96 9696 962009-05-04 23:52:14 / rev bb931e4b905eChapter 6Dimensions6.1 Power of multinationalsThe first example shows what happens when people take no notice of di-mensions.Critics of globalization often make this argument:In Nigeria, a relatively economically strong country, the GDP [grossdomestic product] is $99 billion. The net worth of Exxon is $119 bil-lion. ‘When multinationals have a net worth higher than the GDP ofthe country in which they operate, what kind of power relationshipare we talking about?’ asks Laura Morosini. [Source: ‘Impunity forMultinationals’, ATTAC, 11 Sept 2002, [url:nigeria-argument],retrieved 11 Sept 2006]Before reading further, try to find the most egregious fault in the compar-ison between Exxon and Nigeria. It’s a competitive field, but one faultstands out.The comparison between Exxon and Nigeria has many problems. First, thecomparison exaggerates Exxon’s power by using its worldwide assets (networth) rather than its assets only in Nigeria. On the other hand, Exxon canuse its full international power when negotiating with Nigeria, so perhapsthe worldwide assets are a fair basis for comparison.A more serious, and less debatable, problem is the comparison with GDP,or gross domestic product. To see the problem, look at the ingredients inhow GDP is usually measured: as dollars per year. The $99 billion for Nige-ria’s GDP is shorthand for $99 billion per year. A year is an astronomicaltime, and its use in an economic measurement is arbitrary. Economic flows,which are a social phenomenon, should not care about how long the earthrequires to travel around the sun. Suppose instead that the decade was the97 9797 97Chapter 6. Dimensions 972009-05-04 23:52:14 / rev bb931e4b905echosen unit of time in measuring the GDP. Then Nigeria’s GDP would beroughly $1 trillion per decade (assuming that the $99 billion per year val-ue held steady) and would be reported as $1 trillion. Now Nigeria towersover the puny Exxon whose assets are a mere one-tenth of this figure.To produce the opposite conclusion, just measure GDP in units of dollarsper week: Nigeria’s GDP becomes $2 billion per week. Now puny Nige-ria stands helpless before the might of Exxon, 50-fold larger than Nigeria.Either conclusion about the relative powers can be produced merely bychanging the units. This arbitrariness indicates that the comparison is bo-gus.The flaw in the comparison is the theme of this chapter. Assets, or networth, are an amount of money – money is its dimensions – and are typ-ically measured in units of dollars. GDP is defined as the total goods andservices sold in one year. It is a rate and has dimensions of money pertime; its typical units are dollars per year. Comparing assets to GDP meanscomparing money to money per time. Because the dimensions of these twoquantities are not the same, the comparison is nonsense! A similarly flawedcomparison is to compare length per time (speed) with length. Listen howridiculous it sounds: ‘I walk 1.5 meters per second, much smaller than theEmpire State building in New York, which is 300 meters high.’ To pro-duce the opposite conclusion, measure time in hours: ‘I walk 5000 metersper hour, much larger than the Empire State building at only 300 meters.’Nonsense all around!This example illustrates several ideas:• Dimensions versus units. Dimensions are general and generic, such asmoney per time or length per time. Units are the instantiation of di-mensions in a system of measurement. The most complete system ofmeasurement is the System International (SI), where the unit of massis the kilogram, the unit of time the second, and the unit of length themeter. Other examples of units are dollars per year or kilometers peryear.• Necessary condition for a valid comparison. In a valid comparison, the di-mensions of the compared objects be identical. Do not compare applesto oranges (except in questions of taste, like ‘I prefer apples to oranges.’)• Rubbish abounds. There’s lots of rubbish out there, so keep your eyesopen for it!• Bad argument, fine conclusion. I agree with the conclusion of the arti-cle, that large oil companies exert massive power over poor countries.98 9898 9898 6.2. Pyramid volume2009-05-04 23:52:14 / rev bb931e4b905eHowever, as a physicist I am embarrassed by the reasoning. This exam-ple teaches me a valuable lesson about theorems and proofs: judge theproof not just the theorem. Even if you disagree with the conclusion,remember the general lesson that a correct conclusion does not validatea dubious argument.6.2 Pyramid volumehbbThe last example showed the value of dimensions in economics. Thenext example shows that dimensions are also useful in mathematics.What is the volume of this square-based pyramid? Here are severalchoices:1.13bh2. b3+ h23. b4/h4. bh2Let’s take the choices in turn. The first choice, bh/3, has dimensions ofarea rather than volume. So it cannot be right. The second choice, b3+ h2,begins with a volume in the b3term but falls apart with the h2, which hasdimensions of area. Since it adds an area to a volume – the crime of dimen-sion mixing – it cannot be right. The third choice, b4/h, has dimensionsof volume, so it might be correct. It even increases as b increases, whichis a good sign. However, the volume should increase as h increases – aproportional-reasoning argument – whereas this choice indicates that thevolume decreases as h increases! So it cannot be right.The final choice, bh2, has correct dimensions and increases as h or b in-creases. Does it increase by the right amounts? Imagine drilling into thepyramid from the top and dividing it into thin cores or volume elements. Ifthe height of the pyramid doubles, then each vertical volume element dou-bles in volume; so the volume of the pyramid should double. In symbols,V ∝ h. But bh2quadruples when h doubles, so that choice cannot be right.The requirement that V ∝ h together with the requirement that V havedimensions of length cubed means that the missing item in V ∝ h is anarea. The only way to make an area from b is to make b2perhaps times adimensionless constant.