"Essential Aspects of Austrian Economics" or "Neodoxy's Low Content Economics Thread"

Economics Economics Questions
1

Well frankly I’m disappointed with the level of economic discussion that is going on with these forums (ever since Dave’s been in and out I have no one to argue economic theory with and the only other decent discussion that’s been going on is the discourse between myself and Jargon concerning the importance of interest and the production structure). Therefore I’m going to make a highly informal post about what I see as some of the essential aspects of Austrian economics in the way that these ideas have been floating around in my head over the past few months. Being able to contrast the Austrian approach to economics with the mainstream view has given me a much greater ability to contrast and appreciate certain aspects of AE (I’d also like to know Wheylous’ views on this).

Within this list I will also include how these factors relate to capitalism, since I think that Austrainism (in the way it has manifested itself, not in its base methodology) can be accurately characterized as a description of the social function of the market economy within society. Indeed the entire path to becoming a supporter of free markets can be described as a path from considering the market as a random, purposeless, and elitist institution, to a spontaneous and purposeful entity which, above all, exists for a reason.

In no particular order:

  1. Human action is purposeful behavior

All human institutions and practices within society exist for a reason and people act in certain ways for specific reasons. It does not mean that these reasons are always positive or wise, but it does mean that the market economy emerges for a reason, and this extends to the entirety of the market economy given time.

This contrasts with non-Austrian views of economics which, even though they don’t usually explicitly state “people don’t act purposefully”, nonetheless marginalize the importance of this fact and are inherently predisposed to elitism and viewing the market as a positive entity

  1. Human action occurs in the face of an uncertain future and is therefore imperfect.

Neither firms, consumers, nor governments act perfectly, however each individual constantly strives to reach a more perfect state of affairs. This means that the market economy doesn’t fit to the relatively static equilibrium that lives within most mainstream conceptions of the market economy, instead the economy is continuously being destroyed and rebuilt. As consumer demand as well as objective productive considerations vary, different processes of production and different quantities of output need to be produced. The invisible hand does not perfectly “grasp” on to a position, rather the invisible hand is forever groping under dim light to find the most perfect level of prices and production.

This is one area where most models of the economy are wholly lacking. Some of the more astute critics of mainstream economics, and therefore capitalism, often base their arguments off of the fact that markets don’t act "perfectly", and therefore the government must come in and correct this state of affairs. Yet this critique only works if one believes that the only defense of markets is that they are indeed perfect, or meant to be perfect. This is not the case, however. No human institution is perfect, but few have the inherent tendency towards establishing a more perfect state of affairs like the market economy does. In reality the existence of this imperfect world just means that well functioning markets need to exist in order to react to these changes as smoothly and swiftly as possible.

  1. The market economy is the “democracy” of the consumer

This is, in my opinion, the ultimate justification of the market economy. The fact is that companies must ultimately service the consumer or else they will go out of business. There’s no way of getting around this. In order to make profits a businessperson has to produce something that consumers want at a reasonable price. What the consumers want are produced, and those companies which are better than this than others receive larger profits. Foolish entrepreneurs go out of business and are replaced by superior counterparts. Even the most vicious of monopolies has an incentive to provide quality products and customer service to increase the demand for what they are selling. Despite the fact that in the short run the capitalists and entrepreneurs control production, it is ultimately buyers on the market who control the entire economy.

If one looks at the companies at the top fortune 500 companies; Mobil, Ford, Walmart, AT&T, Microsoft, Apple, General Electric, CVS, ETC, then one will notice that these are almost exclusively household names that cater to the common man (banks are very high on this list but presumably these are funding other companies such as the ones above), not to the rich and powerful. The market will always tend to allocate resources where they are most desired by consumers, and therefore resources flow from where they are least wanted into where they are most wanted. This inherently means that the market economy is a social system, it is a series of trade that is ultimately not based off of the capitalist, but the consumer and mutual interactions that provide what is desired in larger quantities.

This is not an element that is necessarily lacking in mainstream economics, but the true wonder of it is not emphasized. This is where socialism and statism can result, from the assumption that production is “automatic”, and that it will always occur in the way that we have become accustomed to in our vaguely capitalistic society, when ultimately it is only so long as the consumers democracy and effective market structures are maintained that consumers are properly seen to and the most desired goods are ultimately produced. Just as with all human phenomena it is the result of definite human action that will not occur under all conditions, rather a state of very definite conditions continually allow it to occur.

  1. Production is a process that always occurs over the course of time

This is an absolutely essential aspect of the Austrian model of the economy, particularly because of the school’s emphasis upon the importance of capital (which I’m just going to lump in here). Time is important, time is valuable. I don’t care what will be made available for me to consume in a thousand years, and I’m less interested in hearing what you will be able to do for me in 20 years than in a month. Therefore because production takes a decent period of time (as anyone who has ever seen a construction project knows) the market time preference is an essential coordinating factor within the economy. Furthermore capital, one of the only things which separates the economies of modern and primitive societies (the other primary factor being education) takes time to produce. Therefore by taking a longer period of production we can enjoy a greater living standards in the future.

This is why ABCT should really be just filling in intuitional gaps. Since the interest rate is an essential economic indicator in the accumulation of capital, fiddling around with this rate willy-nilly should ring some alarm bells in anyone’s head. Since the interest rate is the price of time, and time is one of the few omnipresent aspects of the entire economy, large and unexpected alterations within this may obviously cause large problems.

This is one area where mainstream economics generally overlooks the matter. They overlook the importance of savings and production over time. Beginning with the Classical economists and emphasized by Frank Knight the production structure is generally viewed as “instantaneous” and the way that GDP is calculated (including the emphasis put on this measurement) overlooks the importance of savings in the economy. Capital is also considered important within the mainstream viewpoint, but it is not considered as essential and the only consistent means to a higher living standard, as it is within the Austrian view.

Well those are some of my brilliant insights for the evening. This will also serve as a place for other random rants on these sorts of topics that I might have.

I also invite anyone with casual economics questions to post things here, so as to entertain a bored and lonely young economist.

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Why do people double post threads here and on LibertyHQ?!??!

PICK ONE!!!1!

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“PICK ONE!!!1!”

I swing both ways on this issue. I’m a transforumite

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Indeed the entire path to becoming a supporter of free markets can be described as a path from considering the market as a random, purposeless, and elitist institution,

As you pointed out - It’s always funny how the opposition will use hierarchy and elitism both to criticize the market and defend itself, anyway that is a topic for another thread: main addition I want to make to this is Ludwig Lachmann used the phrase to describe the market process as a “circulation of elites” - a phrase I intend to steal.

An analogy that I want to start using in someway to contrast the market mentality are the Egyptian pyramids - an inflexible,massively bureaucratic, hierarchy in “perfect equilibrium” that is built off a crazy master/slave mentality with Shackle’s kaledic world

5

“main addition I want to make to this is Ludwig Lachmann used the phrase to describe the market process as a “circulation of elites” - a phrase I intend to steal.”

That is very true, and it fits in very well with Mises’ full conception of the consumer democracy, where the “leaders” of the market economy are inevitably indirectly appointed by the consumers themselves

6

Here are some as listed by Peter Klein:

“In short, the core concepts of contemporary Austrian economics – human action, means and ends, subjective value, marginal analysis, methodological individualism, the time structure of production, and so on – along with the Austrian theory of value and price, which forms the heart of Austrian analysis, all flow from Menger’s pathbreaking work.” (Essay: “Menger the Revolutionary”)

What is missing? The notion of exact (as opposed to empirical) laws of social phenomena. The notion of exact, universally valid, laws of social phenomena is the central concern of Austrian economics and is what praxeology is all about. As Menger wrote:

“The aim of this orientation, which in the future we will call the exact one, an aim which research pursues in the same way in all realms of the world of phenomena, is the determination of strict laws of phenomena, of regularities in the succession of phenomena which do not present themselves to us as absolute, but which in respect to the approaches to cognition by which we attain to them simply bear within themselves the guarantee of absoluteness. It is the determination of laws of phenomena which commonly are called “laws of nature,” but more correctly should be designated by the expression “exact laws.” (Investigations Into the Method of the Social Sciences, 1985, p. 59)

This approach is what Menger referred to as theoretical exact science (analogous to mathematics, geometry, formal logic, etc.), what Mises referred to as praxeology, and what Hayek referred to as the Pure Logic of Choice. Mises’s praxeology is an extension and elaboration of Menger’s social science of exact laws. What separates Misesian and Hayekian Austrian economics is Hayek’s contention that exact laws (a priori propositions) are not possible with respect to market phenomena. Thus, the notion of exact laws is central to Austrian economics, not only because Menger defines exact laws as “the aim of this orientation” and not only because they are the primary focus of Mises’s praxeology, but also because the scope of the application of exact laws in social theory constitutes the main dividing line between the Misesian and the Hayekian schools of Austrian Economics.

7

Found this article from the Concise Encyclopedia of Economics to be interesting, Austrian economists are seen as similar to flat earthers because they reject optimization. Optimization basically means that everyone acts as if they had perfect knowledge right?

http://www.econlib.org/library/Enc1/NeoclassicalEconomics.html

I read an article asking why neoclassicals don’t just advocate having the economy run by a really big computer

From the article:

Interestingly enough in Human Action Mises said that mathematical economics was the reason people were fooled into believing socialism could have economic calculation. It is not just Marxism or Keynesianism it is the mechanical view of economics that creates so many fallacies. A mechanical view of economics also supports Keynesianism because GDP makes any economic activity look good even wasteful misuses of resources. Why not just run the economy with a really big computer like the Zeitgeist movement advocates

8

@Adam

If what you’re saying is true, then isn’t Hayek’s attempt at constructing this exactness effectively useless since it can’t actually be applied to the market economy?

@Gravy

That’s a very good point. One of the things that I’ve never understood is the idea that neo-classicism and Keynesianism were somehow at odds. Keynesianism is a perfectly logical application of neo-classical economics into the macroeconomic realm with the assumption of sticky wages. Indeed Alfred Marshall was one of Keynes’ major teachers.

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Here’s Adam’s article on Hayek and Praxeology. It’s great.

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“If what you’re saying is true, then isn’t Hayek’s attempt at constructing this exactness effectively useless since it can’t actually be applied to the market economy?”

No, because an individual could apply a set of exact laws to his own “individual action” such as: “If I walk toward a location (action A) I know I will be walking away from a different location (exact law and certain consequence B).”

Thus, Hayek writes that it is only the logic of individual action that is a priori (i.e., to which exact laws apply). The moment you pass from this to the interaction of many people (i.e., market theory) you enter the empirical field (exact laws no longer apply). According to Hayek, in market theory, only empirical laws apply: If X is done, then Y may happen or Y may not happen.

(see Hayek on Hayek, p. 72)

(see also Hayek, “Economics and Knowledge”)

11

What would his rebuttal to the argument that he is wrong on the grounds that market interaction is just the outcome of the individual behaviors of many people?

Also, how exactly would Hayek apply such laws in an actual field of study? It seems like it would only work with things like maybe consumer theory.

12

“What would his rebuttal to the argument that he is wrong on the grounds that market interaction is just the outcome of the individual behaviors of many people?”

I explained Hayek’s reasoning in this Mises Daily article of last November:

In the article it explains in more detail Hayek’s argument that praxeology can’t be applied to the interactions of a number of people.

“Also, how exactly would Hayek apply such laws in an actual field of study? It seems like it would only work with things like maybe consumer theory.”

Well, I assume Hayek didn’t think that much important could be learned from pursuing the Pure Logic of Choice in order to understand or describe social phenomena. Because it seems he didn’t pursue this line of thinking except for his mention of it in a few essays. Maybe he believed that the praxeological aspect of social science was merely a foundational prerequisite for doing empirical field study; a logical exercise we do before our actual empirical social science. One should perhaps ask a qualified Hayekian how Hayek envisioned applying his Pure Logic of Choice to social phenomena.

13

I always see a lot of threads and even journal articles where Austrians note how they really “get” uncertainty and dynamics and that all of this stuff is ignored in mainstream econ models. On the whole, I think this is totally untrue, but say that it was not. What are you really saying in this thread?

You are saying that Austrian economists incorporate better assumptions into their models than mainstream economists. They assume humans act purposefully in a world where outcomes are uncertain and production takes place over time. And that this strikes you as more realistic.

But, if changing these were truly different assumptions and these differences really mattered, Austrian models should yield different results than mainstream models, right? Are you sure that is the case?

Stick to microeconomic models, say the model for consumer behavior. The mainstream model of consumer behavior predicts that if you increase the price of a product, the quantity demanded for that product will decrease (except under very special circumstances). By contrast, some Austrian economists say that if you increase the price of a product, the quantity demanded for that product will decrease (except under ver special circumstances).
http://web.missouri.edu/~kleinp/misc/giffen.pdf

Of course, note that I choose my language carefully there. I only say “some Austrian economists say” and not say “the Austrian model of consumer behavior”, because no such model exists. As best I can tell, there are lots of economists that self-identify as Austrian, each with their own way of analyzing consumer behavior that may contradict each other. For example, as I’ve noted in several other threads, some Austrians claim there is never any circumstance where an increase in price will lead to an increase in quantity demanded.

But let’s move on from consumer behavior. What about how prices are determined in a competitive market? The mainstream model says price is determined by the intersection of supply and demand. By contrast, the austrian model says price is determined by the intersection of supply and demand. GROUND SHATTERING DIFFRENCES!

Now, I know Austrians like to pat themselves on the back and say they focus more on the “process” of how price is determined. But this undeserved self-congratulations. If you read MES, Rothbard’s explaination for how prices are determined doesn’t sound much different from the story you hear in undergraduate textbooks where the process is compared to an auction (consumers “bidding up” prices and all that). And at higher levels of study, mainstream economists certainly move beyond this simple story.
http://mises.org/rothbard/mes/chap2b.asp#_ftnref2

Now this post is way too long and it is way too late. But my point is that you can make sweeping statements about how Austrians don’t assume away the complications of reality and that they actually deal with uncertainty and all that more directly than mainstream economists. All I know is that when I look at individual microeconomic models, the differences are less striking than the similarities.

I guess what I am really saying is don’t tell me how Austrians make different assumptions. Tell me how making those different assumptions MATTERS. Where does it lead Austrians to different conclusions than mainstream economists??? If you had to constrict yourself to microeconomics, I think you would be hardpressed to find major differences between mainstream and Austrian econ.

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  1. That is analogous to saying calculus is unimportant, because 2+2=4 in arithmetic, and 2+2=4 using calculus.

  2. I thought it well known that the micro is the same more or less, the differences being in macro.

  3. In any case, here’s a difference with regard to supply and demand curves. The quotes are from Prof. Shostak.

Using the supply-demand framework for a particular good, mainstream economists proceed further and introduce supply and demand curves for the whole economy. They hold, for example, that if the economy is underperforming, then what is needed is a bolstering of demand by means of fiscal or monetary policies. For a given supply curve, they contend, this will push the demand curve to the right, thereby lifting overall output. Needless to say, the supply-demand framework provides the rationale for government and central bank interference with businesses.

In other words, the mainstream here is ignoring the fact that humans act, that human action is what determines economics. Just shift the demand curve to the right with some govt spending and you’re good to go. Supply will increase automatically, magically, as written in the Holy books. You can’t argue with a graph. It’s pure mathematics, man, sheer science.

We’ll quote how Prof Shostak introduces the human action understanding of economics to blow the magical thinking out of the water:

Also, we have seen that, in reality, it is producers that initiate the introduction of new products. They set in motion increases in goods and services, and not consumers as such. Producers present new products, so to speak, to consumers who, in turn, by buying or abstaining from buying, determine the fate of products. Hence there is no such thing as an autonomous demand that somehow triggers supply.

  1. I remember seeing as well that one of the fundamental tools of economics [is it the law of supply and demand?] has no proof in mainstream economics, but provable rigorously with AE. Anyone able to help me out here?
15

You can prove 2+2=4 using calculus? Interesting.

Anyways, I want to make one thing clearer about my original post for anyone else that may care to comment (since looking back it wasn’t as clear as I wanted to be). Note that I explicitly say that I do not think the assumptions Neodoxy mentioned are that different from the assumptions that mainstream economists make in their models. It is true that the way he (and most Austrians) state these assumptions is much vaguer and less informative than the way mainstream economists would state them, but I will get to that in a second.

For the sake of argument, I say let’s assume that mainstream economists ignore all these things he mentions. Then I ask, do the conclusions of Austrian models differ from their mainstream counter parts? I show at least two examples where they don’t. The point of that comparison isn’t exactly to say assumptions don’t matter. Part of what I’m trying to say is that you should double check you are actually making different assumptions than your mainstream counter parts.

Of course, another part of what I was saying is that it is often hard to tell if Austrians are making different assumptions, because they often describe their models in such vague terms. For example, Neodoxy says that Austrians recognize we live in a world of uncertainty. Mainstream economists recognize this too, but they also recognize that sometimes uncertainty is not a dominant feature of the behavior we are trying to explain. I think this is especially true for modeling simple consumer choices like how much jam to buy at the store. In the real world, there is some uncertainty about the quality of the jam you are buying and the likelihood you will actually make it back to your house alive and not die in a car accident. But, mainstream economists explicitly abstract away from these uncertainties because for many choices the uncertainty is just too small to really matter for what we are concerned with answering. Of course, Austrian economists do this too. It just harder to know what they are assuming because they don’t have to spell it out. IMO this is probably why there isn’t really a consensus around an Austrian model of consumer behavior. How can you build a consensus if you are not 100% sure what the other person is saying?

But I was not just saying that Austrians make the same assumptions as mainstream economists. I do recognize that there are cases where Austrians make explicitly different assumptions than mainstream models. For example, I have heard people complain that the amount of goods being bought and sold are discrete and not in continuous quantities (i.e. you can only buy 1 jar of jam not 1.153 jars). This is true. But does this assumption actually lead you to different conclusions than if you assumed continuous quantities? In some cases maybe. But in most cases not really.

So what do you gain from making this assumption except that you feel like your model is “more realistic” (except for all the other ways your model may be unrealistic)? Anyways, just somethig to think about.

16

Can anyone explain what optimization is because I thought it meant that people act like they basically know everything sort of like dispersed knowledge. In the article I posted one of the main reasons Austrians are seen as flat earthers is because they don’t believe in optimization

After all I would assume to do something optimally you would have to have perfect knowledge of all factors involved

17

You can prove 2+2=4 using calculus? Interesting.

In many ways. For example. The integral of the constant function F=1 from zero to four equals the integral from zero to two plus the integral from two to four. [This does not assume two plus two equals four, but follows from the linearity of the integral functional]. The integral form zero to two is two, and from zero to four is four. [This too, does not assume two plus two equals four, but is a property of they integral functional].

The integral from two to four can be evaluated from first principles by dividing it into two equal squares of dimension one by one. [This assumes one plus one equlas two, but not does not use the fact that two plus two equlas four]. Put all the above together and get 2 plus two equals four.

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19

Student,

I’ve always wondered when we’d finally clash on a real economic topic, so I’ve pulled out a lot of stops for you, although what came out was generally incoherent. Let it begin.

I think that you make some good points, but I also feel as though there’s a problem on both ends on the matter of what the real consensus of the various schools necessarily are. In particular I think that it’s very important to realize that there is no such thing as a perfect consensus in any school of economic thought, and therefore I believe that whatever either of us points to as “the consensus” is going to be somewhat arbitrary and anecdotal. Furthermore I have to say that unfortunately I have a very limited idea of what “modern Austrianism” really means, particularly in terms of the knitty-gritty aspects of certain issues. While I’ve read enough by or about the big Austrians, I’ve read relatively little modern theory, although I want to take a look into recent publications from the Journal of Austrian economics to get a better idea of what it is that modern Austrians really do. Meanwhile I have been exposed fairly thoroughly to modern mainstream economics, so I feel that I have a much greater grasp on what that is, although what I’ve noticed is that there’s a much less consensus on a lot of issues in the mainstream than there is within what I perceive to be Austrianism itself.

I should also note that within the umbrella of what we consider mainstream economics there are a lot of essential elements of AE that are “somewhere” within neoclassical economics, however these are not given nearly the emphasis that they are in Austrian economics. One thing that you can say about Austrianism is that it does start from the basics and work up, the basics being economic methodology. This is one thing that I’ve never seen emphasized in any economics course I have ever taken. Never has the question been answered “how can we know anything about economic phenomena”. This leads to the first obvious difference that Austrians (with a few notable exceptions) avoid mathematical models, whereas mainstream economics revolves around mathematical models. If you want one major way that the Austrian opinion of uncertainty plays into things, then this is one major way that you have it. Furthermore the fact is that, as I stated in my original post, I have never seen a mainstream professor present the market economy as something that struggles towards equilibrium and that equilibrium is never something that is fully reached, that the economy as a whole is always struggling to reach equilibrium and that this is continually broken, and I have also seen this uncertainty surrounding real values and the uncertainty of these values in the market used as the basis for attacks on the capitalist economy. These attacks appear to me to be, ironically, wholly valid when they are levied at the moderate neoclassical economists who usually advocate for a moderation of the market economy with government intervention, but these attacks are pretty useless when they are used against the libertarian Austrian school that fully embraces capitalism.

I also think that it’s unfair to sweep aside Austrian macroeconomic theory when talking about differences between the two schools since, well, that does account for about half of all economic studies (I consider all mainstream macroeconomic schools of thought (with the possible exception of real business cycle theory) to be derivatives of Keynesianism which is in turn a derivative of neoclassical economics). Since production structure analysis is a massive part of Austrian theory, it’s unfair to overlook it (I’m assuming that this is classified under Austrian macro?). Another important area where neoclassical and Austrian economists adhere to different beliefs is in the area of economic calculation. In mainstream models it’s never really addressed, which is a massive failure since calculation is what makes the entire economy possible (see the socialist calculation debate) and there are very real outcomes of interfering with this form of calculation. This could well be considered an outcome of Austrianism’s massive emphasis upon subjectivism. This in turn leads to differing views in areas of social welfare, particularly when we look at the insane mainstream belief that we can actually calculate things like the real value of public goods in the absence of displayed preference. Interpersonal utility analysis also appears to be a real thing within neoclassicism, although I could be wrong there.

So I suppose what you could say is that a huge amount of the difference just comes from where emphasis lies, and this has deep political implications on things, and since I’m probably waay too into the political side of economics, this is a large part of what I see. I would also like to note that what you are looking at, such as price determination, is an area where Austrianism and neoclassicism are practically identical in a lot of respects. In particular the sections that look at consumer theory, production functions, and cost curves would in no way be out of place within a microeconomic textbook, albeit a very good microeconomic textbook. The very attitude of economic inquiry within Rothbard’s text is also extremely different than what I’ve seen within any sort of neoclassical work. while Rothbard’s book is filled with life and a more fluid look at human behavior, what I’ve seen out of neoclassicism seems to be extremely rigid, mechanistic, and non-human. This might not even lead to any differences in the description of events, but it is a difference. Indeed the fact that Austrians and neoclassicals would have similar views on the matter of price determination, since this was the “epicenter” of the neoclassical synthesis. Indeed much of the subjectivism and marginalism that exists within neoclassicism originated with Austrianism. I’ve even seen Austrian economics called a type of neoclassical economics, something that I don’t necessarily disagree with (beyond the matter of titles).

Well now, I don’t even know how one would go about addressing this post, so I’ll be interested to see any responses. Nontheless these are some of my random thoughts on your challenge.

Dave,

On point 3. I think that that’s really a macroeconomic issue, or at very least the necessary microeconomic “fix” to allow for the macroeconomic issue to be a thing. I’ve never heard a microeconomics professor or text talk about a chronic “underdemand” for any particular good. The idea of insufficient aggregate demand originate in macroeconomics. Therefore this must imply that on a microeconomic level demand is too low, it’s not really a matter of microeconomic supply/demand theory except insofar as it leads into macroeconomics.

I think it depends on what you mean by supply and demand (in terms of whether or not this can be proven in mainstream econ). I think that a proof similar to what an Austrian would say wouldn’t be out of place in a neoclassical description of the matter. I’ll crack open my principles book later on and see what it says on the matter. I also find your second response amusing when we consider the breadth of Student’s post.

20

Dave,

In many ways. For example. The integral of the constant function F=1 from zero to four equals the integral from zero to two plus the integral from two to four. [This does not assume two plus two equals four, but follows from the linearity of the integral functional]. The integral form zero to two is two, and from zero to four is four. [This too, does not assume two plus two equals four, but is a property of they integral functional].

The integral from two to four can be evaluated from first principles by dividing it into two equal squares of dimension one by one. [This assumes one plus one equlas two, but not does not use the fact that two plus two equlas four]. Put all the above together and get 2 plus two equals four.

Unfortunately, you haven’t proven that 2+2=4. You have simply demonstrated the additive property of integrals (i.e. the integral of a function from a to c is the same as the sum of the integral from a to b and the integral from b to c).
http://www.math.pitt.edu/~sparling/052/23052/23052notes/23052notestojan14th/node6.html

This is not the same thing as “proving” 2 + 2 = 4. The additive property of real numbers is a totally different beast. When I took real analysis several years ago we talked about how you could prove that the set of real numbers satisfied this and other properties, but it is a bit beyond me now. In any case, I don’t need to prove the additive property to make my main point. Which is that you need the additiive property of real numbers (among other properties) for calculus of real functions to even work.

Specificaly, I am really not sure how you can say you’re not assuming that 2+2=4 when you take the integral of this function. You should look a bit more into how the integral is actually computed. I mean, just look at the limit definition of the integral itself (see link below)!! It is DEFINED as the infinite SUM! You will see what I’m saying more directly if you use the limit definition to take the integral of F=1 from 0 to 2 and the integral of F=1 from 2 to 4. If you do you will see that you are not doing anything different from adding 2 and 2, you are really just rewritting it in more cumbersome notation without realizing it! In other words, your post doesn’t prove 2 + 2 = 4, it works BECAUSE 2 + 2 =4. The exercise I suggest you do is essentially done for you in the website below (click on problem #1 and change F=5 to F=1).
http://www.math.ucdavis.edu/~kouba/CalcTwoDIRECTORY/defintdirectory/

Anyways, I don’t want to get into this, since I can’t think of a worse thing to argue about. So you can have the last word if you still disagree. But I will just conclude by saying you need a better analogy. And maybe you should pick up Bartle and Sherbert’s Introduction to Real Analysis. :stuck_out_tongue: