 34). Interestingly, however, Marshall urged us to extract the real content of the term, rather than to analyze it as something that has no foothold in actual economic affairs.

13An illustration of this can be found in Chapter V of Book V, where Marshall discusses the manner in which the normal supply price of wool is estimated (ibid., 365). The entrepreneur reckons his supply price level for the amount he is willing to offer, according to the conditions that have prevailed in the recent past, while at the same time he makes “allowance for any change that would be likely to affect the supply in the immediate future” (such as weather conditions). The entrepreneur can take into account those forces of “sufficient regularity”, i.e. forces that are expected to make several appearances throughout a period of time and which are apparently directly related to the estimated feature, which is in this case the supply price. Some events may occur although they cannot be foreseen, while others are extremely hard to imagine and are left out of conditions of normality. Even within this context of analysis, time would remain a difficulty that cannot be overlooked, since the events that affect the supply price in different ways do not occur simultaneously. Not that the dating or sequencing of effects brings us closer to reality itself. One might also suggest that it could actually take us further from it, but in this case, an abstract notion of time is called forth. Marshall then resorted to his famous ceteris paribus, a method with which to investigate the result of a tendency in isolation while all the others are assumed to be unchanged (Cartwright, 2001) in order to determine the value of the commodity. The sense of realism that was earlier attributed to his theoretical conception of normality is now taken back, subjugating the time element for the sake of implementing the technique of differential calculus. “The more the issue is thus narrowed, the more exactly can it be handled: but also, the less closely does it correspond to real life” (Marshall, 1961, 366). Not surprisingly, this is where one of the most exceptional misconceptions lies, since Marshall here admits that what he just did does not correspond to reality:

[W]hen “normal” prices are contrasted with temporary or market prices, the term refers to the dominance in the long run of certain tendencies under given conditions. [I]t is true however that the condition that time must be allowed for causes to produce their effects is a source of great difficulty in economics. For meanwhile the material on which they work, and perhaps even the causes themselves, may have changed … (ibid., 36).

14In a way, Marshall exposes both approaches. He used the former, which uses the tools of mathematical analysis, to describe economic phenomena while the latter, the interpretation of the true course of events, is conceived only when the “element of time” intervenes to alter the context within which causal links operate.

2. Analysis, Synthesis and Time

2.1 An ‘Obscure’ Phrase

15In one of his most frequently quoted passages, which appeared in a letter to Arthur Bowley, Marshall vividly described his view as to the proper way in which economists should employ mathematics. His six-step rule is as follows:

(1) Use mathematics as a shorthand language, rather than as an engine of inquiry. (2) Keep to them till you have done. (3) Translate into English. (4) Then illustrate by examples that are important in real life. (5) Burn the mathematics. (6) If you can't succeed in 4, burn 3 (lt. 840).

16The Victorian economist approached mathematics as a thought-stimulating tool that could help “a person to write down quickly, shortly and exactly, some of his thoughts for his own use” (Marshall, 1961, x). He seemed to have admitted a priori that his economic theory would depend only loosely on exact reasoning, since “the argument in the text is never dependent on them [mathematics]” (ibid., x). Krugman (1998), on the other hand, in his interpretation of the Marshallian view, claimed that according to Marshall “one should only use math to check conclusions, not to arrive at them”, an idea that, if carefully examined, breaks the sequence of the above rule. If this were true, then mathematics would have become the ultimate basis through which to verify knowledge while, on the contrary, the touchstone of mathematical truths had to be the use of an example that was important in real life. It is to this particular issue, and to Marshall’s educational environment, that we now turn in an attempt to deepen our understanding of the above reference and to offer an account of the crucial role played by mathematical analysis.

  • 5 In a footnote, Whittaker added: “The aspect of Newton’s methods that Marshall had in mind remains o (...)

17It may at first seem strange, but the idea of beginning with the legacy of Isaac Newton, as it had come down in the mathematical discussions of 19th-century Cambridge, did not arise from an arbitrary caprice but was indirectly dictated by Marshall himself. The phrase that he added immediately after his six-step rule—“I believe in Newton’s Principia methods, because they carry so much of the ordinary mind with them” (lt. 840, 130)—and which his editor found to be “obscure”5, encouraged us to examine the work and ideas of “Cambridge’s greatest son” (Keynes, 1951, 310) much more closely, as seen through the eyes of his most prominent commentators. It is on this specific matter that I aim to shed some light.

  • 6 For a complete understanding of the Newtonian Method see Guiccardini (2009). My inquiry on Newton’s (...)
  • 7 “[T]he ideas, now in the minds of contemporary mathematicians, lie very remote from any notions whi (...)

18Marshall referred to Newton only once in the Principles (Marshall, 1961, 216). The latter’s influence is of course more easily traceable in Smith (Skinner, 1965; Montes, 2006). Marshall’s historical era corresponds to a time when the Newtonian mathematical method6 was gradually going out of fashion even before Newton’s death, while the ideas of a new mathematical formulation, devoid of notions immediately derived through our senses, were gaining new ground.7 Ideas, however, seem to evolve at a slow pace. Marshall’s training for the Mathematical Tripos examinations involved some particularly modern (for his time) and demanding topics, whereas his knowledge of the emerging technique of differential calculus, which was applied in economic theory to a great degree, proved to be thorough.

The new analysis is endeavouring gradually and tentatively to bring over into economics, as far as the widely different nature of the material will allow, those methods of the science of small increments (commonly called the differential calculus) to which man owes directly or indirectly the greater part of the control that he has obtained in recent times over physical nature. (Marshall, 1961, xvi)

19In the first part it is suggested that economics was once separate from the method of differential calculus. In the second part, however, two other intriguing questions are implied. To what extent does the nature of the economic material allow for the science of “small increments” to be implemented? Do we, in this way, therefore gain control over it? Such is the difficulty of the questions that arise as we read the Principles that many readers have chosen to skip the greater part of the work, quickly moving instead on to Book V.

20In Marshall it has been understood that traditional and ethical values, woven together through habits and beliefs, are only gradually reformed and seldom displaced by modern ones. Well-rooted customs endure silently for decades, like inertial forces of the past that are continually but slowly fading away or otherwise resurging, to exist side-by-side with the new ones that are unceasingly becoming dominant. This certainly applies to the educational practices and ideals of the past, and Cambridge’s insistence on Newtonian rather than the Leibnizian mathematics of the Continent was not necessarily founded upon some void and frivolous prejudices. As a figure, Alfred Marshall represented one of the most exceptional examples of a period that was undergoing an immense transition from Victorian scholarship towards the new era of 20th-century academia, which was marked politically by the establishment of utilitarian ideology. In order to understand the significance of his social and academic environment, a small digression into the ideas that captured the minds of the most brilliant intellectuals of 19th-century Cambridge is required. This, I suggest, is the only secure way in which to place Marshall within his proper context, while it also enables a reassessment of the methodological considerations discussed within his works, linking to his ideas on Time.

2.2 Newton’s Principia Method and Cambridge

21The “mathematical way” advocated by Newton’s geometrical exposition (Guiccardini, 2006) should not necessarily be associated with modern formalization, so as to represent certain knowledge. Instead of being taken as arbitrary hypotheses, Newton’s inferred postulates should primarily enable us to find observational counterparts to the phenomena presented to our senses. Newton was a fervent adversary of hypothetical engagement (Redman, 1997, 43) and his apologetic stance at the beginning of Book III gives us a clear picture of his interpretation:

I have not been able to discover the cause of those properties of gravity from phenomena, and I frame no hypotheses; for whatever is not deduced from the phenomena is to be called an hypothesis; and hypotheses, whether metaphysical or physical, whether of occult qualities or mechanical, have no place in experimental philosophy. In this philosophy particular propositions are inferred from the phenomena, and afterwards rendered general by induction. (Newton, [1687] 1846, 506)

22Induction follows. Gravity’s primary cause could not have been presented to our senses, while Newton was frustrated with the idea that the pillar of his treatise should remain occult. In that sense, Schumpeter ([1954] 1986, 20) is right to claim that the word “hypothesis” in Newton had the meaning of “inadequately substantiated speculation”.

23Newton’s appreciation of the Greek mathematician and compiler Pappus of Alexandria grew throughout his later years. Pappus’ Collection—written circa 340 AD—gathers together some important methodological guidelines, which attracted Newton’s attention. There, a now devalued although still meaningful distinction between Analysis (“resolution”) and Synthesis (“construction” or “composition”), sets the basis for a differentiated understanding. Due to Pappus’ authority:

...in analysis we assume what is sought as if it has been achieved, and look for the thing from which it follows, and again what come before that, until by regressing in this way we come upon some of the things that are already known, or that occupy the rank of a first principle. We call this kind of method “analysis”, as if to say ἀνάπαλιν λύσις (reduction backward). In synthesis, by reversal, we assume what was obtained last in the analysis to have been achieved already, and, setting now in natural order, as precedents, what before were following, and fitting them to each other, we attain the end of the construction of what was sought.

There are two kinds of analysis: one of them seeks after the truth, and is called theorematic: while the other tries to find what is demanded, and is called problematic. In the case of theorematic kind, we assume what is sought as a fact and true, then advancing through its consequences, as if they are true facts according to the hypothesis, to something established, if this thing that has been established is a truth, then that which was sought will also be true, and its proof the reverse of the analysis; but if we should meet with something established to be false, then the thing that was sought too will be false. In the case of the problematic kind, we assume the proposition as something we know, then, proceeding through its consequences, as if true, to something established, if the established thing is possible and obtainable, which is what mathematicians call “given”, the required thing will also be possible, and again the proof will be the reverse of the analysis; but should we meet with something established to be impossible, then the problem too will be impossible (Cuicciardini, 2006, 1722, my emphasis).

  • 8 See Guiccardini (2006, 1727).

24Analysis was used as the process of transforming a geometrical problem into an algebraic one. This would also mean that finding the roots of some formulated equations would not correspond to the problem’s solution. Guiccardini (2009, 5) explains that a solution would call for a geometrical construction of the appropriate line segments through synthesis, if certainty of the knowledge acquired was the aim. A line segment is subject to observation. As an outcome of his anti-Cartesian8 stance, which developed during his later years, Newton zealously advocated in favor of the need for algebraic notions to be depicted geometrically. Rigorousness, the means of establishing knowledge, could be fully ensured through geometry, whereas confusing the two had made his fellow mathematicians lose sight of “the Simplicity in which all elegance of Geometry consists” (Newton, 1967, 429). In a nutshell, the process of analysis may serve as a primary step towards the discovery of potential truths that could solely be certified by the other process of synthesizing, which should be able to stand on its own: “Analysis does lead to Composition: but it is not true Composition before it is freed from Analysis ... Composition in it self is perfect, and far from a Mixture of Analytick Speculations” (ibid, 250).

25Both the above attributes contribute emphatically to conforming the idea that suggests that questions of ontological interest or of true and perceptive content should not be extraneous to mathematical inquiry. The certainty of knowledge in the mathematical program encouraged by Newton did not reside in symbolic algebraic reasoning, the results of which could have well been detached from any perceptive correspondent. The results of ancient (Euclidean) geometry, of which Newton considered himself a guardian, were above all interpretable by the human senses and its truth was largely based upon them.

26A key figure of the early 19th century was William Whewell, Master of Trinity College and executive member of the Analytical Society. As an “academic leader” and an “environment-creating individual” (Schumpeter, 1954, 448) he advocated that any proper definition ought to offer a descriptive rather than a formal basis of mathematical truth. This, we might say, constitutes an alternative way of expressing the critical fifth step of Marshall’s approach to Mathematics. Accordingly, Whewell challenged the position shared by some of his contemporaries that mathematics should be approached mainly as an arbitrary knowledge deduced by hypothetical axioms, offering examples of the necessary conceptual ability of the human mind that we must evoke, so that our reasoning is meaningful. “[I]t would appear, that a definition, to be admissible, must necessarily refer to and agree with some conception which we can distinctly frame in our thoughts” (Whewell, 1837, 149)—which is simply another way of saying that the mere forms of deductive reasoning are far from what is needed to assess the soundness of mathematical knowledge. Whewell was concerned that a field of knowledge whose truth did not appear to our minds as adequately or necessarily certain could not serve as a basis for educational purposes. “To be significant, the truth of mathematical statements had to lie in the subject matter being described; it could not be confined to empty manipulations of terms and figures which were merely subjectively defined” (Richards, 1988, 33).

27During the years of his intellectual maturity, Whewell seems to have regretted being a member of a society that was called “analytical”. “The Analyst does not retain in his mind, in virtue of his peculiar processes any apprehension of the differences of the things about which he is supposed to be reasoning. When he has once placed before him these equations of co-ordinates, he no longer thinks at all about the special nature of the things originally spoken of. His reasonings are operations upon symbols; his results are equations” (Whewell, 1850, 45). The manipulated symbols permit the mathematician to abstract from any specific feature of his or her investigated phenomenon. By definition, moreover, a pair of equations is attributed solely to unique set of values and it is only because of them that equations are consistent.

28A constructive combination of these two apparently different views on Mathematics was offered by one of Whewell’s students, Augustus De Morgan. This renowned mathematician largely accepted his teacher’s philosophical objections and he elaborated them in his own peculiar style. Formal analysis is not enough, according to De Morgan, who insisted on a second step that should also be taken, apart from to define the terms to be used as parts of an equation, in relations to the other parts, as being internally consistent. In addition, and for such an approach to be complete, one would require the explanation of the algebraic results, so that particular meanings can be attached to symbols in order for them to become interpretative and descriptive. A mathematician’s basic material does not consist only of several scattered parts of quantitative and measurable aspects but also of qualitative differences that should also guide the scientist who endeavors to reunite them. Edward Ruth, Alfred Marshall’s private tutor who prepared him for the 1865 Mathematical Tripos, was one of De Morgan’s ablest students.

2.3 Remarks on Marshall’s Mathematical Thought

  • 9 For a discussion on the use of curves and mathematics between Marshall and Jevons, see also lt. 62, (...)

29On the day he was elected a professor at Cambridge, Alfred Marshall publicly declared that “never was there a Science more urgently in need of all the work that all her best sons could give her than Economics is now” (Pigou, 1925, 152). One might suggest numerous reasons to explain Marshall’s hesitant stance towards Utilitarian philosophy (see Shove, 1942, 717-719), which at that time was reaching its peak. Marshall’s theoretical stance diverged substantially from that of Walras, Jevons and the other marginalists. The founder of the Lausanne School is scarcely mentioned in the Principles, whereas Jevons managed to attract the penetrating lights of Marshall’s criticism and annoyance (see lts 17, 18 and 31). Jevons’ “brilliant but hasty, inaccurate and incomplete brochure”, which was “as far removed as possible from the painstaking, complete, ultra-conscientious, ultra-unsensational methods of Marshall” (Keynes, 1951, 155) was built upon the idea that economics “deals throughout with quantities” while an economist’s endeavors should be concentrated towards the transformation of our discipline into a “mathematical science in matter if not in language” (Jevons, [1879] 1965, vii)9. To each of these points—the definition of economics, the characteristic feature of “utility” and the application of mathematical analysis to the social sciences—Marshall had developed a more complex view, in which the element of time becomes the epicenter.

30Largely inspired by Cairnes (1861), Marshall follows a clear methodological distinction between premises and conclusions in Political Economy. The former are to be traced within the real facts of life. An economist is undoubtedly bound by empirical evidence, for he deals with “man as he is: not with an abstract or ‘economic’ man; but a man of flesh and blood” (Marshall, 1961, 26). Economic laws or “norms”, inferred by its premises, correspond to “the effects which will be produced by certain causes … subject to the condition that other things [are] equal” (ibid., 1961, 36). Once again, Marshall strongly echoes Hume’s criticism of scientific inquiry by posing the necessary conditions for the method of induction to be legitimate, as he implies that the instances of the causal relations must—at least—be observable. Compare that with Hume, when he says that: “all reasonings from experience are founded on the supposition, that the course of nature will continue uniformly the same. We conclude, that like causes, in like circumstances, will always produce like effects” (Hume, [1740] 1965, 15). Such a fundamental hypothesis, i.e. a ceteris paribus, sets a logically accepted formal basis for the general methods of the exact sciences to be broadly applied to economics, while its application bears the cost of overlooking a major difficulty: “the condition that time must be allowed for causes to produce their effects … For meanwhile the material on which they [economists] work, and perhaps even the causes themselves, may have changed” (Marshall, 1961, 36). The questions thus raised are complex. Can such a hypothesis be substantiated in any way other than observation? Are we obliged to choose pure mathematics at the expense of a theory that does not incorporate time?

31In Marshall’s work, realism appears as a process of reasoning in time, without being driven by the exact context of mathematical rigor. To regard men and women as largely acquiring latent and quantified habits underrates the “deep and rapid changes in human nature”, as it gradually progresses from one historical phase to another, reinforced by a more favorable distribution of income. The difficulty of economics resides in the fact that it attempts to blend two mutually exclusive tensions. While the measurement of human motives on the basis of money provides the necessary quantitative form so that the broad concepts of “mathematical machinery” (Whitaker, 1996, 307) can be applied, the incessant volatility of the “forces of human nature” manifests their analytical insufficiency. These forces are “numerous, less definite, less well known, and more diverse”, and human nature from which they spring is “uncertain and less homogeneous” (Marshall, 1961, 772). Accordingly, a “student of social problems” is frequently asked not to be blinded by his fervent and unappeased desires, as these emerge in his theoretical ideals, and instead to “accept mankind as he finds them; and must base his estimates on that which is practicable. He must nourish the ideal in his heart: but his actions, his conversation, and even his thought must be occupied mainly with the actual: he must resist every temptation to make a short cut to the ideal” (Pigou, 1925, 84).

32Marshall placed human nature directly alongside the classical cost-based theory of value as a supplementary missing part through which time was appreciated. The chapter “On Wants and their Satisfaction”, which took its final form only after the publication of the 4th edition of the Principles (Marshall, 1961, Vol. II, 234), can be seen as an attempt to demonstrate the unilateralism of the classical view of the theory of value, but not its supremacy over it. If one were to choose between them, then we would prefer the classical view, for “it is the science of activities and not that of wants”, given that the former stands for “the interpreter of human history” (Marshall, 1961, 90). As for the idea of utility, it cannot be thought of as a particular quantity whose level is known to the individual beforehand, but should exclusively be thought of as a varying “‘prospective’, that is dependent on the benefit which will at a future time accrue from its possession” (Pigou, 1925, 93). Utility is inextricably linked to the passage of time as it is mediated between the moment that the individual expresses his or her need and the moment of its possession, i.e. after the necessary effort for its attainment has been expended. In a short reply to Edgeworth, Marshall raises the need for “a mode of exposition in which the dynamical character of the problem [of Hedonical Calculus]” (lt. 80) is appreciated. Inversely, the price of a commodity can be attributed to an individual’s marginal utility, but not to anyone else’ marginal utility, since interpersonal comparisons are not allowed. The demand schedule that corresponds quantities to prices in the form of a list can be compiled under “a given time and under given conditions” (Marshall, 1961, 100). This corresponds to a stationary state diagram in which the various pairs of quantities and prices would be made, under ceteris paribus. As Shackle ([1972] 1992, 265) has pointed out: “We are saying, then, that in a non-timeless world, a world of earlier and later, there can be no pre-reconciliation of choice, therefore no fully rational choice, therefore no rigorous analysis of conduct as reasoned response to fully-known circumstances”. Such a world corresponds to that of Marshall.

33The complementarities that exist between the early-19th-century’s static view of man and its accompanying economic theory were fully reflected in the use of mathematics, whose “subject-matter is constant and unchanged in all countries and all ages” (Marshall, 1961, 764). It was indeed in this context that Marshall expounded his criticism of Jevons’ Theory as having made no “important proposition which is new in substance” (Pigou, 1925, 93). The basis of this remark was that Jevons’ persistence on statical mechanics did not allow for the functional determination of any quantity by others, maintaining instead a favorable distance between economics and the problems of real life. As a consequence, Marshall, astute mathematician as he was, should be approached as a descendant of a long-standing Cambridge theoretical tradition. His views on economic theory certainly diverged from the growing dominance of marginalist thought, which began to prevail in the first decades of the 20th century.

3. Evidence from his Correspondence

34Until the mid-19th century, the authority of Euclid’s Elements was largely unquestioned in England. This situation was gradually changing, however, as non-Euclidean geometries were opening up new possibilities for analytical interpretation (Richards, 1988, ch. 2). In opposition to the emerging analytical counterpart, the subject mater of Euclidean geometry was fundamentally descriptive and constructive in relation to the capacity of the human mind to interpret its results in three dimensions. As we have already mentioned, Clifford was also in search for different geometrical spaces, other than the Euclidean ones. Geometry’s interpretative advantage, which preoccupied many of the most eloquent mathematical figures of that period, was particularly conceptual. Analysis, on the other hand, was seen as having the potential to surpass the obstacles posed by the need for a student’s mind to be able to offer a conceivable interpretation of its results from experience. What is more, it was also understood as an attempt to broaden the perspectives of mathematical investigation beyond that of our classical approach to space. Nonetheless, unless mathematics was to remain a truth-seeking field, its analytical conclusions, since they were not limited by commonsense spatial qualities, still had to be interpretative.

35While at Oxford, Marshall developed a lasting friendship with the influential academic Benjamin Jowett, who also played a crucial role in his career (Whitaker, 1972). They enjoyed long conversations together about the nature of scientific knowledge and also about the book Marshall was already considering writing. However, it was not before Marshall had left Oxford that Jowett wrote to him about his sense of unease towards the use of mathematics in Political Economy. We read:

Will you be surprised at my attacking you about Symbols? … I seem to see that various persons such as De Morgan and Boole, have tried to applied mathematics to subjects which did not admit of their use & have rather deluded themselves & others … Now I do not object to their application to Political Economy, provided they are not regarded as a new method of discovery, but only as a mode of expressing a few truths or facts which is convenient or natural to the few whose minds easily adopt such symbols. Political Economy is human & concrete & should always be set forth in the best literary form: the language of Symbols may be relegated to notes and appendices (lt. 148)

36Once again, the issue of the nature of economic material is raised and we find some striking similarities in the way Jowett and Marshall doubt about the services provided by mathematical analysis to Political Economy. The quoted part constitutes a synopsis of the assessment being espoused by many academic circles in mid-19th-century England, while it also functions prophetically, since the Principles were eventually written in the manner described in the last sentence. Diagrams were to be put in footnotes while mathematical analysis went in a separate Appendix.

37Marshall himself had already been independently thinking over such methodological issues. As we may conclude from his correspondence with Edgeworth, whose idea of economics adhered in an exaggerated fashion to the instructed callings for exact mathematical methods (lt. 92), Marshall reveals a reluctant, occasionally gentle although not ultimately neutral stance on the matters they discussed. He occasionally appears to take a straightforward position, such as when he wrote: “His book [Mathematical Psychics] though it shewed great vigour, yet on the whole disappointed me” (lt. 98). He confessed this sentiment to Foxwell, although kept it from many of his other correspondents. Nonetheless, in one of his first letters to Edgeworth, he openly replies to him that:

As regards the applications of geometrical rather than analytical reasoning I have not such decided views as you suppose.

When tackling a new problem I generally use analysis, because it is handier. And in the book which I am just going to begin to write, I shall retain (in footnotes) a little mathematical analysis, for questions which I can’t reduce under the grasp of curves. But—partly because, curves require no special training, partly because they bear more obviously on the science of Statistics, I intend never to use analysis when I can use geometry. (lt. 81)

38After the Principles had made their first appearance in print, Edgeworth focused his attention on the “Mathematical Notes”, where Marshall had collated the analytic results for his theory along with an attempt to interpret them in a realistic way. Whenever he was unable to do so, or when a realistic counterpart seemed to have been exceptionally imperceptible, Marshall provided a full description of the requisite assumptions if the results were to acquire any particular meaning. This was also the case with Note V (Marshall, 1961, 841), the algebraic expression of the discounting of pleasure, to which Edgeworth seemed to object (lts. 340, 354 & 355). R–t, Marshall expounded, represents a uniform discount rate that obeys the exponential law. In other words, it is supposed to be “the same for each element of time” (ibid.) and does not correspond to any real situation in which the process of “discounting” future pleasures follows the irregularities that occasionally occur when a person chooses to postpone the satisfaction of a pleasure or where the pre-considered frequency is not realized. In the 2nd edition of the Principles, and following his correspondence with Edgeworth on that issue, the phrase “this result belongs to Hedonics, and not properly to Economics” was added to this Note, so as—we presume—to indicate his distance from the approach taken by Edgeworth.

  • 10 Marshall refers to Cunningham (1892), Some Improvements in Simple Geometrical Methods of Treating E (...)

39Mathematical expressions should of necessity acquire economic meaning. In his reply to Edgeworth about the “successive cost curves” used by Cunningham10 (lt. 406), Marshall persists on the issue of technical solidity with the understanding of real economic processes, admitting that a diagram cannot correspond to a situation that is depicted with successive states of production unless it was to be meant temporarily. On another occasion, Edgeworth wrote to him that:

There seems to me to be a sort of transition or “second intention” as the schoolmen say—in the meaning of marginal; The primary signification as I have always understood not relating to time, but simply to quantity in general corresponding to a first differential of a continuous function … with well-known relations to equilibrium and … a position of maximum (lt. 427)

40Marshall responded:

I can see no double use of “marginal”: to my mind it always has the same meaning, but the details are different for different trades [and] for different periods of time. That seems to me not a technical, academic, or arbitrary arrangement; but one inherent in the nature of things … Your references to maxima problems seem to me always to raise an impenetrable cloud because you do not state what all the terms used mean. (lt. 428)

  • 11 For an exhaustive analysis of this subject and the “cost-controversy” of the 1920s, see Hart (2012) (...)

41There is a simmering tension between the two views that is to be found primarily in the way the economic meaning of mathematical expressions is recognized. Marshall also judged Cournot along the same lines when, in a letter to Flux, he added that “My confidence in Cournot as an economist was shaken when I found that his mathematics re I.R. [regarding Increasing Returns] led inevitably to things which do not exist and have no near relation to reality” (lt. 564).11

42That Marshall had certainly incorporated the old debate about what analysis means in particular and its use in economic inquiry becomes apparent in his correspondence with Foxwell, wherein they discuss a proper educational schedule for economics. In one letter Marshall writes that Theory cannot be ranked higher than History since analytical considerations are applied in a major way in the former and “analysis is unprofitable when separated from the study of facts”. The classification of economic studies should commence with an “Unanalytical acquaintance with leading facts as a basis” (lt. 515, my emphasis). It should by now be clear that Marshall’s main concern was on the real study of the “ordinary business of life”, detached from any algebraic reasoning that may lead to non-economic conclusions. Deep knowledge and analytical skills made him understand the very nature of the emerging techniques that were capturing the hearts and minds of the young.

43Finally, two more citations from his correspondence can further elucidate my position on the methodological status of Marshall. Regarding the widespread OLS method, he straightforwardly stated: “I think that harm has been done by treating the results as ‘economic’…” (lt. 634). In an extremely short reply to Walras, right after the latter had sent him one of the latest editions of his Elements of Pure Economics ([1926] 2010), Marshall points out that “the right place for Mathematics in a treatise on Economics is the back ground. But I think it is most desirable that different seekers after truth should take different routes” (lt. 273). Marshall’s route certainly did not pass through mathematical analysis.

4. Analysis, Synthesis and Time

44A consequence of what distinguishes Marshall from all other neoclassical thinkers is his interpretation of Time in a synthetic rather than an analytic fashion. This means that in Marshall, Time is taken as a given (data) and as a condition subject to which a great part of his theory is constructed. Short and long periods, temporary equilibrium, partiality in the view of the markets, indeterminateness of the equilibrium position, stability instead of existence, increasing returns and biological analogies, history and epistemology, all spin around in that special category of Time.

45As we have already seen, the descriptive approach to analysis raises a vast number of problems and renders time as a misapprehended spatialized variable, t, which can move forwards and backwards and that is void of all the qualities that make it unmanageable and unperceivable. In his Appendix C, entitled “The Scope and Method of Economics”, Marshall favors the idea that a person’s actions in society are too broad to be analyzed by a single field, making it not possible to gather all the appropriate knowledge. Furthermore, he finds John Stuart Mill’s point that economic forces can be conjoined mechanically to be rather exaggerated. In his view, the chains of deductive reasoning, especially the long ones, belong to laboratory inquiry and not to reality, since individuals change their characters and tastes in a rather arbitrary way, making it difficult—if not unattainable—to formalize them. As a consequence, predictions of the future cannot rely upon them.

46The economist’s analytic considerations are performed in a deductive fashion even though their basis remains inductive. It so happens that for Marshall, past and future, the two inaccessible aspects of Time, call for an explanation and a prediction respectively. The two operations are not that different in essence. The former follows the route from effect to cause, whereas the latter follows that from cause to effect. In the first two editions, Marshall had cited the following paragraphs from Schmoller, which were removed from the 3rd edition:

What we call the inductive process starts from particular, from observation, and seeks the rule, which declares to be true of a class, what has been found to be true of the observed instances. The more complex a phenomenon, and the even more imperfect our observation of the complex conditions, resulting as they do, from the combination of the most various causes, the more difficult is it to discover the true rule, the more often do we get no further than hypotheses, and provisional guesses as to the uniformity of the result. But we apply even these to obtain further conclusions.

  • 12 From an article by Schmoller in Volkswirtschaft, which Marshall cites several times in the Principl (...)

Deduction, which rests on the same tendencies, the same beliefs the same needs of our reason as induction, consists in the wider application of those rules as to causality which have been obtained by induction. What was true in accurately observed cases, must be true in all exactly similar cases; the rule it sought for only in order to apply it further. Every rule attributes a predicate, a mode of action, a characteristic to a class of things; an analysis of the notion of the subject and predicate shows what is contained in the rule in question, to what purposes it applies, what cases come under it, what it can account for (quoted in Guillebaud, 1961, 767)12

47This section seems to suggest that hypotheses are needed whenever the observed material is complex and that, according to the deductive process, the flow of Time does not change the observational instance. Otherwise, we would not be able to broaden the application of the rule.

48That the future cannot be predicted, is almost a common ground. Marshall however, points out that even the Past cannot be explained with certainty. Schmoller’s quotation blends together our own account with the functions of induction and deduction, in a manner that strongly resembles Pappus’ account of analysis and synthesis. It seems that induction and deduction stand as a special case of analysis and synthesis, for they concern the steps we take to get from the general to the specific, and vice versa.

49Once again, Hume’s influence can also be acknowledged here. Marshall approaches causality not as an inner property of historical events, which, even if they are chronologically categorized, the causal relations we apply still correspond to our explanation of them. The generalization of the rule through which history is explained pre-assumes that distant events should be similar to each other in the abstract. “Both the difficulty of analysis and the need for it are increased by the fact that no two economic events are exactly alike in all respects” (1961, 774). Elsewhere he adds: “for a while deduction plays the chief role … and then calls on induction again to do the main share of work” (ibid., 781)—an idea which is strongly Newtonian.

50This can also be seen in a letter he sent to J. N. Keynes (lt. 321) when Keynes asked him his opinion of Chapter 9 of his Scope and Method to Political Economy ([1890] 1999). In his reply Marshall made two very important observations, which he followed faithfully in his own work at least. He actually offers two critical comments. One is about the mutual dependence between induction and deduction, where he argued that a link exists between the latter and analysis, since if we followed the opposite path this would most probably lead us to conclusions that seem not to be true. In his second comment, he attempts to distinguish between analysis and theory: the former is immune to facts and can stand on its own reason, whereas theory is inextricably linked to facts.

I have only two points of any importance to urge. One is an old point. You make all your contrasts rather too sharply for me. You talk of the inductive and the deductive method, whereas I contend that each involves the other and that historians are always deducing, & that even the most deductive writers are always implicitly at least basing themselves on observed facts.

And in consequence you first allow to the inductive method pure & simple more by far than I should allow to it, & similarly for the deductive method: and afterwards, take back a great of what you have allowed by saying that after all deduction involves induction & vice versa. Thus in the end you come to pretty nearly the same result as I should... but you start by saying things that seem to me not true. It is a mere question of arrangement: but I think it is a very important one particularly.

I think the right order is first to emphasize the mutual dependence of induction and deduction & afterwards to show in what kinds of inquiry the economist has to spent a greater part of his time in collecting and arranging and narrating facts, & in what kinds he is chiefly occupied in reasoning about them & trying to evolve a general process of analysis & general theories which shall show the Many in the One, the One in the Many.

My second point is that you continually use the word theory where I should use analysis. This seems to me in itself to cause confusion which is increased by the fact that later on you exclude modern facts from history; & yet you do not boldly say they are parts of theory. If they are then I agree with you that a study of theory should come before a study of history. But I do not myself like to put the case in this way (lt. 321, emphasis in the original).

51Mathematics prompts our thinking to produce long chains of reasoning that seem to be inappropriate for economic theory. The tacit assumption is that when such practices are done by economists then the same causal relations hold for several different cases within real history, or in historical time. Marshall vividly opposed the idea of generalizing a particular case by making it to be true for different moments in history. This bold logical misstep cannot survive historical facts and is also to be found in his reluctance to render the temporary equilibrium of a particular market to a general one. Moreover, he does not miss the opportunity to emphasize the conditions that ought to remain unchanged in order for them to produce reasonable results.

52The “analytical method to economics” is contrasted with the “historical” one, which relies primarily on facts (we could also say data, i.e. givens instead of constants) and observation (ibid., 30). Neither of these, however, is in itself complete. Economics needs to “adapt itself closely to the real phenomena of the world; but none the less must it have a firm backbone of careful reasoning and analysis” (ibid., 769). The view of the economy as an evolving organism that constantly changes its frame and institutional basis goes together with a theoretical perspective that resembles Biology more than Mechanics (whether Statical or Dynamical). Analysis is a general concluding relation, a terminal point, from which syllogisms need to depart in order to end at something that can hold within reason. In other words, analysis operates deductively, for it forms a general rule that is applicable to many particular cases.

Analysis is nothing else but breaking up a complex conception into scraps, so that they may be easily handled and thoroughly investigated. … Science must study facts, ascertain which of them are representative and normal and then analyze and reason about normal conditions … Scientific analyses like the operations of machinery are in their first attempts always clumsy and often a little ridiculous. They are however, changing the face of the world: because their progress is cumulative throughout the whole life of the race, while each man’s common sense, like his skill in handicraft, dies with him (Marshall, 1961, 501-502).

53The “complex conception” must be given in advance and “common sense” as it is determined by the continuing alteration of the social context cannot be inherited. Economics, quite like any other science, undertakes the study of effects, i.e. reasoning’s final acts; but not any kind of effects, only those produced by certain causes. Long chains of reasoning are based upon the same Humean metaphysical suggestion that Marshall indicated at least twice in his work. The first occasion was when he wrote that:

History says an event happened “at the same time as the cause”, or “after this cause”, and reason infers “therefore because of this cause”. It suggests post hoc (or cum hoc) ergo propter hoc (Marshall, [1885] 1925, 176).

54While later he wrote:

By merely placing things together in a certain order, and consciously or unconsciously suggesting post hoc ergo propter hoc, the historian takes on himself some responsibility as a guide (Marshall, 1961, 775).

55History is a factual garb of time. It is substituted by reasonable connections, which are solely based on the grounds of an irrational jump from a succession of events to causality between them. For to prove the latter, one should be able to reason enough. It thus follows that deductive links between cause and effect can only be patented for small intervals of time, i.e. they can work effectively only for temporal situations. Otherwise, the repetition of a particular chain of events ought to be assumed and, in this way, the process and successive links of thought resemble that of production by machinery.

…when there are any processes of investigation or reasoning in which the same kind of work has to be done over and over again in the same kind of way; then it is worth while to reduce the processes to system, to organize methods of reasoning and to formulate general propositions to be used as machinery for working on the facts and as vices for holding them firmly in position for the work … A wise nurse, who reads her patients by instinctive sympathy may give better counsel on some points than a learned physician (Marshall, 1961, 779).

56The scientific method of economics is not denied, but it is its complementary aspect—induction—which is framed as necessary in order for theoretical considerations to attain a real substance. Time’s real consequences are reflected through History, which unfolds unmediated by reason.

Conclusion

57Martins (2012) has expounded the view that the distinction between geometry and algebra that is to be found within the works of Newton most definitely also have influenced Marshall and others in Cambridge. Moreover, it is certainly to be found in the manner in which Keynes treated the paradigm of general equilibrium, which in his own account he treats as a special case of a broader concept. In this article, however, the purpose was twofold. First, to show that there is a particular view about the workings of mathematical knowledge and the truths they encapsulate and, second, to offer a suggestion as to what Marshall really meant with his “obscure” reference to Newton’s method and connect that with his ideas on Time. Utilitarianism in economic theory was boosted by the development of differential calculus, or by what was later to be called mathematical analysis. Within this framework, time was treated as an analytical variable, as an arithmetical entity, to be included in functions or as an index used to date discrete quantities. Analytical thinking, however, was not completely and widely accepted.

58Even so, it appears that Alfred Marshall had great respect for the alternative strand of inquiry, which travels from the specific to the general and from that back again to the specific: “the one in the many and the many in the one”, as he called it. A specific market, or a partial one, even if it exhibits signs of equilibrium possibilities, corresponds to local results that cannot be rendered general unless additional hypotheses of ceteris paribus are introduced. However, this is in complete disagreement with reality itself, and it is within Marshall’s work that the distinction between static and stationary states was meaningful. It seems that among the different hypotheses to be considered Time was of greatest importance, for it was due to its constant flow that the conditions of real life were incessantly changing. But this is not all. The choice of mathematical method is not neutral to value judgments that reflect the precise meaning we give in each variable we deal with.

Various aspects of this article appear as Chapters 3 and 5 of my thesis dissertation (Missos, 2015). Nicholas J. Theocarakis, associate professor in the University of Athens, has aided and advised me at every stage of this article and I am deeply indebted to him for his ongoing support. Among those who have helped me on special issues I would also like to thank the two anonymous reviewers for their concrete and accurate points they have raised. Responsibility for errors remains my own.

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Notes

1 For a view on the reasons why Marshall may have abandoned all efforts to have his promised second volume of the Principles published, see J. K. Whitaker (1990) as well as Keynes’s biographical note.

2 24th March 1908 (incomplete), letter n. 899. References from Marshall’s correspondence will henceforth be cited using the counting system introduced by Whittaker (1996), as “lt. ##”.

3 Temporal sequence, in Hume’s Treatise of Human Nature (part 3, section 2), was considered an essential property of causality itself. He explicitly maintained that there is a “priority in time in the cause before the effect”.

4 Marshall was absorbed with Kant’s philosophical inquiries during his vacations on the Continent. This fact is well known from Keynes’ biographical article as well as from Pigou’s reminiscences of Marshall: “He used to wander about Switzerland, carrying on his back, not, as some of us do, apparatus and food to enable us to climb high mountains, but Kant’s Critique of Pure Reason” ([1781] 1998, 209). In Kant’s Critique, the German word “Wechselwirkung” essentially means “interaction” or “mutual determination” as Marshall has it, rather than “mutual causation”, as Hicks has suggested.

5 In a footnote, Whittaker added: “The aspect of Newton’s methods that Marshall had in mind remains obscure.”

6 For a complete understanding of the Newtonian Method see Guiccardini (2009). My inquiry on Newton’s thought in the section below is limited to the purposes of this article.

7 “[T]he ideas, now in the minds of contemporary mathematicians, lie very remote from any notions which can be immediately derived by perception” (Whitehead, [1920] 1963, 25).

8 See Guiccardini (2006, 1727).

9 For a discussion on the use of curves and mathematics between Marshall and Jevons, see also lt. 62, in Whitaker (1996).

10 Marshall refers to Cunningham (1892), Some Improvements in Simple Geometrical Methods of Treating Exchange Value, Monopoly and Rent, Economic Journal, 2, 35-22.

11 For an exhaustive analysis of this subject and the “cost-controversy” of the 1920s, see Hart (2012).

12 From an article by Schmoller in Volkswirtschaft, which Marshall cites several times in the Principles.

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Vlassis Missos, « Marshall on Time and Mathematical Analysis », Œconomia, 7-1 | 2017, 61-86.

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Vlassis Missos, « Marshall on Time and Mathematical Analysis », Œconomia [En ligne], 7-1 | 2017, mis en ligne le 01 mars 2017, consulté le 21 juillet 2017. URL : http://oeconomia.revues.org/2572 ; DOI : 10.4000/oeconomia.2572

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Vlassis Missos

University of Athens, bmissos@econ.uoa.gr

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