No, it does not.

“Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind, although practical applications for what began as pure mathematics are often discovered later.”

]]>” First you start out assuming that some axioms are true. ”

Please… I am not disagreeing with you that methodologically, Mathematics is a set of abstract ideas. However, my point is that epistemologically, Mathematics evolves from our attempts to understand the world around us. I am just saying that the axioms of Mathematics often lie in reality and that we try to grasp that reality through our conceptual faculty. For instance, take the simple concept of the Axiom of Identity.

1 = 1, not because I randomly choose it to be but because in the real world, 1 apple remains 1 apple (unless of course something happens/is done to it).

There is also the fundamental point ‘What is 1?’. It is a concept of quantity evolved because man can see that one of the differences between objects in his surroundings is in terms of quantity.

I am not knowledgeable in any of the areas of Mathematics you have identified, but would wager that the source of the validity of the axioms would become visible in the linkages these areas have to other fundamental areas of Maths or in the application of these areas to reality.

That apart, let me also attempt to bring this discussion back to the original point. I was just saying that rational people acting in their long-range self-interest will clearly see securing easements on their own property is the best way to advance their own Liberty without violating that of others. That’s what, in my opinion, makes the article fundamentally incorrect. It assumes a scenario in which none of the land owners on the hypothetical planet is rational enough to see what is obvious even to me.

]]>” I am saying logical reasoning is impossible with the language of mathematics in the field of economics. ”

I agree with you completely on this, but see no need to revisit my concept of measurement. I was just trying to say why I think you are right. I was just identifying the point that human action is not amenable to mathematical measurement because of the inherent subjective nature of value and the fact that it all starts with “choice” which does not have to and probably will never be the same for all people.

Hope that clarifies.

]]>As for this:

“But is there any real reason for me to explain the generalisations of addition and multiplication to illustrate my point that mathematics is NOT the “science of measurement”?”

Try P-adic numbers, FiniteGroups, Coding Theory, Sudoku, Rubic-cube, Graph Theory, Ring Theory, Fractional Ideal, Manifolds, Quaternions, Set Theory, infintiy, …

What those things have to do with “measurement” remains a big enigma to me.

First you start out assuming that some axioms are true. Then by using logic you can arive to certain conclusions and statements. Some statement is mathematically correct because you have a proof for it, not because you can “measure” it. You know that your conclusion must be “true” if the fact is given that your assumptions were right.

That’s why Newton was not “really right” with his law of gravity. His assumptions concerning reality proved to be false…

]]>Is it just me, or is this the first time Austrian Economics has been accused of being too mathematical??

As for mathematics, I may not have gotten beyond calculus, but I have developed a theory about math–maybe you more knowledgeable mathematicians would like to critique it: essentially, mathematics is a system of logic. Thus, a common statement like 2 + 2 = 4 is an analytical statement: it is true because it has been defined so, as the numbers and the equation have been derived.

However, like any logical system, its application in reality is very much dependent upon the premises used as mathematical input. We can say that 2 apples + 2 apples equals 4 apples, and nobody will seriously disagree. But if we try to add 2 apples to 2 cans of applesauce, we’re not really going to be sure what that equals, because we have not defined or found the relationship between apples and applesauce, and don’t have a common unit to work with.

Thus, a logical system like math can and is indeed be very valuable in the real world, as long as it is applied properly. And this is the essence of the Austrian critique of mainstream economics’ use of mathematics–it is often based upon a fallacious premise, like Keynesian consumerism, or it’s mixing apples and applesauce, and not really telling us anything useful. ]]>

Mathematics is not about physics

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I never said it was.

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(By the way, algebra is not about “x + y = z or cosÂ² x + sinÂ² x = 1″).

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Algebra is the study of the rules of operations. Both examples you mentioned are specific examples of algebra.

But is there any real reason for me to explain the generalisations of addition and multiplication to illustrate my point that mathematics is NOT the “science of measurement”?

I find it is better to stick to simple examples with simple language if I have a point of view I wish to communicate. After all this is a website about Austrian economics, not complex mathematics.

]]>One should not look to mathematics when looking for scientific mistakes, but rather in ourselves, mankind, which has all to often made terrible mistakes and has time and time again tried to misused mathematics to cover those mistakes up.

]]>Yes Yannick you’re right. This statement is very poorly worded. I see and completely agree with your objection to it.

I not sure how to state what I actually meant by that statement. “known quantities” is certainly not my intended meaning.

Do you understand my intended meaning of that statement?

I wasn’t really prepared when I entered this discussion and typed my post in haste.

My apologies.

I’ll have a longer think about it and reword that statement in a more correct way (but later … gotta sleep now).

P.S. Yannick, maybe you should tone down the arrogance a bit … I know a lot more about mathematics than you think.

P.P.S. While typing this post I just saw your latest post. I’m not entirely sure what you mean by it. I am clearly arguing that Mathematics is NOT the science of measurement … but it looks like you’re criticising me.

]]>Mathematics is not about physics.

(By the way, algebra is not about “x + y = z or cosÂ² x + sinÂ² x = 1″).

If you wan’t to get an impression of what it’s possibilities are… check it out. You can wander around for the rest of you’re entire life, not having to see even one subject twice.

]]>It is not because of the fact that monkeys can ride bicycles, that only monkeys can ride bicycles.

It does not take a genius to figure that out I think.

]]>“Unfortunately, one can only reason in the language of mathematics using known quantities.”

This is simply not true. Logic(and not only logic) is not about quantities…it really is about playing abstract “entities”. In mathematics, I really have a professional edge on you guys, but I even so, I don’t claim, that such and such is the absolute truth about mathematics (for instance formalism). I do not rule out the possibility that I even might be wrong.

Stop raping the science of mathematics all the time… it is a wonderfull tool, like a kind of Swiss Knife, you use the specific substool of mathemtics which helps you the best in the specific task you wish to accomplish.

Mathematics has an enormous repertoire of different tools to offer us. It is not because you or Bala know only very little about mathematics(and believe me you both don’t know not even one tiny little fraction of the different kinds of tools en tecnics there are out there), that this implies that there is nothing more then your eye can see.

You guys are simply not qualified to judge on this specific matter.

Hint: Only make absolute statements about those things you really know well.

]]>When Newton developed the concept of a force, he defined F=ma (Force equals mass multiplied by acceleration) and he defined acceleration as the rate of change of velocity, ie a = v/t (acceleration = velocity divided by time).

A constant force F applied over a period of time t is Ft (Force multiplied by time). If you now use logical deduction to see what happens you get the following:

F=ma

F=mv/t

Ft=mvt/t

Ft=mv

Ft/m=v

Therefore, assuming the mass is initially at rest, one can conclude that if you apply a constant force F for a time period t to a fixed mass m, the mass will be travelling at velocity v.

I am not “measuring” the velocity v. I am logically deducing it. If the fundamental assumptions are true then the mathematical conclusion must also be true. It does not matter what units of measurements I use as long as the same units are used consistently.

I could use units of time called qwerts and units of mass called yoips and units of distance called hilks. Then as long as I measure velocity as hilks per qwerts and Force as yoip hilks per qwerts squared, then the equation will always work regardless of the amounts assigned to qwerts, yoips and hilks.

What was Newton measuring here? He wasn’t measuring anything. He was simply using the language of algebra to arrive at logical conclusions to his fundamental assumptions.

(Note: I greatly simplified the example above for clarity. I felt it wasn’t necessary to go into calculus to illustrate my point)

]]>I am saying logical reasoning is impossible with the language of mathematics in the field of economics. There is a big difference between measurement and reasoning.

I can measure things using English too… that table is very long compared to that other table, my finger nails are very short compared to most people’s finger nails, my computer is quite slow compared to my friend’s computer… but as you can see it’s quite a poor language to use for that purpose.

To be accurate, measurement requires mathematics. But that does not mean that mathematics is exclusively the science of measurement.

Mathematics is used every day in a non-measurement way.

If I have $30 and I spent $18 on a meal then I will have ($30 – $18) = $12 left to spend on my lunch tomorrow.

I am not “measuring” anything here. I am using the language of mathematics to logically deduce if buying the meal I want now will allow me to also purchase the meal I want tomorrow.

This is not measurement. This is reasoning.

” Unfortunately, one can only reason in the language of mathematics using known quantities. Hence mathematics can not be used for the logical reasoning involved in Austrian Economics because we cannot quantify the unknown constantly changing subjective values of individuals in society. ”

What you are saying is essentially that measurement becomes impossible or perpetually incorrect in Economics because one cannot fix a standard of measurement that can be applied to all objects being measured, i.e., acting humans.

What you said reminded me of the movie Matrix where the human being making choices is described as a singularity by the Architect of the Matrix who believes in the omnipotence of mathematical modelling.

]]>” And 1 meter, 1 yard, what kind of measurement do you have in that department. ”

Ask yourself the question “1 meter or 1 yard. Of what?”

” How “much” is 1 meter? ”

Once again, ask yourself the question “How much is 1 foot?” and the answer will be obvious.

The point is that you are confusing the methodological and epitemological aspects of Mathematics. Methodologically, Mathematics is an abstract science. Epistemologically, it is the “science of measurement”.

” The origins of natural numbers finds its roots theory of sets. ”

Try doing this with pre-schoolers. I know how that works because I deal with them. Incidentally, you are (probably inadvertently) debunking all the fantastic work done by Maria Montessori in the teaching of Mathematics to little children.

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I think Einstein was wrong. Mathematics originates in man’s never-ending work of measuring the world around him. After all, what is Mathematics if not the “science of measurement”? Measurement always presupposes measurement of something. In this case, that happens to be the real world.

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Mathematics is not a science. Mathematics is a language.

The usefulness of Mathematics is that there are no ambiguities, no double-meanings and no contradictions. Therefore logical reasoning can occur with exact precision and without bias or underlying false assumptions.

Unfortunately, one can only reason in the language of mathematics using known quantities. Hence mathematics can not be used for the logical reasoning involved in Austrian Economics because we cannot quantify the unknown constantly changing subjective values of individuals in society.

Therefore we have no choice but to reason logically in another language. (In my case English)

Mathematics is the perfect language to use when we take measurements. Logical reasoning becomes so much simpler to perform. Complex calculations become extremely simple if expressed in the language of mathematics.

But this does not make Mathematics the “science of measurement” as you say. Mathematics is simply the most practical known language to accurately express measurements with.

Mathematics is a language. Nothing more, nothing less.

]]> Interesting post, but I would like to hear you explain why events would necessarily play out the way you have described. It may be possible that they would play out that way, but why is that the only scenario you consider? If you would bother to investigate the libertarian position I think you might be surprised to find that it consists mostly of describing a way to achieve a social structure that avoids precisely the scenario you describe.

I would also challenge you to identify even one monarch whose wealth was not based on the forcible appropriation of other peoples property, in other words theft.

After all, if a new continent was found there’d be a land rush, the new continent is filled up with l’il sovereign landowners, some succeed in their endeavours whist others don’t and those who don’t will sell their land to those who have succeeded thus the continent experiences fewer and fewer private landowners, most people keep wealth in the family thus private landowners will bequeath their land to their children thus creating an aristocracy, to help unify the rules between fellow aristocracies the aristocrats create a federation council and vote one aristocrat to be a central figurehead who acts as an arbitrator between debates with fellow aristocrats (i.e. the early feudal era where a king was a central figure but had no real special powers over fellow lords).

It seem as though what might happen in Anarchtopia will follow pretty much what already has happened. Maybe Libertarians feel society went off the rails when aristocratic wars see the emergence of an absolute monarch and the serfs who have no respect for property rights overthrew the monarchies and replace it with Socialistic structure where power is held by a council and the members are determined by the workers themselves otherwise known as Democracy.

]]>“I think Einstein was wrong.”

I think that’s a pretty ambitious claim to make, If Einstein was wrong, and you are right, then why haven’t you figured out string theory yet?

I honestly don’t know for sure, you can’t know everything, but life just ain’t that simple, clear cut.

With a al do respect, but mathematics is not about “counting”. The origins of natural numbers finds its roots theory of sets.

And 1 meter, 1 yard, what kind of measurement do you have in that department. I have never seen a definition that gives me a 1 meter out of mathematics that always allows me to find the right length (or “measure”) so you want. When we need a “unity”, we just pick one (at random).

How “much” is 1 meter?

]]>I can see why we differ so widely in our conclusions. If you really believe that Mathematics is just a set of abstractions, just ask yourself a simple question. What is the concept “1″? The very concept “number” is an abstract concept. However, any number is an abstraction from reality. A number refers to the quantity of a certain unit that has been identified as being distinct from its surroundings. You could say “4 men” or “3 women” or “5 children”, but in each case, the number refers to a concrete number of units of the type being observed. In fact, that is why 3+4 is always 7. 3 men and 4 men together can only form a group of 7 men. That’s also why we have the saying “You can’t add apples and oranges”.

All Mathematics ultimately has its roots in reality. The sole purpose of Mathematics is the measurement of the real world. Higher Mathematics describes a number of nuances involved in the interactions of quantities, but ultimately has its roots in reality.

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