Economics — a rogue branch of applied mathematics

1 September, 2015 at 20:50 | Posted in Economics | 2 Comments

A lot of people complain about the math in economics. Economists tend to quietly dismiss such complaints as the sour-grapes protests of literary types who lack the talent or training to hack their way through systems of equations. But it isn’t just the mathematically illiterate who grouse. New York University economist Paul Romer — hardly a lightweight when it comes to equations — recently complained about how economists use math as a tool of rhetoric instead of a tool to understand the world.

math_logo_35Personally, I think that what’s odd about econ isn’t that it uses lots of math — it’s the way it uses math. In most applied math disciplines — computational biology, fluid dynamics, quantitative finance — mathematical theories are always tied to the evidence. If a theory hasn’t been tested, it’s treated as pure conjecture.

Not so in econ. Traditionally, economists have put the facts in a subordinate role and theory in the driver’s seat. Plausible-sounding theories are believed to be true unless proven false, while empirical facts are often dismissed if they don’t make sense in the context of leading theories. This isn’t a problem with math — it was just as true back when economics theories were written out in long literary volumes. Econ developed as a form of philosophy and then added math later, becoming basically a form of mathematical philosophy.

In other words, econ is now a rogue branch of applied math.

Noah Smith

Indeed.

No, there is nothing wrong with mathematics per se.

No, there is nothing wrong with applying mathematics to economics.

amathMathematics is one valuable tool among other valuable tools for understanding and explaining things in economics.

What is, however, totally wrong, are the utterly simplistic beliefs that

• “math is the only valid tool”

• “math is always and everywhere self-evidently applicable”

• “math is all that really counts”

• “if it’s not in math, it’s not really economics”

• “almost everything can be adequately understood and analyzed with math”
 

Knut-WicksellOne must, of course, beware of expecting from this method more than it can give. Out of the crucible of calculation comes not an atom more truth than was put in. The assumptions being hypothetical, the results obviously cannot claim more than a vey limited validity. The mathematical expression ought to facilitate the argument, clarify the results, and so guard against possible faults of reasoning — that is all.

It is, by the way, evident that the economic aspects must be the determining ones everywhere: economic truth must never be sacrificed to the desire for mathematical elegance.
 

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  1. At last, mathiness problem settled
    Comment on ‘Economics — a rogue branch of applied mathematics’
    .
    Noah Smith writes: “Traditionally, economists have put the facts in a subordinate role and theory in the driver’s seat. Plausible-sounding theories are believed to be true unless proven false, while empirical facts are often dismissed if they don’t make sense in the context of leading theories. This isn’t a problem with math …” (See intro)
    .
    Exactly so. The problem has never been math but that economists have never grasped what science is all about: “Research is in fact a continuous discussion of the consistency of theories: formal consistency insofar as the discussion relates to the logical cohesion of what is asserted in joint theories; material consistency insofar as the agreement of observations with theories is concerned.” (Klant, 1994, p. 31)
    .
    The problem can be exactly located here: “As with any Lakatosian research program, the neo-Walrasian program is characterized by its hard core, heuristics, and protective belts. Without asserting that the following characterization is definitive, I have argued that the program is organized around the following propositions: HC1 economic agents have preferences over outcomes; HC2 agents individually optimize subject to constraints; HC3 agent choice is manifest in interrelated markets; HC4 agents have full relevant knowledge; HC5 observable outcomes are coordinated, and must be discussed with reference to equilibrium states.
    .
    By definition, the hard-core propositions are taken to be true and irrefutable by those who adhere to the program. “Taken to be true” means that the hard-core functions like axioms for a geometry, maintained for the duration of study of that geometry.” (Weintraub, 1985, p. 147)
    .
    The material refutation consists in the refutation of these behavioral axioms, that is, HC1 is vacuous and HC2 to HC5 lack material consistency. This alone puts an end to the traditional research program.
    .
    The formal — i.e. mathiness — refutation consists in: “Thus not all axiomatic theories need to be phrased in terms of set theory but much more conveniently and intelligibly rather in terms of some advanced mathematical structures.” (Schmiechen, 2009, p. 367)
    .
    Solution of the mathiness problem: economics has to move from Debreu’s set theoretical approach to an advanced formal structure (2014). Genuine scientists always understood — and economists never got it — that a theory consists of TWO vital elements.
    .
    “A scientific deductive system (“scientific theory”) is a set of propositions in which each proposition is either one of a set of initial propositions … or a deduced proposition … which is deduced from the set of initial propositions according to logico-mathematical principles of deduction, and in which some (or all) of the propositions of the system are propositions exclusively about observable concepts (properties or relations) and are directly testable against experience.” (Braithwaite, 1959, p. 429)
    .
    Note that the hard core propositions HC1 to HC5 lack observable concepts. This has always been the pivotal problem, not math per se.
    .
    Egmont Kakarot-Handtke
    .
    References
    Braithwaite, R. B. (1959). Axiomatizing a Scientific System by Axioms in the Form
    of Identifications. In L. Henkin, P. Suppes, and A. Tarski (Eds.), The Axiomatic
    Method, pages 429–453. Amsterdam: North-Holland.
    Kakarot-Handtke, E. (2014). Objective Principles of Economics. SSRN Working
    Paper Series, 2418851: 1–19. URL
    http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2418851.
    Klant, J. J. (1994). The Nature of Economic Thought. Aldershot, Brookfield, VT:
    Edward Elgar.
    Schmiechen, M. (2009). Newton’s Principia and Related ‘Principles’ Revisited,
    volume 1. Norderstedt: Books on Demand, 2nd edition. URL
    http://books.google.de/books?id=3bIkAQAAQBAJ&printsec=frontcover&hl=
    de&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false.
    Weintraub, E. R. (1985). Joan Robinson’s Critique of Equilibrium: An Appraisal.
    American Economic Review, Papers and Proceedings, 75(2): 146–149. URL
    http://www.jstor.org/stable/1805586.

  2. ” . . . In most applied math disciplines . . . mathematical theories are always tied to the evidence. . . . Not so in econ. . . . economists have put the facts in a subordinate role and theory in the driver’s seat. . . . empirical facts are often dismissed if they don’t make sense in the context of leading theories. This isn’t a problem with math . . . “

    This isn’t a problem with math. !!!
    Noah also says, “Plausible-sounding theories are believed to be true unless proven false” but doesn’t say what it means to be “proven false” when mere facts are subordinate to “plausible” speculation. I think it often means that economists enjoy collecting “models” as a hobbyist might collect coins, not for their transactional value in intercourse with the world, but to admire for their curious features and histories. Nothing is ever really wrong in economics; models are said to answer particular questions. Money can be neutral in the long-run, but not-neutral in the short-run — different models, you see and no bother about a common reality — and no sees the contradiction or wonders about the properties of the actual economy.


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