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Mathematics

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solon posted on Tue, Mar 24 2009 4:11 PM

Maybe some Austrians might be interested in a "fresh start" from an empirical point of view with Alfred Marshal as the basis (the inventor of supply and demand). He insisted that logic and mathematics are inseparable (and also saw mathematics as potentially redundant but useful for thinking and prediction).

 "(1) Use mathematics as shorthand language, rather than as an engine of inquiry. (2) Keep to them till 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 I do often."

Empiricism rules since Newton, so some sort of model for human action may be necessary for the Austrian school to ever become mainstream.

 

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solon:

Maybe some Austrians might be interested in a "fresh start" from an empirical point of view with Alfred Marshal as the basis (the inventor of supply and demand).

Adam Smith, David Ricardo, and Karl Marx made use of the term "supply and demand" years before Marshall was even born. The only thing that Marshall did was draw two lines and set an abritrary "equilibrium point" which does not exist in real markets.

He insisted that logic and mathematics are inseparable (and also saw mathematics as potentially redundant but useful for thinking and prediction).

Mathematics is a subsection of logic. How verbal logic is inseperable from mathematics when it comes to subjects which require no advanced mathematics is beyond me.

Empiricism rules since Newton, so some sort of model for human action may be necessary for the Austrian school to ever become mainstream.

Empiricism the way you describe it is a self defeating proposition in social sciences. The idea that you can take past experiences that are not controlled for various factors (it's impossible to control "experiments" in social sciences), form them into theories, and then apply those theories to future events is just plainly absurd.

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solon replied on Wed, Mar 25 2009 1:28 AM

Since Austrians have identified money and capital to be primary factors, perpahs such a computational model could focus on their behaviour (maybe would also make it less likely that it could be used for socialist purposes). The problem with praxeology as I see it is that it's not formalized enough to have general credibility (and Austrians also disagree among themselves on some conclusions, as this story illustrates http://mises.org/story/2936).

 

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Answered (Not Verified) Sinewave replied on Wed, Mar 25 2009 10:27 AM
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krazy kaju:

ecoli:
what's so wrong with describing axioms as a set of equations?

It's impossible to do so. You might prefer to and to c, but you might prefer c to a. Mathematics cannot describe this. Likewise, you might prefer one apple to a banana, but you might, at the same time, prefer a dozen bananas to a dozen apples. Then again, you don't think "oh, well the marginal utility of an apple is [insert ridiculous equation here] so I'll pick the bananas." There is no way to use mathematics to describe these relationships.

@krazy Usually when some one does not know what they are talking about, they admit to that. They don't proceed to then state the exact opposite of the truth.

 

When a > b and b > c implies that a > c, we say that the relationship ">" is transitive. It has already been documented by mathematicians that the "utility function" is intransitive. Luckily, there already existed, even before this discovery, of such things as intransitive logics and intransitive relationships in mathematics. You may have already seen some of this in your first year of college. Although I guess you didn't, obviously.

 

You could get a detailed explanation on any book in category theory or logic. You can also get a somewhat more cursory mention in popular staples of the mathematics literature as Topics in Algebra by Hernstein, or Principles of Mathematical Analysis by Rudin.

 

Actually, explaining how the utility function (among others) can arise is the subject of many papers, usually coming to the same conclusion in different ways. Here is an example: "Gambling in a Malthusian Universe: A Game-Theoretic Approach to the Paradoxes of Expected Utility" by Gregory B. Pollock and Keith A. Lewis, published in Rationality and Society, Vol. 5 No. 1, January 1993, pp. 85-106.

 

Basically mathematics is a language to describe things, and if you can't model it then you can't falsify it. I wouldn't expect you to know about these particular references and the particular research that I mention here if you hadn't sought them, but then again I wouldn't have expected you to act like you did.

 

 

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Sinewave replied on Wed, Mar 25 2009 10:46 AM

ladyattis:
Mathematics can only model that which has some quantity or invariant relationship.

I'm glad you mention this! I should mention that invariant relationship is a flexible concept, but I would like to quote what you said first:

ladyattis:
But if you're attempting to model the price of a particular product or service or model what the next technological boom will be, you're out of luck as these things depend on information that is particular to many individuals who often are not in direct communication with each other (due to time, space, interests, and etc). As such, mathematics for modeling their actions is logically impossible. 

While I agree with what you say here in one sense, technically (and for more interesting reasons, to me at least) I have to disagree. So I agree that a model will not tell you exactly who what when where why or how the next "big thing" will be. But a lot of mathematics is not about giving exact numbers, but about detailing the specifics of the whole situation. Let me make this more concrete with what I have in mind:

In general, there is a plague with people who use mathematical models. In essence a lot of people are not using them correctly. A brief summary of the most common pitfall: A lot of people will assume that their mathematical model is correct, and assume that all errors in the model are normally distributed. This means that they are due to outside influences and will cancel themselves out in the long run.

In mathematics we say that our errors are "gaussian" and we put a gaussian probability distribution in to our equations that reflects these errors.

The problem with this, of course, is as you mentioned the next "big thing." There are other probability distributions that deal with this, for example the "power law distribution". These can be used in a model, but a lot of people don't like them because it incorporates the idea in to your model that there are big changes that can come and you don't know when. There is a problem in the business world that if you tell you boss you don't know what's going on, he or she may not think highly of that. This is the subject of a book fairly recently published called "The Black Swan" by Nassim Taleb. He discusses events that are really rare. But what's special about a "Black Swan" event is that if one of these events occurs once, it changes the game forever. For example one can think of the advent of computers, writing, agriculture, as these sorts of unpredicted black swans that have had unpredictable effects on the economy.

I would also like to say that I liked your post.

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solon replied on Wed, Mar 25 2009 12:44 PM

I don't necessarily believe in pure mathematical modeling myself. Keynes himself wrote

"It is a great fault of symbolic pseudo-mathematical methods of formalising
a system of economic analysis, such as we shall set down in section VI of
this chapter, that they expressly assume strict independence between the
factors involved and lose all their cogency and authority if this
hypothesis is disallowed; whereas, in ordinary discourse, where we are not
blindly manipulating but know all the time what we are doing and what the
words mean, we can keep 'at the back of our heads' the necessary reserves
and qualifications and the adjustments which we shall have to make later
on, in a way in which we cannot keep complicated partial differentials 'at
the back' of several pages of algebra which assume that they all vanish.
Too large a proportion of recent 'mathematical' economics are mere
concoctions, as imprecise as the initial assumptions they rest on, which
allow the author to lose sight of the complexities and interdependencies
of the real world in a maze of pretentious and unhelpful symbols."

But I do believe Austrian economics needs to be more formalized in order to become more mainstream, as I don't believe in a general mainstream conspiracy but mostly errors in thinking (no doubt there are also politicized economists, but from all sides of the spectrum it seems to me). The general equilibrium theory for example leaves out money altogether as far as I know although corrections have been attempted. A computational model I suppose could even mix boolean or fuzzy logic and mathematics. This way you can perhaps get some falsifiability that would be generally acceptable.

 

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what do you fellows actually want to model? can you give an example of a scenario that you feel there is value in 'mathematically modelling'?

Where there is no property there is no justice; a proposition as certain as any demonstration in Euclid

Fools! not to see that what they madly desire would be a calamity to them as no hands but their own could bring

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Then just go ahead and do it? I've heard this stated several times, that Austrian econ needs to be cast in formal terms. That's fine. There's nothing wrong with it given that it's mere formalisation. I'm not sure if it'd yield any positive returns, but it'd make Austrian econ more elegant if it succeeded. But either you or someone else interested in doing so should actually carry the task out then. There's others on here who've expressed an interest in so doing, so get in touch with them and get on to it.

Freedom of markets is positively correlated with the degree of evolution in any society...

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krazy kaju:
You might prefer to and to c, but you might prefer c to a.

According to basic Austrian economics this is impossible.

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according to austrian economics its not demonstratable

Where there is no property there is no justice; a proposition as certain as any demonstration in Euclid

Fools! not to see that what they madly desire would be a calamity to them as no hands but their own could bring

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ecoli replied on Wed, Mar 25 2009 2:49 PM

I'm doing just this.

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ecoli replied on Wed, Mar 25 2009 2:51 PM

scineram:

krazy kaju:
You might prefer to and to c, but you might prefer c to a.

According to basic Austrian economics this is impossible.

Huh?  The first chapter of theory of Money and Credit is all about this kind of thing (subjectivity of marginal utility). 

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ecoli:
scineram:

krazy kaju:
You might prefer to and to c, but you might prefer c to a.

According to basic Austrian economics this is impossible.

 Huh?  The first chapter of theory of Money and Credit is all about this kind of thing (subjectivity of marginal utility). 

Suppose an agent's options are restricted to a, b, and c, with a>b>c, '>' being a preference relation for the agent.

Suppose further that c>a.  Without loss of generality, let the agent choose b, meaning that he prefers b to the other two (which is how action is defined).  This contradicts the premise that a>b, since preference is anti-symmetric.

Like scineram said, this is basic stuff.

Diminishing Marginal Utility - IT'S THE LAW!

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ecoli replied on Wed, Mar 25 2009 6:05 PM

I don't see what the problem is.  Besides for the fact that transitive modeling is possible, the point of modeling isn't to describe an individuals preferences but the outcome of many people with different preferences.

Arrow's impossibility theorem and the Pareto efficiency has some of this.

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Solomon:

Suppose an agent's options are restricted to a, b, and c, with a>b>c, '>' being a preference relation for the agent.

Suppose further that c>a.  Without loss of generality, let the agent choose b, meaning that he prefers b to the other two (which is how action is defined).  This contradicts the premise that a>b, since preference is anti-symmetric.

Like scineram said, this is basic stuff.

The problem is that preferences in action are based on value, and value is subjective.

I'm hungry > I'm thirsty > I'm tired

Sure I'll eat before I'll drink. But, even if I'm really really really thirsty, I may decide to take a nap and get a drink later.    

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this just shows that at time 1, you demonstrated a preferrence for  eating over any other course of action(which may have included drinking and sleeping options..)

at time 2, you demonstrated a preference to  sleeping over any other course (i.e. drinking, or eating, etc)

and so on...

 

Where there is no property there is no justice; a proposition as certain as any demonstration in Euclid

Fools! not to see that what they madly desire would be a calamity to them as no hands but their own could bring

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ecoli replied on Wed, Mar 25 2009 6:40 PM

And how is that proof that human action can't be modeled?

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