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Utopia and mathematics

from David Ruccio

In a recent article, Dan Falk [ht: ja] identifies a fundamental problem in contemporary physics:

many physicists working today have been led astray by mathematics — seduced by equations that might be “beautiful” or “elegant” but which lack obvious connection to the real world.

What struck me is that, if you changed physics and physicists to economics and economists, you’d get the exact same article. And the same set of problems. giphy2.gif

Economists—especially mainstream economists but, truth be told, not a few heterodox economists—are obsessed with mathematics and formal modeling as the only correct methods for achieving capital-t truth. Mathematical modeling for them represents the best, most scientific way of producing, disseminating, and determining the veracity of economic knowledge—because it is logical, concise, precise, and elegant.* In that sense, mathematics represents what can only described as a utopia for the practice of modern economics.**

Mathematical utopianism in economics is based on elevating mathematics to the status of a special code or language. It is considered both a neutral language and, at the same time, a language uniquely capable of capturing the essence of reality. Thus, economists see mathematics as having both an underprivileged and overprivileged status vis-à-vis other languages.

Let me explain.  On one hand, mathematics is understood to be a neutral medium into which all statements of each theory, and the statements of all theories, can be translated without modifying them. Mathematics, in this view, is devoid of content. It is neutral with respect to the various theories where it is applied. Partial and general equilibrium, game theory, and mathematical programming are concepts that serve to communicate the content of a theory without changing that content. Similarly, mathematical operations on the mathematized objects of analysis are considered to be purely formal. Thus, as a result of the conceptual neutrality of the methods and procedures of mathematical formalization, the objects of analysis are said to be unaffected by their mathematical manipulation. On the other hand, mathematics is considered to be uniquely capable of interpreting theory in its ability to separate the rational kernel from the intuitional (vague, imprecise) husk, the essential from the inessential. It becomes the unique standard of logic, consistency, and proof. Once intuitions are formed, mathematical models can be constructed that prove (or not) the logical consistency of the theory. Other languages are considered incapable of doing this because the operations of mathematics have an essential truth value that other languages do not possess. Mathematical statements, for example, are considered to be based on the necessity of arriving at conclusions as a result of following universal mathematical rules.

It is in these two senses that mathematics is considered to be a special language or code. It is more important than other languages in that it is uniquely capable of generating truth statements. It is also less important in that it is conceived to have no impact on what is being thought and communicated.

The notion of mathematics as a special code is linked, in turn, to the twin pillars of traditional epistemology, empiricism and rationalism. The oversight of mathematics implied by its underprivileged status is informed by an empiricist conception of knowledge: mathematics is considered to be a universal instrument of representation. It is used as a tool to express the statements of a discourse that already, always has an essential grasp on the real. It is the universal language in and through which the objects (and the statements about those objects) of different economic and social theories can all be expressed. In other words, the role of mathematics is to express the various “intuitive” statements of the theorist in a neutral language such that they can be measured against reality. The underprivileged position of mathematics that is linked to an empiricist epistemology contrasts sharply with the overprivileged status of mathematics. This overprivileged conception of mathematics is associated with a rationalist theory of knowledge wherein the subject-object dichotomy is reversed. Here the subject becomes the active participant in discovering knowledge by operating on the theoretical model of reality. In this sense, the logical structure of theory—not the purported correspondence of theory to the facts—becomes the privileged or absolute standard of the process of theorizing. Reality, in turn, is said to correspond to the rational order of thought. The laws that govern reality are deduced from the singular set of mathematical models in and through which the essence of reality can be grasped.***

The conception of mathematics as a mere language contains, however, the seeds of its own destruction. The notion of language as a simple medium through which ideas are communicated has long been challenged—since language is both constitutive of, and constituted by, the process of theorizing. The use of mathematics in economics thus may be reconceptualized as a discursive condition of theories, which constrains and limits, and is partly determined by, those theories. Mathematical concepts—such as the equilibrium position associated with the solution to a set of simultaneous equations, the exogenous status of the rules of a game, or the definition of a series of overlapping value functions to optimize an overall goal—partly determine the notions of relation and causality among the theoretical objects designated by the theories in which the means of mathematical formalization are utilized. They are not the neutral conceptual tools to which the propositions of different theories can be reduced. Similarly, the rationalist idea of abstraction, of simplification, also leads to a fundamental problem. It implies that there is a noise that ultimately escapes the “fictional” mathematical model. It implies an empirical distance between the model and its domain of interpretation, the empirical concrete. And that distance is conceived to be part of the empirical concrete itself. There is a part of reality that necessarily escapes the model. Thus, rationalist deductions from the model cannot produce the truth of the real because something is always “missing.”

So, what’s the alternative? As I see it, there is a double movement that involves both the rejection of mathematics as the discovery of an extra-mathematical reality and the critique of the notion that mathematics merely expresses the form in which otherwise nonmathematical theories are communicated. Thus, for example, it is possible (using, e.g., the insights of Ludwig Wittgenstein and Edmund Husserl) to argue that mathematics is a historical, social invention, not a form of discovery of an independent reality; it is not discovered “out there,” but invented and reinvented over time based on rules that are handed by mathematicians and the actual users of mathematics (such as economists). By the same token, we can see mathematics as introducing both new concepts and new forms of reasoning into other domains, such as economics and for that matter physics (which is exactly what Gaston Bachelard has argued).

This double movement has various effects. It means that there are no grounds for considering mathematics to be a privileged language with respect to other, nonmathematical languages. There is, for example, no logical necessity inherent in the use of the mathematical language. The theorist makes choices about the kinds of mathematics that are used, about the steps from one mathematical argument to another, and whether or not any mathematics will be used at all. Different uses (or not) of mathematics and different kinds of mathematics will have determinate effects on the discourse in question. Discourses change as they are mathematized—they are changed, not in the direction of becoming more (or less) scientific, but by transforming the way the objects of the discourse are constructed, and the way statements are made about those objects.

Ultimately, this deconstruction of mathematics as a special code leads to a rejection of the conception of mathematics as a special language of representation. The status of mathematics is both more representational and less representational than allowed by the discourse of representation. More, in the sense that mathematics has effects on the very structure of the mathematized theory; mathematics is not neutral. Less, to the extent that the use of mathematics does not guarantee the scientificity of the theory in question; it is merely one discursive strategy among others.

One alternative approach to making sense of the use of mathematics in economic theory is to consider mathematics not in terms of representation, but as a form of “illustration.” For economists, mathematical concepts and models can be understood as metaphors or heuristic devices that illustrate part of the contradictory movement of economic and social processes. These concepts and models can be used, where appropriate, to consider in artificial isolation one or another moment in the course of the constant movement and change in the economy and society. Mathematics may be used, then, to illustrate the statements of economic theory but, like all metaphors (in economics as in literature and other areas of social thought), it outlives its usefulness and then has to be dismantled.

As I see it, this conception of mathematical models as illustrative metaphors does not constitute a flat rejection of their use in economic theory. Rather, it accords to mathematical concepts and models a discursive status different from the one that is attributed to them in the work of mathematical economists. It accepts the possibility—but not the necessity—of using mathematical propositions as metaphors that are borrowed from outside of economic theory and transformed to teach and develop some of the concepts and statements of one or another economic theory.

Deconstructing the status of mathematics as a special code has the advantage of transforming both the way economics is done within any particular theory and the way the debate between different economic theories itself is conducted. It undermines the Truth-effect associated with mathematical utopianism and focuses attention, instead, on the conditions and consequences of different ways of thinking about the economy.

That debate—about the effects of different languages on economics, and the effects of different economic theories on the wider society—has its own utopian moment: transforming economics into a space not of blind obedience to mathematical protocols, but of real theoretical and political choices.


*From time to time, there have been a few admonishments from among economists themselves. Oskar Morgenstern (e.g., in his essay “Limits to the Uses of Mathematics in Economics,” published in 1963) and, more forcefully, Nicholas Georgescu-Roegen (especially in his 1971 Entropy Law and the Economic Process), Philip Mirowski (e.g., in More Heat Than Light, in 1989), and Paul Romer have indicated some of the problems associated with the wholesale mathematization of economics. However, even their limited criticisms have been ignored for the most part by economists.

In recent years, students (such as the members of the International Student Initiative for Pluralism in Economics) have been at the forefront of questioning the fetishism of mathematical methods in economics:

It is clear that maths and statistics are crucial to our discipline. But all too often students learn to master quantitative methods without ever discussing if and why they should be used, the choice of assumptions and the applicability of results. Also, there are important aspects of economics which cannot be understood using exclusively quantitative methods: sound economic inquiry requires that quantitative methods are complemented by methods used by other social sciences. For instance, the understanding of institutions and culture could be greatly enhanced if qualitative analysis was given more attention in economics curricula. Nevertheless, most economics students never take a single class in qualitative methods.

Their pleas, too, have been mostly greeted with indifference or contempt by economists.

**As I see it, the current fad of relying on randomized experiments and big data does not really undo the longstanding utopian claims associated with mathematical modeling, since the formal models are still there in the background, orienting the issues (including the choice of data sets) taken up in the new experimental and data-heavy approach to economics. Then, in addition, there is the problem that others—such as John P. A. Ionnidis et al. (unfortunately behind a paywall)—have discovered: most economists use data sets that are much too small relative to the size of the effects they report. This means that a sizable fraction of the findings reported by economists are simply the result of publication bias—the tendency of academic journals to report accidental results that only appear to be statistically significant.

***Economists often move back and forth between the two otherwise diametrically opposed conceptions of mathematics because they represent two sides of the same epistemological coin: although each reverses the order of proof of the other, both empiricism and rationalism presume the same fundamental terms and some form of correspondence between them. In this sense, they are variant forms of an “essentialist” conception of the process of theorizing. Both of them invoke an absolute epistemological standard to guarantee the (singular, unique) scientificity of the production of economic knowledge.

  1. Geoff Davies
    August 4, 2018 at 2:03 am

    Hoo boy, on and on this goes.

    Mathematics is a collection of abstract logical structures.

    Science is the search for stories that provide useful guidance on the behaviour of the observable world.

    Mathematics is sometimes useful in the practice of science – it is a tool – but mathematics is not science.

    Science requires continuing iteration between ideas-theories-models on the one hand and observations on the other. A successful theory is not Truth, it is a story that provides USEFUL guidance in a particular context.

    Can there be a science of economies?

  2. Ron Goldring
    August 4, 2018 at 6:56 am

    To say some statement about people (and Economics is about people, isn’t it?) is true because the statement is expressed by a mathematical expression, is akin to saying it is true because it is expressed in English/Hebrew/Arabic/any other language you choose.

    One can say anything ‘elegantly’. It has nothing to do with truth.

  3. Robert Locke
    August 4, 2018 at 1:55 pm

    In this dispute about the usefulness of mathematics, we need to ask what education is for. Is it just something to improve our understanding of science? Or is the purpose of education about providing moral guidance to people destined to wield power through accumulation of wealth and state authority. Marianne Wertz, in 1999, discussed Wilhelm von Humboldt’s view of the superiority of a classical education.

    Humboldt appreciated mathematics but wrote in a letter to his wife Caroline, “It is only through the study of language that there comes into the soul, out of the source of all thoughts and feelings, the entire expanse of ideas, everything that concerns man, above all and beyond everything else, even beauty and art.”

    He held that “Language is deeply entwined in the intellectual development of humanity itself, it accompanies the latter upon every step of its localized progression or regression; moreover, the pertinent cultural level in each case is recognizable in it. … Language is, as it were, the external manifestation of the minds of peoples. Their language is their soul, and their soul is their language. It is impossible to conceive them ever sufficiently identical… . The creation of language is an innate necessity of humanity. It is not a mere external vehicle, designed to sustain social intercourse, but an indispensable factor for the development of human intellectual powers, culminating in the formulation of philosophical doctrine.

    As to the study of classical Greek, Humboldt, in an autobiographical fragment written when he was nearly fifty, emphasized the role that the study of the classics had in his own development: “I have always had a revulsion against interfering in the world and an urge to stand free of it, observing and examining it. This led me naturally to feel that only the most unconditional self-control might give me the standpoint outside the world that I should need… . These notions were first awakened in me by antiquity, later they kept me in relation to the ancients for evermore.”

    In commenting on the importance of studying classical Greek for the Prussia of his day, Wertz notes “Humboldt’s words are equally appropriate for our own time: The study of the characteristics of Greek culture is especially beneficial in an epoch when, for countless reasons, attention is more focused on masses of men than on individuals, more on external values and uses than on inner worth and enjoyment, and when a high and variegated culture has deviated very far from the earlier simplicity. … When the [Greek] nation had not yet entirely raised itself out of primitive circumstances, it already possessed an uncommonly subtle feeling for everything beautiful in nature and art. … The broad diffusion of the feeling for beauty among the entire nation is especially admirable; and nothing can be more important for our world than a comprehension of this characteristic feature.”

    His intent, he said, was to “inoculate the Germans with the Greek spirit.”

    Mathematics, whether its useful to the study of economics, seems to be useless in the cultivation of the innerself, that Humboldt sought. It seems to me that an education that would cultivate the innerself is more important in the education of economists, e.g., language of the
    humanities, than mathematics.

  4. Edward K Ross
    August 5, 2018 at 12:54 am

    Thank you Robert Locke for your clear explanation of where the conversation needs to start not only about the usefulness of mathematics, but also about the more important question of
    “what is education for, is it just something to improve our understanding of science?. Or is the purpose of education about providing moral guidance to people destined to wield power through accumulation of wealth and state authority.”

    Here I add that a truly democratic society requires the masses to be informed in simple terms how they can be informed that they have a right to influence the holders of power to respond to their concerns. From my humble position of experience in the real world I find your use of Humboldt’s ideas really resonates with me, particularly the paragraph;

    “He held that language is deeply entwined in the intellectual development of humanity itself.”
    Because this paragraph expresses the importance of “language as the expression of the soul
    [and then] the inner self is more important in the education of humanities, eg., language of the humanities, than mathematics,
    . Again from my logic formed from experience in the real world I think Robert Locke with the use of Humbolts ideas has pointed out where the economic conversation needs to start before engaging in theoretical conversations often based on false assumptions that ignore real people in the real world. Ted.

  5. Helen Sakho
    August 5, 2018 at 2:52 am

    Thank you David and other colleagues.
    Yes, this how the SPECIFIC becomes the CONTEXTUAL, and they swing in between going backwards and forwards.
    If we cannot explain the “What” and the “How” of a phenomena, can we ever pinpoint the “Why” of it, and leave the student to contemplate and formulate their own critical analyses? I do not think so.
    In every language, every culture and every community endless examples exist to illustrate and to plausibly argue that in any world (human, animal, plants) all phenomena are intrinsically intertwined. Once you segment knowledge, you will focus of a specific segment of a “new discovery” and be guided by funding regimes to propagate what needs to be investigated urgently, you will be selling useless information over and over again. Such is the key difference between knowledge and information.
    We need a new curriculum. Books such as “The NO-Nonsense guide to International Migration by Stalker (2001),” “Big Business POOr People” by Madeley (1999), and a very small booklet for a succinct listing of ‘the Universal Declaration of Human Rights” by Waterstones and Amnesty International” should be starting points. As should a highly academic publication I had the pleasure of reviewing a long time ago by the University of Dar es Salaam (1982) ” Debates on class, state and imperialism” – Published in Tanzania and Printed in India.
    The WORLD really does include the MENA region, and our students should learn something from that vast place, what has happened to it, just as the whole world receives our, no doubt, worthy contributions.

  6. David Harold Chester
    August 5, 2018 at 7:46 am

    When mathematics is applied to the various “engineering” sciences one should not always expect the results to be equally successful, yet when this is done the implied and expected results usually are so success oriented that there is extreme doubt cast on the theory when the results are unsatisfactory. Are we better justified in saying that the assumptions are wrong rather than that the analysis is no good? I believe so and also that not all kinds of mathematics are suited for solving the different problems we are likely to encounter in economics. Better to describe the problem in terms of what needs to be assumed rather than which theory is most easily applied.And then when the terms are properly defined we can proceed to apply our math to what structures are outstanding.

  7. August 5, 2018 at 2:12 pm

    So, my takeaway from this is simply: “Use mathematics more for generation of hypotheses and less for their validation”.

  8. Helen Sakho
    August 6, 2018 at 2:17 am

    For an even better “Takeaway” I would stick to Amazon delivery. It is guaranteed. It is online, it is fast. Deliveroo may break your heart when you meet at your doorstep who delivers your food, but Amazon will deliver goods that do not even exist, including books you withdrew from the marketplace years ago, unless you wait for their release in Australia! It will simply inform you that they are not currently available, so you never lose hope as a potential future customer. Yesterday’s villains are tomorrow’s heroes and today is somewhere in between. It is this “In-between-ness” that is of real concern. It is nowhere, but everywhere. And do keep up with news. Headlines will do. In a decade or so, perhaps we can ask our scholars to remember the last week of August 2018, outlining key developments that changed the whole world, and compare and contrast to roughly speaking 2028 (their time). It will be a miracle if they have the slightest idea what we are talking about, unless, they have, to quote an old time favourite quote “suffered a little”. Alternatively, we can rephrase the question (just to be kind and merciful) to something like compare and contrast which famous token victims of a not too distant past made the headlines as a glamour model and which former glamour model denounced her former admirers as dysfunctional. Turning this into a mathematical model is easy, and I think the latter might be a better way of ensuring that some things are remembered accurately. And, if, and only if, no-one can remember how to read, write or calculate under exam conditions by then, perhaps they can just illustrate their case with an authentic photograph or two. The Latino street-vendor kids never had/have a chance of any education, but they sure do know how to calculate the pittance they earn from passers by with gratitude.

  9. August 7, 2018 at 12:02 pm

    David, I concur with your contention that mathematics is a historical, cultural invention, not a form of discovery of an independent reality. That mathematics is not discovered “out there,” but invented and reinvented over time based on rules invented and passed along by their inventors – mathematicians, certainly but more often by ordinary users of mathematics (including economists, as well as accountants, merchants, census takers, builder, etc.). Mathematics reveals no truth to its users, but it is a useful tool for depicting and illustrating how these users believe the world works. Take calculus as an example. The form and foci of physics and the other initial sciences developed from the concerns considered most important at that time. These were the problems in mechanics, hydrodynamics, aerodynamics, ballistics, optics, metallurgy, etc. posed by the nascent capitalism, as well the related problems of navigation, poverty, class conflict, and governing. These tied science and later calculus to a specific form and range of actions. Science became the revealer of “reality.” Thus rejecting “constructing” a better or even different life for humans. And calculus was one of the tools for that work. Thus, from its inception western science rejected the “science” of people like Bernardin de Saint-Pierre that asserted natural phenomenon must be studied in environmental context. Mathematics, particularly calculus was the line dividing those who favored contextual science from those who did not. Understanding the relationships between human actions and the physical world is impossible with non-ecological science. It’s only in the last 50 years that western science has returned to ecological science and moved away from mathematical science that accepts to contexts, but only universality.

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