What's Good for General Motors : Matrix Additivity and Conservation [MT114]

David Noel
Ben Franklin Centre for Theoretical Research
PO Box 27, Subiaco, WA 6008, Australia.

What's Good for General Motors Is Good for the U-S-A.
-- Charles E Wilson, to Congressional Committee, 1952

Counting on Our Fingers

No detailed understanding of the components of a complex system seems possible unless numbers or quantities can be applied to give an idea of the scale of its differerent parts. We need to be able to count the things we are dealing with.

In the human world, our normal numbering system is based on the number ten. We are still counting on our fingers -- in an alien world where the inhabitants had six fingers on each of two hands, their base number would probably be twelve. Even our word for the symbols involved, the digits, actually means fingers.

If we now try to apply conventional counting and measurement procedures to the sort of topics we have looked at in Matrix Thinking, we must do so with caution. There are a number of reasons for this.

First, some of the assumptions which are imbedded in the ordinary sort of counting we do in daily life may not apply in the Matrix area. In MT105, I suggested (Proposition 105I) that matrix qualities may not be subject to the same laws as linear qualities. When we move, as now, to try and handle matrix quantities, it would be wise to check whether our ordinary counting procedures still apply. Often they will not.

It Just Doesn't Add Up

Adding ordinary numbers is 'commutative'. If we have 3 bananas and add 2 bananas, we end up with 5 bananas. If we start with 2 and add 3, going in the reverse order, the answer is the same.

This seems absolutely simple and obvious. That it is not so, is shown by dividing instead of adding. If we divide 3 by 2, we get a different result than we would if we divided 2 by 3.

And of course the additive principle assumes that the items added are identical in nature. Often, in a real as opposed to an abstract world, they are not. In a New Guinea market, the 5 bananas you buy may be intended only for cooking, and quite unsuitable to add towards the fruit salad you want to make.

Suppose you want to see how many job vacancies there are in Australia. You add together the figures for the different states and territories, 100,000 in New South Wales, 20,000 in Western Australia, and end up with a total of, say, 400,000. Is that an accurate procedure?

The answer is no. Some of the vacancies will be counted more than once, they are the same vacancy, 'offering itself' in more than one state at the same time. Other will be very localized, existing only in a very restricted locality, say a distant mine in outback WA. These will be of no relevance to someone seeking work in Sydney. And obviously each vacancy will have its own requirements for filling, the unemployed accountant just cannot take up the plumbing vacancy and go from there.

The deduction from this is that we cannot just add together matrix quantities and necessarily expect the simple total to make any sort of sense.

Proposition 114A**. Matrix quantities are not necessarily additive

Moving Across Systons

The second cautionary aspects concerns adding up things which lie in different areas. Simple mathematical addition embraces a second assumption: a single, continous range along which numbers are added. If we add 26 and 3 and get 29, we automatically assume that if we add 1026 and 3 we will get 1029 -- the numbers are all living on the same linear scale, and working at a point further along the scale should only displace the answer, not alter it.

In the real world, things are not so simple. If a man is walking along at 5 kilometres per hour in a town, roughly how many houses will he pass in a minute? The answer obviously depends on the building pattern, but with a typical Australian street frontage of 10m, with houses both sides, the answer is about 17.

All right, but suppose the surface the man was walking on was the corridor of a train moving at 60 km/h? Obviously the number of houses passed would be far more. And even on the same house spacings, the detailed number would depend on whether he was walking in the direction the train was going or the opposite way.

That is not an example of a second facet of matrix quantities, it is only a parallel. What that facet actually involves is the realization that matrix quantities are not necessarily additive across or over systons.

Proposition 114B**. Matrix quantities are not necessarily additive across or over systons

Take, as another parallel, the population of a shire or county. A local authority may conceivably have a record of how many people live in each of its rateable properties -- it may operate under a poll tax system, for example. If it adds all those numbers together, it will get a total which represents the number of people living in the shire.

Now that is a procedure which is obviously not watertight -- it omits people living in nonrateable properties, for example. But that is a Proposition 114A limitation.

The Proposition 114B limitation comes in when you try to calculate the population of the State, and do this by adding together the populations of its constituent shires. Here, even if the shire counting method was exact, the State count would not be, because some of the population will have houses in more than shire -- perhaps a holiday home, or a farm property managed by someone not living in the farmhouse.

When you move into the more subtle areas of syston makeup, the limitation becomes more apparent. Calculating the number of sports club supporters in a State by adding up the individual club numbers would obviously be useless -- many of those involved will support more than one club, cricket and football, for example.

What's Sauce for the Goose: The Goose's View

Finally, the matter raised in the quotation at the head of this article. I have put this quotation in, not because I believe it is true, but because I think it is a readily-assumed principle which is often completely false. Here is my view:

Proposition 114C***. Things which advantage a particular syston will usually disadvantage a wider syston which contains it

What this Proposition is saying, in effect, is "What's good for General Motors is bad for the USA". At first sight this assertion seems most unlikely to be valid. But let us apply a little linear thinking to it, and then move on to MT analysis.

On the standard view, many of the things which go to make up society -- jobs, money, resources, and the like -- are assumed to be 'conserved'. Here, 'conserved' is used in the scientific sense, that is, the total amount of a given resource is assumed to remain the same, although it may be changed in form.

If we have a resource of 100 million tonnes of coal in the ground at some place, it will stay there unless we use it. If we use it up at a rate of 1 million tonnes a year, by changing it into heat or some or other form of energy, it will last 100 years. All very straightforward.

If the Government lets in 100,000 migrants, that means there will be 100,000 less jobs for Australians. If overtime was banned, that would create huge numbers of new jobs for those at present unemployed. If the Government didn't spend a billion dollars on armaments, it could put the money into the health or education systems.

These last things are not so straightforward. Nevertheless, they are the sort of assumptions which are at the base of much linear thinking.

Of course such assumptions also underlie one of the fundamental feelings in society -- the idea of equity, of Fair Shares for All. This is an aspect of what, in MT109, I referred to as 'tight-banding'. It applies in both directions, down and up.

The Tall Green-Eyed Poppy

Australia is notorious for what is called the 'Tall-Poppy Syndrome', the urge to drag down those who make a lot of money or become very prominent in some area. It is a sort of envy. When it is an expression from a complex syston, rather than an individual -- and usually there will be syston equivalents to all individual urges -- we can call it syston-envy, in MT terms.

Envy at any level is usually reckoned as Bad. Particular instances in the past have been justified on the grounds of another urge, usually reckoned as Good -- the idea that things should be shared out fairly, that is evenly.

There is nothing in the Matrix Thinking approach to support this view. In fact, MT would regard it, like any other instance of tight-banding, as leading to a reduction of infocap, normally associated with a disadvantage to wider society.

Proposition 114D**. A syston is not advantaged by attempting to share its resources equally among its systels

How about the other direction, that is, how about somebody who is very rich sharing out his wealth with lots of others? The Proposition just stated applies equally here, too. But, how about the very poor, doesn't this principle imply that it would be a mistake to top up their resources and bring them closer to the average?

It is not the intention of Proposition 114D to suggest that those who have very little should not be helped to improve their lot. But, it must be admitted that we have arrived at a sticking point in our MT analysis. We will not be able to achieve more clarification of the situation until we arrive at the concept of Threshold Levels, as in MT116.

Readers will have noted the relation between the last two Propositions and Proposition 113A, which suggested that a syston is disadvantaged by discrimination between its systels. At first glance, the various principles suggested might seem in conflict. The conflict disappears, however, when the clear distinction is made between discrimination, applying to inequality of opportunity or treatment, and sharing-out, applying basically to physical possessions.

But this article was to look at the nature of matrix quantities. Before we end it, and go on to look at attempting to measure these quantities in MT115, we should formally put forward a basic facet of matrix quantities, as a reminder of one aspect of Proposition 106D.

Proposition 114E***. Matrix quantities are not conserved

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