Mathematical Analysis as a Source of Mainstream Economic Ideology
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The paper contends that neoclassical ideology stems, to a great extend, from mathematical analysis. It is suggested that mainstream economic thought can be comprehensively revisited if both histories of mathematical and economic thought are to be taken collaboratively into account. Ideology is understood as a “social construction of reality” that prevents us from evaluating our own standpoint, impedes us from realizing our value judgments as well as our theories of society and nature. However, the mid-19th century’s intellectual controversies about the validity of mathematical thought, truth and knowledge can procure new interesting insights concerning the ideological stance of the first marginalists. In this respect, the methodological categories of analysis and synthesis serve as the basis for the crucial distinction between old geometry and mathematical analysis, indicating that the discipline of mathematics has its own history of fundamentally unresolved disputes. Lastly, this approach may also explain Alfred Marshall’s peculiarly reluctant attitude towards the use of mathematical analysis
The purpose of this paper is to demonstrate that the use made of mathematics in economics itself reflects a social construction of reality rather than an abstraction from such constructions. It is argued further that social construction implicit in mathematical economics depends on the type of mathematics employed. The paper draws on a series of historical debates, with a particular focus on the mid-nineteenth century. This focus provides a particularly interesting comparison between analysis and synthesis, or ‘mathematical analysis’ and ‘old geometry’ as a case study of the way in which different mathematical approaches are value-laden.
This is an excellent project which makes a useful contribution to what is already a considerable discussion of the role of mathematics in economics. It addresses head on the dominant view of mathematics as simply a neutral language which not only allows greater precision than verbal argument but also facilitates a separation of the normative from the positive. This view provides the (usually implicit) basis for regarding mathematical economics as more ‘scientific’ than alternatives.
The paper is rich in conceptual and historical detail, and is long, so it is important to make sure that the arguments emerge really clearly from the detail. The first such issue is conceptual clarity and relates particularly to the term ‘ideology’ itself. I have no objection to its use, but this kind of term is so emotive, and yet the meaning given to it so circular, that there needs to be a more explicit discussion of it, and the implications of different meanings, at the start. In my view, the explanation of meaning as it stands doesn’t adequately take account of the scope for misunderstanding.
Vlassis Missos uses the term ‘ideology’ to refer to the underpinnings of economic knowledge, mathematical and otherwise. This sense of ideology refers to social constructions of reality – the interface between the ontological the epistemological and the political. But ‘ideology’ generally tends to be understood much more narrowly, e.g. in terms of value judgements governing policy preferences. It complicates matters that different meanings of the term themselves often reflect ideology in the sense used here, where ideology tends to be unexamined – hence the scope for misunderstandings.
Another aspect which would benefit from more explicit discussion is how the paper relates to ongoing critiques of the use of mathematics in economics. It is an important argument of the paper that deductivist mathematics was promoted on the grounds that it abstracted, not only from the values/perspectives of individual economists, but also from quantification. Yet these are often the very grounds for critique. It is shown that mathematics itself, in a particular form, has actually played an important role in appearing to validate a sharp separation between (mathematical) theory on the one hand and ideology (understood as value judgements) on the other. Showing that this separation is itself ideological is to make a valuable contribution.
The detailed discussion of different approaches to mathematics is an important element of the argument. This could also benefit from more clarity. A contrast is drawn between analysis and synthesis, where synthesis is related to ‘old’ geometry. But then analytical mathematical economics in the mid-to-late nineteenth century employed the analytical ‘new’ geometry. Again some clearer signposting would be helpful, e.g. finding a different subtitle, rather than ‘Analysis and geometry’ (which implies a simple dichotomy).
Finally, at several points there are rather cryptic statements which need explanation, e.g. the reference to Blaug on page 2, para 2, or the reference in footnote 2 to early Scottish enlightenment philosophy as being neo-Platonic, or mention on page 4 para 1 of debate between Lawson and O’Boyle and McDonough. Further there are early references to Newton, but no explanation of his approach until page 12.
These comments all refer more to presentation and contextualisation than to content, based on a concern that the arguments are conveyed as effectively as possible. The paper is already long, and the suggestions made above would add to that length. Length is less of an issue if the argument is signposted clearly. But one option would be to split the paper in two. One paper could address more fully and directly the meaning and role of ideology and address the modern debate over the role of (different kinds of) mathematics in economics. The other paper could focus on the detailed history of thought. (While I’m suggesting more work, I can’t resist pointing out that there is scope also for someone to take the coverage back to the eighteenth century.)
I would like to thank Professor Sheila Dow for taking the trouble to read my essay and to make a series of extremely helpful comments. Her insights and recommendations are very much appreciated and will certainly help me to revise the paper in various respects. It goes without saying that my intention is to incorporate as many of the comments proposed as possible, so as to improve the way my arguments are presented, but a full-scale embodiment is, I think, very difficult to accomplish.
To begin with, among the comments made, one that has taken a central position is that which stresses the need for ‘conceptual clarity’ especially in the way ideology is treated as a term. Prof. Dow suggests that “there needs to be a more explicit discussion of it [ideology] and the implications of different meanings”. Unarguably, there is considerable truth to it, and I agree that the paper’s contents could have been arranged differently, so as to include an extensive discussion on ideology.
The main reason why this path was not chosen is that I wanted to present some critical advantages we might gain when both histories of thought – that of mathematics and of economics – are combinedly examined. The ideological issue is thus taken as an aspect of this collaborative examination.
In addition, as the complex title suggests, the paper is an attempt to shed some light not on the relation between neoclassic thought, mathematics and ideology in general, but on mathematical analysis as a source of mainstream economic ideology in particular. In other words the idea is to present why I believe the early 19th century debate on mathematical analysis [not any other approaches to mathematics] is important for a better understanding of neoclassical ideology.
To achieve this, a bold decision was made that one of the three main aspects of my essay (mathematics, ideology and mainstream thought) had to be suppressed. Ideology was chosen as the one which I had less to contribute while at the same time, the existing literature on ideology is too large and could have extended the length of the paper, making it inappropriate for publication. The debate on mathematical analysis on the other hand, is I think vital while the references to the economic thinkers/writers are helpful to grasp the ideological essence of choosing the right mathematical theory instead of another.
This is also obvious in my Introduction where a “survey” on ideology has been indirectly evaded by endorsing Prof. Macfie’s (1963) view that ideology is “partial in scope and prejudiced in direction”. Some additional comments on ideology also exist in the beginning of the second section of the paper though, I understand, they are not able to clear up the misunderstandings noted by Prof. Dow.
Having said that, to lessen the level of misunderstanding generated by the absence of “a more explicit discussion”, a new version of the paper can be provided that would more adequately incorporate a discussion on ideology. More specifically, an additional part on ideology can be included in the middle of my Introduction, extending it by two or three paragraphs – if the reviewer finds it sufficient.
As far as the “cryptic statements” are concerned I ll do what is necessary to make them as clear as possible.
Thirdly, changing the subtitle seems to be a great idea. The same part can also be improved so the argument would make itself more apparent. Prof. Dow is right on indicating the role of ‘new’ or non-Euclidean geometry. The analytical concepts generated in the mid-to-late 19th century have based on the old names/terms – for example to that of “geometry” – by abolishing the need for an experience-based knowledge.
Lastly, for reasons already noted, I’ m not sure that splitting the paper in two different parts would fit its general purpose. I can see that its length can make it a bit tedious, but in its current form its main purpose of mixing the histories of the two fields is I believe, better served.
I read this paper and I enjoyed it, I think it would be good to consider publishing. I am interested in the relationship between economics and ideology (if the editors don’t mind, I will recommend my comment on Donald Gilles’ article “Ideology and Science in Economic Theory” in this same forum – I don’t want to repeat my comments done there, even though I think they are relevant, but I would recommend the author to read them and Gilles’ article).
The most enjoyable thing about this paper is how it ties both history of mathematics and history of economic thought, because mathematics is just accepted as the language of economics – Lucas even said “Economic theory is mathematical analysis. Everything else is just pictures and talk” –. Ever since reading Nancy Cartwright (1983) book on methodology of physics, it surprised me how little economists try to understand how physics actually work, and how its debates are developed and “solved”. I have the feeling it’s not different for mathematics as well, how many economists are aware of this debate between analysis and geometry?
One point I approached in my thesis (Almeida, 2019) was on the separation between political economy and economics. Even today, the term “political economy” is still reserved to topics in the periphery of the mainstream, such as the use of economic tools to political problems, new institutionalism, economics of culture and so on. Very productive scholars, whom we would not dare to contest their adherence to the neoclassical tenets, are part of this movement such as Alberto Alesina, John List and Daron Acemoglu, and yet they are not considered fully “orthodox”, and some would go as far as to call them “heterodox” (Almeida, 2019; Guimarães, 2019), exactly because their problems are hard to fit neatly in the mathematical paradigm. This might be an indication that the more orthodox economists still consider their work to ideological (cf. Lucas’s distinction between “science” and “ideology”, in which non-formalized models were “ideology”, even if it doesn’t need to be pejorative, see De Vroey, 2011).
Did it work? Most economists seem to believe so, but historians and methodologists would cast some doubts, as you mentioned. Though, I must admit, while I am glad of research that intersects both history and methodology of economics, I am not sure what your focus is. In other words, are you using both history of economics and history of mathematics to understand if economics is ideological/value-laden or are you just exposing how mathematics alone cannot eliminate ideology/turn economics value-free?
The article is already dense itself, so I wouldn’t recommend adding much (I could ask about an intersection with the history of physics, but many economists have already written on it, so I appreciate you go through the history of mathematics instead, and it is better to leave it for another time – though, as a bias of mine, the history of early econometrics might have potentially interesting material to your argument). I read your exchange with Prof. Dow and I understand you cannot separate the articles, but I would recommend focusing a little more on one argument (say, the historical one) and preparing the other (the methodological-sociological one) to be developed in more details in another article, because I did not see the link between the history and the “social construction” very well in the conclusion.
As a final commentary, Vernon Smith is working on the same issues of this transformation in economics caused by mathematics (e.g. Smith, Inoua, 2019), I have the feeling we might see a book of him published in the next few years on this topic. Take a look at the working papers of the economics department of the Chapman University – as far as I can see, he takes a more whig view, supporting the “turn into ideology-free” narrative, so it would be interesting to contrast.
Also, check p. 6 for a wrongly written reference to Dobb.
Almeida, Rafael Galvão de. Dreaming of unity: essays on the history of new political economy. Thesis, Doctor in Economics. Belo Horizonte: UFMG, 2019.
Cartwright, Nancy. How the laws of physics lie. Oxford: Oxford University Press, 1983.
De Vroey, Michel. Lucas on the relationship between theory and ideology. Economics: The Open-Access, Open-Assessment Journal, v. 5, p. 1-39, 2011.
Guimarães, Bernardo. Qualis as a measuring stick for research output in economics. Brazilian Review of Econometrics, v. 31, n. 1, 2011.
Smith, Vernon; Inoua, Sabiou M. Cournot marked the turn from classical to neoclassical thinking. ESI Working Paper 19-14. Orange: Chapman University, 2019.
I would like to thank Dr. Rafael Galvão de Almeida for making the trouble reading and commenting on my article. All suggestions made are very helpful (especially the literature proposed) and I really appreciate the time he has devoted.
I have already read the discussion/comments made to prof. Donald Gilles’ article and it was a bit of surprise to see that there is a vivid interest on the issue of ideology. No doubt my paper was sent in the right journal.
The history of physics – as the reviewer has already pointed out – has been linked with that of the mainstream thought is several occasions. However, most writers focus on the common concepts and methodology. The ideological aspects of Philosophiae Naturalis and what followed from that, need further research which has not been made adequately yet.
One of the crucial questions raised concerning the main purpose of this article. I would say that the central aim is to expose “how mathematics alone cannot eliminate ideology” and that the first mainstream political economists were completely aware of this issue. In revising the paper, I intent to make this point as clear as possible.
This is a very interesting paper which deserves to be published. Its key merit lies in its assertion that all theoretical systems, -including mathematics – have socio-historical roots that help to shape their development and influence the nature of any resulting output.
In other words, conta Schumpeter and Samuelson, the positivist ideal cannot stand up to scrutiny even on its seemingly most advantageous ground – mathematics.
To defend this important assertion, the author looks at the history of mathematics in mid 19th century Britain and links this with the development of political economy around the time of the Marginal Revolution.
Here they usefully distinguish a more abstract (Cartesian) mathematics associated with Analysis from a more geometric-empirical (Newtonian) mathematics associated with Synthesis.
This is a highpoint of the paper as the epistemological differences between these two systems are linked to debates that were current at the time.
There is also an argument that Analysis took hold earlier on the continent, but was delayed in Britain due to the immense authority of Issac Newton. Be-that-as-it-may, analytic mathematics gradually gained traction -particularly as Non-Euclidean geometry emerged – and this was taken up by marginalist thinkers like Jevons in a way that contrasts with the more empirically minded Marshall.
All of this is interesting and deserves careful follow up investigation, particularly the social history which isn’t as well developed.
There are, however some issues that I would like to address
(1) Although the paper defends the idea of rooting ideas in their social and historical context, it actually interprets this in quite a narrow way. Specifically, it focuses overwhelmingly on academic debates around the nature of knowledge in a way that excludes other influences on the development of Marginalism. Classical Political Economy was replaced by Marginalism for a mixture of reasons that included ideas internal to knowledge, but also ones that related to the class struggle that was visible in the former system and screened out of the latter. If we accept that every thinker is influenced in ways they do not always appreciate and that ideas have socio-historical roots, then the influence of bourgeois society must play a part in this story. In my opinion by far the more important part.
(2) This leads to my second point which is that the conception of ideology used in this paper is neither developed enough (a point made by Sheila Dow above) nor broad enough to capture the influences I refer to above. Economic theory developed as the world-view of the rising capitalist classes and its mainstream has always defended the commodification of our lives through capitalist free markets.
In our paper, referred to by the author, we take up the complex links between ideas and their social context noting the ways that abstract symbols can carry political and social import. In other words, the ideology of neoclassical economics -including its mathematics- doesn’t just distort the world and bring in prejudice, it helps to legitimise and reproduce the dominant relations of the society we live in. Understanding ideology in this broader sense, would, in my opinion, add to the epistemological influences brought to light by the author.
This brings me finally to the reference to our work. If interpreted narrowly and technically then it is true that Newtonian mathematics wasn’t key to marginalism in the way that the more abstract form of Analysis was. This is something important I will take from this paper and confirms the need to be very specific when thinking about these matters. Yet it is also possible to see this point as complimentary to our own. We argued that despite being used as legitimation, Newton’s more empirically based scientific method was severed from his mathematics (the calculus) in the Marginal Revolution. This should now be tweaked to include the point that even the mathematics were not strictly Newtonian.
That said, it is also important to remember that the general Newtonian world view was kept and in many ways foregrounded. Here I refer to the idea of atomised units of matter brought into harmonious interaction by underlying forces. These were interpreted fruitfully by Newton in terms of mass and gravity and ideologically by the marginalists in terms of individual subjective utility. This leads me back to the key point I have been making which is the need for a wider lens that takes in key social influences from bourgeois society to compliment the narrow lens on epistemolgical ideas brought out in this paper.
The points raised by Dr. Brian O’ Boyle are of great importance and require a deep understanding of history, economic theory and philosophy. I deeply thank him for offering me his insights and I am glad to see that – apart from his critiques and observations – he also acknowledges the complementarity of this paper to his own work (O’ Boyle & McDonough 2017).
Truly, this is an extensive article (as Sheila Dow has pointed out) with a narrow scope – to expose one of the many controversial aspects that exist in the history of mathematical thought and to represent how this crucial debate was reflected upon the ideas of some of the leading academic scholars of the late 19th. This – I tend to believe – gives potential to new interpretations in the history of economic thought.
Dr. Brian O’ Boyle is absolutely right to observe that the influence of social history/context on marginal theory is essentially absent from my narrative. Its role has been acknowledged but only in passing. The main reason I could offer for this is the following:
One of my main aims was to expose the debate of the scholars themselves and not to criticize it from an external point of view. It is important to understand that the views of the late 19th century scholars on mathematics are deeply differentiated and this is not just of technical importance. These historically long rooted traditions are able to convey a variety of different interpretations on what is the qualitative basis of our knowledge and of our conclusions. We – economists – have learned to refer to mathematics as a solid field of knowledge, as if there is no history in that field, but only a cumulative process destined to excellence based on proofs and unargued solutions. Well, as you have already mentioned in your paper, there is a branch of “mainstream mathematics” and historians of economic thought ought to acknowledge. Many scholars of the late 19th were totally aware of these debates and the Cambridge tradition had ontological and ideological reasons resisting against the rising analytical trend.
On his second comment Dr. Brian O’ Boyle stresses that my approach on ideology is limited. He says: “the ideology of neoclassical economics -including its mathematics- doesn’t just distort the world and bring in prejudice, it helps to legitimise and reproduce the dominant relations of the society we live in”. For once again, I couldn’t agree more. However, I should observe that this view corresponds to the role of the “dominant ideology” and not on ideology in general. The synthetic reasoning of old geometry – for example – was excluded from the “mainstream mathematics” and today, it is only referred to as an outdated way of generating mathematical solutions. The way the nature of knowledge is understood includes something which is deeply ideological, no matter if it is the dominant or the submissive.
The important sociological and historical reasons why the synthetic reasoning was abandoned is an unexplored issue that waits to be researched. For example, what the most textbooks of mathematics offer about Newton is the Leibnizian interpretation of his calculus and not the essential features of his “fluxions”.
Lastly, as I have already explained in my reply to Sheila Dow, my view on ideology had to be suppressed. In the revised version, i have tried to make some room for it in the introduction. However, ideology is still disproportionately treated.