On mathematics and economics

22 Aug, 2013 at 10:57 | Posted in Economics, Theory of Science & Methodology | 6 Comments

Neoclassical economic theory today is in the story-telling business whereby economic theorists create mathematical make-believe analogue models of the target system – usually conceived as the real economic system. This mathematical modeling activity is considered useful and essential. Since fully-fledged experiments on a societal scale as a rule are prohibitively expensive, ethically indefensible or unmanageable, economic theorists have to substitute experimenting with something else. To understand and explain relations between different entities in the real economy the predominant strategy is to build mathematical models and make things happen in these “analogue-economy models” rather than engineering things happening in real economies.

Formalistic deductive “Glasperlenspiel” can be very impressive and seductive. But in the realm of science it ought to be considered of little or no value to simply make claims about the model and lose sight of reality.

With his profound knowledge of mathematics, Keynes realized the limits of its applicability to the real world — and that it was certainly not enough for a relevant social science to prove things about thought up worlds:

But I am unfamiliar with the methods involved and it may be that my impression that nothing emerges at the end which has not been introduced expressly or tacitly at the beginning is quite wrong … It seems to me essential in an article of this sort to put in the fullest and most explicit manner at the beginning the assumptions which are made and the methods by which the price indexes are derived; and then to state at the end what substantially novel conclusions has been arrived at … I cannot persuade myself that this sort of treatment of economic theory has anything significant to contribute. I suspect it of being nothing better than a contraption proceeding from premises which are not stated with precision to conclusions which have no clear application … [This creates] a mass of symbolism which covers up all kinds of unstated special assumptions.

Keynes to Frisch 28 November 1935

Neoclassical economists often hold the view that criticisms of its one-sided, almost religious, insistence on mathematical-axiomatic-deductivist modeling as the only scientific activity worthy of pursuing in economics, are the conclusions of sadly misinformed and misguided people who dislike and do not understand much of it. This is really a gross misapprehension. To be careful and cautious is not the same as to dislike. And as any perusal of the works of people like for example John Maynard Keynes or Tony Lawson shows, the critique is put forward by respected authorities. I would argue, against “common knowledge”, that they do not misunderstand the crucial issues at stake. Quite the contrary. They know them all too well – and are not satisfied with the validity and philosophical underpinning of the assumptions made for applying its methods:

The fundamental problem of modern economics is that methods are repeatedly applied in conditions for which they are not appropriate … Specifically, modern academic economics is dominated by a mainstream tradition whose defining characteristic is an insistence that certain methods of mathematical modelling be more or less always employed in the analysis of economic phenomena, and are so in conditions for which they are not suitable.

tony-lawsonFundamental to my argument is an assessment that the application of mathematics involves more than merely the introduction of a formal language. Of relevance here is recognition that mathematical methods and techniques are essentially tools. And as with any other tools (pencils, hammers, drills, scissors), so the sorts of mathematical methods which economists wield (functional relations, forms of calculus, etc.) are useful under some sets of conditions and not others.

The specific conditions required for the sorts of mathematical methods that economists continually wield to be generally applicable, I have shown, are a ubiquity of (deterministic or stochastic) closed systems. A closed system is simply one in which an event regularity occurs. Notice that these closures are as much presupposed or required by the ‘newer’ approaches to mathematical economics, those often referred to as non-linear modelling, complexity modelling, agent-based modelling, model simulations, and so on (including those developed under the head of behavioural or neuro- economics), as they are by the more traditional forms of micro, macro and econometric modelling.

The most obvious scenario in which a prevalence of such closures would be expected is a world 1) populated by sets of atomistic individuals or entities (an atom here being an entity that exercises its own separate, independent, and invariable effect, whatever the context); where 2) the atoms of interest exist in relative isolation (so allowing the effects of the atoms of interest to be deducible/predictable by barring the effects of potentially interfering factors). Not surprisingly the latter two (ontological) presuppositions are easily shown to be implicit in almost all contemporary economic modelling contributions …

However, explicit, systemic and sustained (ontological) analysis of the nature of social reality reveals the social domain not to be everywhere composed of closed systems of sets of isolated atoms. Rather social reality is found to be an open, structured realm of emergent phenomena that, amongst other things, are processual (being constantly reproduced and transformed through the human practices on which they depend), highly internally related (meaning constituted though [and not merely linked by] their relations with each other – e.g., employer/employee or teacher/ student relations), value-laden and meaningful, amongst much else …

Clearly if social phenomena are highly internally related they do not each exist in isolation. And if they are processual in nature, being continually transformed through practice, they are not atomistic. So the emphasis on the sorts of mathematical modelling methods that economists employ necessarily entails the construction of economic narratives – including the sorts of axioms and assumptions made and hypotheses entertained – that, at best, are always but highly distorted accounts of the complex phenomena of the real open social system … It is thus not at all surprising that mainstream contributions are found continually to be so unrealistic and explanatorily limited.

Employing the term deductivism to denote the thesis that closed systems are essential to social scientific explanation (whether the event regularities, correlations, uniformities, laws, etc., are either a prior constructions or a posterior observations), I conclude that the fundamental source of the discipline’s numerous, widespread and long lived problems and failings is precisely the emphasis placed upon forms of mathematical deductivist reasoning.

Tony Lawson


  1. Quantitative social science is more often pseudoscience than not.

    I also consider the people who claim that their critics do not understand maths, to be the very people who do not “get” maths, they may be good at the labour of maths, but clueless as to understanding it.

    What makes maths successful and relevant in natural science are entirely due to constants. Without any constants it would be completely useless even in natural science. Not understanding this is to not “get” maths.

    Social science would be much better served by people being out there looking for a single constant or generating an actual datapoint, instead of sitting in their chambers doing maths on system responses, which just is a form of intellectual masturbation.

  2. On Intellectual Craftsmanship
    Oxford University Press

    “TO THE INDIVIDUAL social scientist who feels himself a part of the classic tradition, social science is the practice of a craft. A
    man at work on problems of substance, he is among those who are quickly made impatient and weary by elaborate discussions of
    method-and-theory-in-general; so much of it interrupts his proper studies. It is much better, he believes, to have one account by a
    working student of how he is going about his work than a dozen ‘codifications of procedure’ by specialists who as often as not
    have never done much work of consequence. Only by conversations in which experienced thinkers exchange information about
    their actual ways of working can a useful sense of method and theory be imparted to the beginning student. I feel it useful,
    therefore, to report in some detail how I go about my craft. This is necessarily a personal statement, but it is written with the hope
    that others, especially those beginning independent work, will make it less personal by the facts of their own experience.

    In many academic circles today anyone who tries to write in a widely intelligible way is liable to be condemned as a `mere literary
    man.’ or, worse still, `a mere journalist’ Perhaps you have already learned that these phrases, as commonly used, only indicate the
    spurious inference: superficial because readable. The academic man in America is trying to carry on a serious intellectual life in a
    social context that often seems quite set against it. His prestige must make up for many of the dominant values he has sacrificed
    by choosing an academic career. His claims for prestige readily become tied to his self-image as a ‘scientist’. To be called a `mere
    journalist’ makes him feel undignified and shallow…
    To overcome the academic prose you have first to overcome the academic pose. It is much less important to study grammar and
    Anglo-Saxon roots than to clarify your own answers to these three questions: (1) How difficult and complex after all is my
    subject? (2) When I write, what status am I claiming for myself? ( 3 ) For whom am I trying to write? (1) The usual answer to the first question is: Not so difficult and complex as the way in which you are writing about it. Proof of
    that is everywhere available: it is revealed by the ease with which 95 per cent of the books of social science can be translated into English.
    But, you may ask, do we not sometimes need technical terms? 9
    Of course we do, but `technical’ does not necessarily mean
    difficult, and certainly it does not mean jargon. If such technical terms are really necessary and also clear and precise, it is not
    difficult to use them in a context of plain English and thus introduce them meaningfully to the reader.
    (Those who understand mathematical language far better than I tell me that it is precise, economical, clear. That is why I am so
    suspicious of many social scientists who claim a central place for mathematics among the methods of social study but who write
    prose imprecisely, uneconomically, and unclearly. They should take a lesson from Paul Lazarsfeld, who believes in mathematics,
    very much indeed, and whose prose always reveals, even in first draft, the mathematical qualities indicated. When I cannot
    understand his mathematics, I know that it is because I am too ignorant; when I disagree with what he writes in non-mathematical
    language, I know it is because he is mistaken, for one always knows just what he is saying and hence just where he has gone
    Perhaps I can best summarize what I have been trying to say in the form of a few precepts and cautions:
    ( 1 ) Be a good craftsman: Avoid any rigid set of procedures. Above all, seek to develop and to use the sociological imagination.
    Avoid the fetishism of method and technique. Urge the rehabilitation of the unpretentious intellectual craftsman, and try to
    become such a craftsman yourself. Let every man be his own methodologist; let every man be his own theorist; let theory and
    method again become part of the practice of a craft. Stand for the primacy of the individual scholar; stand opposed to the
    ascendancy of research teams of technicians. Be one mind that is on its own confronting the problems of man and society.
    ( 2 ) Avoid the Byzantine oddity of associated and disassociated Concepts, the mannerism of verbiage. Urge upon yourself and
    upon others the simplicity of clear statement. Use more elaborated terms only when you believe firmly that their use enlarges the
    scope of your sensibilities, the precision of your references, the depth of your reasoning. Avoid using unintelligibility as a means
    of evading the making of judgments upon society-and as a. means of escaping your readers’ judgments upon your own work.
    ( 3 ) Make any trans-historical constructions you think your work requires; also delve into sub-historical minutiae. Make up quite
    formal theory and build models as well as you can. Examine in detail little facts and their relations, and big unique events as well.
    But do not be fanatic: relate all such work, continuously and closely, to the level of historical reality. Do not assume that
    somebody else will do this for you, sometime, somewhere. Take as your task the defining of this reality; formulate your problems
    in its terms; on its level try to solve these problems and thus resolve the issues and the troubles they incorporate. And never write
    more than three pages without at least having in mind a solid example.
    ( 4 ) Do not study merely one small milieu after another; study the social structures in which milieu are organized. In terms of
    these studies of larger structures, select the milieux you need to study in detail, and study them in such a way as to understand the
    interplay of milieux with structure. Proceed in a similar way in so far as the span of time is concerned. Do not be merely a
    journalist, however precise a one. Know that journalism can be a great intellectual endeavor, but know also that yours is greater!
    So do not merely report minute researches into static knife-edge moments, or very short-term runs of time. Take as your time-span
    the course of human history, and locate within it the weeks, years, epochs you examine.
    ( 5 ) Realize that your aim is a fully comparative understanding of the social structures that have appeared and that do now exist in
    world history. Realize that to carry it out you must avoid the arbitrary specialization of prevailing academic departments. Specialize your work variously, according to topic, and above all according to significant problem. In formulating and in trying to
    solve these problems, do not hesitate, indeed seek, continually and imaginatively, to draw upon the perspectives and materials, the
    ideas and methods, of any and all sensible studies of man and society. They are your studies; they are part of what you are a part
    of; do not let them be taken from you by those who would close them off by weird jargon and pretensions of expertise.
    ( 6 ) Always keep your eyes open to the image of man-the generic notion of his human nature-which by your work you are
    assuming and implying; and also to the image of history-your notion of how history is being made. In a word, continually work
    out and revise your views of the problems of history, the problems of biography, and the problems of social structure in which
    biography and history intersect. Keep your eyes open to the varieties of individuality, and to the modes of epochal change. Use
    what you see and what you imagine, as the clues to your study of the human variety.
    (7) Know that you inherit and are carrying on the tradition of classic social analysis; so try to understand man not as an isolated
    fragment, not as an intelligible field or system in and of itself. Try to understand men and women as historical and social actors,
    and the ways in which the variety of men and women are intricately selected and intricately formed by the variety of human
    societies. Before you are through with any piece of work, no matter how indirectly on occasion, orient it to the central and continuing task of understanding the structure and the drift, the shaping and the meanings, of your own period, the terrible and
    magnificent world of human society in the second half of the twentieth century.
    ( 8 ) Do not allow public issues as they are officially formulated, or troubles as they are privately felt, to determine the problems
    that you take up for study. Above all, do not give up your moral and political autonomy by accepting in somebody else’s terms the
    illiberal practicality of the bureaucratic ethos or the liberal practicality of the moral scatter. Know that many personal troubles
    cannot be solved merely as troubles, but must be understood in terms of public issues-and in terms of the problems of historymaking. Know that the human meaning of public issues must be revealed by relating them to personal troubles-and to the
    problems of the individual life. Know that the problems of social science, when adequately formulated, must include both troubles”

    Click to access Mills_Intell_Craft.pdf

  3. […] Lars Syll has linked to a really interesting paper by Tony Lawson amidst a discussion about maths and modelling in economics. The paper really is worth a read in its entirety. It is entitled Mathematical Modelling and Ideology in the Economics Academy: Competing Explanations of the Failings of the Modern Discipline? and can be found for free download here. In it Lawson deals with what makes mainstream economics so desperately poor and he ultimately undertakes an examination of what I called “Brain-Slug Economics” elsewhere. […]

  4. Lars, You might be interested in an advance summary of Lawson’s forthcoming paper. I think it;s a big deal:


    • Thanks Philip. Tony’s argumentation is indeed very interesting, and I hope to get some time off later this week for writing a post relating to it.

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