Physics Envy and All That

9 January 2007 at 12:44 pm 50 comments

| Steven Postrel |

We often hear (sometimes on this blog) that mainstream economics suffers from an excess of mathematical modeling. Supposedly, math is distracting, or misleading, or limits the questions one can study. Occasionally it is asserted that math serves the purpose of disguising the triviality of one’s thoughts, or that it serves as a guild’s protectionist barrier against the worthy but unschooled. In my view, all of the same critiques may apply to any use of technical language (say in philosophy); one can find examples of all of these pathologies even when no math is involved.

Our problems, when they occur, do not lie in our tools but in the quality of our ideas, and our honesty in expressing them. And given the extreme difficulty in thinking clearly or precisely without mathematics about things like supply and demand, or optimal investment, or contingent contracts, or network structure and growth, I’m more than willing to entertain mathematical approaches. At least I can figure out what people’s assumptions are. (Of course, once you have the mathematical intuition down, it’s a good idea to try to translate your new understanding into verbal form, as long as everyone understands that something is always lost in translation.)

Critics of mathematics in economics, however, rarely come to grips with these specific issues of problem solving and understanding. (I have yet to hear, for example, an explanation of how diminishing or increasing marginal returns makes sense in anything but a mathematical context.) Instead, they employ a tactic that I think Phillip Mirowsky pioneered, accusing economists of suffering from “physics envy.” The idea, phrased to have an insulting Freudian resonance, is that economists aren’t smart enough to be physicists but like to play dress-up by adopting (what they think are) similar formal methods as their more-prestigious colleagues. Every now and then we hear the Feynmanesque “cargo-cult science” accusation that economists are like primitive Pacific islanders building grass control towers (mathematical models) and expecting real airplanes (scientific truth) to land.

Some people appear to think that the historical process by which math came into economics has a bearing on the accuracy of this smear. If Walras was influenced by contemporary works in physics, then the charge of physics envy is supposed to be proven somehow. This is a simple genetic fallacy, since the context in which someone gets an idea (say general equilibrium) has little bearing on the idea’s validity. The structure of the carbon atom supposedly came to its discover in a dream, yet historians of chemistry don’t talk about “shaman envy.”

But beyond this fallacy, the physics envy accusation is silly on its own terms. John von Neumann knew a fair amount of math and physics, and it didn’t seem inappropriate to him that economists should be using math — he complained that they didn’t use sufficiently sophisticated math (back in the 1940s), and would have laughed at the idea that the subject was somehow not suitable for formal modeling. Do ethologists have “economics envy” when they apply game theory or individual optimization methods to animal behavior?

The whole point of math is that lots of substantively different problems have similar formal properties. Exponential functions describe compound interest and they also describe radioactive decay (obviously, with a sign reversed). It would be an absurd example of Harold Bloom’s “anxiety of influence” if economists abjured exponential discounting to avoid accusations of physics envy. (Since exponential growth was one of the first formal ideas to be used in economics, by Malthus, it is an appropriate example.) Black and Scholes didn’t know that their partial differential equation for option pricing had something to do with the physics of freezing lakes — it just turned out that way.

So the next time someone pulls out the “physics envy” card, don’t be intimidated. They’re bluffing and playing a weak hand.

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50 Comments Add your own

  • 1. David Gordon  |  9 January 2007 at 2:34 pm

    Philip Mirowski doesn’t oppose the use of scientific models in economics. His complaint is not that economists are guilty of “physics envy” — a phrase he did not coin — but that they rely on outdated science. Von Neumann is a hero of his Machine Dreams, and he thinks economics should imitate up-to-date “cyborg” science.

    Harold Bloom’s “anxiety of influence” concerns a writer’s troubled relationship to a strong predecessor. The notion doesn’t seem relevant to the point being made here.

    It is surprising that Malthus’s use of exponential growth is cited in a post supporting the use of math in economics. Malthus’s model of population growth is generally regarded as radically mistaken.

  • 2. Lasse  |  9 January 2007 at 3:22 pm

    I, for one, openly admit to physics envy. I envy them their scientific success. Don’t we all?

  • 3. spostrel  |  9 January 2007 at 6:45 pm

    David: Thanks for the clarification about Mirowsky not using the term “physics envy” himself. I never said that he opposed the use of science, but that he opposed the use of mathematical models of the sort economists routinely deploy. If he’s now onboard with game theory because it’s newer, great, but he is invoked regularly to attack the modeling practices of economics. It’s also possible that most of the people who cite him are vulgarizing him, which is a danger for any influential writer.

    I was hoping for a less literal-minded attitude toward the “anxiety of influence.” The idea in poetry is that the younger writer struggles to find a new way to express things to escape the shadow of his great predecessor. He tries to avoid simply aping this elder, hence engages in various misreadings and transformative strategies. By analogy, an economist might treat the body of physical theory as a strong predecessor and go out of his way to be original. I don’t think that that would be a sensible to way to make scientific progress. I was looking for a little poetic license.

    Malthus wasn’t wrong about the mathematical form of population growth. He was wrong about a) the stability of fertility at much higher levels of per capita income and b) the potential growth rate of agricultural productivity due to technological innovation (he didn’t foresee McCormick or Borlaug coming along).

    Lasse: I don’t envy physics its successes. When I can understand them, I enjoy them. (What I envy is the clarity of their empirical problems and the stability and uniformity of the systems they study.) I might even see if some trick of theirs might apply to what I do. Unfortunately, when they have to work with complex systems and non-equiilibrium processes, like snow crystal growth (very good cover story on this in the American Scientist on the newsstand now), they grope around a lot, too. And that’s with the ability to perform controlled experiments. Some problems are just really hard.

  • 4. steven postrel at O&M « orgtheory.net  |  9 January 2007 at 11:36 pm

    [...] Note, Steven Postrel is guest-blogging at O&M. [...]

  • 5. Carl  |  10 January 2007 at 5:42 am

    It was benzene, not carbon per se, but close enough.

  • 6. Michael Greinecker  |  10 January 2007 at 11:55 am

    Ecellent post!

    Let’s take a look at people that have worked in mathematical economics: Stephen Smale and John Milnor are fields medalists. Abraham Robinson is one of the most important model theorists of the 20th century. Roko Aliprantis is an excellent functionial analyst how simply learned that his math can be used in economics…

    So where does the physics envy come in?

  • 7. Peter Klein  |  10 January 2007 at 12:20 pm

    I agree that the term “physics envy,” as used by critics of mathematical economics, is designed to have a certain rhetorical bite. However, I don’t think the critique of formal methods is as ad hominem as Steve’s post implies. Critics have raised a number of serious issues which — to be fair — the defenders of formal methods have generally ignored or dismissed. (Indeed, I think it’s fair to say that most of the insults have gone the other direction — “You don’t understand math”; “That’s just literary economics, not real science”; etc.) And, in response to Michael’s comment, some of the critics are accomplished mathematicians as well.

    In short, Steve is right that how formal models came into economics, how good particular economists are in math, what kind of psychological problems we economists have, etc. isn’t all that important. But it doesn’t follow that there are no serious issues involved, or that the critics haven’t made good arguments.

    Regarding the literature, a good place to start is Kenneth Boulding’s review of Samuelson’s Foundations in the JPE (56:3, June 1948). Hayek’s _Counter-Revolution of Science_ is also good (though lengthy). In these and other works you’ll find thoughtful discussions of the advantages and disadvantages of formal methods and quantitative analysis in the social sciences, all delivered insult-free.

  • 8. Cliff Grammich  |  10 January 2007 at 5:31 pm

    To what extent, if any, is there an issue in economics regarding possible mismatches between the sophistication of (1) methods used to collect data and (2) that for analyzing it? I’ve been involved in collecting some (social science but non-economics) data sets that have been analyzed by others and am sometimes surprised at some of the advanced methods I’ve seen used to analyze them. To be sure, I think many aspects of data collection generally (and of the data I’ve helped gather in particular) have improved over time, but sometimes I fear I’m seeing diamond cutters use their tools on products assembled with the use of sledge hammers. Or maybe I mean sausages served with wines far too fine . . .

  • 9. spostrel  |  10 January 2007 at 8:42 pm

    Cliff is right that we have precision outrunning accuracy at times. Something like the Consumer Price Index is really fraught with all sorts of judgments about what is a new product, how to value improvements, do Wal-Mart’s lower prices entirely reflect lower shopping amenity value, etc. And these judgments may lead to systematic biases if the economy changes structurally.

    The most convincing empirical papers using statistical data start out with very simple analyses and then elaborate them to deal with all of the confounding issues. Unfortunately, journals often don’t want to give that kind of space.

  • 10. Cliff Grammich  |  10 January 2007 at 9:30 pm

    That’s a good point about journal space. I suspect that if I were to see all (and not just the published) deliberations of authors whose articles have arched my eyebrows then I would not be so surprised. And the point about precision outrunning accuracy makes more succinctly the one I tried to make.

    But is there also an issue in economics about what can be quantified, or maybe better put, what should be analyzed with sophisticated statistical methodologies? Within the “softer” social sciences, for example, I’m not sure how much about religion or–going back to an earlier discussion on this blog–sexual activity can or should be analyzed with such methodologies. I’ll hazard a guess about an economics issue: “Entrepreneurialism” can in many ways be quantified (e.g., new business starts). But can all the variables influencing it?

    Obviously I’m asking such questions as a complete outsider to (or completely ignorant of) the debates within economics. (If I were concerned about making a fool of myself doing so, then I wouldn’t be yapping here!) But I was struck by Steve’s earlier comment about seeing if some trick of physicists might be applicable to his work. I’ve often felt the same way about economists (hence several econometrics courses I took while getting degrees outside the field). In fact, what he calls “physics envy” I’ve often thought of as “economics envy.”

  • 11. salbando  |  11 January 2007 at 4:31 am

    You mention some cases of models that work, and it would be interesting to know where you think modeling doesn’t work. Or put another way, what is the alternative to modeling? I recently read David Warsh’s book about economic modeling, and I kept asking myself, if the economists want to understand economic growth, why don’t they just study the fastest growing companies and try to figure out what accounts for the growth?

  • [...] Steve’s first guest post tackles the question of “physics envy” in economics and the use of mathematical tools in economics. His opening statement captures some things that I myself have said at various points [...]

  • 13. Rafe  |  13 January 2007 at 7:29 am

    “I have yet to hear, for example, an explanation of how diminishing or increasing marginal returns makes sense in anything but a mathematical context.”

    What about the end of year exam situation where you have to decide how to allocate study time across a range of strong and weak subjects? It generally pays to allocate more time to weak subjects rather than the ones where you are batting 90+ anyway. I seem to recall working this out at school (or being told) without reference to maths and long before I ever heard about marginal returns. Sure, it refers to numbers (scores) but not to equations or functions. It is more like the kind of calcuations that Menger used, based on his time as an observer and reporter on markets.

    For a number of criticisms of the abuse of maths in economics and sources of other critical commentary, see the Appendix to Chapter 4 in Bill Hutt’s book “The Keynesian Episode”.

    See also Schwartz on ‘The pernicious influence of mathematics in science’ in “Logic, Methodology and Philosophy of Science” eds Nagel, Suppes and Tarski, 1962. To summarise, he argued that even in physics the maths can often obscure as much as it illuminates, and he suspected that the problem is even more intense in the social sciences. Perhaps the main reason for the success of physics, at least celestial mechanics, was less the use of maths than the incredible good fortune to have a stable and isolated system to work on, with a convenient scale, so it only took earth a year to do a circuit and not several decades a la Halley’s comet.

    Finally, there is a fascinating book by two Italians that describes the origins and evolution of mathematic equilibrium theory in economics. There is a summary here.

    http://oysterium.blogspot.com/2006/02/maths-invades-economics.html

  • 14. spostrel  |  15 January 2007 at 7:47 pm

    Rafe: The studying-for exams-problem is a classic resource allocation problem. Of course, you can always work out numerical examples instead of constructing a general equation. The principle behind each numerical solution, however, the generalization beyond the specific example, has to do with equating returns on the margin (if the problem has enough convexity). One’s intuition about this problem is therefore inherently mathematical. Whether a specific act of formalization is worthwhile is a case-by-case determination.

    I read the link describing the two Italians’ book on the origin and evolution of mathematical equilibrium theory. It is not impressive because 1) it apperars to subscribe to the genetic fallacy mentioned in the post, 2) overstates the importance of general equilibrium theory in the intellectual and status structure of economics, 3) displays defective judgment in rating economists (I”ve read a whole bunch of Morgenstern’s papers, and while he was entertaining and occasionally a perceptive critic, his understanding of basic economic ideas he critiqued, such as supply and demand curves, was sometimes defective), and 4) doesn’t seem to have better answers about how to handle specific problems than the theory it critiques. Maybe the book is better than the summary, though.

  • 15. charles austin  |  15 January 2007 at 8:31 pm

    Hmm…, has someone come up with something better than mathematics to model the universe? Or human behavior?

    There would seem to be a corollary here somewhere about those who speak of their vocations as being more art than science.

  • 16. Jason  |  15 January 2007 at 8:52 pm

    Those who complain about the application of mathematics towards any subject should be forced to study mathematics until they are sufficiently conversant to understand said application.

    Lemma: There is no subject towards which a sufficiently advanced understanding of math cannot be applied.

    Proof of lemma : With sufficient regression, all subjects are physics. Physics requires math. QED.

  • 17. David OHara  |  15 January 2007 at 9:04 pm

    Being a physicist, I have never heard of “Physics Envy” and cannot imagine it. I mean, we dont have cute groupies unless you like nerdy geek guys. OTOH, Economists actually make good money compared to physics types and you get to discuss worldly things. There is nothing more boring than a bunch of physicists
    I go to all these conferences and its always geeky men like myself but economists go to conferences with gorgeous women, arab sheiks, and politicos.

  • 18. Memechose  |  15 January 2007 at 10:10 pm

    At least the mathematics used in physics collides with experimental evidence from time to time, and gets shown to be either right or wrong.

  • 19. Peter Klein  |  15 January 2007 at 10:46 pm

    Steve, in your response to Rafe above you refer to “inherently mathematical” reasoning. I think your concept of mathematical, in this sense, is broader than what the critics of mathematical economics (e.g., Boulding and Hayek, cited in comment 7 above) have in mind. These critics do not deny the need for abstract reasoning, or logical precision, or general formulations. It is the language of mathematics per se, and the attempt to quantify what they see as essentially qualitative phenomena, that they object to. (Note I’m talking here about economic theory, not applied economics or economic history.)

    Part of the confusion in this discussion, I think, is a limited sense of the alternatives. Consider three alternatives to conventional neoclassical economic theory:

    1. A-theoretical, historical studies, the writings associated with the German Historical School and the “Old Institutionalists” (Veblen, Commons, Mitchell).

    2. Post-modern, narrative, “critical” studies, e.g. McCloskey’s “rhetoric,” feminist economics, some post Keyensian stuff.

    3. Logical, rigorous, general theory expressed in words rather than symbols and numbers.

    To us modern economists, it’s hard to conceive of category 3. But that’s what economic theory was until the mid-twentieth century.

  • 20. Eric S. Raymond  |  15 January 2007 at 11:11 pm

    I’m an ex-mathematician who spends a lot of time thinking about economics. (Yes, Steve, the same theoretician of open-source software that you’ve met a few times; I’m posting here partly because I ghave good memories of a couple of conversations with you and Virginia.)

    I’m here to take the contrarian position that while many of the regularities in economics are well-captured by mathematical abstractions, rushing to formalize them as explicit equations is a mistake that often obscures more than it clarifies. And I say this from the position of someone who is fully capable of doing that sort of formalization when it’s appropriate.

    The problem is that when one writes these things down down as equations, one is all too tempted to (a) way overestimate the degree of quantitative accuracy that can be gotten out the other end, and (b) forget that one’s fudge factors are, in fact, fudge factors.

    So, even though I am mathematically literate, I prefer to do my economic argument in English. It”s a valuable discipline — forces me to keep the exposition clear and stick to the sound qualitative insights, rather than fooling myself into thinking I know mre than I do.

  • 21. spostrel  |  15 January 2007 at 11:17 pm

    Peter: I have no trouble imagining 3. I’ve seen it with Coase (although he uses a lot of numerical examples), Demsetz (although there are some implicit references to curves and so on), Williamson (although he also uses mathematical diagrams) and so on. Sowell’s Basic Economics is a heroic effort to explain lots of econ without a single equation or graph. Then there are the agent-based computational folks, and I guess also any Wolframites who might be out there in econ land, who feel that computer algorithms rather than equations are the proper medium for economics.

    My own thinking is verbal as much as it is mathematical. It’s just that it’s often clarifying to have to build a formal model with clear assumptions and rules of procedure. Many an apparent intuition has gone down in flames once I tried to model it.

    Most people aren’t clear enough thinkers to engage correctly in non-formal chains of complex reasoning. Fallacies of equivocation, forgetting what’s endogenous, missing forces that go in the opposite direction from the one you initially focus on, arbitrage opportunities that must be explained away, etc., are all rife when trying to do econ without math.

    For example, people talked about limit pricing for 50 years before formal game theory made it clear that such a thing could only work if entrants were uncertain about the incumbent’s costs in very particular ways. (Now that we know the mechanism needed, its empirical plausibility is lower, in my opinion.)

    Working with models of various types over time builds a certain kind of intuition. You can get a glimmering or sense of what kinds of outcomes are possilbe from different sorts of models even without working them out formally (although you should always check if it’s something you’re going to build on). To my mind, those intuitions are just as mathematical as the formal machinery.

    I suppose this puts me a bit uncomfortably into the camp of Gian-Carlos Rota, with a phenomenological view of math. It does seem more accurate than a purely logico-deductive account. The goal of math is correct intuition; proof is a discipline, or sometimes a scaffold, for building that intuition.

    In Hopp and Spearman’s nifty operations textbook Factory Physics, they state explicitly that their objective in laying out all the formulas and models is not to reduce the management of factories to a mechanical process; real factories are too complex for that to be possilbe. Instead, they use the formulas and models to build students’ intuition about what happens when you push on the system in various ways. That seems right to me.

  • 22. spostrel  |  15 January 2007 at 11:43 pm

    Eric: I agree with you in two senses and dissent in part.

    First: There are lots of topics, mainly relating to complex systems but also to problems of heterogeneous beliefs and bounded rationality, that we don’t know how to model very well at all. Nevertheless, these are important topics and deserve as fruitful and careful a treatment as we can give, and I would be the last person to say that we shouldn’t talk about them. Hayek was definitely on to something in his thoughts about complex systems and the role of economic analysis. Asking us not to comment on why economic central planning is likely to be a bad idea simply because we can’t model the crucial issues is censorship, not methodology.

    Second: I agree with you in that I also have no problem with translating things into clear English. Alfred Marshall advocated this, famously, and I’ve already cited Sowell’s efforts in this direction.

    But: I heard Debreu give a speech after he won the Nobel where he proved the first fundamental theroem of welfare economics without any math. He pointed out, though, that he could only do that because the mathematical intuition behind the results was very well understood. The point is that translatiion is involved.

    The idea of a demand schedule is an abstract, mathematical idea. You may picture it as a curve in quantity-price space and talk about it without using any formal apparatus, but if your audience doesn’t have that same picture in their minds, your translation is a very partial thing.

    Even when people are exposed to the math and the pictures, as well as to extensive verbal discussion with lots of examples, etc., it may not sink in for those who lack the requisite mathematical intuition. When I taught basic micro to MBA students, a significant bifurcation occurred in week one between those who could intuitively grasp the concept of marginal revenue and the distinct minority who couldn’t (I didn’t realize the nature of the problem until later).

    The idea that if you charge one price to everybody, selling an additional unit implies losing some revenue on all the units you would have sold anyway, is not something that everyone gets naturally. And without that essentially mathematical idea, you can’t think clearly about profit maximizing output, incentives, price discrimination, or just about any other topic in business micro.

  • 23. dave eaton  |  16 January 2007 at 12:26 am

    I’m a physical organic chemist. Some of us have physics envy, but to echo postrel, it’s the envy of more well-defined problems with fewer moving parts- there’s no need to envy the methods, which I steal with glee, when I have the chops to understand and use them.

    Economics is fascinating stuff, which is why I frequent econ sites, but I confess I find it intellectually intimidating precisely because it is so experimentally uncontrolled, and therefore, there aren’t enough constraints on the mathematics.

    Similarly in chemistry- a good portion of what I care about is too damned complicated to be captured very well by the mathematics, but that rarely dissuades anybody from trying. In contrast to econ, in chemistry the experiments generally give clean, if not clear, results. It’s usually pretty easy to tell when you are wrong, anyway.

  • 24. Carl Pham  |  16 January 2007 at 12:53 am

    Well, as a theoretical chemical physicist (or physical chemist), I’m one of the worst of the lot. Never published a paper with less than 75 equations or so, alas.

    However, I avoid math to the best of my abilities. It’s a last resort, when an attempt to explain it in words ends up with too many invented portmanteau words and phrases with adjectives as far as the eye can see. If I simply can’t explain it in words, it’s usually wrong. The useful and good ideas usually can be explained pretty simply, in a sentence or two. Of course, it may take a lot of math to prove the idea, or work out its consequences. But the central idea, if it’s any good, is usually easy to state. I think that’s even true of the big ideas in physics from Newtonian gravity to strings (if you believe in them).

    I took a little undergraduate economics at MIT in the 80s, and based on that experience I can see one place where economists might well envy us physical scientists: typically if I make a statement about — oh, say the freezing of water — anywhere, be it in a cocktail party (if I went to any) or a conference, then no janitor at the back of the room is going to stand up and say You’re wrong, because I’ve seen water freeze lots of times, and it’s obvious to me that…. People may still say I’m wrong, but they’ll be sure they have PhDs in relevant fields and understand ever equation in my theory before they do.

    Not so in economics, is my impression. I get the impression that every third Joe Average figures he’s got a “practical” PhD in economics because he spends money and stuff, and so feels perfectly competent to criticize whatever one of you might have come up with, working six years of nights through eight tons of complex data. That doesn’t seem right to me he’ll say, so you must be wrong. I don’t need to understand the math.

    I can see that would be frustrating in a Rodney Dangerfield way. So maybe some economists would like to have some heavy math to drop on Joe Average’s head to get some respect.

  • 25. M. Simon  |  16 January 2007 at 12:56 am

    One of the big defficiencies I see in economic theory is insufficient use of control theory.

    The supply demand curve being equivalent to “gain” and lags being lags.

    With it you ought to be able to figure out stability and oscillations (bubbles).

  • 26. M. Simon  |  16 January 2007 at 1:17 am

    re: My #25,

    Maybe gain ought to be profit.

    You can tell I’m an engineer not an economist. Still it seems like an economy is a bunch of interacting amplifiers.

    Oscillations in oil prices ought to be possible to model using the right kind of control system equivalent, the appropriate gain curves, and the lags inherent in the rise of prices vs. the rise of production.

  • 27. spostrel  |  16 January 2007 at 1:27 am

    Carl: The enforced humility of dealing with fellow citizens who have strong opinions can be an annoyance, but the necessity to explain the fundamentals in open public debate ultimately strengthens the profession. I am extremely uncomfortable pulling rank the way natural scientists do, and I think most economists have a similar feeling. We tend instead to get all pedantic and try to explain ourselves, which only works a fraction of the time but does keep us slightly more honest than we would otherwise be.

    When you recall that some of the Ur-works in the field–Wealth of Nations, for example–were aimed at public persuasion, you can see why the “shut up, you crackpot” response is not in our collective DNA (with a few exceptions not to be called out here). In a polemical context, I might get sharp with someone I think is being deliberately misleading, but economists really do want to persuade–one another, primarily, but the public and the leadership, as well.

    I lurk at some science blogs, and I’m struck by how much mental energy is spent on policiing who is a crank, etc. The ideology of science is that it is an open system where all can have their evidence assessed impartially, but I see a lot of practical pressure not to say things that upset one’s colleagues. The calls for shunning and ostracism because someone does or does not follow someone elses’s definition of what is respectable are offputting.

    And I’m not talking just about outsiders or just about extraordinary claims. I’m talking about whether one highly credentialed physicist thinks string theory is a good thing to work on or another one doesn’t. It’s amazingly nasty. What happens when a respected person goes off the reservation even a little bit is even more ugly. Read Louis Frank’s The Big Splash and some of the later stuff on his website to get a taste of it (regardless of the merits of his views, look at how he was treated).

    I guess I think of science like the old saw about the stock market–in the short run it’s a voting machine but in the long run it’s a weighing machine. The voting part can be pretty rough if you’re on the wrong end of the count. Eventual vindication is small consolation if the long run exceeds your career or your life. But eventually reality does adjudicate the dispute.

  • 28. Frank Dobbs  |  16 January 2007 at 4:01 am

    Steve is getting close to the key insight, but is not quite there yet.

    Demand curves do not exist. They are a metaphor for economic insight about supply, demand and price. But there is no demand curve anywhere in the real world. Even calling it a curve is a distortion. It is not like the curve that plots acceleration of an object falling under gravity. That has a curve that corresponds to a real phenomenon that is both measurable and conceivable. The demand curve is a way of expressing an economic insight, but it is a dangerous and deceptive way of so thinking.

    The falling object forms a curve because it’s phenomenon has continuity. There is no continuity in transactional economics. All there is is discrete transactions. Nothing else can be measured, and, really, nothing else exists. The instant one treats the demand curve as a mathematical entity, as opposed to a metaphor, and tries to do mathematical operations on it one drifts further and further from sound ecomic intuition.

    Traduttore, Traditore, as they say. The math is but a translation of a verbal –or rather, an economic insight. Nothing wrong with that –except for the tendency to treat the metaphor as a thing of value in and or itself, which of course it is not. It is a form of sloppiness to take the simple mathematical metaphor instead of the complexity of real world transactions as the basis of one’s thought.

    It shows a lack of humility and austerity, without which insight is impossible.

  • 29. Don  |  16 January 2007 at 6:30 am

    There’s still heaps of confusion/discussion/argument over what Keynes Really Meant when he published the General Theory, and we’re some 70 years on. There’s no similar confusion over what Nash Really Meant, or Debreu, or Aumann, or any of the others who used math to express their insights.

  • 30. Jay Manifold  |  16 January 2007 at 8:31 am

    Wow. Can’t believe somebody else hasn’t already posted this Heinlein quote: “Anyone who cannot cope with mathematics is not fully human. At best he is a tolerable subhuman who has learned to wear shoes, bathe, and not make messes in the house.”

    Also, just for fun, see Friedrich Kekule and the Benzene Ring.

  • 31. Rob Perelli-Minetti  |  16 January 2007 at 8:34 am

    For economists (and most others) mathematics can perhaps best be seen as an alternative language in which to discuss, and to refine, economic concepts. Although I don’t do economics for a living, I did a couple of years of graduate work in mathematical economics some 30 years ago and found it exceptionally interesting and stimulating.

    One of the biggest complaints one heard at that time (and may still for all I know) from students trying to understand economics is that the elementary models used are some overly simplistic and do not capture the “real world.” The assumptions required to make the (relatively) simple mathematics of (for example) elementary price or demand theory (thing Stigler’s Theory of Price) [nonsatiety, functions are continuous and twice differentiable] are too much for many to swallow.

    As an aside here, I once tried to teach a prominent law professor elementary economic theory because he wanted to understand enough economics to think seriously about law and economics work done by people like Posner. The man could simply not work on the basis that the assumptions were true to look at the model, and didn’t have the mathematics to relax the assumptions.

    From my outdated and no doubt simplistic viewpoint, I regarded (and regard) the use of sophisticated mathematics (e.g. topology and measure theory in general equalibrium theory) as simply ways to progressively relax the strong assumptions made at the elementary level to see if the results still hold with the weaker conditions. Seems they do.

    The problem, of course, is that the mathematical price of admission to relax the assumptions is high. One is reminded of the famous quotation from Alain Enthoven (I found it in the forward to Armen Alchian’s University Economics) that:

    the economic theory we are using is the theory most of us learned as sophomores. The reason Ph.D.’s are required is that many economists do not believe what they have learned until they have gone through graduate school…. (orig: American Economic Review 53 (May 1963), 422)

  • 32. ajacksonian  |  16 January 2007 at 9:18 am

    One of the few things I put out awhile back was trying to give definition space to the sciences and then examine what is and is not a science and why. Economics, by having a very generalized feedback system is working towards the hard sciences, but mathematics is only a part of the necessary toolkit for examining postulates and results. The given assumptions for such postulates based on society viewed, timeframe, measured temperament and societal outlook I would think all play factors in the validity or non-validity of economic models. Addressing those in a qualatative fashion can only yield generalized, qualatative results for a set of given assumptions that are not well defined.

    The refinement of the actual understanding of what the assumptions are, putting actual measurement capabilities against same that can be verified and such testing duplicated then leads to a harder foundation for things built upon such quantative results. With that said the actual definitions and measurments of humans as individuals and societies has not been approached with any rigor nor view towards measurement and understanding and working towards a framework of that understanding that has testable basis. A better set of actual, real measurements of those things we consider to be human nature and society can then be examined as variables with *limits* in economic models and traced back to the known quantities and interactions of those things. Putting forth a postulation based on those things then allows for actual results to be measured and asessed as to the validity of underlying principles and measurement systems and then their application capability in multiple spheres of reference.

    I really don’t see much of ‘physics envy’ in that… what is seen more is the complaints of the tools having no hard and fast tie-backs to basic understanding of human temperament, outlook, societies and social capability. That is because no one wants to touch those things with an aim towards making such things testable, verifiable and useful in a quantative way based on qualatative description. If you don’t know what you are measuring and can describe it and why it is a valid set of measurements for a given postulation space, then those measurements are just ‘taking readings’ that may or may not have a tie back to what you are trying to measure.

    There is a certain humility that comes with putting forth ones ideas, then taking hours or days or weeks or months of ‘readings’ and then finding out you were measuring the wrong thing or had set up the experiment in the wrong way to get valid measurements. Lots of readings, but no meaning because the set-up was invalidated.

  • 33. Eric S. Raymond  |  16 January 2007 at 9:31 am

    Steve: I see merit in your argument about English being a poor channel for expressing some core ideas. Nevertheless, I think I’ll stick with natural-language-centered exposition, if for no other reason than that equations would lose my typical audience.

    It may also be that I’m more influenced than you are by Thomas Sowell and David Friedman, two economists who both seem to treat careful argument in English not just as a means of public persuasion but as a form of constructive self-discipline.

  • 34. Bill  |  16 January 2007 at 9:55 am

    Interesting discussion. Prof. Postrel, I especially liked your observation that:

    “When I taught basic micro to MBA students, a significant bifurcation occurred in week one between those who could intuitively grasp the concept of marginal revenue and the distinct minority who couldn’t…”

    Teaching intro micro w/o using basic differential calculus seems unecessarily onerous. As many have noted here, one must understand the mathematics in order to explain the economics qualitatively. When I took intro micro in the Chicago MBA program (in which the curriculum doesn’t shy away from heavy quant lifting) my prof ditched the calc in order to reach those students w/o any math background. I found that made grasping the economic intuition much harder than it would have been using the math. Not a problem one finds in a sample of pure econ majors, one assumes, instead of a heterogenous group of MBA students.

    After 12 years working in financial economics I side with von Neumann (and current mathematicians observing finance such as Nassim Taleb and Benoit Mandelbrot) … the problem is too little math in the field and a lack of mathematical sophistication where it might be useful. I addition, too few economists exist who are eager to translate the math into qualitative descriptions accessible to the general public.

    Regarding Black-Scholes… the story I read is that Black and/or Scholes were wandering around Chicago’s campus c. 1972 trying to figure out how to solve the final equation to which they had reduced their thoery. One of them mentioned this to a friend who was a physics or math prof/grad student who, upon looking at the equation, chuckled. He saw that they faced the basic two-dimensional heat equation from second-semester undergrad physics. Back then, economists evidently weren’t that well-versed in diff eq. Today one finds finance profs with undergrad degrees, and sometimes doctorates, in engineering and physics.

    The math in finance that is most powerful and widely useful coprises stats, linear algebra and, for more esoteric derivatives work, stochastic calculus. Friedman was brilliant in stats in his day; I wonder how much advanced calculus he understood and how that might have been useful to his work. Does anyone here know?

    M. Simon… I’m a former engineer; I like your idea about control theory. Unfortunately, the systems under consideration aren’t analogous in meaningful ways to the well-specified physical phenomena that control theory describes. The system, such as it is, seems too complex and multivariate for one to make useful predictions using the analogies of gain and feedback. I am curious, however, whether economists have looked deeply into this. I haven’t seen the techniques used in the financial literature.

    My uncle recently retired after 45 yrs as an econ prof. He’s a macro guy who claims that his field hasn’t seen a truly useful model since Hicks’ IS-LM formulation in 1937. Do any readers here have a reaction to that?

    Thanks to all for their comments. I learned quite a bit.

  • 35. D J  |  16 January 2007 at 10:57 am

    I am sorry if this seems rude. But economics cannot become a real science so long as economic models are based on false assumptions, on assumptions simplified to the point they ignore what is real. Examples of false economic assumptions: all consumers have perfect knowledge. If this were true then there is no need for laws to punish insider trading. Another example: all consumers are rational. I believe this assumption is often not true.

  • 36. dsquared  |  16 January 2007 at 11:04 am

    The specific charge of “physics envy” in Mirowski (and I think it is there, in “More Heat Than Light”) is that economists have, in a number of important areas, appropriated models wholesale from physics without thinking hard enough about whether it makes sense to do so. In particular:

    1) too many implicit assumptions of reversible processes. Far too much use of either one-period models or pseudo-dynamic models based on the Bellman equation. The way in which the vast majority of economics treats time is wholly unsatisfactory; it’s nearly always an abstract, reversible, “physicist’s time” rather than actual historical time. Any model which contains “T-subscript-0″ is likely to have a bit of physics envy in it, because it’s likely to be ignoring the manner in which we got to T-subscript-0. Expected utility theory is kept on because it offers dynamically consistent solutions to problems, which is only a virtue in non-historic time.

    2) Far too much mathematical effort and too many assumptions used in protecting analogies with perfect-gas models. The Arrow-Debreu results, for example, depend on commodities and individuals being so homogeneous as to raise genuine questions about whether they’re distinct at all. None of the main general equilibrium results can be made to work if relative preferences change as income does. Mass substitutability is the big intellectual vice of economics.

    Mirowski goes through these things in detail in his books, explaining why they’re wrong, why theyir wrongness matters (in that it makes us say potentially badly wrong things about economics), and gives specific historic and institutional detail about how these mistakes came to be made out of a conscious desire by economists to imitate physicists.

    M Simon:

    [Oscillations in oil prices ought to be possible to model using the right kind of control system equivalent, the appropriate gain curves, and the lags inherent in the rise of prices vs. the rise of production.]

    This is the socialist calculation debate. Oskar Lange thought the same thing as you; that it would be possible to gather enough information about the economy as a whole to make an input-output matrix which could be solved for a set of optimal prices (or shadow-prices; of course, if you had such a system you wouldn’t necessarily need a market economy at all).

    Friedrich Hayek disagreed, believing that 1) there was so much interaction in the economy that no model of any sub-system of the economy could be stable and 2) it would be absurdly and disproportionately costly to get the information needed as inputs.

    Hayek ended up getting the best of this one, although it did rather predate large computers and the Internet. Some people today occasionally have a go at reviving the socialist calculation side. However, Hayek also had two other objections:

    3) that the vast majority of the knowledge needed to construct the relevant input and output functions was “tacit”, non-propositional hands-on knowing-how rather than knowing-that, and couldn’t even in principle be converted into mathematical form.

    4) that if economic agents knew that they were providing input into a planning model, they would change what they did in order to maximise their own function.

    3) is the distinctive Austrian critique of the use of mathematics in economics, and is arguable either way – I suspect that Steve and the McKinsey organisation would disagree with the extent to which even complicated processes can be Taylorised and measured.

    4) is a real problem though; there’s a Godelian argument in von Neumann’s book which proves that an economy of even moderate size has no equilibrium under quite general kinds of assumptions about the interactions between agents. Of course, just ignoring this problem by making a lot of homogeneity and independence assumptions to prove the theorems of modern economics, doesn’t make the problem go away.

  • 37. Kent  |  16 January 2007 at 11:11 am

    Interesting post.

    When I reflect on the Freudian allusion, though, I have to wonder: Is Mirosky saying that we physicists are bunch of d—s?

  • 38. dsquared  |  16 January 2007 at 11:17 am

    [Regarding Black-Scholes… the story I read is that Black and/or Scholes were wandering around Chicago’s campus c. 1972 trying to figure out how to solve the final equation to which they had reduced their thoery. One of them mentioned this to a friend who was a physics or math prof/grad student who, upon looking at the equation, chuckled]

    It was Robert Merton. He had a BSc in Engineering Mathematics and an MS in Applied Mathematics, but at the time of this incident, had been an economist for three years. The analogy to the diffusion equation wasn’t the difficult part, btw – if it had been, Black would surely have solved it. The difficult bit was the use of Ito’s lemma to derive the original Black-Scholes differential equation. This is all from memory so I’m not sure exactly who did what when, but the point is that Merton was already bringing stochastic calculus into economics at that time, and that stochastic calculus was by no means a commonplace of the physics or engineering worlds either.

  • 39. Hans Pew  |  16 January 2007 at 11:29 am

    I’m a physicist, and unlike David OHara above (comment 17) I have heard of Physics Envy and I believe that it is a real phenomenon in the social sciences. It isn’t envy of physicists, rather of the success of the discipline of Physics in quantifying, describing and even predicting observations. Physics Envy manifests in an attempt to use mathematics even though the assumptions are numerous and highly uncertain and will have a strong effect on the results, or when the statistics are so poor that the results are – as a practical matter – meaningless.

    Now, I don’t pretend to be an expert on Economics, so I won’t try to assess whether or not Physics Envy is common problem in the field. From what I do know about the subject, though, I don’t think that there is any question that there are many useful – even essential – uses for mathematics. There’s also some definitely some potential for Physics Envy abuses, but my impressions are that Economists generally do a decent job of policing themselves – most of them seem to understand the mathematics they are using.

  • 40. Yevgeny Vilensky  |  16 January 2007 at 11:33 am

    Frank Dobbs:

    There is substantial evidence to suggest that the theory of “space” as a continuous object is incorrect. There is now much work being done in loop quantum gravity to show that space does have irreducible pieces and that objects move through space discretely and not continuously. So in that sense, objects would not have continuous curves.

    Furthermore, from a statistical physics perspective, while the curves may even be continuous they are certainly not smooth. The motion of particles, though appearing smooth on time scales we are accustomed to observing, are actually very rough and jagged, sort of like Brownian motion, as a result of millions of discrete bombardments from smaller particles.

  • 41. Bill  |  16 January 2007 at 12:24 pm

    DJ… I don’t mean to be rude either. Please learn some introductory microecon and then read Friedman’s “The Methodology Of Positive Economics.” Yours is a time-worn, utterly discredited critique of the field.

    d-squared… Thank you. I didn’t realize that Merton was the one who helped Black; I had also long since forgotten that Ito’s lemma was involved.

  • 42. profbiker  |  16 January 2007 at 1:14 pm

    dsquared is correct about Ito’s lemma. As a research associate at the UofC at the time, part of the foucs was on recruiting undergrads with math and/or science undergrads to take advantage of the math skill.

  • 43. Neil Ferguson  |  16 January 2007 at 1:20 pm

    Regarding Scholes and Black: considering the fate of Longterm Capital Management, are they the most convincing names to bring up when discussing the legitimacy of mathematical models in Economics?

    Cordially,

  • 44. Adrian Smith  |  16 January 2007 at 8:21 pm

    Carl: The enforced humility of dealing with fellow citizens who have strong opinions can be an annoyance, but the necessity to explain the fundamentals in open public debate ultimately strengthens the profession. I am extremely uncomfortable pulling rank the way natural scientists do, and I think most economists have a similar feeling.

    I think the problem is that the achievements of physics (or their technological byproducts) are to some extent accessible to the man in the street, whereas to appreciate the achievements of economics you have to more or less become at least an amateur economist, at which point the sunk costs effect kicks in and your inclination to take potshots at the foundations of the discipline is probably going to decline.

    My own difficulty is that in the hands of some, economics seems to be used largely to provide “scientific” justification for various ideological positions, though of course it’s more palatable when I happen to like the ideology concerned.

  • 45. Zzyzx  |  16 January 2007 at 9:25 pm

    This is an interesting discussion.

    I am a biologist who makes theories about gene regulation. I spend a lot of time telling biologists NOT to have physics envy: The set of things that can be calculated in physics is smaller than one might think, and often involves very specialized situations. For example, the Lamb Shift caluculation, accurate to about one part in 100 million, applies to a single hydrogen atom, but hydrogen atoms come in pairs, not one at a time. In biology, on the other hand, the genetic code is accuarate to about the level of the lamb shift calculation, is used everywhere, and is expressable
    by a simple table without equations.

    I think that the problem with math in ecomomics is not in the equations, but rather the state variables used in the equations. The varibles in a theory need to have a fairly fixed mapping to the real world. As a poster noted above, this works well in celestial mechanics,
    where the absurd approximation of representing an
    entire planet by a point with mass but zero size works well because the distances between planets are large compared with their diameters. This breaks down if two planets collide, obviously. In biology it’s trickier—
    sometimes you can use chemical concentrations but sometimes you have to do statistics on individual atoms.

    In economics the state variables represent human beings. If they are “well-informed” and “rational” then average behaviors follow certain laws. Sometimes, however, an individual knows something no one else knows or does something crazy or unpredicable that has major consequences. For example, it seems to me that a complete theory of economic dynamics must predict the future evolution of economies, but such efforts (Malthus and Marx?) have failed because ultimately economics couples to the whole of human activity.

  • 46. Bill  |  17 January 2007 at 9:57 am

    Neil… considering that the Black-Scholes equation transformed risk management and enabled the creation of transparent insurance where previously none or opaque remedies existed I’d say that LTCM is irrelevant to our discussion.

    Goddard’s rockets occasionally exploded on the launch pad; that hardly negates his contributions to science.

  • 47. spostrel  |  17 January 2007 at 12:23 pm

    Zzyxx: I agree with the thrust of your view, especially about the difficulty of applying reductionist laws to complex systems (even the hydrogen molecule). I wholeheartedly agree with your point that “The variables in a theory need to have a fairly fixed mapping to the real world.”

    In fact, if a theory is strong enough on that dimension, is also tractable, and produces intersting results, we can bet heavily on its success even before empirical testing. Think about Shannon’s communication theory. The mapping is so clear between the theory and the systems it aims to describe that mathematical derivations are almost conclusive. An empirical “test” of the theory is silly. Close, but not quite as locked in, would be something like the arbitrage theory of options pricing cited in this thread many times already. The mapping isn’t as clear as in communications theory, e.g. because there are questions about whether you can at every moment execute the trades you need to make the arbitrage work, but the basic structure is pretty compelling and maps pretty clearly to actual financial markets.

    So a good filter in doing any kind of theory is to economize on the number of variables whose defintion and conceptual measurement are equivocal or ambiguous. (I’m not inveighing against hidden variables, just ones whose nature isn’t clear in your own mind.) You often can’t eschew these messy variables completely, at least at first, but it’s a good idea to treat them as having a high shadow price.

  • [...] throughout this and future posts, I do not want to be misunderstood as leveling the “physics envy” critique at economists, e.g. that economists use modeling as a thinly veiled attempt to make [...]

  • 49. Mark Pawelek  |  29 November 2009 at 4:58 am

    [quote]Supposedly, math is distracting, or misleading, or limits the questions one can study.[/quote]

    You seem oblivious to what the critics are actually saying. Which particular critics have you read deeply?

    [quote]Our problems, when they occur, do not lie in our tools but in the quality of our ideas[/quote]

    Exactly. But the emphasis on maths has allowed generations of PhD students to get their doctorates without ever having to even understand the arguments against economic orthodoxy (the Monetarist-Neo-Classical, psuedo-Keynesian orthodoxy). The emphasis on maths allows economics the pretence of science. One can teach economics without really having to get to grips with ones own critics (the heretic economists). When Paul Krugman gave a lecture at the LSE recently guess how many LSE professors turned up to listen to him, refute him, etc? Precisely zero. They live in a bubble universe.

    [quote]Critics of mathematics in economics, however, rarely come to grips with these specific issues of problem solving and understanding[/quote]

    Having studied maths and science I can tell you that, based on false or incoherent axioms you can do ‘anything’ with maths (or science). Nothing you do will be of any use but you could, easily spend your life doing it. That’s why foundations are so important in science. Even today, we spend so much time and energy trying to get the foundations of science right. The critics of economics are saying that economic orthodoxy (such as the 64 professors who ignored Krugman when he journeyed to visit them, at their own manor) have fraudulently presented their subject to the rest of us.

    I advise you to spend time studying the philosophy of science. Einstein’s critique of Newton, especially the history leading up to it is particularly useful and the reformulated axioms the lead to his Special and General relativity is illuminating. No need to actually understand relativity for this. All you need to know is why Newton was ‘wrong’.

    Economics is at a pre-scientific stage. If it actually admitted as much I would give it far more respect. Outside of a scientific discipline I find it impossible to see any use in mathematical models. That’s the critique being levelled led at economics. ‘Physics envy’ is just a metaphor.

  • 50. Alexander Repiev  |  20 November 2010 at 2:32 am

    Perhaps this will be of some interest:

    “’Physics envy’ – physics abysmally misconstrued!” – http://www.repiev.ru/articles/Physics-Envy.htm

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