Dani Rodrik has two recent posts which explain why in his mind we need to use mathematics in economics. See here and here. I wrote the following as a response to his original post, but it was too late as he was already writing his seccond post. So I am lifting it from the comments section and putting it up here.

Dani,

Two points. First, Alfred Marshall had a wonderful saying for economists and it was "burn the mathematics." His basic idea was that mathematics is a very useful servant, but a horrible master. He did not argue that we shouldn't use mathematics to check our logic. Instead, he argued for the use of mathematics along the same lines you lay out. But then he turns around and says once you have tested your argument against mathematics you should "burn the mathematics" and commnicate the argument in the clearest use of language possible.

Second, there are some instances where the important questions defy mathematical rendering. Here I recommend Kenneth Boulding's 1947 review in the JPE of Samuelson's Foundations where he argues that the flawless precision of mathematical economics may prove impotent in addressing the complexity of the social world --- a world that was better analyzed with the literary vaguness of economic sociology.

Again, mathematics can be a good servant to thought, but not a very good master in the realm of economic discourse. Think of all the issues that were raised by non-mathematical scholars over the years that have proven essential to understanding development and social order: institutions (North), alternative legal systems in a positive transaction cost world (Coase), constitutional retraints on predation (Bucuahan), enterpreneurial activity (Schumpeter, Kirzner and Baumol), power and collective action (Olson), cooperation and coordination (Schelling), and spontaneous order (Hayek).

Especially in the field of development economics scholars such as Hirschmann and Streeten said a lot of valuable things without the use of mathematics. Should we now only believe them if we can put it in a model?

So again I think it is a matter of having some perspective on the issue of mathematics in economics. Is it a good signal of intelligence? It certainly can be. Does it guarantee good economics? Certainly not, we can prove much economic nonsense using higher mathematics. Being a good economist is about a lot of things, mathematical acuity may be on the list, but certainly doesn't exhaust it. History, philosophy, political theory, languages, demography, technology, etc. would be on my list of the skill set that usually is found in good economists.

Samuelson used to argue that ambiguity in thought results from either using the same words to mean different things, or using different words to mean the same thing, and that mathematics would eliminate this source of ambiguity. But as Rothbard argued in Man, Economy and State, this argument doesn't quite work in economics because we are caught in an act of double translation --- we take an argument in words, translate it into an argument in math, and then translate it back to words to communicate with others.

What is your favorite argument against the abuse of mathematics in economics?

My favorite argument?

The central causal explanatory element in economics is open-ended learning/judgment within the context of changing relative prices and ownership relations. This causal explanatory element cannot be capture in _any_ mathematics. The math/logic of valuation relations is an aid to the perception of the structure of price relations, but it is not an accurate "model" of those relations -- and it is no explanation of the central causal element explaining the ordering processes of the market -- which is the changing judgment & learning of entrepreneurs within a changing and individually perceived structure of price and social relations.

So appeal to pure math makes a theorist utterly blind to the central causal process at work in the market economy.

That's my argument

Posted by: Greg Ransom | September 06, 2007 at 11:56 AM

I would say mine is Mises argument against the use of calculations to find equalibrium. If I understand his argument correctly, using numbers from an economy in disequlibrium it would make it absurd to force them into a perfect equilibrium model. Even if we could calculate equilibrium it would do us little good because the economy will not stay in equilibrium for very long.

Maybe that's Mr. Ransom's point?

One that I never understood is the idea of being able to calculate utility. The closest way to calculate how much I value a given item is the price. But even with that it is impossible to know how much I truly value it. Someone maybe selling an item at a much lower price than I would pay for it.

Posted by: Matt C. | September 06, 2007 at 02:22 PM

The argument that maths gets over the problem of multiple meaning of words is completely bogus, it does not even start to address the issue. The way to get over that problem is to write clearly and directly so the message comes from the sentence or the paragraph and the meaning of the words is unambiguous in the context.

Another way is to minimise the use of over-loaded words like "rational", for example try "deliberative" for a considered action and "effective" for an action or a policy that produces the desired effect.

For great non-mathematical economists add Hutt and Bauer.

It is possible that Talcott Parsons went astray after he re-invented praxeololgy/situational analysis because he thought that the sciences of human action needed a language of their own that would be the equivalent of maths in the natural sciences so that the high point of theoretical development would resemble a big closed system of equations. I think he misread the role of maths in science, on top of other errors.

For a crit of abuse of maths in hard science check out J Schwartz 'The pernicious influence of mathematics on science' in "Logic, Methodology and Philosophy of Science" eds Nagel, Suppes and Tarski 1962. One of his points is that maths demands clearly defined systems and that calls for heroic assumptions about the world. To be appropriately used in science the assumptions must be correctly chosen from a larger point of view invisible to mathematics itself.

Posted by: Rafe Champion | September 06, 2007 at 06:34 PM

My favorite are arguements by non-economists such as Stephen Leacock, Picasso and even Einstein who all (at times) questioned the limits of abstracting interacting and changing beings into rigid deterministic models. Especially when public policy was then crafted according to that abstract and then imposed upon a already changed real world. Stephen Leacock's book Winnowed Wisdom starts out describing a average man and woman in the US (here is the first few paragraphs):

"In point of residence, it seems only logical to suppose that the average man lives at the centre of population, in other words, in

the United States he lives at Honkville, Indiana, and in Canada at

Red Hat, Saskatchewan.

In the matter of height the average man is five feet, eight inches,

decimal four one seven, and in avoirdupois weight he represents 139

pounds, two ounces, and three pennyweights. Eight-tenths of his

head is covered with hair and his whiskers if spread over his face

could cover it to the extent of one-tenth of an inch. This ought

to be a promising sign to a reader.

The average man goes to church six times a year and has attended

Sunday School for two afternoons and can sing half a hymn.

Although it thus appears that the average man is rather weak on

religion, in point of morals the fellow is decidedly strong. He

has spent only one week of his whole life in the penitentiary.

Taking an average of theft and dividing it by the population it

appears that he has stolen only two dollars and a quarter. And he

never tells a lie except where there is some definite material

advantage.

The average man is not, by statistics, a great traveller. The poor

fellow has been only sixty-two miles away from his own home. He

owns nine-tenths of a Ford car, punctures a tire once every twenty-

two days, and spends, in the course of his whole life, a month and

a half underneath his car."

Posted by: Dan Smith | September 06, 2007 at 10:44 PM

Now as much as Professor Boettke advises his students against methodology, I cannot help myself here, because I think it has a lot to offer us on this question.

What I would first like to point out is the belief among many practicing economists of the scientifically true, verifiable and inevitable nature of mathematical proofs. In order to prevent falisification, or escape criticism, it seems very convenient to include complicated mathematics, because an appeal to authority can come from no greater place.

So the real question should not be over whether the use of mathematics is appropriate for economists. We are already conceding far too much, namely the infallible nature of mathematics. Read Kurt Godel, and Karl Popper. Truth can never be grasped in its entirety. Also, a theory (even a mathematical one) can never be decisively verified. Learning is an open-ended process. What is true today may be falsified tomorrow.

A priorism (and its appeal to logic and mathematics) fares no better. A priorism is a fundamental feature of scientific activity -- but it too is not immune from error. MISES WAS WRONG! Every human affair is fallible. We must always remember that.

How was that for a student's stab at economic methodology? Is there a reason why we young economists are not encouraged to apply methodology to economic problems-- is it because the science of economics would become like the science of philosophy? A big cluttered field of relativism, anti-foundationalism and nihilism?

In my view, what are we waiting for?

Posted by: matthew | September 07, 2007 at 12:33 AM

Dan Smith,

Stephen Leacock was an economist. I was shocked when I learned.

http://en.wikipedia.org/wiki/Stephen_Leacock

Posted by: Dan in Euroland | September 07, 2007 at 01:03 AM

Not exactly against mathematics, but i believe Nassim Taleb in his 'Fooled by Randomness' has criticized economists for having physics envy, i.e. following the wrong kind of math.

Posted by: cvj | September 07, 2007 at 03:10 AM

All boat seems the same, but the math could help you find out which boat for which condition and it should purposed.

Same lenght beam and draft are subject to hullspeed but it's doesnt mean it would have same speed otn the OCEAN OF IGNORANCE.

That why Math could help you with their statistical toolls for all design ratio which could sweets meet your skipper skills.

SAILING ANARCHIE

http://www.sailinganarchy.de

Posted by: donny tedjo | September 07, 2007 at 10:45 AM

Math is the ultimate condenser of ideas:

f=ma, e=mc^2, etc...

Simple formulas that capture very complex situations.

The mathematics that is used in economics seems to be mainly PDEs, i.e. Calculus, and some game theory, probability and stochastic analysis. But that's not all of math. Maybe one day we'll see other math concepts spill over into econ: such as non-commutative geometry (where points disappear), conformal field theory, category theory, sheaves, cohomology theory, bizarre algebraic structures, etc....who knows.

Personally, I count the language of mathematics as being an extension of our usual language, but it doesn't hurt to provide a simultaneous translation if the audience happens to be math-illiterate.

Posted by: Unit | September 08, 2007 at 12:56 AM

Why is it shocking that Stephen Leacock was an economist? All sorts of unlikely people have done economics, like J K Galbraith and Mick Jagger (at the LSE). When he left, his teacher suggested it was a bad career move because there was no money in skiffle bands. http://khufu.openlib.org/~tchecndg/archive/1994/0699.html

I had an idea that the literary laureate V S Naipaul might have been at the LSE as well (wrong) but the search turned up another fascinating laureate from the Caribean - Sir Arthur Lewis (1979) who acknowledged support from Arnold Plant and Hayek although he did not think along the same lines. http://www.caricom.org/jsp/projects/uwicaricomproject/caribbean_nobel_laureates.jsp

Posted by: Rafe Champion | September 08, 2007 at 08:38 AM

I think the primary advantage of math is that it keeps the post-modernists at bay.

The main problem with math? To paraphrase Tom Lehrer, "A mathematical model is like a sewer -- what you get out of it depends on what you put into it." If you don't include a variable to stand for a particular margin of adjustment, then your model will not predict any change on that margin. For example, if you're modeling rent control, and your model does not include a variable to represent bribes, you will fail to predict bribery. If your model does not include variables for aspects of quality, then you will fail to predict reductions in the quality of housing (such as frequency of maintenance). Because of the open-ended and creative character of human action, it is impossible to specify all margins. As a result, any mathematical model is necessarily incomplete. But that doesn't mean it's not useful, just that you have to take it with a grain of salt.

Posted by: Glen Whitman | September 08, 2007 at 03:16 PM

I think the primary advantage of math is that it keeps the post-modernists at bay.

The main problem with math? To paraphrase Tom Lehrer, "A mathematical model is like a sewer -- what you get out of it depends on what you put into it." If you don't include a variable to stand for a particular margin of adjustment, then your model will not predict any change on that margin. For example, if you're modeling rent control, and your model does not include a variable to represent bribes, you will fail to predict bribery. If your model does not include variables for aspects of quality, then you will fail to predict reductions in the quality of housing (such as frequency of maintenance). Because of the open-ended and creative character of human action, it is impossible to specify all margins. As a result, any mathematical model is necessarily incomplete. But that doesn't mean it's not useful, just that you have to take it with a grain of salt.

Posted by: Glen Whitman | September 08, 2007 at 03:17 PM

Dr Boettke,

I was reading a response to another post on math in econ by Rodrik by Dr. Cowen.

Dr. Cowen mentions Rodrik's new book with an Amazon link. The book description says the following:

To most proglobalizers, globalization is a source of economic salvation for developing nations, and to fully benefit from it nations must follow a universal set of rules designed by organizations such as the World Bank, the International Monetary Fund, and the World Trade Organization and enforced by international investors and capital markets.

Am I being simplistic here? I never considered those supra-national institutions like the WTO and IMF to be integral to globalization. Am I wrong here?

Posted by: John | September 08, 2007 at 07:20 PM

I've always thought that losing otherwise good people from the ranks of great economists is an argument against the excessive mathematization and formalization of economics. Some great potential economists do not matriculate in economics because of the current mal-focus on mathematics in all formal economics programs.

If important economic problems exist that do not lend themselves to mathematical solutions, and if outcomes -- in this case, the progression of economic science and even occasional Kuhnian paradigm shifts as needed -- are partially endogeneously determined by the spontaneous order of the participants in the scholarly debate, then we are losing the perspective of people who could significantly add to mix. They are simply not at the table because they are not allowed by our current hierarchical programs at the good schools.

Posted by: Kirk | September 09, 2007 at 07:40 PM

This is not from an econ blog, but here is a good post explaining the importance of simple algebra in understanding not only economics but a variety of fields.

http://scienceblogs.com/gnxp/2007/09/a_little_knowledge_can_go_a_lo.php

Posted by: TGGP | September 10, 2007 at 03:28 PM

Rafe Champion wrote:

"Another way is to minimise the use of over-loaded words like 'rational', for example try 'deliberative' for a considered action and 'effective' for an action or a policy that produces the desired effect."

But the ambiguity of a term like "rational" doesn't come from math, but from its conversational use. Rationality in economics is understood as a mathematical construct and is quite specific:

For a given preference ordering, rationality means the ordering is complete and transitive. These are mathematical applicatives and are completely unambiguous.

Posted by: saxdrop | September 11, 2007 at 01:43 PM

Thanks saxdrop, but how does your mathematical definition of rationality relate to economic phenomena, like the contents of my shopping basket?

Posted by: Rafe Champion | September 14, 2007 at 02:58 AM