I was in an econometrics class yesterday where our tutor was walking through a statistical problem concerning an independent normal distribution with us. We calculated 2 probablitlty values for the likelihood a member of the set would be in 2 regions. The next question asked what would be the value the probability would be above 50(the range was between 0 and 100). Now even if this distribution wasn’t normal the answer would have been trivial, since the probability of lying in the other regions(covering the range below 50) summed to 1/2.
So, the probability of the value being above 50 was (shockingly!), 1-1/2=1/2. I shouted out the answer before he finished writing the question. He at first fumbled, and then agreed with me, but said that you would not get full marks unless you demonstrated this to be the case presenting a longwinded calculation similar to the one he used to calculate the other 2 values.
This got me thinking. For a while I’ve had the feeling that economists don’t seem to use maths to generate much genuine insight, but merely to pretentiously show off that they know how to play with Greek symbols. When I was a physics student, if I answered in the above way, a professor would either be indifferent or actually enthusiastic that I was using my brain to actually find smarter, easier ways to solve problems than regurgitating from rote memorisation. This doesn’t seem to be the case with economists, indeed I am beginning to think this might be why my marks on the econ course I’m currently doing are not as high as they could have been, though they are not bad.
#1. econometrics != mathematical economics. econometrics is essentially using statistics to study economic data. any problems you have with econometrics you would have statistics in all of its forms.
#2. i personally don’t think mathematical economics (using math to model economic phenomena) is pretentious at all. primarily i’d say it is a tool for clearing one’s own thoughts but maybe more importantly a tool for communication. expressing economic theories through mathematics forces one to make every assumption explicit. that may not sound like a big deal, but there is a reason you won’t see any papers that ask “what kenneth arrow really meant”–mathematics allowed him to make it perfectly clear to begin with! considering all the papers written on interpreting and re-interpreting mises and hayek, i can see why many austrian economists don’t like the idea of using math–it could put them out of businees!
now, i am sympathetic to some arguments that there may be an over emphasis on math these days (paul krugman actually had a good blog post recently on the value of less technical econ), but i think arguing that it is merely showing off is missing the point.
Krugman writes in a number of places that without the math he could never have figured out and properly thought through his contributions to trade.
Math offers a useful way to think in an organized, systematic, consistent fashion. Using only words can lead to sloppy thinking in a manner impossible with math (unless you just suck at math).
I do not really have a problem with including math with economics as a tool to help illustrate theories but I oppose the using of math as the only means to explain a theory. If one cannot explain economic law using words and instead uses math, that is a fallacy.
Sure, math is language, and it can always be translated to english, although obviously “2 + 2” is often easier than “two plus two” in many situations. Nevertheless, math is a particularly useful language for being organized, systematic, consistent, and logical, much like how French might be a particularly useful language for love or German might be a particularly useful language for doing philosophy.
As for my qualifier, obviously if one cannot do math well one’s thinking will appear sloppy–2 + 2 = 5. Some people might do better with words. They might also not have much to contribute to economics.
Symbolic logic is to logic as mathematics is to economics.
Essentially, they wanted to get rid of the ambiguity and equivocation and ‘unclear-ness’ of wordyness. Both are important, but in the case of symbolic logic, you better well have an appropriate foundation in logic before you move on.
Likewise, you need an appropriate foundation in ‘word’ economics before you move on–otherwise you’re apt to get totally confused.
If you’re asking me, then symbolic logic is used to diagram incredibly complex arguments that are otherwise misconstrued when represented any way other than symbolically.
For instance: ~{[(p v q) → (r → s) * ~x] v z}
That’s just a small example, but when p, q, r, s, etc., stand for arguments, then it becomes near impossible to represent any other way.
I have an amateur interest in advanced mathematics (particularly, algorithmic information theory) and I have to say that as an outside observer, I would extend OP’s criticism not only to most modern economics but also physics itself.
If you take first year physics (I have), you will learn things like conservation of momentum, and the explanation you are given will be much like that in the linked Wiki and you will learn something like MV = mv. Now, MV=mv is a handy mnemonic, a great tool for calculating momenta in physical problems. But absent the physical experiments and chain of reasoning that led to the equation MV=mv, you have no real comprehension of what you’re talking about. You’re just an arithmetic monkey.
I read the first chapter of Ernst Mach’s Science of Mechanics, in which he presents Galileo’s physical argument for the conservation of momentum. Galileo rolled a ball down an incline of angle A and measured how far the ball rolled up an adjoining incline of angle B. He found that the height to which the ball rolled up the second slope depended solely on the height at which the ball was released from the first slope and independent of the angles of the slopes. Therefore, if you fix an angle A, as you decrease the angle B (neglecting the effects of friction) the ball will roll farther and farther in order to reach the height at which it was released on the initial slope. In the limit, the ball will travel infinitely far. Hence, a body in motion remains in motion (Newton’s first law or the Law of Inertia from which conservation of momentum follows).
Mathematics is simply a compressed form of natural language. Mathematical objects, therefore, are mental objects, that is, they exist only in the minds of human beings as categories of human thought and language. The only things which mathematical objects fully emulate are themselves, that is, the properties of a mathematical object only correspond to the mathematical object itself. Only pure numbers are perfectly modeled by pure numbers. Correlation between numbers and real objects is dependent on a human brain (or, in the computer age, a machine built by a human brain) to translate some set of observed physical states into a mathematical description which can then be manipulated according to the rules of mathematics. The result of these manipulations can then be translated back to a set expected physical observations. Any information lost in the translation may result in error and the human brain is a fairly “lossy” computational device.
For example, if you lay out 7 rows of 6 rocks each, you can compute the total number of rocks without having to count them one-by-one. First, you translate the phsyical perceptions of the number of rows and the number of rocks in each row (including the physical fact that each row is equinumerous). Once you have “6” and “7” in mind, you can follow the rules of multiplication to arrive at the conclusion that 6x7=42. Finally, you may translate the “42” back to the expected physical perception of having counted 42 rocks one-by-one in order. If you were to count the rocks one-by-one, this is the physical state of affairs that you expect to obtain (to count 42 of them).
Stranger’s point is probably the number one reason for the importance of math.
To which somebody answered " If one cannot explain economic law using words and instead uses math, that is a fallacy." This completely misses the point, the problem isn’t that these insights can’t be “translated” into words, it’s just that ex ante some insights of economics would be missed if the math weren’t used to trace things through carefully.
Yes, to the OP’s point there is a lot of mental masturbation done in the form of useless math, but, this doesn’t necessarily lead to perverse outcomes. If we assume that publishing in good journals is an important part of your average economists utility function and that mathematical sophistication is a good way of signalling aptitude in economics (even if it is a noisy signal) then the outcome is likely that people will try to be more novel in their use of math. Which, is probably beneficial for economics.
The argument that “the economists of math sucks” can either lead to the conclusion “don’t use math” or “use math in a more sophisticated way”.
I was speaking to a professor of mine who told me that there’s been a move away from theory (read useless mathematical masturbation) to empirical work lately.
You’re understating the case. Mathematical (and logical) formalism is the result of a long philosophical search for “irrefutable truth” or “absolute truth.” The liar’s paradox easily shows that the possbility of finding absolute true or false through ordinary human language is impossible since one may always construct statements that cannot be evaluated into either the category of “true” statements or the category of “false” statements… they belong to a third, unexpected category of “meaningless” or “paradoxical” statements.
Beginning in the 19th century, logicians and mathematicians sought to be more rigorous in their proofs and the idea emerged that perhaps it is possible to establish “provability” or “unprovability” rather than “true” or “false.” In other words, perhaps the problem with true and false is that they are “meaningful” concepts and if we just avoided anything to do with meaning and concerned ourselves with the empty manipulation of symbols, we can establish for any given string of formal symbols whether it is provable or not. Then, through a rigorous search of proofs, it would be possible to discover heretofore unknown mathematical truths and use formal methods to vet long-standing mathematical ideas to be sure they are correct.
To this end, Russel and Whitehead wrote their epic Principia Mathematica. It was the first swipe at absolute truth through formal methods. Once completed, it would provide a foundation that could be extended and refined. But then Kurt Godel - inadvertenly - threw a grenade into the whole affair with his 1931 paper On Formally Undecidable Propositions of Principia Mathematica in which he outlined what are today called Godel’s First and Second Incompleteness Theorems. In layman’s terms, Godel showed that you can’t escape the Liar’s Paradox by restricting yourself to formal systems. He used a clever trick to encode statements in an artificial mathematical language as numbers (we’re used to this nowadays so it doesn’t seem strange but, at the time, it was a completely novel technique) and then performed proofs on the encoded statements. In particular, he encoded the statement that says, “This statement is not provable” and showed that the statement - by virtue of what it says about its own provability - is not provable in a consistent mathematical system and he showed that this kind of sentence can be constructed in any mathematical system powerful enough to express the natural numbers (in other words, in any interesting mathematical system). This means there are always true statements which cannot be proved to be true, which makes any formal mathematical system incomplete.
Later, mathematician Gregory Chaitin would go on to show that this is not just an isolated problem that exists in a tiny niche of the mathematical universe of interest only to pedants and quibblers. Rather, incompleteness is the rule in mathematics. As Chaitin says it, “Almost all true mathematical statements are true for no reason.” You can watch a lecture on YouTube where he explains this in his own words. It’s humorous and engaging:
I think you have misunderstood what I was alluding to. This was not really a thread to discuss mathematical economics or its utility, though I have strong reservations about the latter, the case against it would deserve to be made in a separate thread with its own supporting argument (not filed in the category “General”). Neither do I have a problem with statistics, which is all econometrics is, aside of the different conclusions many econometricians have compared to other users of statistics as to what can be concluded from them for open systems.
My point was much more narrow, about the way economists seem to do and use maths. Hence for instance as in the above case, the grinding out of long winded calculations, to gain a trivial result one could gain otherwise by simply using one’s logic, is encouraged over doing the latter, for whom marks are penalised in a test. This is by no means the only case, I noticed that my microeconomics professors often made a big deal about “proofs” that displayed nothing more than circular reasoning arriving back at the terms they started their manipulations with(unlike a genuine mathmeatical proof that gains a theorem with some new insight and unpacked informational content). But again, I’m fairly sure such pointless mental masturbation is rewarded and encouraged(and this is definitely not the case in physics; even in theoretical physics, which is my background, I was precisely penalised by one of my viva examiners for this in my first “practice” presentation of my masters thesis).
But again, I’m fairly sure such pointless mental masturbation is rewarded and encouraged(and this is definitely not the case in physics; even in theoretical physics, which is my background, I was precisely penalised by one of my viva examiners for this in my first “practice” presentation of my masters thesis).
The mental masturbation is rewarded because it serves a purpose. It’s stops people bullshitting and equivocating about important things in order to find some theoretical support for their preconceived beliefs as many accuse the Austrians of doing (and sometimes quite rightly in my opinion). That’s the thing, economics is very complex and almost everybody allows their political values to influence their economic beliefs at some point, so it’s absolutely necessary to force people to state their assumptions clearly and to be explicit about the definitions their using. Mathematical formalism allows people to do such a thing.
By the way, I read this the other day and your post reminded me of it
EconomistInTraining, are you really sure that being able to make up equations and graphs stop one of bullshitting?
Are you going to tell me that Paul Samuelson bullshit didn’t stink because it was supported by heavy use of math? He used math to tell that war creates prosperity, it’s impossible to have inflation with unemployment and that the Soviet Union would be successful.
What bullshit of Samuelson’s are you talking about? His contributions to economics are numerous so it’s important to distinguish between those contributions which reveal his true insight and those areas where he was, perhaps, simply being partisan. The thing is the Soviet Union example, as far as I can tell, was simply an extrapolation of bad Soviet data into the future, whatever your opinions on formalism there is always need for some sort of speculation but also some numerous difficults involved. But your inflation and unemployment example is precisely an example where mathematical formalism has helped remove controversy. The fact is that the crude Keynesian models of the time (as far as I’m aware) didn’t use anything like maximising agents and weren’t particularly explicit about expectations, this only came with Lucas.
The ABCT in the other hand is being credited for every business cycle from the big bang to all the way in the future. Precisely because it’s so easy and tempty to make ad hoc changes to what you mean by ill defined terms like “higher order goods” and even the business cycle in and of itself. By the way, in an of itself formalism only assures that the theory is internally consistent, not that it really corresponds to anything “out there”. More empirical work is needed to reach the latter result.
You have an incomplete understanding of Austrian claims about the market - Austrian theory, including business cycle theory, eschews any claim of predictions about the future state of the market. Business cycle theory itself is an ex post facto explanation of why past business cycles have happened in the past but explicitly rejects crystal-ball gazing regarding future business cycle events.
The Austrian view is that the only individuals who actually engage in meaningful prediction about the future state of the market are entrepreneurs. Academic economists can offer nothing more than an analysis of cause and effect in past real economic events or “praxeological” theories explaining human choice under conditions of voluntary exchange (“catallactics”). Thus, any academic economist who says, “If you follow policy X, employment will increase by X%” is just a charlatan. This agnosticism regarding the real outcomes of government policies is what motivates Austrian criticism of any government policy intervention in the market. Since we cannot know ex ante what the real outcome of any government policy will be, no government policy can be justified on the supposed benefits it will provide to the market.
One deficiency I see in Austrian theory as it stands is that it does not talk about political entreprenuerialism, that is, the entrepreneurial aspect of political means. This is becuase the political means fall outside the scope of voluntary exchange and, therefore, do not conform to the axioms of catallactics. However, I believe it would be possible to formulate a set of axioms that describe human choice where coercion can be given and received and this would greatly assist in analyzing the behavior of the political class. For example, governments are themselves subject to evolutionary forces similar to those which govern the free market - governments which adopt policies that are too aggressive or oppressive tend to be replaced by governments which adopt more reasonable policies and government which adopt policies that do not extract sufficient resources from their subject population or allow their subjects too much liberty tend to be replaced by governments which are more austere in disciplining and subjugating the population.
Not exactly the mainstream way of analyzing government action but mainstream political analysis is more useless than theology.