# Mathematics, science and the Cambridge tradition

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## Abstract

In this paper the use of mathematics in economics, and the way in which mathematics contributes, or not, for economics to realise its potential as a science, will be discussed, by comparing two approaches to mathematics, a deductivist (algebraic) approach to mathematics, and a realist (geometrical) approach to mathematics. The differences between these approaches will be discussed in the context of the Cambridge tradition, while arguing that the work of key authors of this tradition was initially characterised by tension between their philosophical vision and their mathematical method, most notably in the case of Marshall, and that Keynes resolved this tension by focusing on uncertainty and probability relations instead of certainty and exact relations, while privileging a realist approach to the open and interconnected nature of economic phenonema.

Martins’ paper “Mathematics, Science and the Cambridge Tradition” clearly has the makings of a fine publishable paper, but I have found a number of omissions and some possible misinterpretations. Also in the early pages he does over repeat himself – I’m sure he could edit them and remove this.

Detailed comments:

P3: Lionel McKenzie should be given credit alongside Arrow and Debreu.

P4 line 5: for

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P6: Astronomy does not construct laboratorial situations yet the author uses it as an example of a natural science?

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P9 para 3 line 2: for , otherwise the author is a one-handed economist.

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Last line: why is a relation a distance?

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P13 top line: but it is Marshall through Pigou

Para 2: Marx not S and R, was the most important of all in Sraffa’s view.

P15 para 2: the most thorough and convincing account of this is in Neil Hart Equilibrium and Evolution AM and The Marshallians Palgrave 2012

P16 para 2 line 2: either Arthur Cecil or AC Pigou

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This discussion needs references to the work of Gay Meeks, Flavio Commin and John Coates.

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P20 para 1: McKenzie

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Quote from Lawson line 5: omit

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P22 para 2 What does Groenewegen say about this?,

PS had excellent philosophical reasons for this, see for example, Velupillai’s tribute to Krishna Baradwaj and Stephanie Blankenburg’s PhD, Ch2.

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Para 3: What of the type of maths they used – on this again see Velupillai’s papers.

P23 para 1: But see the counter arguments by eg. Partha Dasgupta and the very strong practical case exemplified by his very short Introduction to Economics OUP.

Para 3: Where is a reference to O’Donnell?

P24 para 3: What about Marx’s ideal versions of actual modes of production? for (twice)

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Para 5 line 1: Actually these are due to JM Keynes, see my paper “Theoretical methods and unfinished business”, first published in 1987 and reprinted in Sardoni (ed) 1992 On Political Economists and Modern Political Economy, Routledge

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I’m not sure I agree – the whole greater than the sum of the parts is exemplified in the fallacy of composition to be guarded against in macroeconomic analysis.

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See Hart 2012

P27 para 1: Where are references to Gay Meeks and Rod O’Donnell??

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P28: Frank Hahn would say the object is to see how far inferences drawn from this approach illuminate what is observed and also to establish conditions that have to be satisfied for conjectures to be true. not

Again see Hart 2012

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May I urge the author to read my 1987/1992 paper and my essay on Economics and Maths in GCH Capitalism, Socialism and Post-Keynesianism, Edward Elgar, 1995.

The author states in the introduction: “The mathematical methods are deductivist in the sense that they presuppose constant conjunctions in the form ‘if event X, then event Y’.” This, though, is a description of causality. Deductive reasoning is in the form ‘if antecedent X, then consequent Y’ and has nothing to do with causality:

“… deductive chains of reasoning cannot on their own establish the existence of causal processes in the real world.” (Hodgson, 2001, p. 76)

The author seems not to realize that ‘deductive’ refers to mathematics and ‘causal’ to physics.

“It is a well-known jest that ‘a mathematician is a scientist who knows neither what he is talking about nor whether whatever he is talking about exists or not’.” (Cartan, quoted in Ronan, 2006, p. 70)

Nobody has ever criticized mathematicians for this. Quite the contrary, the plain fact that products of pure deductive reasoning correspond in numerous cases admirably to the objects and processes of reality has puzzled physicists, philosophers, and the mathematicians themselves since the Greeks (Wigner, 1979), (Kline, 1982, pp. 328-354), (Velupillai, 2005), (Penrose, 2007, pp. 613-614).

The distinction on p. 1 between a Cartesian a priori approach and a Newtonian a posteriori approach is artificial and misleading. Newton applied the deductive method strictly in the spirit of Euclid:

“But it was a second and more important quality that struck readers of the Principia. At the head of Book I stand the famous Axioms, or the Laws of motion: … For readers of that day, it was this deductive, mathematical aspect that was the great achievement.” (Truesdell, quoted in Schmiechen, 2009, p. 213)

The author oversimplifies the scientific stances of Newton and Descartes as antithetic. Yet:

“Despite the hostile attitude to Cartesian science, Newton’s debt to Descartes appears again and again in his Principia. To the cognoscenti the presentation of Newton’s laws as “Axiomata, sive Leges Motus” … would have had rather obvious overtones of Descartes’s Principia, … It was from Descartes’s Principia that Newton learned the law of inertia …” (Cohen, 1999, p. 46)

With regard to the relation between analysis and geometry Newton himself contradicts the author:

“The Propositions in the following book were invented by Analysis. But considering that the Ancients … admitted nothing into Geometry before it was demonstrated by Composition I composed what I invented by Analysis to make it Geometrically authentic & fit for the publick.” (Newton, quoted in Cohen, 1999, p. 122)

With regard to “intuition and creativity” (p. 10) in Cambridge the author’s description seems to be a bit too idealistic:

“In patriotic duty bound, the Cambridge of Newton adhered to Newton’s fluxions, to Newton’s geometry, to the very text of Newton’s Principia … Thus English mathematics were isolated: Cambridge became a school that was self-supporting, self-content, almost marooned in its limitations.” (Forsyth, quoted in Mirowski, 2004, p. 355)

“Marshall … was a product of this system, which valued … down-to-earth geometry over continental analysis, and regimented conformity in puzzle solving and humble prostration before timeless truths over originality.” (Mirowski, 2004, p. 355)

The crux with mathematics in theoretical economics is that it is borrowed ready-made from the math or physics department and then forced upon reality. The calculus is a case in point. A profit maximum exists only with decreasing returns. Whether returns decrease or increase is an empirical question and we may well find out that, say, decreasing, constant, and increasing returns are equally distributed among the firms in an economy. Hence, if one is determined to apply calculus, and this is the whole point of marginalism, one has no choice but to posit a well-behaved production function whatever the facts may be. Thereby, one is from the very outset destined for missing two-thirds of reality — independently of whether the behavioral hypothesis is plausible for the rest or not (granted, for the sake of argument, that well-behaved production functions actually exist).

“The mathematical language used to formulate a theory is usually taken for granted. However, it should be recognized that most of mathematics used in physics was developed to meet the theoretical needs of physics. … The moral is that the symbolic calculus employed by a scientific theory should be tailored to the theory, not the other way round.” (Wittgenstein, quoted in Schmiechen, 2009, p. 368)

Newton developed calculus in order to come to grips with velocity and acceleration. The defining characteristic of economic equilibrium is the absence of any change whatever. The mathematics is the same but something does not fit, that much is obvious. Yet:

“Here, just in the art of painting, we should not blame colors for the horrible works some ‘painters’ may produce with them.” (Georgescu-Roegen, 1979, p. 323)

References

Cohen, I. B. (1999). The Principia; Mathematical Principles of Natural Philosophy, chapter A Guide to Newton’s Principia, pages 11–370. Berkley, CA, Los Angeles, CA, London: University of California Press.

Georgescu-Roegen, N. (1979). Methods in Economic Science. Journal of Economic Issues, 13(2): 317–328. URL http://www.jstor.org/stable/4224809.

Hodgson, G. M. (2001). How Economics Forgot History. The Problem of Historical Specificity in Social Science. London, New York, NY: Routledge.

Kline, M. (1982). Mathematics. The Loss of Certainty. Oxford, New York, NY: Oxford University Press.

Mirowski, P. (2004). The Effortless Economy of Science?, chapter Smooth Operator: How Marshall’s Demand and Supply Curves Made Neoclassicism Safe for Public Consumption but Unfit for Science. Durnham, London: Duke University Press.

Penrose, R. (2007). The Road to Reality. New York, NY: Vintage.

Ronan, M. (2006). Symmetry and the Monster. Oxford: Oxford University Press.

Schmiechen, M. (2009). Newton’s Principia and Related ‘Principles’ Revisted, volume 1. Norderstedt: Books on Demand, 2nd edition.

Velupillai, K. (2005). The Unreasonable Ineffectiveness of Mathematics in Economics. Cambridge Journal of Economics, 29: 849–872.

Wigner, E. P. (1979). Symmetries and Reflections, chapter The Unreasonable Effectiveness of Mathematics in the Natural Sciences, pages 222–237. Woodbridge, CT: Ox Bow Press.

This paper is a welcome contribution to the ongoing debate about the role of mathematics in economics. The particular contribution stems from not treating mathematics as a homogeneous category. Rather Nuno Martins distinguishes between deductivist Cartesian mathematics, which support marginalist models of constrained optimisation, and Newtonian mathematics, which support a pluralist methodology. It is argued that it is the latter which characterises the Cambridge tradition. The approach, as in the particular specification of open and closed systems in the introduction and the identification of the Cambridge tradition in ontological terms in the conclusion, is critical realist.

The distinction between types of mathematics is important for qualifying any tendency to criticise mathematical economics per se. It is therefore particularly useful to have a detailed account of a non-deductivist use of mathematics and its history in economics. But I would have liked to see the argument be made more concrete in the sense of making more clear what is involved for economics in adopting a Newtonian approach. It is only towards the end of the paper, for example, that it is made specific that it requires mathematics to be used alongside other methods. More clarity on what is involved would also help illuminate the issue (noted on page 23) of exclusivity. The heterodox critique of orthodox economics is generally put as an objection to formal mathematics as the exclusive method. Here, as I understand it, the argument is that it is in the nature of the deductivist approach to be exclusivist while the Newtonian alternative is by its nature not exclusivist. This is a very important argument which could usefully be brought much more to the surface.

The distinction is drawn between theory and method, with the question raised in section 2 as to whether mainstream economics is better characterised by its method than its content. The discussion then proceeds to an argument that it is not mathematical method per se, but specifically deductivist mathematics which defines mainstream economics. It is suggested (pages 4-6) that marginalism facilitated the rise of deductivist mathematics in economics, but that the insistence now on deductivist mathematics does not necessarily entail marginalism. But the nettle is not really grasped as to how far there is still interplay between theory and method, such that deductivist mathematics encourages (or indeed requires) certain types of theory.

While the sweep of the paper is wide, there is one literature which is missing that would have helped the author to concretise Newtonian methodology and also fill a gap in the history of the Cambridge tradition. The interpretation of Newton (a substantial subject in itself) given here is closer to the one current in eighteenth-century Scotland than in France (noted on page 11) or in England until much later. Smith challenged Descartes’s deductivism, employing instead Newton’s experimental methodology which sought to build theory by building on principles drawn from detailed observation. But these principles were provisional and theory was specific to context, developed through a process of abduction. Further this methodology was based on an epistemology Smith shared with Hume, who was in turn a direct influence on Keynes’s theory of probability. This epistemology was formed in relation to the Scottish philosophy of common sense, which influenced Moore. Further, the way the paper is organised means that the discussion of development of ideas is not always chronological; tracing the development chronologically by including the route from Newton to Hume and Smith and then through them back to Cambridge might be more satisfactory.

This paper covers a lot of ground: the critical realist critique of deductivist mathematics, the history of the two types of mathematics being discussed, a history of deductivist mathematics in economics and of Newtonian mathematics in Cambridge, an account of the Cambridge tradition in terms of tensions between the appeal of certainty against the appeal of realism, Keynes’s resolution of this tension and, finally, the Cambridge philosophical tradition. It may be that the paper is trying to do too much. But it certainly addresses controversial questions. With some streamlining, so that there is a clearer focus on the main strands of argument (avoiding controversy where it is not central to the argument) and filling in some gaps as suggested above, this could be a good contribution to the literature.

In her comment, Sheila Dow summarizes:

“Rather Nuno Martins distinguishes between deductivist Cartesian mathematics, which support marginalist models of constrained optimisation, and Newtonian mathematics, which support a pluralist methodology. It is argued that it is the latter which characterises the Cambridge tradition.”

This contradistinction cannot be upheld.

“Newton not only unified a vast number of experimental and theoretical results of Kepler, Galileo, and Huygens but he placed mathematical description and deduction at the forefront of all scientific accounts and prediction.” (Kline, 1982, p. 54)

In point of fact, Newton was even more deductivist than Descartes:

“The abandonment of physical mechanism in favor of mathematical description shocked even the greatest scientists.” (Kline, 1982, p. 55)

And Walras, a supporter of marginalist models of constrained optimisation if there ever was one, followed — not Descartes — but, yes, the founding father of a pluralist methodology:

“Walras’s early reading consolidated his boundless admiration for Newtonian astronomy and the solid edifice of classical mechanics, which he regarded as unequaled models of scientific knowledge throughout his life.” (Ingrao und Israel, 1990, p. 88)

But, then, the Scotts and Adam Smith rescued common sense from the deductivist threat for later application in Cambridge?

“In this regard he [Adam Smith] proceeded consistently with his evaluation of Newton’s approach in the natural sciences (and it may be added in conformity with the recommendations of Hume and the Scottish historical writers). Smith’s admiration for Newtonian method carries over also to the deductive aspects of model building. His “system” relating to the competitive mechanism of investment priorities over time and the theory of value and distribution, are outstanding examples of deductive theorizing ….” (Hollander, 1977, p. 151)

The Post-Walrasians cannot be blamed for trying to be deductivist and the deductive method is not wrong because it has been mishandled by Post-Walrasians.

References

Hollander, S. (1977). Adam Smith and the Self-Interest Axiom. Journal of Law and Economics, 20(1): 133–152. URL http://www.jstor.org/stable/725090.

Ingrao, B., and Israel, G. (1990). The Invisible Hand. Cambridge, MA, London: MIT Press.

Kline, M. (1982). Mathematics. The Loss of Certainty. Oxford, New York, NY: Oxford University Press.

Firstly, I must start by thanking for the comments that have been made to the article. The comments have raised many important issues which I will surely address in future revisions of the article. I was already engaging in some of those issues in a book, where for example I am drawing on the work of several authors mentioned by Geoffrey Harcourt (some of them, like Gay Meeks and Flavio Comim, do indeed appear at two points in the book, first when addressing the issues raised by Harcourt, but also when addressing the contributions of Amartya Sen). But I must surely include their contributions in the article too as Harcourt rightly points out. The issues that Sheila Dow raises concerning the different intrepretations of Newton in Scotland and France are also crucial, and I will include them in future revisions of the article.

If I had only received these two comments, I would be now spending my time revising the article according to those suggestions. However, there was another contributor, Egmont Kakarot-Handtke, who was also kind enough to spend time commenting my article. But unfortunately, this later contribution contains many misrepresentations of my article. Furthermore, Kakarot-Handtke has commented not once, but twice, my article. And the misrepresentations of the first set of comments appear again in the second set of comments. So after reading the second set of comments, I felt I should have replied to the first set of comments earlier, so as to avoid the repetition of the same misrepresentations. Since I can no longer do so, I will do it now.

The first comment by Kakarot-Handtke starts with the following sentences:

“The author states in the introduction: “The mathematical methods are deductivist in the sense that they presuppose constant conjunctions in the form ‘if event X, then event Y’.” This, though, is a description of causality. Deductive reasoning is in the form ‘if antecedent X, then consequent Y’ and has nothing to do with causality:“… deductive chains of reasoning cannot on their own establish the existence of causal processes in the real world.” (Hodgson, 2001, p. 76) The author seems not to realize that ‘deductive’ refers to mathematics and ‘causal’ to physics.”

Now, if Kakarot-Handtke had read a bit further, he would have found that the full sentence of my article actually reads:

“These mathematical methods are deductivist in the sense that they presuppose constant conjunctions of the form “if event X, then event Y”, where “event X” and “event Y” can refer to real or possible events”

The sentence ends noting that “event X” and “event Y” can refer to real or possible events precisely in order to avoid the misunderstanding that an empirical claim about “the existence of causal processes in the real world” is being made. In fact, I believe that the truncated version of my quote that Kakarot-Handtke presents would already make this clear, since it is a conditional statement. But because some commentators tend to misinterpret this statement, Mário da Graça Moura suggested, when we were writing an article on Tony Lawson’s contribution back in 2007, to emphasise that “event X and event Y” need not be actual events, so as to leave no excuse for those who want to misrepresent Lawson’s stance on deductivism. I have followed this advice since then. But for this qualification to be effective, it is necessary to read the full sentence.

Kakarot-Handtke finishes by saying, based on the truncated version of my quote he presents, that “The author seems not to realize that ‘deductive’ refers to mathematics and ‘causal’ to physics”. I believe it is clear now that this conclusion springs from a misunderstanding. Furthermore, the conclusion that Kakarot-Handtke draws renders itself to other misunderstandings, for I see no reason why the term “deductive” must be used only in mathematics, and why the term “causality” is limited to physics. But since these other misunderstandings are related to the author’s own conception of deduction, causality, mathematics and physics, and not to my article, I will not discuss them any further.

Kakarot-Handtke also writes that “The distinction on p. 1 between a Cartesian a priori approach and a Newtonian a posteriori approach is artificial and misleading. Newton applied the deductive method strictly in the spirit of Euclid”, that “The author oversimplifies the scientific stances of Newton and Descartes as antithetic.”, and that “With regard to the relation between analysis and geometry Newton himself contradicts the author”

Now, if Kakarot-Handtke had read also the section that comes after the section where the distinction between a Newtonian approach and a Cartesian approach is made, he would have found that my argument is not that Newton had invariably a clear position on these issues. Thus I write:

“However, as Guicciardini (2006: 1736) also notes, there was a tension in Newton’s perspective, for while “Descartes was the champion of an impious mechanistic philosophy (…), Newton conceived himself as a restorer of an ancient, forgotten philosophy according to which nature is always open to the providential intervention of God”, and thus, this “led Newton into a condition of strain, since his philosophical values were at odds with his mathematical practice, which was innovative, symbolical, and – pace Newton – deeply Cartesian.”

Concerning the supposed “artificiallity” of the distinction between the Newtonian approach and the Cartesian approach, I can understand why Kakarot-Handtke perceives it in such a way. For the literature he refers to in the many quotes he presents is mostly part of a tradition of economists who see mathematics in the Cartesian deductivist approach. The point of the article, however, is precisely to go beyond that approach. I use the distinction in a sense which is not used in that economic literature, and is used by the mathematician Michael Atiyah. As to that distinction, and as to Kakarot-Handtke’s observation that Newton used the “deductive method strictly in the spirit of Euclid”, let me quote Atiyah again (from page 655 of the work of Atiyah referenced in my article), in order to avoid misunderstandings:

“Geometry and algebra are the two formal pillars of mathematics, they are both very ancient. Geometry goes back to the Greeks, and before; algebra goes back to the Arabs and Indians, so they have both been fundamental to mathematics, but they have had an uneasy relationship. Let me start with the history of the subject. Euclidean geometry is the prime example of a mathematical theory and it was firmly geometrical, until the introduction by Descartes of algebraic co-ordinates, in what we now call the Cartesian plane. That was an attempt to reduce geometrical thinking to algebraic manipulation. This was of course a big breakthrough or a big attack on geometry from the side of the algebraists. If you compare in analysis the work of Newton and Leibniz, they belong to different traditions, Newton was fundamentally a geometer, Leibniz was fundamentally an algebraist”. (Atiyah 2005: 655)

Atiyah then goes on to argue that the reason why “Newton was fundamentally a geometer, Leibniz was fundamentally an algebraist” is because Newton was concerned with the empirical study of the laws of nature, while Leibniz had the aim of “formalising the whole of mathematics, turning it into a big algebraic machine” (Atiyah 2005: 655, this is the same quote I cite in my article). That is, in Newton there is a concern with empirical reality, something which, I argue, is absent from mainstream economics.

The point of my article is precisly that if we want to understand what mathematics is, and its role in economics, we must broaden our view of mathematics. This means that we cannot be confined to a literature on mathematics and economics which interprets mathematics only in one perspective. The reason why Kakarot-Handtke sees the division between the Newtonian approach and the Cartesian approach as an articificial one is because he did not realise that it builds upon a general division between geometry and algebra. In fact, this misinterpretation appears again when Kakarot-Handtke writes, while citing several authors:

“With regard to “intuition and creativity” (p. 10) in Cambridge the author’s description seems to be a bit too idealistic: “In patriotic duty bound, the Cambridge of Newton adhered to Newton’s fluxions, to Newton’s geometry, to the very text of Newton’s Principia … Thus English mathematics were isolated: Cambridge became a school that was self-supporting, self-content, almost marooned in its limitations.” (Forsyth, quoted in Mirowski, 2004, p. 355). “Marshall … was a product of this system, which valued … down-to-earth geometry over continental analysis, and regimented conformity in puzzle solving and humble prostration before timeless truths over originality.” (Mirowski, 2004, p. 355)”

Now, if Kakarot-Handtke had again read a bit further in page 10 from where he selects the words “intuition and creativity”, he would have seen that I was not talking about Cambridge here, for the whole passage reads:

“While in the “geometrical” approach mathematics is concerned with empirical reality (such as space) and relies upon our human intuition and creativity when perceiving an external reality (a point much emphasised by Poincaré), in an exclusively “algebraic” approach mathematics becomes an algorithmic, formalistic, rationalistic and deductivist exercise, divorced from reality, and concerned only with possible realities that can be addressed in a formalistic and deductivist way, as it was the case with the “Bourbaki” school. It is the later approach to mathematics that characterises mainstream economics, which uses mathematics in a deductivist way, in the sense that it attempts to reduce economics to a formalistic approach that presupposes closed system regularities.”

The total passage shows how “intuition and creativity” are discussed here as part of the “geometrical” approach to mathematics, not specifically within the context of the Cambridge tradition, as Kakarot-Handtke suggests when refering to “my” discussion of ““intuition and creativity” (p. 10) in Cambridge”. For the discussion taking place in page 10 is not about Cambridge, contrarily to what Kakarot-Handtke misleadingly suggests. Whether Marshall, or any member the Cambridge tradition, also had “intuition and creativity” when adopting this approach is another issue. The fact that I mention Poincaré in this context, a mathematician who to the best of my knowledge is not related to the Cambridge tradition, as an example, should suffice to show that I am discussing the geometrical approach at a broader level here.

Also, I cannot but fully agree with the quotes presented by Kakarot-Handtke, that “In patriotic duty bound, the Cambridge of Newton adhered to Newton’s fluxions, to Newton’s geometry, to the very text of Newton’s Principia”, or that “Marshall … was a product of this system, which valued … down-to-earth geometry over continental analysis””. My article argues precisely that a Newtonian approach to mathematics which privilleges “down-to-earth” geometry remained in Cambridge when the continent was moving already in a Cartesian direction. What I do not agree is with the assessment that this approach was limited or against originality. Rather, the point is that it had a critical approach towards the use of Cartesian deductivist methods in mathematics which today seems to be missing from economics (and from much mathematics).

Of course, I do not argue that Marshall, like Newton, was not influenced by the Cartesian tradition too. My point is rather that not only Newton, but also Marshall, remained in a condition of strain, since they influenced by the geometrical appraoch, but relied also on Cartesian methods. Thus I write in the article, in page 15:

“There is an interesting similarity between the explanation that Guicciardini (2006, 2009) gives for Newton’s postponement of the publication of his work, and the explanation that Stephen Pratten (1998) provides for Marshall’s failure to finish the second volume of his Principles. Pratten (1998) argues that it was the tension between Marshall’s vision of reality, and his use of a mathematical method which was inappropriate for analysing such reality, that prevented Marshall from achieving a satisfactory second volume, that would reconcile the statical method used in the first volume with the dynamical approach that was to be developed in the second volume.”

The issue at stake here concerns thus, firstly, the geometrical approach, and its emphasis on intuition and creativity, and secondly, the degree to which the Cambridge authors were successful in fully embracing this tradition. While Marshall remained in a condition of strain, Keynes seems to interpret Newton’s approach more fully in terms of the geometrical approach, and its emphasis on intuition and creativity. Otherwise, how would one interpret Keynes’ claim that: “Newton was not the first of the age of reason. He was the last of the magicians, the last of the Babylonians and Summerians” (Keynes 1963, reference in my article).

Kakarot-Handtke then makes his own assessment of the problem of mathematics in economics: the problem is that economics borrowed from a given type of mathematics (for example that of the physics department), and applied it to reality. So in the end, Kakarot-Handtke accepts the crucial point of the article, that the problem is how mathematics is used (although I believe we cannot limit its influence to the mathematics of the physics department, as the case of game theory shows). Kakarot-Handtke ends with the quote:

“Here, just in the art of painting, we should not blame colors for the horrible works some ‘painters’ may produce with them.” (Georgescu-Roegen, 1979, p. 323)

The problem, however, is rather the fact that only a specific type of colours are allowed (those conforming to mathematical-deductivist methods), or that the way in which they are used is imposed by the structure of mathematical-deductivist methods. This is why I felt the need to point out that there are other ways to use mathematics, and also non-mathematical ways to do economics.

The second set of comments from Kakarot-Handtke repeats the same misunderstandings, with a stronger emphasis on the issues concerning Newton mentioned above. For this reason I elaborated upon those issues a bit more in what I wrote above.

Hello Nuno, just a few points:

1. It may be true that Marshall’s approach to mathematics differed from others (something Weintraub uses to criticise Marshall) but isn’t the underlying point that the core argument of the Principles is an extended development of a position based on reductive assumptions that put aside reality at the same time as acknowledging it. The nature of the use of assumptions is one that flows from a kind of mathematically based thinking (Marshall’s famous quote ‘Burn the maths’ is one that makes it clear that one begins from mathematical thought but does not foreground it in representation and can perhaps also dispense with it – but nonetheless begins from it) that is conducive to further elaboration along lines of tractability. There is a reason why Marshall is a founding father (albeit sometimes grudgingly acknowledged) of mainstream economics. It is not simply a distortion of his work in the Principles – it is rather a continuation of some of its core ways of proceeding. In the end it is perhaps the similarities of underlying thinking rather than the differences to subsequent approaches to mathematics that are important – at least if we take important to be a retrospective consideration of a major line of descent from The Principles. Of course the lost opportunity that the Principles also represents is important too…

2. It is an open question whether the idea of science that informs the Principles is biology. However, though the acknowledged Mecca may well be evolutionary this – as Steve Pratten notes – highlights the disjoint in the Principles. If the idea that informs the Principles had indeed been biological then Marshall would surely have found it easier to write the second volume… So, ‘informs’ may not be the appropriate term… I’ve read The Principles many times as part of writing a book on its significance with Heikki Patomaki (not yet finished – I can send you 7 chapters if you are interested) and Marshall’s statements about his method and the need for realism etc always seem more like auto-critique than consistent construction of a single coherent position. Marshall scholars have been arguing about this for years as you know…

3. As with 2. there is no clear evidence in The Principles that it is actually influenced to any great degree by Hegel (the many in the one notwithstanding) – if one had Phenomenology of Spirit in one hand and The Principles in another it would be difficult to see any inspiration in the latter from the former (or any of Hegel’s other major works). Talcott Parsons is (briefly) interesting on this in his QJE 46(2) 1932

Best, Jamie

Dear Jamie,

Many thanks for your comments, which are most helpful, and will certainly be an important input for the revision of the paper.

As for the first comment, yes, it is true, hence the tension between Marshall’s method (followed before burning the maths) and his vision of reality. And I think even evolutionary theory, which you mention in your second comment, provides an example of what you way. For example, the following passage from Book IV, Chapter VIII, of the Principles, shows Marshall’s evolutionary vision (sorry for the length of the quote):

“Before Adam Smith’s book had yet found many readers, biologists were already beginning to make great advances towards understanding the real nature of the differences in organization which separate the higher from the lower animals; and before two more generations had elapsed, Malthus’ historical account of man’s struggle for existence started Darwin on that inquiry as to the effects of the struggle for existence in the animal and vegetable world, which issued in his discovery of the selective influence constantly played by it. Since that time biology has more than repaid her debt; and economists have in their turn owed much to the many profound analogies which have been discovered between social and especially industrial organization on the one side and the physical organization of the higher animals on the other. (…)

This increased subdivision of functions, or “differentiation,” as it is called, manifests itself with regard to industry in such forms as the division of labour, and the development of specialized skill, knowledge and machinery: while “integration,” that is, a growing intimacy and firmness of the connections between the separate parts of the industrial organism, shows itself in such forms as the increase of security of commercial credit, and of the means and habits of communication by sea and road, by railway and telegraph, by post and printing-press.

The doctrine that those organisms which are the most highly developed, in the sense in which we have just used the phrase, are those which are most likely to survive in the struggle for existence, is itself in process of development. It is not yet completely thought out either in its biological or its economic relations. But we may pass to consider the main bearings in economics of the law that the struggle for existence causes those organisms to multiply which are best fitted to derive benefit from their environment.” (Marshall 1920[1890]: 200-201)

Although Marshall refers to Malthus and Darwin here, his account of “differentiation” and “integration” is actually closer to Spencer’s. Quotes like this sustain the idea that Marshall had a vision of evolutionary processes different from mainstream economics.

But there is a most curious footnote in the mathematical appendix (note XI) of the Principles, where Marshall discusses the Taylor series, and he writes:

“There is more than a superficial connection between the advance made by the applications of the differential calculus to physics at the end of the eighteenth century and the beginning of the nineteenth, and the rise of the theory of evolution. In sociology as well as in biology we are learning to watch the accumulated effects of forces which, though weak at first, get greater strength from the growth of their own effects; and the universal form, of which every such fact is a special embodiment, is Taylor’s Theorem” (Marshall 1920[1890]: 694)

This note seems to show your point, that mathematics, before being burned, would drive much of the research. And apparently this happens also concerning even Marshall’s evolutionary theory. This is why I say in the paper, following Stephen Pratten, that Marshall’s vision remained constrained by his method. In Industry and Trade, Marshall again resorts to differential calculus (mentioning Leibniz and Newton) as the basis of his method of focusing on direct effects first (while neglecting indirect effects which only appear later) – Sraffa has a most curious comment on this passage in his unpublished manuscripts, where he says that this was Marshall’s one intelligent contribution.

As for Marshall’s Hegelianism, there are a range of positions on this. The minimalist one is to say that Marshall focused only on Hegel’s Philosophy of History (neglecting other work by Hegel), that Marshall used Hegel making it conform to what Marshall wanted to say, and that this influence is limited to Marshall’s earlier writings and some parts of the Principles. And even the idea that Marshall’s theory would be some kind of Hegelian synthesis between classical political economy and marginalism cannot be sustained if we take seriously the following remark Marshall wrote in a letter to J. B. Clark dated from 24 March 1908:

“One thing alone in American criticism irritates me, though it need not be unkindly meant. It is the suggestion that I try to ‘compromise between’ or ‘reconcile’ divergent schools of thought” (Marshall, as cited in Shove 1942, p. 295).

Other positions would say that Hegel was more important than this. But for the point I want to make, concerning the influence of the notion of “internal relation” in Marshall’s vision (which was nevertheless constrained by his method), the minimalist position on Marshall’s Hegelianism, and what he says on biology, is sufficient. For example, as you know, Marshall writes:

“Perhaps the earlier English economists confined their attention too much to the motives of individual action. But in fact economists, like all other students of social science, are concerned with individuals chiefly as members of the social organism. As a cathedral is something more than the stones of which it is made, as a person is something more than a series of thoughts and feelings, so the life of society is something more than the sum of the lives of its individual members.” (Marshall (1920[1890]: 20-21)

That is, Marshall certainly did have a vision of reality where something like “internal relations” did play an important role. But as Steve Pratten argues, that vision was constrained by the methods he used to develop that vision, as you also say. Nevertheless, even those methods were very much shaped by the Cambridge Mathematical Tripos, which had a stronger emphasis on geometry and mechanics rather than on symbolic algebra. Although the algebraic approach was already gaining ground in Cambridge even before Marshall’s time, there was still a different approach in Cambridge when Marshall did his mathematical training, which was of course fading away already.

Best,

Nuno

References:

Marshall, A. 1920[1890]. Principles of Economics, London, Macmillan.

Shove, G. 1942. “The Place of Marshall’s Principles in the Development of Economic Theory”, Economic Journal, 52, 294-329.

Hello Nuno

Yes, I also use the Shove paper; though it is worth noting of the Clarke quote that one need not read the Principles in terms of Marshall’s subsequent statements regarding it – intent and outcome need not match nor need claim and content… and of course, there is the possibility that retrospection is self-serving…

I take your point re the cathedral quote too, but would note again that most of these kind of statements in the main body of his text do not inform theory, they warn against relying on the primitive form of his theory… nor do they provide a way to consistently build outwards from his basic competitive model… having an idea of what would be more real and then having a consistent account of reality in terms of economic theory are not the same things. Neither point of course detracts from the core argument in your paper.

Dear Jamie,

Thanks again for the comment. As for Marshall’s statements like the cathedral one, and the extent to which they inform theory: I think one should avoid the extreme of saying that Marshall had a very clear theory in his mind, while making parallel statements disconnected from it. Marshall did not always have a very clear theory in his mind (remember for example the appendix H of the Principles, where Marshall confesses many of the uncertainties he has concerning the theory), and this kind of statements were important to complement the gaps and uncertainties he may have felt concerning his theory? They show the “vision”, or overall intuitive picture of reality Marshall had, and to which he would resort to when attempting to develop a realist theory (which had however many inconsistencies with the vision, hence the tension in Marshall’s writings). In fact, that is what a “vision” does, it provides intelligibility and consistency to whatever our theory is? The idea of someone simply following the theory while disregarding reality, and making parallel statements which do not inform theory, I think applies more to mainstream economics than to Marshall? I guess the point is that in Marshall there is still an opportunity for a more realist analysis (not least due to the ambiguities too), which disappears with the evolution of mainstream economics. I suppose this is your point too?

But regarding Marshall’s Principles, and Marshall statements on the Principles which you also mention, of course one must be cautious because of Marshall’s agenda. Sraffa writes in one of his unpublished manuscripts that Marshall wrote to be read by business man because he wanted to have influence on economic life, and that his true intuitions were hidden in the footnotes. I agree with the assessment, and think that it also applies not only to Marshall’s Principles, but also to Marshall’s statements on the Principles.

Hello Nuno, the point on Marshall’s intuitive realism is well taken – it is as you say also a problem to disjoint his work entirely. However, his intuitive realism creates at least three potentials – an attempt to reconstruct what is problematic in regard of reality in relation to the formal potential of theory, an attempt to justify what is problematic as the best available compromise (despite the problems) and a broader attempt to justify in theory/method that the approach is in fact somehow acceptable as an account of reality. I would suggest Marshall vacillates between the latter 2 (for example, in terms of his claims that the Principles is a first volume of necessary work…). Again, I don’t think this damages your argument, but it does raise questions about the way we tend to split Marshall. In the end I don’t think we should be too precious about these things – one can read through the Principles and find many different useful specific aspects, as well as many different lines of development of potentials… Marshall may not be a taxi cab one can take where one wishes, but the Principles is a bus station from which one can buy tickets to a variety of destinations… including your own. I look forward to reading the amended version and citing it in Heikki and my work….

Although I might not be the right person to say that this paper should or should not be published, I therein see a strong and redhibitory weakness. It adresses “reality” which is at the junction of the two mathematics that are supposed to divide economic thought in two streams. Nuno seems to consider that the reality could jump directly into mathematics without any mediation, under the only condition that the thinker cares about it. So was thought geocentrism, for example. And so Newton’s theory of universal attraction, nowadays rejected for the profit of space’s curvature.

Mathematics cannot be held responsible for intellectual failures, and they cannot help theorists finding their concepts. Marshall did recognize the “diminishing returns” opposed by nature to “increasing returns” of mankind industry, but his mathematic method did not even lead him to draw the resultant trend of these two contradictory tendencies.

Is there a huge difference in approaching reality, between deductivist algebraic mathematics and the geometrical “realist” one? Indeed, the former is likely to describe time processing, but it is not true of Walras’s, Debreu’s (and so many more) equations that are instantaneous. Indeed, the latter is likely to describe space processing, but space is not the same for Euclid, Legendre, Poincare, Riemann and Lobachevski. In both cases, the reasoning begins with a set of hypothetic axioms that are supposed to represent a reality we are unable to know as such without resorting to the mediation of our senses and imagination.

The divorce between mainstream theories and reality is not due to deductivist mathematic method, but to an axiomatic basis which calls for an inverted representation of reality: instantaneity, entropy denial, investment due to savings, prices governed by money supply. Pluralistic empiricist approaches that are supposed to oppose mainstream theorems share some of these basic axioms. By failing to treat mathematically the natural constraints, both emphasize behaviourism, individual for the first ones, collective for the others. Pluralistic empiricism may seem non-dogmatic, but it is actually as dogmatic and non-scientific as mainstream theories.

Now, how to know whether an axiomatic basis is or is not consistent to reality? To my mind, that is the question, far beyond any dialectical controversy between mathematic methods. In my paper under review, “Theoretical aspects and political neutralization of labour-cost competitiveness”, I particularly propose the answer of functional isomorphism criterion.

Romain Kroës argues that I seem “to consider that the reality could jump directly into mathematics without any mediation, under the only condition that the thinker cares about it”. The point of the article, however, is to criticise the opposite position, namely the position according to which one may construct mathematical methods while neglecting the nature of the reality that those methods are supposed to describe. I cannot see how reality could jump into mathematics without mediation. As for the question “Is there a huge difference in approaching reality, between deductivist algebraic mathematics and the geometrical “realist” one?”, the answer can be readily seen by studying the radical change in mathematics that occurred during the twentieth century (and started before). The reason to relate space to geometry, and time to algebra, comes from Kant’s claim that geometry springs from our intuition of space and numbers arise through our intuition of time. But the very word “space”, which of course has not meant the same thing in every historical context, is now an algebraic, rather than a geometrical, notion, which now denotes algebraic structures, as one can see by looking at notions such as “Hilbert space” (which is essential to understand contemporary mathematics). The point, however, is not to criticise the use of these algebraic structures, but those that use them while neglecting reality. As I write in page 11 of the article, “Algebra, no doubt a powerful tool, is sterile when addressed only in a rationalistic and deductivist way”. A method that is developed independently of reality may, by chance, turn out to be applicable to some reality, but in mainstream economics there is rarely an attempt to see whether the application is helpful or not. Furthermore, as Jonathan Barzilai convincingly argues in an article which is also under review now in the World Economic Review, the way in which mathematical spaces are used in economics is inconsistent. It is not only reality, but mathematical precision itself, that is neglected in mainstream economics

The article can be summarised in four claims: 1) Mathematical methods used in economics should take into account the nature of economic reality; 2) There is a tendency in the algebraic tradition (which is the mainstream approach in mathematics) to develop mathematical methods independently from the study of concrete phenomena, a tendency which did not exist in the geometrical approach; 3) This tendency is certainly manifest in contemporary mainstream economics; 4) The Cambridge economic tradition emerged in a place where the geometrical tradition lasted longer than in other places, and had a more realist approach.

Romain Kroës writes, in what seems to be a summary of the comment: “Now, how to know whether an axiomatic basis is or is not consistent to reality? To my mind, that is the question, far beyond any dialectical controversy between mathematic methods”. I believe that the question is indeed consistency with reality, but it is not just the axioms, but also the methods, that must be consistent with reality.