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Homogeneity through Time

 

– A Follow-up Paper to Questions at the Royal Society Scientific Discussion Meeting: The Nature of Mathematical Proof

(Monday 18 and Tuesday 19 October 2004)

 

With a Rider on the Dimensions of Physics

 

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by A. C. Sturt

 

 

 

 

 



a. Number

b. Algebra

c. Non-numerical entities which are homogeneous through time

i. discrete

ii. Continuous entities

d. Variables which are non-homogeneous through time

e. Keynes and mathematical economics

f. Generalisation of the concept

g. Homogeneity through space

Rider: The Dimensions of Physics

References

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

a. Number

b. Algebra

c. Non-numerical entities which are homogeneous through time

i. discrete

ii. Continuous entities

d. Variables which are non-homogeneous through time

e. Keynes and mathematical economics

f. Generalisation of the concept

g. Homogeneity through space

Rider: The Dimensions of Physics

References

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

a. Number

b. Algebra

c. Non-numerical entities which are homogeneous through time

i. discrete

ii. Continuous entities

d. Variables which are non-homogeneous through time

e. Keynes and mathematical economics

f. Generalisation of the concept

g. Homogeneity through space

Rider: The Dimensions of Physics

References

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

a. Number

b. Algebra

c. Non-numerical entities which are homogeneous through time

i. discrete

ii. Continuous entities

d. Variables which are non-homogeneous through time

e. Keynes and mathematical economics

f. Generalisation of the concept

g. Homogeneity through space

Rider: The Dimensions of Physics

References

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

a. Number

b. Algebra

c. Non-numerical entities which are homogeneous through time

i. discrete

ii. Continuous entities

d. Variables which are non-homogeneous through time

e. Keynes and mathematical economics

f. Generalisation of the concept

g. Homogeneity through space

Rider: The Dimensions of Physics

References

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

a. Number

b. Algebra

c. Non-numerical entities which are homogeneous through time

i. discrete

ii. Continuous entities

d. Variables which are non-homogeneous through time

e. Keynes and mathematical economics

f. Generalisation of the concept

g. Homogeneity through space

Rider: The Dimensions of Physics

References

 

 

 

 

 

 

During a discussion session at the above meeting I asked a question in which I raised the matter of homogeneity through time when procedures moved from mathematics to computer aided proofs. The concept seemed to perplex at least some of the participants. In effect it is the confounding of variables which manifests itself with the passage of time. The first use of this term of which I am aware was by Keynes in respect of mathematical economics, as described in the paper.

 

I consider the concept to be of such fundamental importance that after thinking it through again I have set it out in full below.

 

The methodology which I use starts with the assumptions inherent in the use of number and mathematics. This requires a detail which some of those skilled in the art may find laborious, but it allows me to add more concisely the successive assumptions which illustrate the concept of homogeneity through time. The descriptions are non-mathematical, because mathematical terms all come with their own definitions and connotations. In some ways this is what is being challenged, and so we need to begin at the beginning.

 

  1. Number

 

In a collection of numbers in no particular order such as:

 

1          2          5          3.2       ……    21

 

the number 1 is a conceptual entity which has unit value everywhere and at all times. It does not grow or shrink or change into something else, whether in the same place or in another place. The most succinct description of this is that it is homogeneous through time and space.

 

For most purposes it is unnecessary to state that something is homogeneous through both time and space, because if it is homogeneous through time, then it is probably also homogeneous through space within our frame of reference. The proof of this is to move to another place an infinitely small step at a time. What starts homogeneous through time, remains homogeneous through time at each step. The exception is that for analysis which addresses the nature of time itself, it may be necessary to consider homogeneity through space in its own right. This is considered below.

 

From the definition of the number 1, it can be seen that the number 2 is simply two units of 1 in additive association i.e. 1,1 or (1,1) or 1+1. The same is true of the other numbers. Decimals are conceptual entities which are defined as identical and equal to one tenth of the unit, and as such are similarly homogeneous through time etc.

 

Describing the number 2 as a symbol which represents 1 in additive association with 1  takes us back to ancient Mesopotamia, when arithmetic evolved to keep accounts on clay tablets of their equivalent of bushels of grain in granaries. One bushel was one stroke, two bushels two strokes etc. However, when the number reached five, it was marked for convenience of reckoning with a diagonal stroke which signified a defined bundle of strokes. This was in effect a new symbol. So

 

 

 

 

 

 


which is not so very different from the notation used by the Romans.

 

 

The notation:

 

            0          1          2          3          4          5          6          7          8            9

 

was a later invention from the Middle East, whence the term arabic numerals.

 

Units may be added or subtracted using either convention. So

 

3 + 2  = 5

or

 

 

 

 

 


On the basis of addition and subtraction, they may also be multiplied or divided. They may be ranked in order of magnitude. They may be used with the power notation i.e. 32 is the same as 3 x 3.

 

Negative numbers may be accommodated by considering them as on the other side of a balance, so that their magnitude is in effect subtracted from the value on the first side. Operations on the other side of the balance are carried out in the same way as on the first side, but the result is retained on the negative side. Double negatives are accommodated by performing the operation on the negative side of the balance, and then bringing it back, so that the result is on the positive side.

 

All this is trivial and generally accepted.

 

b.      Algebra

 

Algebra uses a symbol such as x to represent a number, so that operations may be carried  out without the labour of arithmetic. If the collection of numbers above is represented in algebraic symbols, each number would have to have a separate symbol, such as x1, x2, x3 and so on. Addition and subtraction would lose the convenience of numerical notation, because even if x1 has a value of 2 and x2 has a value of  3, there is no algebraic symbol for 5 and they would have to stay as x1 +  x2 ,which loses the point of the symbols.

 

The solution is to use both arithmetic and symbols in association together as multiplicative entities, and to express all numbers in terms of a number of xs: so 2x, 5x etc. The result is that all the operations which apply to arithmetical number can be applied to an algebraic number. They can be added, subtracted, multiplied, divided, adopt negative values, be expressed as powers etc. The essential point is that both the number and the symbol are treated as numbers multiplied together, even if one is not specified. So:

 

2x multiplied by 3x is the same as 2 multiplied by x multiplied by 3 multiplied by x

 

which by reordering is the same as:

 

2 multiplied by 3 multiplied by x multiplied by x

 

which is

6 multiplied by x multiplied by x

 

which is 6x2.

 

The symbol x is called a variable. Arithmetical values do not have to be substituted for x until the end of the operation. In fact they may not have to be substituted at all to provide meaningful relationships.

 

There may be a symbol y which represents a relationship expressed in terms of x at every value of x, which is described as y = f(x). The values of y for each value of x may then be plotted on Cartesian co-ordinates to show the shape of the function e.g. straight line, parabolic, exponential etc.

 

Since the algebraic symbols represent numbers, algebraic functions are homogeneous through time.

 

c.       Non-numerical entities which are homogeneous through time.

 

Since x is an undefined numerical entity, it is natural to substitute other entities which are homogeneous through time in algebraic equations. Indeed that is where it all began, with Mesopotamian bushels of wheat. Undefined entities fall into two categories: discrete and continuous.

 

i.                     Discrete

 

An example of a discrete entity which is homogeneous through time might be an apple. An apple is a discrete entity if it designated as such conceptually. The description does not mention size or shape or flavour. An apple comes as a whole conceptual unit which is homogeneous through time. Apples, of course, are real entities, and in this form they are certainly not homogeneous through time. But conceptually they are homogeneous through time, since otherwise we would not be able to talk about them without identifying the particular apple in question. ‘Apple’ is the symbol.

 

The arithmetic then deals with 1 apple, 2 apples, 3 apples etc. Numbers of apples may be added and subtracted, and they may take on negative values as above.

 

However, they may not be meaningfully multiplied or divided in their associated form: twice 2 apples means 4 apples, but 2 apples multiplied by 2 apples is a meaningless concept. Similarly (2 apples)2 is meaningless; (22 apples) is 4 apples.

 

The sequence of logic is:

 

-        This calculation is about apples.

 

-        The arithmetic or mathematics is performed to give a number or algebraic expression without the appearance of ‘apples’.

 

 

-        The answer is expressed in terms of a number of apples.

 

 

‘Apple’ is not a substitute for the algebraic x.

 

ii.                   Continuous entities.

 

A continuous entity here means something like kilogrammes of gold. Discrete golden entities would be gold coins. By comparison weight is a continuous variable, even if it is expressed in kilogramme units. (Entity may not be quite the right term here, but no obvious alternative springs to mind).

 

The same analysis applies as to discrete entities. Weights of gold can be added, subtracted, and ranked in order of magnitude. They can be given negative values as on the other side of the balance. They can meaningfully take the double negative.

 

But they cannot be multiplied or divided; 2 kg of gold multiplied by 3.2 kg of gold is a meaningless concept. By comparison 2 kg multiplied by 3.2 is certainly meaningful i.e. 6.4 kg.

 

The reason for selecting gold as the example of a continuous entity is that gold is to all intents and purposes homogeneous through time. It does not shrink or grow or turn into something else with the passage of time, which is one reason why it has always been such a valuable commodity. Its material homogeneity through time is the parallel of the conceptual homogeneity through time of the apple. The result of the calculation is expressed in terms of numbers of kilogrammes of gold.

 

d.      Variables which are non-homogeneous through time

 

The above discussion prepares the way for the main burden of this paper, the mathematical handling of entities which are non-homogeneous through time. The most graphic introduction is an invented example from economics, the consumption of alcoholic beverages in Utopia.

 

The data are:

 

1          2                                                      3                                                    21

 

this time in millions of litres/yr.

 

The same analysis applies as to gold. Litres of beverage can be added, subtracted, ranked and take negative values, always provided in this case that the result of the calculation does not give a negative consumption. But they cannot be multiplied or divided.

 

However, each figures refers to a year. They are a time series, even if the order in which they are written does not necessarily represent sequential years. Nor do they represent a simple concept or material which is homogeneous through time, because they are aggregates with a composition which may change with time.

 

For instance, during the year represented by the first figure, 1, the composition may be 2% wine and 98% bitter beer. On the other hand the figure of 3 million litres a year may represent 3% wine, 27% lager and  70% bitter. If they are consecutive years and not just a random collection, the picture may look as follows:

 

Components

Composition %

1960

1970

1980

1990

2000

Wine

Lager

Bitter

2

-

98

2

18

80

3

27

70

4

36

60

5

55

40

 

These are not so much abstract points as a commentary on the socio-economic dynamics of Utopia over five years. As prosperity increased, more wine was drunk. As thirsty heavy industries declined, and those remaining were relatively less prosperous, less bitter was drunk. Lager consumption increased with the rise of the spectator. Consumption is an aggregate which is non-homogeneous through time.

 

A number of observations distinguish non-homogeneity through time from homogeneity through time.

 

i.                     The analysis of necessity refers to a particular place (non-homogeneous through space).

 

ii.                   There are problems in forecasting the next point in the series. There would be problems even if the trend for consumption were a straightforward mathematical function. The reason is that the 2005 figure, when it comes, has a significant new component in the form of 5% of some completely new beverage.

 

 

iii.                  There are interactions of these data with other series in the Utopian economy, for instance the demand for potato crisps. It may be socially obligatory to have a packet of crisps with a can of lager, in which case consumption of the two are linked.

 

 

iv.                 Changing technology plays a significant role in the changes of composition which occur. The invention of the can with a ring pull makes a considerable difference to who drinks what and where. Similarly the invention of the widget.

 

 

v.                   Aggregates of such series across the whole economy are the basis of Gross Domestic Product (GDP). To produce a variable which can be aggregated, supply and demand are expressed in terms of the money spent or expected to be spent on each item. Money is continuous and homogeneous through time by definition, so that it is analogous to the above example of gold. This device makes the algebraic operations possible, but the underlying socio-economic dynamics remain, hidden beneath the blanket of money.

 

 

vi.                 No relief is to be obtained by breaking the data down to something more ‘fundamental’, say the figures for wine, unless that is the goal of the analysis, because the question then becomes: red or white, wine bar or supermarket, aperitif or table etc? Nor can one specify the consumer more precisely in this sense; there is nothing more unpredictable than the individual consumer. You just have to choose the goal carefully, and live with the imprecision.

 

 

That is not to say the figures are meaningless, or the series not worth recording. It depends on what one is trying to achieve. The point is that calculation may be necessary but it is not in itself sufficient.

 

e.      Keynes and mathematical economics

 

The first use of the term non-homogeneity through time of which I am aware was by Keynes, who was unhappy with the application of algebraic functions to his new economic model because economic “material is, in too many respects, not homogeneous through time” (1, 2). He thought it could not substitute for keeping ‘at the back of our heads’ the necessary reserves and qualifications.

 

I have no way of knowing how far he pursued the idea, but I have extended it to economic theory in my own book The Scale and Scope of Economics (3). Conventionally economic theory considers only economies of scale; the assumption is that processes are scaled up without changing their basic nature, rather like multiplying them by two i.e. they are homogeneous through time.

 

However, this assumption is far from true. Notional economies of scale are often, though by no means always, outweighed by the diseconomies which are introduced by changes of scale, such as difficulty of process control, problems of management of organisations of increased size, increased problems of location and environment, problems of learning to operate larger equipment, problems of security of supply of inputs, and so cost of down-time, problems of  insurance etc.

 

None of these problems is insuperable, if the effort is worth it, but they all have to be taken into account when the costs are reckoned in decision-making. This is in addition to the hardware- or scale-related costs. They are factors which depend on people and time, whether in the enterprise or in the environment in which it is intended to operate, and they may not all be foreseeable. They are non-homogeneous through time. Rather than always using the cumbersome term non-homogeneous through time in this context, I use the neutral term ‘scope’ to distinguish it from ‘scale’. It is scope which changes with time. Hence economies or diseconomies of scope. Hence the ‘scale and scope’ of economics.

 

There is a problem of summarising these two aspects of economic systems mathematically. To describe the economy as a function of time is trivial, because it must be so. However, it may be possible to decompose the system into scale and scope subsystems. Scale applied to operations depends only on current conditions i.e. output varies with current input prices depending on current demand, which varies with time. This is the production function. However, all the aspects of processes which are non-homogeneous through time are continually being addressed to bring improvements i.e. the process is evolving through time; it is changing in scope, another function of time. These two functions, scale and scope, are not independent; each gives rise to the other. Perhaps the best way to describe them mathematically is as some kind of product.

 

Another way of describing this process of evolution is ‘learning system’, as described in my book

A Degree of Freedom, which sets out a new, non-mathematical analysis of systems thinking (4).

 

continued


 

 



problem of computer aided proofs


Keynes

homogeneity through time an important concept

methodology is to start from fundamental non-mathematical principles






 


 



 

 

number homogeneous through time and space

 

 

 

addition







the origins of number






 












 

 

 

 

 

 

 

 

 

 

 

 

 

multiply and divide

 

 

 

negative numbers

 

 

 

 

 

 

 

 

 

symbol

x represents a number

 

 

 

 

 

symbols and numbers used in association

 

 

 

 

 

 

multiplication

 

 

 

 

 

 

 

 

 

 

 

variables

 

 

 

 

 

 

algebra homogeneous through time

 

 

 

 

 

 

 

 

 

 

 

apple as entity conceptually homogeneous through time

 

 

 

apparently algebraic

 

 

but cannot be multiplied

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

‘apple’ is not a variable like x

 

 

 

 

continuous entities can be homogeneous through time

 

 

 

 

 

same limitations: no multiplication

 

 

 

 

 

 

 

 

 

 

variables may be non-homogeneous through time

 

 

 

 

 

 

 

 

e.g. litres of beverage

 

 

 

number and litre are homogeneous through time

 

 

beverages not

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

distinctions between homogeneity and non-homogeneity through time

 

 

forecasting

 

 

 

 

 

interactions

 

 

 

 

 

technology

 

 

 

 

 

aggregates used of necessity

 

 

 

 

cannot solve by breaking down aggregates

 

 

individual consumers even more unpredictable

 

 

 

calculation necessary but not sufficient

 

 

 

Keynes used the term suggesting caution

 

 

Concept extended in my book The Scale and Scope of Economics

 

The term “scope” subsumes non-homogeneity through time

 

 

 

scope of processes changes with time

 

conventionally ignored

 

 

 

 

 

changes of scope enable change of scale

 

in effect economic evolution or learning system as described in my book A Degree of Freedom

 

 

 

Copyright A. C. Sturt 2004

continued on Page 2

 

 

Churinga Publishing

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