Homogeneity through Time – A Follow-up Paper to Questions at the
Royal Society Scientific Discussion Meeting: The Nature of Mathematical Proof
With a Rider on the Dimensions of Physics
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.
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.
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
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.
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
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.
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:
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).
Non-homogeneity through time is the unplanned or unavoidable confounding of variables in a statistical sense which manifests itself only with the passage of time. In a snapshot at any particular time it may simply be an undetected heterogeneity. It is the changing composition of the heterogeneity through time which may cause confusion. Economics is only one, perhaps exaggerated, example.
In practice there are very few systems which may be considered as absolutely homogeneous through time, whatever is planned. People are non-homogeneous through time. Apart from fashion, they respond to what they believe to be threats and opportunities in the environment. They may have an optimistic or pessimistic outlook by nature, they may be influenced en masse by random events beyond anyone’s control. This affects the outcome of all processes in which they are involved, whether in planning, execution or interpretation, whether safety procedures or management or university admissions, whatever the mathematical principles applied. All living things have this susceptibility: animals driven by hunger, plants suffering from drought, and even insects and microbes, the great survivors. Evolution itself is the result of non-homogeneity of populations through time.
Even wider, it applies to everything in the physical world subject to entropic degradation. Non-homogeneity arises from two sources. First, the system being described may itself be non-homogeneous, say a rocky structure containing flaws, a building or a piece of equipment. How these behave may depend on where they are located and how long they have been in existence. This may not become apparent until something snaps. Such phenomena are non-homogeneous by their very nature, like the weather.
Secondly, systems for obtaining inputs may be far from clearly and uniquely defined, especially in an evolving society and with changing technology. Similarly the systems for interpreting the outputs may be far from unambiguous, even if reproducible. Hence the ‘scientific’ or experimental aspect of computer proofs, even if they are free of bugs.
This points up the difference between the conceptual and the actual with respect to proof. Mathematics is always conceptually homogeneous through time and space, because that is how it is conceived. No one develops an axiom or a theorem to be valid only in a specific place or with an expiry date. They may later be shown to be false or they may be extended by the application of further mathematical reasoning, but their proof is what they set out to prove everywhere and at all times: QED.
The consequence is that mismatches must arise when the mathematics is applied to systems which are non-homogeneous through time. Things do not turn out quite as calculated, because they do not match the assumptions on which the proof is made. Even if they do now, they will probably not later. The passage of time will reveal the differences. Under these circumstances the term proof must mean proving the use of the mathematics in the sense of putting it to the test i.e. the percentage ‘explained’.
That is not to cast aspersions on the use of mathematical proofs of any sort. Sometimes the most fruitful analyses are the most difficult because they deal with inputs which are non-homogeneous through time. Rather it is to conclude that the goals and limitations need to be clearly understood at all stages; calculation is not enough. There is no substitute for understanding, experience and insight.
Mathematicians may be allowing themselves a certain amount of satisfaction that at least their theorems are above the hurly-burly according to this analysis; it is only when they have to be translated into systems and perhaps computing that consideration must be taken of the complexities of non-homogeneity through time. However, that may be a little premature.
Sooner or later mathematics has to address change as a function of time. But time is nothing without an event to mark it (Einstein). Events mark the passage of time, whether they are homogeneous or heterogeneous in a static sense, as in a snapshot; it does not matter if the heterogeneity has no time to change. If there are no events, there is no time; there is no way of knowing.
The way of measuring time is to provide a sequence of evenly spread events. Even this implies that you can follow the intervals between events. You watch the pendulum throughout its swing, and note when it passes a fixed point, which is the event. All clocks here on Earth are arranged to produce the same interval of time, the second, whatever physical phenomenon is used.
However it cannot be assumed that any particular phenomenon is homogeneous through space. Indeed it is known that the interval of time measured by a pendulum would vary from place to place on the face of the Earth depending on the force of gravity on which it depends. This is also true more generally. As the clock moves from position to position in space, the phenomenon on which it is based may be influenced by, say, gravity, magnetism etc. This may be true of electromagnetic phenomena as a satellite leaves the surface of the Earth and heads out of the Solar System. The interval of time which it produces may then vary from place to place in space; it may be non-homogeneous through time. Another way of describing this is that space may be differentiated as far as this particular phenomenon is concerned.
These qualifications may be true of the currently used interval of time which is based on an electromagnetic phenomenon, because it is considered to be a fundamental property of atoms and the speed of light, both homogeneous through time. However, there are indications that the standard time interval may not be as immutable as thought; it could turn out to be a ‘gold standard’ in which ‘gold’ was found not to be homogeneous through space. An even more sobering thought is that these days all units of distance are measured in terms of time.
Nor can we assume that this applies only to space ‘out there’. What about the subatomic space which fills atoms? Or nuclei? Atomic clocks provide the phenomenon which we use to describe the macroscopic world i.e. atoms, molecules and the structures we perceive, but the internal structure of atoms also changes with time, whether through the motion of subatomic particles or by some outside interference.
Moreover, there is no reason to assume, as Relativity does, that all physical phenomena used to make clocks would be identically affected as they move out into space. For example, different phenomena might be more, or less, affected by gravity or magnetic fields etc in the same locations in space. The result would be that space is differentiated differently for each phenomenon.
When clocks leave the place where they are all calibrated against each other, those based on different phenomena would then behave in different ways and produce intervals of time of different length. Perhaps it would be better to look at the stars, which is where it all began, but an interval of time derived from them must contain all the celestial movements which are not only elliptical with long time periods but precess to boot: orbits around the Sun, motions of the Sun etc. A second which was a fraction of a year, as at present, would not be the same as a tenth of the same fraction of 10 years, because of annual variation. Nor would it be the same in a different 10 year period.
Finally, one might ask why bother with clocks and physical phenomena at all in mathematics? Why not treat time as just another continuous variable? But time is nothing without an event to mark it (Einstein, again). No clocks, no events, no time! You must have a clock to have a variable.
It seems probable that a phenomenon such as exponential decay of a radioactive isotope would remain exponential throughout time and space, because the number of radioactive nuclei which decay in a population depends only on the number present at the time, as far as is known. The exponential curve as such would remain homogeneous through space as well as time. In this case it would just be the constants which varied (!), because they are all calibrated in Earth intervals of time i.e. seconds. This is because there is no way of timing the rate of radioactive emissions other than with a clock calibrated effectively in seconds.
A possible solution which I have proposed is a radioactive clock with a decay curve which is exponential and therefore homogeneous through time, but which does not rely on measuring rates of decay in seconds with another, conventional timepiece. Such a radioactive clock would give intervals of time which were Absolute, the same everywhere and at all times. It could then act as a Universal standard.
The methodology is as follows: to specify an interval of time in terms of a number of counts; to derive a parameter of decay which is dimensionless because it relies only on number, i.e. it eschews the second and the phenomenon used to measure it; and to obtain successive intervals of elapsed time of the same absolute length by calculation (5).
Such a clock would require, in the first instance at least, extensive evaluation and international agreement before it could become a standard.
Guildford, Surrey, UK
25 October 2004
Physics uses equations such as:
where m is mass, t is time and d is length. The other symbols in the equations are either constants or derived from m, t, and d.
The symbols m, t and d are variable values of what are called the dimensions of the equations for the purposes of dimensional analysis. Dimensions are denoted as follows:
Since the symbols represent numerical quantities of these dimensions, they can be used in all the same operations as x in algebra. They can be added, squared and multiplied. They can be differentiated and integrated.
It certainly works, because these equations of Newton’s enabled him to predict the motion of the planets very precisely. In fact this sort of equation is the working basis of all scientific and engineering calculations.
However, it is worth pointing out the assumptions which underlie the algebraic operations.
Thus 3t means 3 units of time-interval, on the assumption that the time-interval does not vary through time and space. This is reasonable for homogeneity through time because that is how clocks are devised, but what about space? Similarly 2d means 2 distance-intervals, on the assumption that the distance-interval does not vary through time, which is reasonable, because that is how the distance-interval is defined. But again, what about space?
In fact we know that the assumptions break down at velocities through space which approach the speed of light. Acceleration of a mass is no longer proportional to applied force, and in fact there is a limiting velocity beyond which mass cannot be accelerated, the speed of light. No longer a straight line relationship, it appears to be asymptotic i.e. a hyperbola (6).
The Theory of Relativity is based on the hypothesis that the algebraic relationships still hold good for measurements taken by anyone travelling at the relativistic speed,
i.e. they are homogeneous through time and space, but the observed discrepancies in everyday experience occur because the dimensions change with velocity. All the dimensions change together; it is a question of frames of reference.
In Relativity mass increases and time dilates as velocity approaches the speed of light. Clocks slow down, and so seconds grow longer by comparison with the standard time-intervals. Metres grow longer, but only in the direction of travel. It is not the dimensions which are homogeneous through time and space, but the algebraic relationships which depend on them. As pointed out above, these algebraic relationships are composed of variables which in fact signify number. Thus the Theory of Relativity seems to try to have it both ways.
An alternative which I have proposed in my series of papers (7) is that both the dimensions of physics and the fundamental physical phenomena, such as electromagnetic radiation, are homogeneous through time and space. Discrepancies are observed at relativistic speeds because the equations themselves, which have been derived under our local conditions, do not adequately represent physical behaviour under all conditions in the Universe. In other words we have not fully evaluated the nature of the fundamental physical phenomena to which they refer.
Examples of this may be the inverse square law for light and gravity It is suggested in another paper that there may be an unsuspected phenomenon at work, namely deflections of microentities (8). Nor are these likely to be isolated instances. Departures from the equations which we might expect to apply everywhere at all times may be exposed as our focus of attention shifts to the extremes of astronomical distances or the world of fundamental particles.
1 December 2004
1. John Maynard Keynes quoted in Keynes by D.E. Mogridge 1976 Fontana Collins, London and Penguin, New York.
“Economics is a science of thinking in terms of models joined to the art of choosing models which are relevant to the contemporary world. It is compelled to be this because unlike the typical natural sciences, the material to which it is applied is, in too many respects, not homogeneous through time.”
2. The General Theory of Employment, Interest and Money by John Maynard Keynes 1936. Chapter 21, Section III p297:
“It is a great fault of symbolic pseudo-mathematical methods of formalising a system of economic analysis, such as we shall set down in section VI of this chapter, that they expressly assume strict independence between the factors involved and lose all their cogency and authority if this hypothesis is disallowed; whereas, in ordinary discourse, where we are not blindly manipulating but know all the time what we are doing and what the words mean, we can keep ‘at the back of our heads’ the necessary reserves and qualifications and the adjustments which we shall have to make later on, in a way in which we cannot keep complicated partial differentials ‘at the back’ of several pages of algebra which assume that they all vanish.”
3. The Scale and Scope of Economics by A.C. Sturt 1984 http://www.churingapublishing.com/
4. A Degree of Freedom by A.C.Sturt 1993 http://www.churingapublishing.com/ p34.
5. Radioactive Clocks: a Basis for the Absolute Measurement of Time by
A.C. Sturt 30 July 2003. http://www.churingapublishing.com/radio_1.htm. See also patent application.
6. Mass in the Universal Inertial Field – A Revised Version by A. C. Sturt
18 September 2003 http://www.churingapublishing.com/numass_1.htm
7. The Timeless Universe, Light and Mass and A New Model of Physics – Foundations http://www.churingapublishing.com/
8. Light and Gravity Inside and Outside Solar System: Potential Consequences of ‘Particle’ Deflection Hypotheses. Submitted for publication. To be made available on churingapublishing.com website.