Vake recently forwarded us an article criticizing mathematics in economic analysis. At UES we have often discussed this topic, and I thought it would be nice to continue that discussion here.
Anderson begins by rightly stating that just because most economists use mathematics, that doesn’t make it the best approach. Although, since most economists use mathematics, the probability of mathematics being a good approach is close to 1. (A statement Anderson would likely reject.) Similarly he is correct that claims of Austrian mathematical incompetence are weak counterattacks.
Anderson’s second and last claim is that “one cannot quantify human action.” His only supporting evidence is the inability to measure cardinal utility. And this one problem apparently invalidates use of mathematics in all of economics. He ignores the mathematics in producer theory, all of econometrics and empirical economics, etc. And hence I find his argument incomplete at best. If he is to show that mathematics is completely useless to economics, he must reveal greater flaws in all subfields, not just one.
A while back I read an argument against statistics by Rothbard himself. Unfortunately for the Austrians, his argument is quite flawed as well. He claims that statistical inference is useless because most data are not normally distributed. True, but the beauty of statistics is exactly the opposite of what he claims! Some sample statistics follow a normal distribution even when the data are not normally distributed. This is the central limit theorem, a foundation of statistical inference. And hence Rothbard’s confusion over the distribution of data and the distribution of a statistic invalidates his argument. Perhaps claims of Austrian mathematical incompetence are true….
This coming from an economics student who is double majoring in mathematics.
I am not calling the Austrians mathematically incompetent, as I said before those claims are quite weak. I agree with Anderson’s refutation. My final statement was simply a fun jest. I don’t naively think anyone who doesn’t believe what I claim is an idiot.
Next, statistics is a realm of mathematics. Theoretical statistics uses the same formal deduction techniques as formal mathematics. And from the theoretical results (e.g., CLT) we make inferences. Empirical economics works exactly the same way. From the theoretical econometric results, we are able to make inferences. Hence by rejecting mathematics everywhere you reject any empirical inferences based on some theoretical result.
Keynes was a predecessor of modern economics. Economics didn’t become highly mathematical until after 1950, so it makes sense that Keynes’s work wasn’t either. Simply existing before a period of change does not refute that change. The first supply and demand graph wasn’t drawn until 1890, and even then relegated to just an appendix.
As for the subjective theory of value…Sure, most economists agree measuring consumers’ utility values poses a problem. But while it appears that the Austrians throw up their hands in defeat, the rest of us move on. This cardinal utility problem doesn’t appear in many other realms of economics, so I continue to wonder why the Austrians reject mathematics in all realms, yet they cite this sole problem. This oddity I question the most.
Here’s an analogy within mathematics itself. There’s one philosophical camp called the constructivists. They reject theorems which prove mere existence without actually constructing the object, showing us what it looks like. Most mathematicians reject this philosophy. Although the constructivists have a point, knowing something exists, but knowing only that it exists is quite odd. But allowing such statements has been enormously useful to the development of mathematics. Similarly, allowing such ideas as cardinal utility has been useful to economics.
I also did not claim that the absence of mathematics makes something incorrect. I said the enormous infusion of mathematics makes the probability of mathematics being useful high.
I look forward to the rebuttal.
“I am not calling the Austrians mathematically incompetent, as I said before those claims are quite weak. I agree with Anderson’s refutation. My final statement was simply a fun jest. I don’t naively think anyone who doesn’t believe what I claim is an idiot.”
Good enough.
“Next, statistics is a realm of mathematics. Theoretical statistics uses the same formal deduction techniques as formal mathematics. And from the theoretical results (e.g., CLT) we make inferences. Empirical economics works exactly the same way. From the theoretical econometric results, we are able to make inferences. Hence by rejecting mathematics everywhere you reject any empirical inferences based on some theoretical result.”
I will disagree with this statement. Statistics may use mathematical tools but it is not math in itself. Just like physics uses math, so does statistics. True math, in my opinion, is deductive and statistics by its very nature is not deductive. Any empirical analysis, in my view, is simply an effort by economists to bastardize the field into a natural science. Knut Wicksell is a good reference to what I am trying to say. Wicksell used Austrian insights and mixed them with a Walrasian analysis. The end result was Austrian economics fused with calculus and in fact it made alot of sense. But it was not statistics but fundamental, logical, mathematics and calculus. In mathematics, just like logic, the premises must be very clear. 2 2 must equal 4 and A must be A. This may seem very elementary but economic science, unlike natural sciences, can not give mathematics variables their proper roles. Utility can not truly be measured, being one example, therefore it can not be used in mathematical forumlas.
I have very little problem with a Wicksellian take on economics but the use of a natural science such as statistics in economics is not good enough to formulate theories. All statistics can provide is the uncovering of problems in current theories and does not have the ability to formulate them (the reasons why I already discussed in the previous blog post).
“Keynes was a predecessor of modern economics. Economics didn’t become highly mathematical until after 1950, so it makes sense that Keynes’s work wasn’t either. Simply existing before a period of change does not refute that change. The first supply and demand graph wasn’t drawn until 1890, and even then relegated to just an appendix.”
Period changes do not come out of the blue, Matt. Keynes already was alive when this period change was in motion and he rejected it. It isn’t like he wasn’t smart enough to understand how to use statistics in economics but simply did not see a place for statistics in economics. In addition, I believe this period change you described moved economics from pure theoretical calculus to statistical and empirical analysis. My critique of Phelps also has insight as to why the use of empirical data to formulate theories is incorrect.
“As for the subjective theory of value…Sure, most economists agree measuring consumers’ utility values poses a problem. But while it appears that the Austrians throw up their hands in defeat, the rest of us move on. This cardinal utility problem doesn’t appear in many other realms of economics, so I continue to wonder why the Austrians reject mathematics in all realms, yet they cite this sole problem. This oddity I question the most.”
In theoretical analysis, you do not simply “move on” just because the rest of the profession is using it. This last paragraph simply states that because everyone else is using it, we might as well too even though it may be flawed. It is not like Austrians do not understand these things but we use the subjective theory of value to guide our analysis. The subjective theory of value is not simply a claim to people’s valuation but it also a methodology. That is why so many Austrian books read differently than others. It is because we use a logical, subjective approach to economics.
“Here’s an analogy within mathematics itself. There’s one philosophical camp called the constructivists. They reject theorems which prove mere existence without actually constructing the object, showing us what it looks like. Most mathematicians reject this philosophy. Although the constructivists have a point, knowing something exists, but knowing only that it exists is quite odd. But allowing such statements has been enormously useful to the development of mathematics. Similarly, allowing such ideas as cardinal utility has been useful to economics.”
The term “useful” could signify various things. It may be useful to neoclassicists but it isn’t for Austrians.
“I also did not claim that the absence of mathematics makes something incorrect. I said the enormous infusion of mathematics makes the probability of mathematics being useful high.
I look forward to the rebuttal.”
Once again, the Austrian approach to economic theory is one that I believe is the correct methodology. It is one that has given birth to many great theories and has provided answers when other schools have had none. It is why in the mysts of the depression of the 1970s Hayek was given a Nobel Prize in business cycle theory and Ayn Rand began publishing books referencing von Mises. It was at that time, and still today, that the Austrian approach has been the only answer to many of the events that have befallen out economy in the last 100 years. So convincing was the Austrian methods that its base has grown and the dark years of 1950-1970 are gone. Since then the Austrian school has grown in numbers and will continue growing because of our superior methodology. The days of neoclassicism are near and so are the days of the bastardized Keynesian paradigm.
Only time will tell what will happen to the Austrian school but thus far it seems that it is making quite a come back. The school of thought that the Austrians gave birth too, the Chicago school, has virtually disappeared and all those libertarians and truth seekers are naturally heading in our direction. The growth of followers in the future will be. in my opinion, enormous.
In sum, this discussion is simply a difference in opinion but I believe the Austrian approach is the superior one and it has been able to outlive countless other methodologies. Due to the nature of theory and deductive logic, the use of statistics and even Calculus in economics is incorrect. That said, it is difficult to change minds once they have adapted to a specific line of thought. I do not mind that Matt loves statistics and neoclassicism with a naughty taste of Keynesianism, thats his right. That said, when the day comes that a phenomenon has occurred on the market that is unexplainable to your school, I highly doubt that the Austrians had not already figured out an answer before the event even happened- just like the depression of the 1970s.
Reiterating what I said earlier: Theoretical statistics is pure mathematics. There is exact precision and formal logic. The same logic that proves the fundamental theorem of algebra proves the law of large numbers. It is the application of statistics which you refer to, the application of statistical results to economics. But the economic inferences are only as good as their theoretical foundation. Hence my main point: in rejecting mathematics in economics you must necessarily reject empirical economics, whose foundations lie in mathematics. Of course this really doesn’t matter because you reject both, regardless of their relationship.
I don’t recall Keynes rejecting all mathematics in economics. The absence of extensive mathematics in The General Theory doesn’t imply rejection. Is there any correspondence that says otherwise? Alfred Marshall, however, is famously known for turning away from mathematics in his later years: “I had a growing feeling in the later years of my work at the subject that a good mathematical theorem dealing with economic hypothesis was very unlikely to be good economics.” If you can’t translate the mathematics into English and illustrate with examples, burn the whole theory.
Mathematics itself was quite different at the turn of the century. Hilbert’s program only began around the mid 1920’s and Gödel proved his incompleteness theorem in 1931. The General Theory was published in 1936. I don’t know whether Keynes knew of the foundational crisis in mathematics or what he thought of it if so. Also, Arrow and Debreu didn’t prove the general equilibrium theorem until 1954.
And so, although I respect Marshall and Keynes’ opinion, they didn’t have the extensive perspective on mathematics in economics we have today. And hence they are unable to conclude this debate. This is what I alluded to calling Keynes a predecessor to the economics which requires every orthodox graduate student to know advanced mathematics.
Next, I never claimed that I or anyone else “move on just because the rest of the profession is using it.” I said we move on because each person individually sees the idea’s usefulness. And by usefulness I mean useful to the individual. Obviously Austrians don’t find it useful or else we wouldn’t be having this debate.
Also note I have never said the Austrian approach is useless, I think all problems are best analyzed by multiple approaches, which is why I must defend mathematics as one such approach, and why I would also defend the Austrian methods.
There is but one thing left. I would still like to know what exactly it is in “the nature of theory and deductive logic” that makes “the use of statistics and even Calculus in economics…incorrect.” Is the Profit Maximizing Rule MR=MC incorrect? This is a simple application of Calculus. Does this rule not give us a firm’s profit maximizing quantity?
Pedantic point: neoclassical economists generally dropped cardinal utility long ago in favor of ordinal utility. That is to say, neoclassicists generally don’t even bother trying to measure utility, they simply want to order bundles of goods and services in order of preference. Thus we no longer need “units,” as bundles are ranked by the natural or real number line. At the very least, will the Austrians grant that an agent can rank bundles in order of preference, no least by the amount he’s willing to pay?
Given an ordering that satisfies certain properties, we can ultimately deduce a function. Induction is required to get continuity and other nice properties, but all we need for a working neoclassical theory is that 1) an agent can choose among and rank all possible/observable bundles; 2) that if bundle A is at least as good as B, which at least as good as C, then C is at least as good as A. (Weak transitivity.) Boom, from that one preference ordering we can construct infinitely many “utility” functions. Hence the futility of cardinal utility.
“Reiterating what I said earlier: Theoretical statistics is pure mathematics. There is exact precision and formal logic. The same logic that proves the fundamental theorem of algebra proves the law of large numbers. It is the application of statistics which you refer to, the application of statistical results to economics. But the economic inferences are only as good as their theoretical foundation. Hence my main point: in rejecting mathematics in economics you must necessarily reject empirical economics, whose foundations lie in mathematics. Of course this really doesn’t matter because you reject both, regardless of their relationship.”
Again, I do not believe that “theoretical statistics” is pure mathematics. It is a science that uses and relies on inductive or outside information. Calculus does not depend upon this, it simply depends upon the premises being correct. Statistics is inductive, Calculus is deductive, both in theory and in practice.
“I don’t recall Keynes rejecting all mathematics in economics. The absence of extensive mathematics in The General Theory doesn’t imply rejection. Is there any correspondence that says otherwise? Alfred Marshall, however, is famously known for turning away from mathematics in his later years: “I had a growing feeling in the later years of my work at the subject that a good mathematical theorem dealing with economic hypothesis was very unlikely to be good economics.” If you can’t translate the mathematics into English and illustrate with examples, burn the whole theory.”
I never said that Keynes rejected all mathematics but I may have been a little extreme as far as Keynes view. Nonetheless, Keynes believed that fundamentally logic, philosohy and deductive reasoning should form the bulk of economic science. These correspondences are hard to find and I discussed this with Breitenstein when he was last down here. He apparently has also found literature that lends to the belief that Keynes was not too fond of mathematics in economics. One such correspondence could b this one:
http://ideas.repec.org/a/oup/cambje/v14y1990i1p29-47.html
In addition, I believe it was Alfred Marshall that steered Keynes away from a mathematical approach to economics, but I am not sure.
“Mathematics itself was quite different at the turn of the century. Hilbert’s program only began around the mid 1920’s and Gödel proved his incompleteness theorem in 1931. The General Theory was published in 1936. I don’t know whether Keynes knew of the foundational crisis in mathematics or what he thought of it if so. Also, Arrow and Debreu didn’t prove the general equilibrium theorem until 1954.
And so, although I respect Marshall and Keynes’ opinion, they didn’t have the extensive perspective on mathematics in economics we have today. And hence they are unable to conclude this debate. This is what I alluded to calling Keynes a predecessor to the economics which requires every orthodox graduate student to know advanced mathematics.”
Ok.
“Next, I never claimed that I or anyone else “move on just because the rest of the profession is using it.” I said we move on because each person individually sees the idea’s usefulness. And by usefulness I mean useful to the individual. Obviously Austrians don’t find it useful or else we wouldn’t be having this debate.”
Correct-a-mundo!
“Also note I have never said the Austrian approach is useless, I think all problems are best analyzed by multiple approaches, which is why I must defend mathematics as one such approach, and why I would also defend the Austrian methods.”
I do not believe this, I believe logic and deduction is the best methodology for certain sciences including economics but this is a difference in opinion.
“There is but one thing left. I would still like to know what exactly it is in “the nature of theory and deductive logic” that makes “the use of statistics and even Calculus in economics…incorrect.” Is the Profit Maximizing Rule MR=MC incorrect? This is a simple application of Calculus. Does this r”ule not give us a firm’s profit maximizing quantity?
Marginality comes from the marginal revolutionaries (i.e. Menger and Jevons). I am sure you know this information and to not seem like a parrot, Menger used subjective valuation to deduce marginality. Marginal whatever is a product of Menger and Jevon’s efforts.
Most of this seems to simply be a difference in opinion. I use the Austrian method, you use w/e. You say other methods are needed to find a clear solution, I do not believe this. You believe statistics is a logical, deductive science like Calculus, and I disagree.
If you continue to have problems with my point-of-view, a good book to recommend, if you can find it, is Hayek’s Monetary Theory and the Trade Cycle. I believe I currently have the university’s only copy but they might have more. I believe a few other austrian books have good analyses on this subject.