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If a logical argument doesn't assume what it is supposed to prove, then it isn't a logically valid argument. The problem isn't the argument, but what people think "proving" does.

Sorry, I don't get Romer's remarks.

Social contract theorists have been studying rules that humans would follow as an scientific project for some time: http://plato.stanford.edu/entries/contractarianism-contemporary/

Brian Skyrms is probably the most well known philosopher who uses extended game theory to explore social contract formation.

Romer owes us a little more here, even though the skyhook metaphor is cute.

Prof Boettke,

You seem somewhat ambiguous about the use of mathematics in economics, coming out against what you deem its misuse but not its use altogether.

I'm with Mises, clearly and unambiguously condemning its use in economics altogether.

D.G.,

What part was "ambiguous" about what Beottke wrote: "When mathematical techniques are used correctly they are extremely helpful -- perhaps essential -- to clear thinking on many topics."

Seemed clear to me!

Kelly,

Boettke says mathematics is helpful to clear thinking on many topics.

Does that include economics?

Saying that it's helpful is a strange statement.

It isn't helpful but essential to the solution of mathematical problems, and not helpful but vicious when applied to non-mathematical problems.

I don't see any middle ground, how it can be less than essential to the solution of mathematicsl problems and less than vicious in non-mathematical fields.

DG,

There are certainly useful math applications in economics and the other social sciences. Even a GE or max U model can be helpful for understanding the economy, as long as we keep in mind that the economy is not in GE and that max U is restricted to unusually simple choice situations. The problem with these models is not that they are used, but that all problems that cannot be expressed as such are deemed illegitimate/irrelevant to the "economic theorist." An additional problem is the over-emphasis on what happens when there is a tiny (i.e. uninteresting) change in one of the assumptions, in the case where it leads to tiny changes in the conclusions.

PS: If you include statistics in math, then you have even less justification. Statistics is much more than macroeconomic time-series models (the most problematic use of statistics, and thus a strawman).

I agree with Prof Boettke's remarks.

A question arises, however, about whether modern formal economics has outgrown Mises's criticisms against it or not.

From the point of view of DSGE, it appears that it is still impossible to model important topics such as economic calculation, entrepreneurship, capital heterogeneity, consumer heterogeneity, etc. This is because there is the curse of dimensionality (consumption and production must be simple in order to get a workable model) and because some problems are assumed away by construction (entrepreneurship and market process).

I'm even less knowledgeable about game theory than DSGEs, but I often notice, especially in mechanism design theory, that there are problems with taking into account economic calculation. Kreps in his textbooks explicitly states that the theorem behind the revelation principle wouldn't hold if agents weren't capable of computing the final equilibrium, but he adds that no one has ever formalized this criticism. This looks like saying that as long as mathematics doesn't cricitize these theories, they will assume away the main problem.

I was on the other hand positively surprised by some new work by economists such as Robert Axtell, which has written an interesting paper on RAE. In another paper he for instance compared the computational complexity of decentralized and walrasian equilibria and proved that: all decentralized equilibria are stable, non-equilibrium dynamics have wealth effects (profits and losses, I would say), non-equilibrium dynamics are path dependent and never turn back to the same point, and that decentralized equilibria are computationally more efficient for a high number of agents and goods than walrasian ones. I interpreted it by asking if economics was on the way of mathematizing Mises. There is still to much focus on efficiency and equilibrium, maybe, and the model was pure-exchange, but it said what I had always known reading Hayek or Mises or Kirzner, albeit in a puzzingly new way.

I wonder whether these new theories may be capable of putting mathematics into AE and modelling every distinct element of AE which has been overlooked in standard macro. Probably BRICE and ABM economics is more suited than DSGE to mengerize economic thought.

That's exciting, although the fear that mathematics is still more a limitation than an opportunity may not be antiquated yet.

DG - The difficulty here is which are the mathematical fields. Aside from mathematics some physicists and engineers, for example, find mathematics used, but in what sense, or why, are they mathematical fields? I'm not sure what the answer is - any thoughts anyone?

I mostly agree with you though. And perhaps to help others to see DG's objection, could you point us to where in economics you think maths has been useful. Again, I am not ruling out that is has been useful, but I would just like to know which examples you could present to help me see your perspective.

And how about Rothbard's typical argument about Occam's Razor and Mises' elucidations about the epistemological problems of the sciences of human action - mainly the abscence of particular event regularity?

Does not all this render mathematics in economics something very far from helpful or essential?

The problem here is the association of mathematics with naive and simplistic modeling or naive and simplistic empiricism. I think that there is little wrong with using mathematics though.

It's not necessarily even true that if an idea is derived from Mises own thoughts that it is best written without mathematics.

In "The Theory of Money and Credit Mises" writes:
"The demand for money for international trade is composed of two different elements. It consists, first, of the demand for those sums of money which, as a result of variations in the relative extent and intensity of the demand for money in different countries, are transported from one country to another until that position of
equilibrium is re-established in which the objective exchange-value of money has the same level everywhere. It is impossible to avoid
the transfers of money that are necessary on this account. It is true that we might imagine the establishment of an international deposit
bank in which large sums of money were deposited, perhaps even all the money in the world, and made the basis of an issue of money-certificates, i.e. of notes or balances completely backed by money. This well might put a stop to the physical use of coins, and might in certain circumstances tend to a considerable
reduction of costs; instead of coins being used, notes would be sent or transfers made in the books of the bank. But such external differences
would not affect the nature of the process. The other motive for international transfers of money is provided by those balances that arise in the international exchange of commodities and services. These have to be settled by transfers
in opposite directions, and it is therefore theoretically possible to eliminate them completely by developing the clearing process."

Mises could easily write this as a mathematical sum, since that's what he's describing in words.

There is an even more confusing paragraph in that book where Mises describes an equation in words. I can't find it right now though.

The issue of when to use math gets pretty dicey pretty quickly. One problem is that humans are creative in the math they invent and in the way they use math. You can't predict such creativity, so most, maybe all, generalizations about what math can't do are at risk of being overturned tomorrow by some crazy new thing you could not have expected. The only safe statement becomes "We should use math only in appropriate ways," which doesn't exactly narrow things down.

All of that stuff notwithstanding, I have a rule of thumb that might be helpful. Economics is about the unintended consequences of human action. These unintended consequences are patterns that, because unintended, have no "meaning" in the way the actions of individuals have meaning. There is no particular reason math might not be used to describe such patterns. When you turn to the human actions that unintendedly generate the pattern, however, you have a problem about "meaning." Those actions are meaningful and we describe such actions in terms of such meanings. We are not very good at translating "meaning" into math. Thus math is probably going to be a more limited tool at the level of individual action. It's rule of thumb, but perhaps a good one.

Neither Mises nor other Austrians ever challenged the usefulness of mathematics as a tool of logical thought, or its practical applicability in many of the natural sciences.

And, indeed, Mises insisted that to determine whether a method or analytical tool was relevant and applicable to economics one had to be knowledgeable about it.

Indeed, in "The Ultimate Foundations of Economics" (pp. 3-4), Mises argues:

"What is needed to prevent a scholar from garbling economic studies by resorting to the methods of mathematics, physics, biology, history or jurisprudence is not slighting and neglecting these sciences, but, on the contrary, trying to comprehend and master them. He who wants to achieve anything in praxeology must be conversant with mathematics, physics, biology, history, and jurisprudence, lest he confuse the tasks and the methods of the theory of human action with the tasks and the methods of these other branches of knowledge."

And Mises goes on to say, "When I expressed this opinion in a lecture, a young man in the audience objected. 'You are asking too much of an economist,' he observed; 'nobody can force me to employ my time in studying all of these sciences.' My answer was: 'Nobody asks or forces you to become an economist.'"

I think all Austrians should heed the advice of this leading figure of the Austrian School in the 20th century.

Richard Ebeling

Richard:

That's been one of my favorite Mises quotes.

One problem I have with math is it often reflects a more general problem in economic theory: formalism for formalism's sake.

Here is the famous statement of Alfred Marshall in a 1906 letter to the mathematical economist A.L. Bowley:

"But I know I had a growing feeling in the later years of my work at the subject that a good mathematical theorem dealing with economic hypotheses was very unlikely to be good economics: and I went more and more on the rules—(1) Use mathematics as a shorthand language, rather than as an engine of inquiry. (2) Keep to them until you have done. (3) Translate into English. (4) Then illustrate by examples that are important in real life. (5) Burn the mathematics. (6) If you can't succeed in (4), burn (3). This last I did often."

I haven't had an education in economics at a level where mathematics is heavily used, but somtimes I try to translate my intuitive understanding into mathematical terms. I find it a very useful and illuminating exercise, because many logical relations are more obvious when rendered in that form.

Having said that, I don't consider such mathematisation to be anything more than an illustration or approximation.

Dave:

The problem of formalism for the sake of formalism lead James Buchanan to the following statement in 1983, clearly out of a deep frustration with the state of the economics profession:

"Economics, as a discipline, became 'scientific' over the quarter century, but I put the word in quotation marks and I deliberately use it perjoratively here. As it is practiced in 1983, economics is a science without ultimate purpose or meaning. It has allowed itself to become captive of the technical tools that it employs without keeping track of just what it is that the tools are to be used for. In a very real sense, economists of the 1980s are illiterate in basic principles of their own discipline . . .

"Their motivation is not normative; they seem to be ideologial eunuchs. Their interest lies in the purely intellectual properties of the models with which they work, and they seem to get their kicks from the discovery of proofs of propositions relevant only to their own fantasy lands . . .

"Our graduate schools are producing highly trained, highly intelligent technicians who are blissfully ignorant of the whole purpose of their alleged discipline. They feel no moral obligation to convey and to transmit to their students any understanding of the social process through which a society of free persons can be organized without overt conflict while at the same time using resources with tolerable efficiency."

I would suggest that there has been little change within the mainstream of the economics profession since Dr. Buchanan made this lament 25 years ago.

It sometimes bothers me when people talk about macroeconomic aggregates and averages, and it is not because I am opposed to them per se, but rather that people are so often sloppy when interpreting what they mean. It helps, I think, to consider aggregates and averages as a shorthand way to write a huge disjuction of every specific sequence of events from which they can be derived. But people often talk and act as though only the smooth specific sequences are possible, or assume that each is equally probable as any other. This isn't an error of mathematical procedure, but of invalid inference.

For example, consider the price level and the Austrian story of the business cycle. The price level it is like a huge disjunction of all possible price combinations from which it can be derived, and it suffices for when considering many economic problems. The Austrian claim is that, when a central bank inflates the money supply, not all of these possible price combinations is equally probable. In fact, they say that price combinations that show large relative differences are more likely to emerge. In other words, when considering some economic problems, it is necessary to delve beyond the aggregates and averages, and enter into a realm where mathematical tractability is not so great.

"I would suggest that there has been little change within the mainstream of the economics profession since Dr. Buchanan made this lament 25 years ago."

I can't agree with that, Richard. I would suggest there has been a huge change in the economics profession as reflected in, among other things, the prize to Ostrom. BRICE and all that. It's a different profession, one that I enjoy. I remember the 1983 economics Buchanan complained of. Much of it was indeed simply awful technical garbage. Now we have a profession in whose big names include, Greif, Ostrom, Schleifer, Shiller, Kahneman, Vernon Smith, and other interesting people. Buchan got his Nobel shortly after that was written, which helped change the profession away from what he complained of. It's a different game.

The bad old ways have not been completely shaken off and we still have lots of bogus mathphilia. But it just ain't the same profession today. Your can read interesting stuff that matters and you can *do* interesting stuff that matters. Bourbaki is pretty much dead. At any rate, he's not looking very healthy!

Richard:

The Buchanan quote is great, and that was a quarter of a century ago!

Which brings up an observation. Okay, so Krugman rightly chides the profession for being overly mathematical. It has gone too far. But even his benchmark of the "appropriate" level of formalization was that which Buchanan criticized decades ago. So, while it's nice to hear some top economists criticize the current use of math, it has to be interpreted in the context of the level that they actually favor.

Turning back to the 1970's and 80s might be quite welcomed, but even there the change won't be "radical" enough.

Roger:

You said that the patterns of unintended consequences have no meaning in the same way that human actions do. This brings up the concerns I had with Lachmann and esp. Lavoie. Lavoie interprets Lachmann as trying to unearth "the meaning" of institutions, when those very same institutions are the product of spontaneous order.

If they are right, then your argument that math can appropriately come to terms with unintended consequences is weakened. But, still, I am not so sure what "the meaning" of money, for example -- it's an unintended institution -- is.

Like you said, it gets dicey quite quickly.

Roger,

I didn't see your last post. A response: how much of the good stuff is in the AER, JET, etc? Still a very small percentage, no?

Boniface Kiprop wrote,

"could you point us to where in economics you think maths has been useful. Again, I am not ruling out that is has been useful, but I would just like to know which examples you could present to help me see your perspective."

Boniface,

I absolutely and positively guarantee you that the mathematical fakers will never, I repeat, never, comply with your request. They will never present you with an example of "mathematical economics," for the simple reason that there is no such thing. All of their talk of mathematics in economics is hot air, and you can feel it, but you'll never see it.

Never!

Hayek thought that Hicks' Value and Capital was a great work. Who here, besides me, agrees with Hayek? I think especially that his contribution to optimal choice theory -- indifference curves and all that -- is second to none, even though I prefer the subjectivist approach to choice under ignorance and uncertainty, its attempt to come to terms with meaning, and plans, and so on. At the very least, it is a book that economists ought to respect.

Boniface,

Let me amend the foregoing slightly.

They'll present you with the pretense of mathematical economics, with symbolic allusions to numbers, xs and ys, but never any actual numbers, the real raw material of mathematical operation.

Dittoe on Hicks' book, Dave, and much of GE, including dear old Walras. I think Mises' ERE is pretty clearly a verbal rendering of Walras or, perhaps, Cassel. Hoteling's article on Edgeworth taxation paradoxes is just one, somewhat randomly chosen example of a great piece of economic reasoning that simply cannot be separated from the math. Whenever you use simple supply and demand, you're doing "mathematical economics." For two (related) recent policy-relevant examples of high-tech math used to support an economic argument that cannot be made without that math see:

1)da Costa, N.C.A. and F. A. Doria. 2005. “Computing the Future,” in Velupillai, Vela. ed., Computability, Complexity and Constructivity in Economic Analysis. Oxford: Blackwell.

2)Velupillai, V. (2007). The impossibility of an effective theory of policy in a complex economy. In M. Salzano & D. Colander (Eds.), Complexity hints for economic policy. Springer.

For a user-friendly treatment of these two articles see

1) Koppl, Roger. “Thinking Impossible Things: A Review Essay,” Journal of Economic Behavior and Organization 2008, 66: 837-847.

2) http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1248409

Game theory. The list goes on.

Allow me to add that Hicks was "Marshallian" in the context of the quote provided by Mario. He offered a handful of graphs but put it all in words -- the bulk of the book is just that. The math he placed in the back appendices. Of course, outside the textbooks, the profession focused exclusively on the math after that.

Roger,

Your idea of what is "in" in the profession is much wider than mine. The main reason to quote Romer is just how radical it is to have someone of his stature write what he wrote. But also remember that he has retired.

Read the reaction at Harvard to Williamson, or the reaction reflected at other places to Ostrom.

This should be a moment of excitement for all of us, but do keep in mind that in the reaction we can also see how little we have moved from the world that James Buchanan described in 1983. The edges are blurry, and we are able to occupy a place at the edge (thankfully so due to the hard work of scholars such as Buchanan, Coase, North, V. Smith, and Lin Ostrom). But the sort of 'humanistic' economics --- even with the fancy dress of computability --- isn't cutting into the core. At the edges, yes, in the core, NO. But let me be clear -- by the core I mean: Harvard, Princeton, Chicago, MIT and Stanford; and the conversation in the AER, JPE, QJE and Econometrica. Outside of that we are talking the edge --- from close in edge (Berkeley) to outer edge (GMU). But the edge is not inside the core of our profession.

The Nobel was a victory for those at the edge, and a shocker to those in the core. What Buchanan is talking about in 1983 is the core, not the edge; and when you and I talk this is where we fail to communicate --- you (and David Colander and Barkley) have a broader definition of the core than I do, that includes those on the edge.

Boniface,

You see what I mean?

Boniface:

Like I already said, Hicks' Value and Capital.

"Atlantis puts a tariff t on the importation of each unit of the consumption commodity p, the world market price which is s. If domestic consumption of p in Atlantis at the price s+t is a and domestic production of p is b, b being smaller than a, then the costs of the marginal dealer are s+t. The domestic plants are in a position to sell their total output at s+t. The tariff is effective and offers to domestic business the incentive to expand the production of p from b to a quantity slightly smaller than a. But if b is greater than a,...."

Mises, Human Action.

This is hardly mathematical economics but, DG, even here it is discussed in terms of, as you say, "symbolic allusions to numbers, xs and ys, but never any actual numbers, the real raw material of mathematical operation."

Mises had no problem with that.

Prof. Boettke,

You wrote,

"Roger,

Your idea of what is "in" in the profession is much wider than mine."

I, for one, don't care what is "in" in the profession, but only in the science. And, after over forty years of looking for it, I still don't see any mathematics in it.


Prof. Prychitko (Dave from now on),

I agree that there's nothing wrong, in principle, with using numbers or symbols to illustrate a point. I have done so myself.

But, as you yourself said, that in itself, is hardly mathematical economics.

And that is what I'm still waiting to see.


Dave,

And, by the way, I have great admiration for Mises, but now when he wrote like that.

DG,

I'm pretty sure there's a problem of lingo here. I get the sense that you have a specific and, probably, idiosyncratic definition of "mathematical economics" in mind.

You say, "but never any actual numbers, the real raw material of mathematical operation."

If the "actual numbers" you intend are measured or estimated from life, then manipulations on them would normally be called "statistics" or "econometrics," wouldn't they?

Anyway, I don't think it's true that numbers are the "raw material of mathematical operation." There are no numbers in Hilbert's geometry, for example, and that usually counts as math I think. A better candidate for the "raw material" would be sets. You know, zero is the empty set, one is the set containing only the empty set, two is the set containing 1 and 0 as just defined, and so on. So in the usual way mathematicians build these things up, sets are at a stratum, as it were, "below" numbers. The real truth, IMHO, is that there are no "raw materials" of math or "mathematical operation."

Anyway, maybe you should clarify what you mean by "mathematical economics."

Pete,

We keep having the same old fights. Anyway, Greif is at Stanford, Schleifer at Harvard, and so on. I'm not saying there isn't work to do, just that it really ain't 1983. I'm not sure I see where I'm so off with that rather mild comment.

Roger,

You wrote,

"If the "actual numbers" you intend are measured or estimated from life, then manipulations on them would normally be called "statistics" or "econometrics," wouldn't they?"

You're confusing the passing data of econometrics with the eternal truths of economics. Since there is none of the passing data in the eternal truths, there are no quantities, and no mathematics. However essential to econometrics, it is irrelevant to economics, and, its only effect, to obscure it.

You wrote,

"Anyway, I don't think it's true that numbers are the 'raw material of mathematical operation.' There are no numbers in Hilbert's geometry, for example, and that usually counts as math I think."

I don't know about Hilber's geometry, but Archimedes' was a science of measurement, which implies magnitudes or numbers. And to lump non-numerical along with numerical concepts into it is to deprive mathematics of its meaning, and admit that you don't have any.

You asked what I mean by mathematical economics.

Mathematical fakery.

And when are you going to face the challenge posed by Boniface above, and give him your example of economics informed by your "sets"?


Roger,

You wrote,

“A better candidate for the ‘raw material’ would be sets. You know, zero is the empty set, one is the set containing only the empty set (in other words, one means zero. DGL), two is the set containing 1 and 0 as just defined (two implies all the quantities less than itself. DGL), and so on. So in the usual way mathematicians build these things up (one and one is two, two and two are four. DGL), sets are at a stratum, as it were, ‘below’ numbers (numbers are below numbers. DGL). The real truth, IMHO
(IM Hopeless . DGL), is that there are no ‘raw materials’ of math or ‘mathematical operation.’”

But the operations, nonetheless, without the materials for them.

Thanks for the enlightenment.

Koppl says, "humans are creative" even with math. Any answer that appeals human creativity appeals to me.

There would be a certain irony if math, which was justified in its attempt to eliminate fudging the verbal practice of social science were now nothing but a mask for some pretty basic normative assumptions about the way the world works.

I think it just serves as a filter, like a cartel. A shibboleth. The in-group pays low cost for playing with math. The out-group doesn't speak the language. We need some filter, why not a functional one like mathematical ability? However, this assumes complimentarity and not substitution. If a mathematician cannot speak competently about social science then what is the value of their ability to create an equation to explain it. The gap between the laity and the clergy can only yawn so far. At the end of the day their has to be a firm relationship between supply and demand.

The next question is: Who are the consumers?

Michael,

You wrote,

"The next question is: Who are the consumers?"

In other words, if I'm interpreting you correctly, who is economics for the benefit, the teachers or the students?

For the first hundred years after Smith, for the benefit of the students. For the next hundred, for the benefit of the "teachers." But the Internet is bringing economics back to its roots, a science of, by, and for the students.

My condemnation of the profession in general is not to deny that there are pockets of it, and this is certainly one, at which great work is done.

So, hats off to you rare professsional economists, the honorary amateurs.

My condemnation of the profession in general is not to deny that there are pockets of it, and this is certainly one, at which great work is done.

So, hats off to you rare professsional economists, the honorary amateurs.

Hmmm. Pete seems to want me to say something. Regarding the math issue, well, some math is more useful than others, and it depends on the question and situation. The big problem is when someone has a theoretical model that depends on some assumption that does not hold in reality, and then they go around asserting that reality resembles their model. There is a lot of this unfortunately.

I do have an observation that for many people their attitude towards the use of math depends on if they are familiar with the math or not. If they are not, then the math is to be condemned. Of course as with other things, the more comprehensible to a wider audience something can be, the better, which means that people should avoid making math-based arguments more esoteric than necessary, although there is also a lot of this done for showing off purposes.

Regarding what is "conventional," I note that Colander and Holt and I distinguish between "orthodox" and "mainstream." We say that the former is an intellectual category, while the latter is a sociological category that consists of dominating elites, who may not have a consistent intellectual standpoint. Someone orthodox will have some sort of consistent viewpoint, presumably that can be labeled "neoclassical" more or less, although that is a matter of some controversy.

"Heterodox" is both intellectual (anti-orthodox) and also non-mainstream in the sense of being alienated from the power centers of the profession. In any case, and I think this is what Pete is getting at and I guess does not like, we allow for there to be people who are "non-orthodox mainstream."

"Atlantis puts a tariff t on the importation of each unit of the consumption commodity p, the world market price which is s. If domestic consumption of p in Atlantis at the price s+t is a and domestic production of p is b, b being smaller than a, then the costs of the marginal dealer are s+t. The domestic plants are in a position to sell their total output at s+t. The tariff is effective and offers to domestic business the incentive to expand the production of p from b to a quantity slightly smaller than a. But if b is greater than a,...."

Mises, Human Action.

There is an error here right? Draw this as you would in an Intro Micro class. If the marginal seller has costs = s + t where is there any incentive to expand production from b?

Am I missing something?

Or does Mises mean b is domestic production at the world price s?


All trade whatsoever without exception only occurs because that which is received is valued more than that which is given away in exchange. In strict mathematical terms person 1 trades good A to person 2 for good B. Person 1 values good B greater than good A, and Person 2 values good A greater than good B. Therefore, B is greater than A and simultaneously A is greater than B. Mathematics destroyed. And this is the only reason why any and all trade occurs.

I like that Monximum says mathematics destroyed while invoking "greater"

A serious answer to my Mises query would be much appreciated. I am sure I am missing something (& thanks to DG Levsic in advance for not wasting my time, his time, your time, by replying)

The mathematics used by mathenomisists (the calculus) was developed to solve problems in physics. It assumes that smooth curves describe the movement of physical matter in space. As Mises points out in a passage in HA, this doesn't apply to human action. Class dismissed.

Rosser says, "...which means that people should avoid making math-based arguments more esoteric than necessary, although there is also a lot of this done for showing off purposes."

Which strikes me as a good insight. Why then do we see math as a positional good? Where did demonstration of mathematical sophistication replace something like speaking multiple languages, or being able to digest something as dynamic as the book "Capitalism, Socialism, and Democracy" or "Theory of the Leisure Class?" These seem to be equally good at selecting for a minority of particularly gifted individuals.

The only things that I can think of is the relative ease of graduate testing (GRE and it precursors) for Mathematical ability and the appeal of an indecipherable mathematical cant to government budget offices.

Life must be so easy when you are a know-nothing libertarian activist like Bill Stepp.

People here seem to be taking "mathematics" as one thing. However, it isn't really. It is grounded in logic, if examined sufficiently carefully it becomes clear that it is a branch of logic.

Despite what DG Lesvic says it is quite possible to talk about and use mathematical equations without quantifying their parts. I gave an example above where Mises avoids using an equation where one would be helpful. If an equation were given that would not in any way imply that the parameters of the equation could be quantitively evaluated. It doesn't even imply that in physical sciences, though it is often the case. "Rothbards Equation" which is given in Joe Salerno's paper on money prices is a good example.

Bill Stepp, the problems with using Calculus in economics are not the same as the problems in using mathematics. Calculus is only a part of mathematics.

Add Barkley Rosser to the list of mathematical economists who will not give us an example.

Monximus, Bill Stepp, and Michael: you said it all for me. Thank you.

Perplexed: and you always will be.

Grumpy, you wrote:

"Life must be so easy when you are a know-nothing libertarian activist like Bill Stepp."

Well, then, nothing is more than you know.


And, you notice, still not one example of mathematical economics. Not one!

So how long is this farce going to go on?

DG - please explain Mises's argument to me. You are the great authority it appears so please please enlighten me as to why there is no error in what he says.

Many thanks

Current writes:

the problems with using Calculus in economics are not the same as the problems in using mathematics. Calculus is only a part of mathematics.

Agreed with the second sentence (I took math beyond calculus in school), but calculus is the main math tool used in economics. My point about calculus and economics applies mutatis mutandis to other types of math beyond calculus.
As far as statistics goes, assuming somewhat heroically that it can be used to glean some empirical insights, this would be economic history (or maybe applied economics) but not economic theory proper, at least as Mises would have used that term.

Perplexed,

As I had said, I was a great admirer of Mises, but not when he wrote like that. I will not be able to understand what he was saying without a great deal of effort, but, since you have asked for it, you will get. I'll try, but am not guaranteeing anything.

Perplexed,

You have a right to be perplexed at anything written like that. That was Mises at his worst. After laboriously translating all that from logistics into plain English, where it should have remained in the first place, I don't see any error in his reasoning, as far as Dave related it to us.

And I do think, yes, he was referring to domestic production at the world price.

But I think the best thing to do with horrible writing like that is throw it right back at the author.

Mises was head and shoulders above anyone else in the last 100 years, but he was human too.

Bill Stepp,

You sure hit the nail right on the head.

"And I do think, yes, he was referring to domestic production at the world price."


So the marginal seller can't have costs equal to s + t.

If what you say is true, then the marginal seller would have costs equal to s.

This is my final post on this topic. The blog is very enjoyable

Just for kicks, here's my refutation of econometrics, which was published under the nom de guerre Anne T. Positivist in a libertarian student journal.

Prove the fallacy of econometrics.

1. Econometrics is mathematical history.
2. Mathematics is a priori knowledge.
.:. Econometrics is a priori history.
3. A priori history is false.
.:. Econometrics is a fallacy.

Perplexed,

Just as I said I would throw that kind of writing back to its author, even Mises, I'm throwing it back to you.

Speak English, and I'll be glad to respond.

Bill Stepp,

I'm afraid you've lost me. I have no problem with econometrics, in its place, just in place of economics.

When a businessman tracks his sales, his costs, and his returns, and calculates his profit, that's economic measurement, or econometrics.

It just isn't economics.

Underpinning a lot of talk about the role of maths and related issues in methodology is a very unhelpful change in the perception of science. The word used to just refer to an organised body of knowledge and to be scientific simply meant to be methodical or systematic in any field of endeavour from chemistry to cooking.

However Newton's triumph precipitated a change from science to Science which was the Truth, as revealed by the inductive method and complex sums. Then to handle the conjunction of Newton's Truth and Humes Skepticism, Kant came up with the idea of a priori knowledge that spawned the a priori/a positori debate, that was eventually killed by the conjectural turn introduced by Popper in the 1930s (the positivists may have finally got the point from Quine).

The recent upshot of the Newtonian turn, as noted by the Petes in 'Was Mises Right' was that people like Samuelson and Friedman argued that in order for economics to be a real science it would have to take a formal and mathematical turn. Consequently people became obsessed with the formal qualifications of disciplines, instead of focussing on the capacity of theories to solve problems, explain things, stand up to tests, help policy-makers etc etc.

Unfortunately Popper's thoughts on falsification were recruited into this round of the science wars, to no good effect. He may have made too much out of the idea (more research required).

On a helpful reading Popperism prompts two questions (a) are you taking evidence seriously? and (b) are you using evidence to test your theories?

The point of (a) is to minimise time spent debating true believers who don’t care about evidence. The point of (b) is to correct the positivist obsession with verification and meaning.

On this reading it is absurd to think that "falsificationism" represents anything like a full and complete account of Popperism and it is amazing that this view spread far and wide, usually with a collection of arguments to demonstrate that this so-called falsificationism (and hence everything else that Popper ever wrote) is inadequate and out of date.

Does this matter for Austrians? Maybe it does. In the same way that a lot of rubbishy reasoning about economic policy can be traced to introductory test books, so a lot of anti-Austrian philosophical presuppositions can be traced to text books in philosophy.

End of Rant for the Day:)

Champ,

Sounds interesting.

Wish I knew what you were talking about.

Of course I know roughly what you're talking about. I just don't get your point.

Are you for the apriori or ex posteriori approach?

There seems to be a lot of confusion here. As someone who does microeconomic cross-sectional econometrics, I get somewhat irritated every time Austrians imply that econometrics refers to models of supposedly representative aggregates. Econometrics need not imply macroeconomics.

And math does not equal calculus. There have been some recent attempts to model the economy using graph theory, which is much more compatible with Austrian theory (especially the stuff by Jason Potts).

The point is that math and statistics are ok, but only if the methods have the ultimate aim of illuminating real life, rather than being self-contained abstractions that only refer to itself and perhaps other models.

Something completely different: why is it so important to consider what is done by the majority of economists at Harvard, MIT etc. Even there, there are exceptions, and the exceptions sometimes become Nobel laureates, which shows that mainstream economics is not shorthand for mainstream science. It is as if some people think that a certain political party is the criterion of success, and one can only be really successful if being approved by the leaders of that party. What's wrong with creating a new party?

The point is that the philosophy of science became focussed on physics and, worse than that, it focussed on a particular conception of physics that does not do justice to physics. This is something that von Mises did not understand, at least in some moods he was prepared to accept that positivism was ok in the natural sciences. But it was not.

Realising this does not by itself advance the study of economics but it has implications for the people like Mark Blaug and others who have spent years writing about falsification, for the prima facie purpose of helping economists, inspired by the Lakatosian misreading of Popper.

David,

You wrote,

"There seems to be a lot of confusion here."

And especially after your posting.

Are you talking about economics or economic history?


Champ,

If that "Lakatosian misreading of Popper" still bothers you tomorrow morning, I'd take the good old-fashioned Ex Laxativian remedy for it.

DG,

You're implying that the Misesian view of a priori theory vs. economic history (everything else, including everything that's quantitative) is uncontroversial and generally accepted by people commenting here. It is not.

DG writes,

When a businessman tracks his sales, his costs, and his returns, and calculates his profit, that's economic measurement, or econometrics.

It just isn't economics.
--
You're right that it's not economics, but it isn't econometrics either. It's called accounting, which has nothing to do with econometrics. Accounting is an entirely different subject, which is usually taught in business school. Accounting is productive and quite useful unlike econometrics.

Might I suggest that elements of this thread are pushing the bounds of constructive discourse? Can we try to treat serious arguments with serious responses and treat silly arguments with either silence or serious responses, rather than ad hominems or even more silly responses?

Steve, the comments here make me sad, because this is something I would really like to see a good discussion about.

David EA, were your comments about aggregates in mathematical economics in reaction to my previous comment? I didn't mean to suggest that the use of math in economics must involve aggregates; however, it is very commonly the case. My comment was merely intended to express an general objection I have to how such aggregates are frequently interpreted.

D.G., I think you ought to consolidate your multiple comments, because they can be disruptive to the conversation.

Bill Stepp,

You're a great commenter, and are probably right, but I'm just not completely convinced yet.


David,

How does arguing against an opinion imply that there is no such opinion?


Prof. Horwitz,

Would you exclude the wit and riposte of a Churchill or Mencken? I fear that in trying to keep us within "the bounds of constructive discourse" you will just keep us deadly dull, and eliminate an essential part of "constructive discourse," the despatch of fools.


Lee Kelly,

You're absolutely right, and I am sorry about that, but sometimes, it just can't be helped.

Wish I were perfect, but, at the risk of shocking you, I'm not.

DG,

I find it interesting, and ironic, that you think I'm against dispensing with fools, given your history around here.

If I may add an incidental history of thought footnote. Mises actually thought very highly of Francis Edgeworth.

When Edgeworth's collected writings were published in (as I recall) in 1926, Mises wrote (in German) a very favorable, indeed, admiring review of the body of Edgeworth's contributions to economics. And, obviously, much of Edgeworth's economic analysis was presented in the, then, cutting edge mathematical economics.

So, clearly, what Mises was more concerned with was that the economic reasoning and persuasiveness and realism was not "lost" in the mathematical technique. And that the math did not straightjacket the proper pursuit of various economic problems in a preconceived mold of any of the methods developed in another discipline.

Richard Ebeling

Prof Horwitz,

Touche!


Prof. Ebeling,

I don't know what Mises wrote in German, but I could hardly imagine a greater misreading of what he wrote in English.

By the way, you had previously cited Mises as demanding a mastery of mathematics as a precondition for the study of economics. He also demanded a mastery of German, and by the same logic, of physics, chemistry, and biology.

And only then could economics be "the main and proper study of every citizen."

And only after all of that, I suppose, could he concern himself with making a living.

Well, I’m sorry, but even with only a nodding acquaintance with mathematics, German, physics, chemistry, and biology, and while still trying to make a living, I have some questions about economics for the both of you.

DG Lesvic,

Do you have an objection to the quantity equation MV=PT?

Sure it is a tautology, a definition of V. But, isn't it a useful tool for discussion?

Bill Stepp,

Calculus is only one part of maths, Statistics is only a part too.

Current,

Sorry, but I don't understand your equation.

Also, I believe Bill Stepp already responded to your challenge to him.

To make a long story short, are you ever going to present us with any of the actual numbers of your mathematical economics, rather than just symbolic allusions to them?

And let me remind you of what I said at the outset.

"I absolutely and positively guarantee you that the mathematical fakers will never, I repeat,never...present you with an example of "mathematical economics," for the simple reason that there is no such thing. All of their talk of mathematics in economics is hot air, and you can feel it, but you'll never see it.

Never!"

D.G.,

For some period, suppose that total spending is $5000 and the money supply is $1000.

Question: how do we calculate the average number of times that a unit of money has been spent?

Answer: 5000/1000 = 5

Let's call this money velocity. Now we can get the following equation:

money velocity * money supply = total spending

That is, the average number of times that a unit of money is spent multiplied by the money supply is equal to total spending.

Now, let's do another. For some period, suppose that total spending is $5000 and the total number of goods sold is 500.

Question: how do we calculate the average price of the goods that have been bought?

Answer: 5000/500 = 10

Let's call this the price level. Now we can get the following equation:

price level * goods bought = total spending

Let total spending be TS, money supply be MS, money velocity be MV, goods bought be GB, and the price level be PL. Then we get the following result:

TS = MS*MV = PL*GB

Using this equation, it is easy to see how changes in the supply of money, demand for money (velocity), and the supply of goods, all contribute to changes in the price level.

I am guessing this is basically what the quantity equation current mentioned is talking about. I just made up the above when trying to formalise my own intuitive thoughts, and I considered it a very enlightening exercise.

Lee,

You're wasting your time. You'd be better off banging your head against a wall. The odds of the wall changing into a pillow are slightly higher than convincing some people that their ideas are, in fact, in error.

DG,

You wanted an example of what? An interesting or alternative math econ model? I am a fan of the sort of agent-based computational models that Rob Axtell at Mason does, the last student of Herbert Simon. You can find a somewhat similar approach in a paper by me with Gallegati and Palestrini that is linked to in the Wikipedia entry for "period of financial distress."

I found this as one of the most interesting topics in economics; the use (or mis-use) of mathematics in economics.

I agree that the concerns are not so much on mathematics per se but on a wrong use of it. Certainly mathematics is one of the most important tools we have. There are even places in economics where math is important and almost an unavoidable tool, for example econometrics and finance (very related, if not part of, economics).

I think, however, that it is not the best tool to deal with the problems economic theory has to deal with.

One of the arguments I heard most is that mathematics is just another language, and that therefore there is no reason to not make use of it. I see mathematics more as a symbology than as a language. If it were the latter, then we should be able to read any "model" or "mathematical" expression without the author having to explain what he means by each mathematical symbol. The symbol "x", for example, it's empty of meaning until attached to a concept or idea, for example "apple". It is true that the the symbol "apple" is also attached to a specific concept, but the mathematical symbols need to recur to another symbol to be meaningful. They do not have meaning by themselves (except, maybe, specific cases of mathematics).

I think this is important, because if that is the case, then mathematical symbology cannot be more precise and unambiguous than language (a point raised by mathematician Karl Menger). That is, at most, mathematics can be as clear and precise as language. Just an example to illustrate, and not to be a general case, in what sense "Two plus two equals four" is less scientific or precise or clear than "2 + 2 = 4" if "2" is attached to "two", "+" to plus and "=" to "equal"? Which symbols give meaning to which ones?

In other words, anything said in mathematics can be translated to language, but the other way around does not hold. Not everything said in language can be translated to mathematics.

This does not imply, of course, numerous cases where mathematical symbology is more convenient than plain discursive logic; like solving a quadratic equation (historically first done by discursive methods rather than mother symbology). Mathematics is a tool, but to choose mathematical symbols, discursive writing, or even pictorial representations may affect the length of a demonstration, but this choice does not affect the fact that it constitutes a demonstration.

To write a scientif text is not the same as to write a love or mystery book. The scientific writer has to be as precise with his writing as the mathematician has to be with his mathematics. And the reader has to be as careful when reading a discursive scientific text as he is when studying a mathematical problem. A bad use of language is unambiguous just as careless use of mathematics is, but what should be compared is a proper use of both methods. But if math requires to recur to language for their symbols to have meaning, language, as a more rich and flexible tool may be more precise and detalied when studying complex phenomenon than a more rigid and structured tool as math.

Another aspect, maybe related to the previous one, is that mathematics per se, at least as commonly used in economics, does not imply causation, but proportional relations. The expression y=f(x) does not have causality in it. Causality has to be added by the individual before writing y=f(x). If he is wrong and writes x=f(y) he won't notice just because he wrote it in mathematical symbology. If we have two variables, x and y, we cannot solve the question of who causes who just by math. That question, the causality problem, has to be solved outside mathematics. Mathematics, then, does not help to avoid the problem of swaping cause and effect and writing UM = f(p) [Marginal Utility as function of price]. The use of mathematics does not guarantee we will avoid this kind of important mistakes. This case might be too obvious, but another example could be price of equilibrium and demand and supply curves. The graph (as the equation behind) does not answer the question of who determines who: a) The curves determine the price of equilibrium or 2) the price of equilibium determines where the curves should be? So, while math may be useful in many aspects, it does not help to answer the causality question and it's not a mechanism that avoids pitfalls in the causality relation (does consumption drive income or is the other way around?; no mathematical solution).

The consistency problem is much more deep. I think Godel's resumes the problem quite well: mathematical economic cannot prove its own consistency. In what sense, then, is more consistent than discursive logic? (In Studies, or New Studies, Hayek takes this problem to human mind as well).

Mathematical economics, by the set of assumptions it usually uses, is like using hiperbolic geometry to deal with an euclidian problem. Regardless it consistency or inconsistency, it is dealing with a different problem even if their theorems are correct. In some way, Mises was the Euclides of economics when he put forward his praxeology approach: The set of axioms economics uses are these... (Of course, he was following previous authors like N. Senior).

Neither logic, nor mathematics, answers the question if the statement is true or false, just if the reasoning is right. If I say a) clouds uses glasses, b) I'm a cloud; ergo c) I use glasses is true for the wrong reasons. But I may also say a) clouds do not use glasses, b) I'm a cloud, ergo c) I don't use glasses, which is logically correct but false. Math, by itself, cannot avoid this problems (true conclusion for wrong reasons and wrong conclusions with consistent reasoning).

Science, and economic science, happens outside mathematics, and the latter is a tool that sometimes is usefull to express certain relations. But causality and scientific explanation (and even demonstrations) does not require mathematics to be scientific, rigorous, or unambiguous.

I apologize for the long post. A difficult topic to deal briefly and correctly (or precise) in a blog. But I find this topic very interesting and I'm looking forward for further comments and thoughts on this!

Bests,
NC

Kelly,

“…it is the great quality of verbal propositions that each one is meaningful. On the other hand, algebraic and logical symbols, as used in logistics, are not in themselves meaningful…to develop economics verbally, then to translate into logistic symbols, and finally…back into English, makes no sense…” Murray Rothbard

First you gave us logistics:

"TS = MS*MV = PL*GB"

Then English:

"Using this equation, it is easy to see how changes in the supply of money, demand for money (velocity), and the supply of goods, all contribute to changes in the price level."

So, you couldn't understand the English without the logistics, and I couldn't understand the logistics without the English.

If you can't understand plain English, and I can't understand logistics, how are we to have this discussion?


Barklely Rosser,

You wrote,

"You wanted an example of what? An interesting or alternative math econ model?"

Interesting, not interesting, alternative, not alternative, just give me an example of a law of economics that was arrived at through a mathematical operation and could only have been arrived at through a mathematical operation.

And no links, please. I've gone on too many of those wild goose chases. So if you have something in OuterCyberspace, you go chase it down and bring it to us.


Nikolas,

I think you're more or less on the right path.


Kelly,

You wrote,

"For some period, suppose that total spending is $5000 and the money supply is $1000.

Question: how do we calculate the average number of times that a unit of money has been spent?

Answer: 5000/1000 = 5"

But that is not a law of economics, for you could just as easily have supposed that total spending was $6000, and the money supply $1000, in which case the answer would have been 6.

You're confusing the illustration of the principle with the principle itself. It is as though you illustrated the principle that he who takes what isn't his'n must sooner or later go to prison with the example of Smith robbing Jones and Smith going to jail for it, and then actually throwing Smith into jail, or, turning it around, throwing Jones into jail.

The answer is not necessarily 5 and the culprit is not necessarily either Smith or Jones. They're just hypothetical illustrations of the principle, and not the principle itself.

I'm still waiting for an example of a law of economics that is not just illustrated by numbers but actually consists of the numbers, of numbers that are not just illustrations of it but the law itself, of constant relations in economics not just between cause and effect but between the magnitudes of the causes and effects.

DG: "I'm still waiting for an example of a law of economics that is not just illustrated by numbers but actually consists of the numbers, of numbers that are not just illustrations of it but the law itself, of constant relations in economics not just between cause and effect but between the magnitudes of the causes and effects."

So far I'm the only one who's DEMONSTRATED the inverse of that. Economics is about PREFERENCES, ordinal rankings. Trade does not occur because of indifference, because of equations of value, but because of INEQUALITIES of value. Trade occurs because that which is received is valued strictly MORE than that which is given away in exchange, including trade of goods for "money". The Austrians have been stuck for 50 years simply because they have failed to grasp that all trade, including trade for "money", is strict barter. Money is always a good, an independently subjectively valued good just like apples and oranges.

Trade occurs because different people value the same things DIFFERENTLY, one values good A greater than good B, and simultaneously, repeat SIMULTANEOUSLY, the other values good B greater than good A. Mathematically A is greater than B, and simultaneously B is greater than A. This is the Law of Trade, and it explains precisely in Newtonian Physics fashion how goods move from person and place to person and place, VOLUNTARILY. Supply, demand, markets, trade, economics only applies to VOLUNTARY exchange. After you grasp this you can rip to shreds the entirety of Keynesian economics in a single paragraph.

Mises came really close to grasping this but came up short, remaining confused about "trade cycles" and "balance of payments". I'm still trying to track down a Mises quote along the lines of paraphrasing "People are either voluntarily trading with one another or they are not. There is no third way possibility."

Monximus,

You started out gangbusters and then messed it up.

"Economics is about PREFERENCES, ordinal rankings."

Exactly!

"all trade, including trade for "money", is strict barter."

No. Money is the medium, not the object of trade. Barter is direct trade between the objects of it. Money trade is indirect trade between them, trade through the medium of exchange.

"Trade occurs because different people value the same things DIFFERENTLY."

Excactly!

"Mathematically A is greater than B..."

No. Logically it is.

Steve,

I am a selfish commenter, I write as much to see what my thoughts look like on paper (or screen) as I do to inform others. Sometimes you really don't know what you're talking about until you try to explain it to someone else. But, in any case, in these kind of conversations you should remember that others are (usually) reading without comment. You might be frustrated that some commneter, such as Nikolaj, will never be swayed by your (or others') arguments, but I have learned a lot from reading them.

D.G.,

Math is a formal language. I suggest you go and learn something about that, and you might just learn some interesting things about logic, too.

DG,

I've been asked offlist not to feed you as you are a publicly identified troll, and so this will be it, and it will not satisfy you, but if that is the case, you are just going to have live with it.

Sources? Pretty much any grad micro theory textbook. If you do not have access to one, you are just going to have to live with it. If your math skills are insufficient to deal with a garden variety grad micro theory textbook, you will just have to live with it.

So, proof that people rationally trying to maximize gains for themselves and accurately forecasting the behavior of others who are doing so also, will not lead to an outcome that is efficient in the sense that one cannot make somebody better off without making someone else worse off. If you actually know any economics, what I am referring to is the proof that the Nash equilibrium in a prisoner's dilemma game is not Pareto optimal.

There is more, but there are no numbers, just theorems about preferences and such. And, no, I am not going to waste other peoples' time here with further elementary expositions of such standard stuff. If you or anybody else does not like this or think it is unimportant or whatever, you are just going to have to live with it.

(Oh, and overcoming the prisoner's dilemma problem just happens to lie at the core of what Elinor Ostrom's work is all about, for those of you who are not quite on top of this.)

Lee and Barkley,

While I appreciate your friendly advice, it is no substitute for what I have been asking for, and, at the outset, predicted I would never get.

Just one wee little example of what you're talking about, mathematical economics.

DG: "No. Money is the medium, not the object of trade. Barter is direct trade between the objects of it. Money trade is indirect trade between them, trade through the medium of exchange."

Money is indeed the object of trade. Money is a good that is independently subjectively valued. If you disagree, feel free to toss $100 bills on the ground and see if anybody picks them up. :P When you "buy" a cheeseburger for $1, you value the cheeseburger greater than $1 and simultaneously the cheeseburger seller values the $1 greater than the cheeseburger. Thus, money is irrefutably an economic good just like apples and oranges with its defining characteristic being that money is merely the most commonly exchanged economic good.

Unfortunately even the Austrians in the 20th century got confused by money. This confusion over money is the only reason why the economics sub discipline of "macro" economics exist. In reality, all of "macro" economics is non existing gobbledygook nonsense.

Why this confusion? Because money evolves slowly. Even fiat paper currencies don't collapse often. But money and money substitutes are constantly almost imperceptibly evolving at the margins. You might see this as money being just gold, to money being gold, silver, copper, nickel, platinum to money subsequently being just gold and silver. But money is a good. The whole "medium of exchange" nonsense is where equations of value were originally inserted into economics methodology. Sure, money might be a "medium of exchange" but so also is all the surplus production of a specialized division of labor society a "medium of exchange" to acquiring other goods.

Since money is a good, discovering more "real" money is always net economically beneficial to society in the exact same way that discovering more oil deposits is beneficial to society, in the exact same way that doubling agricultural production is beneficial to society. Government fiat paper currency is only artificially scarce and would not be market evolving money in the absence of involuntary force. Doubling the supply of real scarce commodity money means more people can save more. The number one ("unwritten"?) macro economics fallacy is that all the money equals all the value of all the other goods (or some multiple). That's as absurd as saying all the particular quantity of one good such as oranges equals the value of all the other non oranges goods and "money". If too much of a single good of "money" is discovered or produced then the market will adjust, will evolve it's units of "money", such as gold and silver being money going to just gold being money.

DG: "No. Logically it is."

Logically AND mathematically. Greater than and lesser than are mathematical concepts. The great surprise is just that in economics trade the value of A is greater than the value of B and simultaneously the value of B is greater than the value of A, which is contrary to fundamental mathematical axioms, but irrefutably axiomatic in economics trade. It also would not surprise me if this insight is the key to the next great advances in physics in the 21st century w.r.t. relative positions movement, but that's just "out there" speculation.

DG,

Can you not read? I just gave you one, you whiner. And it is a big one, not a "wee one." If you cannot figure it out, then that is your problem. I shall not comment on your level of intelligence.

Barkely wrote,

"I shall not comment on your level of intelligence."

I've noticed.

Barkley,

You haven't follow DG all the way down his particular rabbit hole. He requires "a law of economics that was arrived at through a mathematical operation and could only have been arrived at through a mathematical operation." This is not what everybody-else-in-the-whole-wide-world calls "mathematical economics," but it is his standard. Never mind that he didn't articulate it until long after laying down his imperious demand. Given that we are down here in DG's rabbit hole, it's kind of hard to tell whether there is anything that will satisfy his criterion. Presumably, any "law" will have to be universal. But then its got to admit of no possible exceptions, I think. Thus, it has to be "a priori" in some sense. So he wants an apriori truth that cannot be contradicted by experience. A tautology, you ask? Well down here in DG's rabbit hole, I think we still like "synthetic apriori" statements. So to satisfy our scholarly friend, we need to come out with a mathematical operation that yeilds a synthetic apriori statement that refers to human action. It's a tall order. He has a side condition that it should somehow involve "actual numbers," which makes it an even more difficult order to fill. I can't help believing, however, that even if we could somehow do all that, he would invent some new side condition that outlawed our example. DG is speaking a private language.

A famous economic scientist and Nobel (or whatever the prize is called) laureate criticized the Austrians many years ago for their a priori proclivities. Fast forward to 1965 and he writes a paper adducing "Proof that Perfectly Anticipated Stock Prices Fluctuate Randomly." The paper later appeared (I think) in his _Collected Scientific Papers_.

It would have been news to the presumably unscientific traders on Wall Street, as well as to other interested observers of stock prices, (and of commodity prices, in the case of Benoit Mandelbrot, 15 years or so after this scientific proof was presented). Never mind that the scientist offering up the proof didn't use any actual scientific methods, at least none that would have passed muster with physicists and scientists in other fields.
In the grubby world of the hard sciences, including the allied medical sciences, scientists often observe physical phenomena they don't understand well and can't explain with the science at hand. Through experimentation, trial and error, and inspired theorizing, followed by more tests, experiments, etc., they occasionally solve old problems and pioneer new scientific vistas. (The New York Times had an interesting piece the other day about how some medical scientists recently solved a problem this way.)

Benoit Mandelbrot studied commodity prices, and learned that they didn't fluctuate randomly and that their distribution didn't follow a normal distribution. (The rubes on Wall Street already knew this.) He used fractals to explain this in his book _The (Mis)behavior of Markets_. His finding also applies to stock prices, which according to the famous scientist (and to the EMT theoriests who came along later), would follow a bell-shaped curve.

The moral of the story is that whenever someone claims to be a scientist and in the same breath presents a scientific proof aiming to show some empirical result without any empirical evidence, be skeptical and ask for the evidence backing up his proof. If he can't, have a good horselaugh and retire to the local pub to throw darts at his outlined countenance.

Ask not what your scientific method can do for you. But what you can do for your scientific method!

Bill Stepp,

I've liked your previous postings here a great deal, but this one has left me in the dust.

Economic's next top model ends in riot.

http://www.youtube.com/watch?v=Ieeh9GUGN6A

Kelly,

Let me try to express this a little better.

The idea of mathematical economics is not just that mathematics is a better linguistic but better analytical tool for economics, not just a better way of explaining things but of arriving at the truth, that there are truths of economics that are beyond logic and could not be arrived at without mathematics.

And that is what I am asking for an example of.

Really there is no firm difference between logic and mathematics. Mathematics is constructed from logic. It can be entirely described in English language, it's just that doing so is very inconvenient and makes progress difficult.

The important bit of mathematics that we want (in my opinion) is the ability to state relationships more simply and briefly than we can in English.

Current,

They why are you telling me this in English, why not in mathematics?

DG Lesvic: "They why are you telling me this in English, why not in mathematics?"

Because in this case it's easier to state in English. That isn't always the situation.

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