MINISTRY OF PUBLIC BUILDING AND WORKS
BUILDING RESEARCH STATION
NOTE NO IN 55/70
A HIERARCHICAL DIPHASE MODEL OF MATERIAL BEHAVIOUR
by F J Grimer
SUMMARY
This note provides a theoretical framework for a hierarchical model of
material behaviour. The model employs only two elements, a quasi-solid phase
and a quasi-fluid phase. The necessary complexity required to cope with
the complicated behaviour of real materials is obtained by repeated use of these
two elements at different hierarchical levels.
One of the important differences between this hierarchical model and
traditional views of material is that for the hierarchical model, tensile
forces only exist as negations of compressive forces and not as self
sufficient entities. The reasonableness of this view is argued and some of
the consequences discussed.
A17/37/251 Building Research Station
April 1970 Garston Watford WD2 7JR
SE/FJG (SE5/66) Tel: Garston(Herts) 74040
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A HIERARCHICAL DIPHASE M0DEL OF MATERIAL BEHAVIOUR
by F J Grimer
INTRODUCTION
In the course of research into the strength of mechanical properties of
materials in general and concrete in particular it has been found necessary
to alter the conceptual framework of that branch of physics commonly
termed 'strength of materials' in a very fundamental way. Unfortunately,
although the conceptual changes required are logically quite simple they
give rise to considerable psychological difficulties since they involve, for
example, altering the notion of a tensile force from a positive to a
negative concept.The difficulty of a complete switch of view of this type
can be compared with the difficulty of deciphering script written
backwards or interpreting a negative photograph.
It would be convenient if the conceptual changes could have been l
imited to a restricted field such as concrete and justified on pragmatic
grounds in this field alone. However, this is just not possible. Physical
phenomena form a continuum and basic concepts cannot be altered at
one level of phenomena without altering them at other levels involving
similar phenomena; even where this alteration may not be explicit it will
certainly be implicit. The interconnection of concepts between the
various scientific disciplines imposes a severe constraint on their
development since ideas which are historically antecedent exert an
analogous effect to a master patent in the sphere or inventions as far
as the innovation of new concepts is concerned. To make other than
trivial conceptual changes it is necessary not only to justify the
modification of concepts in the particular field under investigation but
also to demonstrate that the modification implicitly required in
interconnected fields is possible and ~reasonable; like the task of
breaking a master patent this prospect is sufficiently daunting to
effectively suppress nearly all innovation.
This note has been written to show that the new ideas proposed for
the study of materials are reasonable in the interconnected fields also
and it sets out a skeleton theoretical framework which emphasizes
certain correct aspects of existing theoretical structures and draws
attention to others which are redundant or wrong.
This framework is vitally necessary before progress can be made
because in the author's experience it is not possible to accept a
model sufficiently to develop it successfully until one has a reasonably
high certainty of its intrinsic truth or to put the position in more
poetical terms, without faith one cannot move mountains. Where the
theory embraces a wide field of application the conviction of its truth
must embrace a correspondingly wide field. This onerous requirement
of credibility has its advantages however since one the correctness of
the view can be psychologically accepted in a general context then
understanding in a particular context becomes very much easier.
HISTORICAL BACKGROUND
The concepts outlined in this note had their beginnings in research carried
out during the late 1950's into the properties of soils and soil-cements.
The research formed part of the programme of the Materials Division of
the Road Research Laboratory which at that time was part of the
Department of Scientific and Industrial Research.
A17/37/251
Note No IN 55/70 - 1 -
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It was noticed that the relationship between the compressive strength of
specimens molded from soil-cement and the average density of the
specimens was an extremely simple one the form
compressive strength = a constant x (density)^n
where n was also a constant.
The author became curious as to how exact this relationship was and set
about refining the experimental techniques of manufacture and testing the
specimens to determine the level of precision to which this relation held.[1]
The maximum refinement possible showed that the power relationship still
held to within the limits of experimental accuracy. The variation of points
about the regression line (1.4%) were sufficiently small to eliminate the
other forms of relationship which have been suggested for strength versus
density at various times.
Examination of other materials such as concrete, coal briquettes and
plastics [2][3][4]indicated that the relation was true within the limits of
experimental error for a wide range of materials and furthermore that it
held not only for compressive strength but for other measures of strength
as well. Different materials and different measures of strength gave rise to
different constants but the form of the relation was a power curve in all
cases. This implied, of course, that in general the relation between
different measures of strength for a material of increasing density will
be of the form:
measure of strength A = a constant x (measure of strength B)^n
where n is also a constant
and this form of relationship has often been observed [5] but has always
been treated as an empirical relation. No attempt ever appears to have
been made to understand why it should take this form.
To the author it seemed seriously amiss that relationships of this type
which had such universality and unifying power should be so
unrecognized and disregarded. It seemed that the processes and materials
which gave rise to these relations were extremely complicated and one
would not a priori expect all the information contained in a set of strength
measurements at a range of densities to be conveyed by a single number.
Could it be that the processes were really much simpler than they looked
and they only appeared complex because we had vast amounts of
redundant information. Were we, in terms of information theory [6]
receiving a signal in the presence of considerable noise. This note is the
result of the author's attempt to answer these questions and some of the
conclusions arising from the answers obtained.
BEGINNING OF THE HIERARCHICAL CONCEPT
On considering the relation....
strength = a constant x (density)^n
....it soon became obvious that density, i.e. mass/unit volume was not the
best parameter to use on the right hand side of the equation since one
could readily conceive of a change in mass occurring without any change
in strength. Therefore the parameter density was replaced by the parameter.
Vs / V
Where Vs = volume of solids
V = total volume
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For a given material Vs is proportional to mass and so the form of the
relationship remains unchanged by this substitution.
The ratio Vs / V is a non-dimensional number less than 1 and it can
be thought of in several different ways, viz.
(a) a volume concentration of solid in space V
(b) a probability of finding the solid part at any randomly chosen point,
(c) a complementary strain; if we regard V as the volume available
to the solid and Vs as the volume it actually occupies we can regard
the solid as strained relative to its space by an amount (1- Vs)/V in
which case Vs/V becomes the complementary strain.
All these concepts are non-dimensional and it is considered that all
three are useful in different contexts.
When this concept of volume concentration/probability/strain is
applied to a material such as soil-cement or clay it is evident it
can be applied at different hierarchical levels. For example,
consider a specimen composed of wet clay lumps which have
been compacted together. At the first level of scrutiny or nth
hierarchical level the volume of the solid component will be the
volume of the wet lumps of clay and the difference between this
volume and the total volume will be the air voids content. At the
second level of scrutiny or (n-1)th hierarchical level the volume
of the clay minerals within the wet lump will be the volume of the
solids and the difference between this volume and the total volume
will be the water voids content. It can be seen that the concept of
solidity is being used in a relative sense and that what is or is not
solid will depend on the hierarchical frame of reference. It can also
be seen that by increasing the level of scrutiny we descend to
successively lower hierarchies.
Now at first sight the introduction of a hierarchical concept of this
kind does not appear to be very important. However, this is far
from being the case, the modification such a concept has on the
way that phenomena are interpreted is profound. The difference
between a non-hierarchical concept of Vs/V and a hierarchical
concept of Vs/V may be compared with the difference between introducing
one rabbit to Australia and introducing two. Like the pair of rabbits
a hierarchical concept of this kind is self reproducing and may be
extended indefinitely.
The concept was first applied to various measurements of strength
made on soil cements and clays and it was found that the relation....
S = k(Vs/V)^n
S = some measure of strength Vs= volume of solid in a particular hierarchy.
V = total volume of same hierarchy k and n are constants.
.....was equally true at the first hierarchical level where the solid
volume was the volume of the wet lump and the remaining volume was air
voids, and also at the second hierarchical level where the solid volume
was the volume of the mineral particle and the remaining volume
was the volume of the water voids(2).
BEGINNING OF SOLID-FLUID CONCEPT
The hierarchical concept seemed such a powerful one that its
implications were examined in greater detail. Firstly, the term
Fluid was adopted for the difference between the Solid volume
and the total volume of the hierarchy. It is evident that in the
examples considered above the term Fluid is appropriate
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since both air and water are fluids. Thus a material can be considered
as divisible into two phases, a Solid and a Fluid at the first hierarchical
level; each phase can in turn be divided into two phases and so on.
The suitability of the terms Solid and Fluid at the lower hierarchical
levels will be discussed later. To distinguish between the hierarchical
terms Solid and Fluid and the ordinary meaning of these words the
hierarchical terms will be denoted by capital letters.
The adoption of a Solid-Fluid model enabled the strength of a material
to be seen as resulting from a very different mechanism to that normally
assumed. A material may be regarded as a skeletal Solid phase in a
continuous Fluid phase with the Fluid phase in a high state of tensile
strain relative to its free condition and the Solid in a state of balancing
compressive strain. With sands, silts and clays the validity of this model
can be demonstrated directly since the tension in the Fluid phase, in this
case water, can be measured directly(7) (8) and shown to attain values
of many hundreds of atmospheres.
Soils are a particularly simple case since the strength, or more generally
the stress-strain properties are determined basically by a single
hierarchical system, i.e. the particle-water system, in which the Solid
phase is a readily identifiable solid (the soil particle structure) and the
Fluid phase corresponds to a readily identifiable fluid (water).
In general, however, the stress-strain properties of a material will not be
dependant upon a single hierarchical system but will be the result of the
combined influence of several such systems at different levels.
Since the total behaviour of a complex material such as soil cement has
been shown to have a simple mathematical form it seemed clear that the
behaviour of each hierarchical system is essentially similar and that
because of this the multitude of systems can be replaced by a single
equivalent system in which the total effect of the different orders of
Solids involved are represented by a single equivalent Solid and the
total effect of the different orders of Fluids involved are represented
by a single equivalent Fluid. At first sight this suggestion seems just too
simple to be true. Fortunately there is a very good precedent for this
type of suggestion, viz. gravitation. In this phenomena the individual
effect of a large number of very different objects of all shapes and sizes,
all materials and structures over a wide range of distances both
stationary and changing can be replaced by a single effect of an
equivalent point object at a particular distance. Nowadays this is taken
for granted but at the time it was made it must have seemed an
incredible suggestion.
Considering for a moment a single hierarchical Solid-Fluid system,
it is clear that the tension compression model is a field model of
behaviour since the particles are pulled towards one another by the
tension in the fluid rather than attracted towards one another by any
intrinsic properties they may possess.
The internal environment of the material, the Fluid, is at a lower
pressure than the external environment and therefore the pressure
outside the specimen is positive relative to the inside and we can
regard the particles as being pushed together by this outside pressure.
This alternative way of looking at the tension in the Fluid phase again
views the Solid component as passive and inert and its internal
properties as irrelevant. The mechanical properties of the Solid-
Fluid system as a whole can be seen as a consequence of the
pressure difference between the internal and external environments.
It is clear that this model of material behaviour views the solid
structure of a material as normally under a state of considerable
compression equal to the pressure difference between the inside
and outside of a material. Thus an applied tension becomes a
decrease in the state of compression whilst an applied compression
becomes an increase in the state of compression. If we apply s
ufficient tension to a material to reduce the pressure difference
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between inside and outside to zero then the structure divides into two or
more parts because there is no longer any net compression force holding
it together. Failure is thus seen as a point of zero total compression,
i.e. an absolute zero point as far as pressure in that particular hierarchy
is concerned.
For the structure then, there is no such thing as tension, there is only
compression. For the structure the phenomena of pressure is given the
same conceptual framework as the phenomena of temperature. Just
as there are only positive degrees of temperature so there are only
positive pressures. It is evident that the way we look at stresses at
present is very anthropomorphic. We take a frame of reference
which is appropriate to us but which is not appropriate to the material
we are studying. This is equivalent to measuring temperatures as
positive and negative from 0degC say and talking in terms of degrees
of heat (compressions) and degrees of cold (tensions). If we adopted
such a procedure in the case of temperature all sorts of simple
relationships between temperature and other phenomena would be
hidden from us. The reason the relation between strengths and
volume ratios is a simple one now becomes clearer.
Measuring the strength of a material involves taking the material down
to its pressure datum level or in other words its hierarchical zero
absolute pressure and so strength is a measure of the total pressure
on a material.
If the above general arguments are accepted then all materials have
to be regarded as under stated of compressive stress and strain
initially. If we apply a compression or tension to a material we merely
alter a pre-existing state of stress and strain and it is the total state
which is relevant to the material properties. This datum shift presents
no difficulty of course from a mathematical aspect; indeed it will
make the mathematics easier since it will eliminate the constant which
would otherwise have to be used to allow for the present false
position of the datum. However, from a conceptual point of view the
datum shift presents considerable difficulties in much the same way
as the datum shift of the centre of planetary motion from the earth to
the sun presented considerable difficulties a few centuries ago. These
difficulties are not of course rational but emotional and can be
expected to increase as the logical consequences of the datum shift
are progressively understood.
APPLICATION OF THE SOLID-FLUID HIERARCHICAL
CONCEPT AND DATUM SHIFT TO OTHER HIERARCHIES
If the concepts outlined above are valid then it follows that any
material whatsoever can be regarded as being a Solid-Fluid (this
will be abbreviated as S-F) system which in turn is composed of
other S-F systems in the manner illustrated in Fig 1. Furthermore
it follows that any S-F system at any hierarchical level can be
treated as a material irrespective of whether it is an extremely low
hierarchy, a nucleus say, or an extremely high hierarchy, a galaxy
say. Just as wet clay can be treated as a S-F system with the S
component under a high state of compressive stress, each
component of the clay in turn can be treated as an S-F system
with the S component under a high state of compressive stress.
One of the components of wet clay is water and so it was
decided to examine the physical data relating to water to
determine whether or not there was any indication of the stress
on the solid structure and whether substitution of total stress for
apparent stress would produce simple relationships between
stress and volume.
Bridgman's[9] high pressure data relating pressure to volume for
water was used and it was found that normal water apparently
only under atmospheric pressure was in fact under a stress of
3750 atmospheres[10]. The reason that water is virtually
incompressible is thus seen as due to the fact that the stresses
we normally apply to water are virtually negligible compared
with the stresses that are already on it.
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Bridgman applied very high stresses to water and at 12,000 atmospheres
the volume had been reduced to 0.8 of the volume at 1 atmosphere.
It was found that adding 3750 atmospheres to the pressure applied by
Bridgman and so getting a total pressure P, led to a very simple
relationship between pressure and volume viz.
P = k(Vs/V)^6 see Fig 2
or to express it in an alternative and more conventional notation
PV^6 = a constant
The agreement between the experimental volumes and the volumes
calculated from the above equation was 1 part in 10,000 which was
of the same order as the experimental accuracy. Examination of
Bridgman's data for other fluids showed that essentially the same
form of relation held for all of them.
LOCATION OF STRUCTURAL COMPRESSION FORCE
The fact that water under an apparent external compression force of
1 atmosphere behaves as though it is in fact under a triaxial compression
stress of 3750 atmospheres suggests that we should regard the tensile
stresses of water incorporated in clay soil not as tensile forces at all but
as reductions in compressions. In effect we should regard the soil
structure as shielding the water to some extent from the pressures to
which it would be exposed if it were free water.
Now comes the most important step of all. Seeing a material as a fluid
in tension and a solid structure in compression is only an intermediate
step towards a much more radical viewpoint of material behaviour.
Though the tension-compression concept is only an intermediate view
which ultimately had to be discarded it is a very necessary step since
it serves as a psychological crutch without which the conceptual gap to
the final view of material behaviour would prove too large. In this the
tension-compression model is analogous to the Tycho Brahe model
of planetary motions in which the planets instead of rotating around
the earth in epicyclic orbits as in the Ptolemaic system, rotated instead
around the sun in circular orbits whilst the combined sun-planetary
system rotated about the earth. Evidently the simpler Copernican
system of planetary motions which was ultimately adopted was just
too large a change of view for Brahe to make in one step.
The conclusion that the intermediate tension compression model
eventually leads to is a very startling one, viz. that materials are not
held together by tension forces acting inside the material but by
compression forces acting outside the material. Water, for example,
in its free state is acted upon by a very large external pressure; a
pressure which resembles air pressure but which is 3750 times
larger; a pressure of which we are as unconscious as we are of
air pressure.
Even though we know of the existence of air pressure we only
appreciate this existence intellectually; we have no emotional
appreciation of the fact that we are externally loaded by a fluid
pressure of 1 ton per square foot and we still for example
emotionally think of a vacuum cleaner as pulling dust up even
though intellectually we are forced to admit that it is air pushing
dust up.
When water is incorporated into a clay particle structure the
compression on the water reduced. We normally look at this as
the water going into a state of tensile strain; in fact the compression
strain already on the water is being reduced. These two situations
are mathematically equivalent but conceptually very different.
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The fraction of the external compression shed by the water is taken by the
clay structure. Thus the clay structure is acting as a semi-permeable
membrane and so allowing a pressure difference to develop between the
inside and the outside fluid.
The fact that this external fluid must exist is indicated by the nature of the
hierarchical system itself. From Fig 1 it is evident that a material can be
made up of a solid and fluid component of the nth order which is in turn
divided up into a solid and fluid of the (n + 1)th order and so on indefinitely.
However, working in the opposite direction, upwards from the (n - 3)
hierarchy say, it is found that when the (n + 1)th hierarchy is reached,
i.e. the object itself, the series seems to stop. This because our existing
concepts of an object suggests that it can exist in nothing, or empty space
which for all practical purposes is regarded as nothing. However it is
evident from the nth and (n + 1)th hierarchical levels that the non-Solid
space is far from empty though is easy to see how it might seem to be
empty to an object in that space.
This section may be summarized by saying that the form of pressure/volume
relationships, the nature of the hierarchical system and a historical appreciation
of the natural tendency to look at physical phenomena from false frames of
reference all suggest that objects are held together from without and not from
within.
NON-EXISTENCE OF TENSILE FORCES
The simplicity of the Solid-Fluid hierarchical system is at one and the same
time both its strongest asset and its greatest liability. It is its strongest asset
because once a phenomena has been fully understood in one hierarchy, it is
understood in all: it is its greatest liability because once an explanation has
been claimed to be correct in one hierarchy it must be claimed to be correct
in all. An explanation is like the general term in a series, it admits no
exceptions; if it is true for the nth term then it is also true for very other term.
It follows therefore that if the hierarchical concept is valid, once a tensile
force has been shown to be the negation of a compression in one hierarchy
then tensile forces in every hierarchy must be negations of the appropriate
compression. One is therefore in the embarrassing position of having a
theory which denies not only the existence of tensile forces at one level but
at every other level as well, atomic, nuclear, sub-nuclear and gravitational.
All so-called attractive forces are manifestations of an external compression
force, an environmental pressure.
Now it can be established that this contention is entirely reasonable.
Firstly, considering electric and magnetic forces. North and south poles or
positive and negati1re charges have the same field pattern as a source and
sink of water at the bottom of a deep ocean. From the Solid-Fluid system
the reason for this is quite obvious. This is just what they are, a source and
a sink at the bottom of an electric and a magnetic ocean respectively.
All we are conscious of are the differential effects as in the case of
atmospheric pressures and we ignore the ambient magnetic and electrical
pressures entirely. Now this contention is testable in principle though
possibly not in practice, since just as the variation of the compressibility
of water with pressure will give rise to asymmetric flow between a source
and a sink when the pressure is large in relation to the ambient pressure,
so similar asymmetries should occur between unlike poles or charges
when the pressure differential is large in relation to the ambient electric
or magnetic pressure. These asymmetries would show whether a north
or south pole was at a higher magnetic pressure and whether a positive
or negative charge was at a higher electrical pressure.
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Secondly, considering co-valent bonding an electron can be thought of as
a relative absence of its corresponding environment, a three dimensional
eddy, a relative hole or bubble. Since the forces of electron bonding are
extremely high and the areas over which they are acting are extremely
small it is evident that the total pressures involved are absolutely
astronomical by normal standards.
Thirdly, considering nuclear bonding, here we are involved with
environmental pressure many orders of magnitude higher than those
involved in atomic bonding with the meson as the relative hole in the
background fluid. Fourth and last we may consider the most famous
attractive forces of all - the force of gravity.
The currently accepted view of gravity, that of general relativity, is
that it is the result of a distortion of space. However, it seems that this
is not a very satisfactory explanation as it transfers the problem from
the word gravity to the word space. It is perhaps significant that the
only really satisfying explanation of gravity was the one given several
centuries ago, viz that it was due to the impact of extremely small
particles to which matter was practically transparent. This leads to a
gravitational radiation pressure which coalesces objects in the same
way that electromagnetic radiation coalesces objects which interact
with radiation. This theory is of course precisely what is suggested
by the Solid-Fluid system as an explanation. It gives rise directly to the
inverse square law since the force pushing two bodies towards each
other is proportional to the area of the mutual shadowing which in turn
is inversely proportional to the square of the distance apart. Moreover,
it indicates that gravitation is a manifestation of a very low level Solid
indeed and acts over areas much smaller than the area of the nucleus.
This in turn suggests that gravitational waves are propagated at
speeds many orders of magnitude higher than the velocity of
electromagnetic radiation.
In the light of the above arguments it is not difficult to see the reasons
behind the failure of 19th century scientists to discover the electromagnetic
ether
(a) they did not realize that there is not one ether but an indefinite number
of different ethers,
(b) they had inherited the false notions of tensile forces from Newtonian
gravitational theory and chemistry together with the false notion of a body
being able to exist in isolation from an environment, i.e. in empty space.
The gross errors in the conceptual foundations of physics proved too much
and the only solution to the perplexing difficulties presented by the
phenomena of electromagnetic radiation seemed that of doubting the
reasonable and obvious solution of the existence of a propagating medium.
This doubt developed into despair under the onslaught of special relativity
and eventually led to a denial of the existence of the ether. This kind of
denial has of course been very common in the history of science. Two
classic examples are the denial of the possibility of any other numbers than
integers and fractions by the ancient Greeks and the denial of the possibility
of splitting the atom by 19th century chemists. A more modern example is
the denial of the possibility of amounts of energy smaller than the quantum.
Unfortunately these kinds of negative assertions have a very bad effect for
whilst believed they act as barriers to all further progress in that direction
by turning an open ended into a closed system with all the consequent
frustration, boredom and destruction of creativity that such a system implies.
In addition, once an unreasonable view has been accepted this provides a
foundation or precedent for the next unreasonable view and soon science is
submerged again by the mysticism it has always striven desperately to
eliminate.
It is not long before history reasserts itself and a new religion is set up
complete with its initiation ceremonies and burnt offerings, temples and
high priesthood who act as intermediaries and interpreters between the
unintelligible tyrant of the nuclear underworld, say, and the common man.
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Having touched upon the main issues which the S-F system raises it is
important to discuss the two most important aspects of all physical
phenomena, viz the nature of space and the nature of time, for evidently,
a proper understanding of both these is required for any universal
system such as the one proposed.
NATURE OF SPACE
The hierarchical Solid-Fluid system gives rise to a new notion of space
which is very different from conventional notions. In the Solid-Fluid
system space is a fluid and a point in space is an element of that fluid.
It is evident that in this system space is not the static grid of points which
is implicitly assumed but not explicitly stated by Newton and Einstein
but a dynamic grid which is constantly changing with time .
Consider for a moment that the fluid is a gas with the individual elements
(molecules) in constant motion relative to each other. These molecules
are the points in space. Thus we can see that space has three very
important properties; it is real, it is quantized, it is dynamic.
It is therefore quite different from the static unreal infinitely divisible
space which is implicit in most descriptions.
Now a dynamic space of this type gives rise to some very interesting
properties as can be shown by the following examples .
Consider an object surrounded by a fluid which for the sake of
simplicity we shall take as a gas .
Firstly consider what meaning can be attached to velocity. Velocity
implies change in position with time relative to some frame of reference.
But the only frames of reference are the molecules of the gas themselves
so there are as many frames of reference as there are molecules.
Let us consider the velocity of the object relative to a molecule moving
to the left at speed Lv. Relative to this molecular frame of reference the
object will be moving to the right at speed Rv. Now by considering each
molecule in turn we can get a complete set of vectorial velocities for the
object. We find that the object is moving in all directions at once at a
wide range of velocities. At first sight this statement seems paradoxical
in view of the fact that the object is obviously' stationary. However, this
paradox, like others, arises because the two conflicting statements have
different reference frames. The reference frames for the statement that the
object has a range of velocities and directions are the molecules
considered as individuals. The reference frame for the statement that the
object is stationary is the molecules taken as a group.
Thus we can see that velocity is a statistical concept and that what
velocity we actually obtain for a body will depend on how we decide
to combine the population elements and. this in turn will depend on what
particular aspect of the population is relevant to the problem we are
trying to solve. In the case of velocity we could combine the elemental
velocities algebraically for example, in which case we would get a
velocity of zero. On the other hand we could combine them non-
algebraically by averaging v2 say which will give us a measure of the
energy of the object relative to the individual particles of the field.
The same line of reasoning can be applied to angular velocities.
If next we consider particles which are actually hitting the object we can
see that during the time interval of impact these particles are accelerating
away from the object. t1ow since the field particles constitute the only
frames of reference there are for specifying the acceleration of the object
one can see that with equal truth the surface of the object is accelerating
away from the particles. In other words the surface of the ball is
continuously accelerating towards its own centre.
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If the essence of the argument has not been properly grasped, this
statement must sound as paradoxical as the statement about velocities;
how can the object be accelerating when it is obviously stationary and
if it is accelerating towards its own centre how is it that it never arrives
at its own centre.
As in the case of velocity the answer lies in the fact that the space in
which the object exists is not a static space but a dynamic space.
When we say that the surface of the object is stationary we implicitly
assume a static space in which points in space are stationary in the
same way as the points formed by the intersection of grid lines on a
sheet of graph paper are stationary. But space only approximates to
this condition when averaged over a large number of points.
At the quantum level of space the points are moving and space is
dynamic, or more simply, space is a particle field.
MULTIPLICITY OF SPACES
Just as there is a multiplicity of hierarchical fluids or particle fields, so
there is a multiplicity of corresponding spaces.
Any real object, since it is a hierarchical body, exists in a multitude of
different spaces. Consequently it can be moving at one speed in a
particular direction in one hierarchical space and quite another speed in
the opposite direction in a complementary hierarchical space .
A good example of this is provided by a cork floating on water.
The two complementary spaces in which the cork exists are the air
and the water. These spaces are particularly useful for explanatory
purposes because they are separated and not intermingled as are
hierarchical spaces in general. Usually the two hierarchical spaces,
the air and the water, will be moving relative to one another.
If we use one space as a frame of reference, the water say, the cork
will be moving at a speed Vw in a direction Ødeg through that space.
If on the other hand we use the air as a frame of reference then the
cork will be moving at a speed v[a] in a direction (180deg + go)
through the air space. The numerical sum of the relative speeds in the
air and water spaces will equal the speed of the two spaces relative
to each other.
The movement of the cork through the two spaces in opposite
directions will set up two drag forces, fw in direction (180 + Ødeg)
in the water space and fa acting in direction Ødeg in the air space.
The ratio va/vw will be governed by the requirement fw = fa.
In the above example two particular hierarchical spaces were
considered and these spaces were complementary, i.e. they were
taken as comprising the totality of spaces in which the object existed.
The spaces were defined as air and water space for ease of
explanation but they could have been defined more precisely as water
and not water space; this definition makes the totality of the space
more apparent.
The general principles which apply to two complementary hierarchical
spaces can be extended to n complementary hierarchical spaces which
can be separate as in the case of air and water or co-extensive as in the
case of air and a cloud of gnats, say. In general a body will be moving at
a different speed and in a different direction in every hierarchical space
in which it exists. The relative speeds will be those for which the algebraic
sum of the drag vectors is zero.
At this point it is pertinent to ask the question, what is the relationship
between a hierarchical space and Cartesian space?
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This question is best answered by drawing a comparison between
Cartesian space and the interval between zero and one. In a unit of
Cartesian space the number of points is unlimited. In a hierarchical space
however the number of points is limited and can be specified numerically.
Thus a particular hierarchical space corresponds to a particular class of
fractions, 1/1000ths say. A different hierarchical space will correspond
to a different set of fractions 1/999 ths say. It can be seen therefore that
we can carve an unlimited number of hierarchical spaces out of Cartesian
space.
The objection to the Cartesian concept of continuous space is that it
involves unlimited numbers and these cannot be specified numerically nor
grasped conceptually whereas the concept of hierarchical space can
because of its discontinuous nature.
This problem of continuity has occurred many times before.
The classic example is of course the conflict between the wave and
particle behaviour of electromagnetic radiation; a more revealing but
less well known example is the one given by communication theory
where it is only by treating a varying signal as a discontinuous process
that problems of information content and coding become tractable( 6) (12).
EXTERNAL AND INTERNAL SPACES
For every identifiable level of an object there will be an external and an
internal hierarchical space. The internal space comprises the external space
of the object at the next identifiable level down. The existence of the
object at any level is the manifestation of the difference between the
external and the internal space. For a hierarchical space where the
difference between these two spaces is negligible, the existence of this
space may be neglected in considering the properties of the object.
If existence at any level is defined as the manifestation of a difference
between the object and its environment at that level then for hierarchies
where the difference between the internal and the external environments
are negligibly small the existence in these hierarchies can in practice be
neglected.
The surface of an object is the boundary between its internal and external
environment. Where there is a sharp discontinuity between internal and
external environment there will be a sharp boundary; where the
discontinuity between internal and external environment is more gradual
the boundary surface will be correspondingly diffuse.
Since an object exists in a hierarchical set of spaces it should have a
corresponding hierarchical set of surfaces. It will be readily recognised
that this is indeed so. Moreover, an object is only a single object at its
highest hierarchical level of existence. At lower hierarchical levels it is a
collection of objects. This raises the problem of the one and the many,
the simple and the complex which will be dealt with in a later section.
Just as an object is in equilibrium in relation to all the forces generated
by existence in and movement through the external hierarchical spaces,
so an element of the surface is in equilibrium between the external and
internal environments.
HIERARCHICAL SURFACES
Consideration of internal and external spaces has focussed attention on
the fact that a body has many different hierarchical surfaces.
We normally consider the surface of an object to be the surface detectable
by sight. In general this surface will also coincide with the surface detectable
by touching the object with another object but many exceptions to this rule
occur. An obvious example
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is that of a plate glass door under the right lighting conditions. A less obvious
example is a pair of magnets the like poles of which are brought together;
in this case the magnetic surface is exterior and much less sharp than the
visual surface and we do not recognise its surface characteristic.
Recognition of the existence of different hierarchical surfaces leads to the
realisation that all repulsive forces between objects are manifestations of the
deformation of a hierarchical surface. In other words, repulsive forces
commence when the two hierarchical surfaces touch. This is shown in Fig 3
in relation to the Condon-Morse curve by dividing the force distance
diagram into three separate regions. When the atoms are separated then the
distance of the atomic surface from the atomic centre is d2. When the atoms
are at their lowest potential energy position, i.e. total force F = 0 then the
distance of the surface from the atomic centre is d1. The zone between d1
and d2 constitutes the surface zone, in other words the surface. This example
draws attention to the importance of time and volume in relation to the concept
of surface.
The importance of time can be illustrated by a simple example. Suppose we
have an aeroplane propeller rotating at high speed. If a goose flies into the
propeller it will soon find that the surface of the propeller is defined by the
swept volume of the static propeller. We may describe this surface as the
dynamic propeller surface (an interesting example of dynamic volumes of this
type is given by the Bènard cells formed by convection currents in a thin layer
of fluid uniformly heated; see Fig 4). However for a bullet fired at the
propeller the surface of the propeller is essentially the same as the surface of
the static propeller. It can be seen that just as different hierarchical surfaces
of an object become manifest as the spacial scale of scrutiny is altered so also
different hierarchical surfaces become manifest as the temporal scale of
scrutiny is altered. The importance of volume in relation to the concept of
surface is that it gives it reality. A surface must not be though of as a
geometric abstraction, as a two dimensional concept with no physical
reality, but as a real entity, dependent on the antecedent existence of the
object and the environment but quite distinct from them. Examples of this
interaction term can be given for every system. The child is an example of
the interaction term in the sociological field. It is dependent on the antecedent
existence of its parents but it is quite distinct from them. In a biological context
the surface or interaction term constitutes a skin or membrane which is a
physical entity in its own right having a definite though usually small thickness.
In terms of political geography the surface or interaction term comprises a
border zone which will generally be found to constitute a small but distinct
region in comparison with the two parent countries. In communications the
surface term corresponds to the communication channel between the sender
and the receiver, a physical entity distinct from either the sender or the receiver.
In sentence structure the surface term corresponds to the verb which relates
the subject to the object. In mathematics the surface or interaction term is the
combinatorial sign between the two numbers. This combinatorial sign has a
conceptual reality equal to the conceptual reality of the number concepts.
THE NATURE OF MEASUREMENT OF PHYSICAL QUANTITIES
An appreciation of the nature of space as outlined above draws attention to
the fact that measurements of any kind can only be made in a relative and
not in an absolute sense. This means that all measurements made in physics
are non-dimensional and therefore numerical.
For example, vie do not measure length, we measure specific length and
provided we know the standard for length it is unnecessary to refer to it
(this fact is utilized in modern engineering drawing practice).
Thus measured length is non-dimensional in precisely the same way that
specific gravity is non-dimensional.
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When we specify the specific gravity of a material we merely specify a
number; it is understood that this number refers to the density of water
as a standard.
In precisely the same way, when we give a measure of length we only
need to quote a number. In fact we do usually give a unit because for
historical reasons there are many different standards of length and we
quote a unit to show which of the standards we are using. However,
if as in our measurement of specific gravity we adopted a single
standard the need to quote it would no longer arise.
In effect the concept of length in the measured length cancels out with
the concept of length in the standard length and only the number
expressing the ratio is left. Though this principle was discovered
independently by the author it is not a new one(11).
Now just as the concept of length is redundant as far as the
measurement of length is concerned, so the concepts of the so called
fundamental dimensions are redundant in measurements of what are
considered as less fundamental properties such as force, acceleration
etc. This can be illustrated by taking the property velocity and showing
how the concepts of length and time are redundant as far as the
measurement of this particular property is concerned.
Suppose we wish to measure the velocity of an object. Normally we
would measure out a course over which the object could travel and
then use a clock to measure the time for the object to travel from the
beginning to the end of that course. We would then express the velocity
as the length divided by the time and because of the way in which it had
been derived it would be natural to think of velocity as having the
dimensions of length/time. However, we do not need to measure
velocity in this way. There are other ways which do not involve the
notion of either length or time, eg if we set up two mirrors facing one
another and send the object and a light ray from one mirror to the other
mirror we can count the number of times the light ray hits the second
mirror before the object hits the second mirror. The velocity of the
object is then the reciprocal of this number. This system of measuring
velocity gives rise to a pure number which, by a suitable choice of scale
constant, is seen to be exactly the same number as the measurement
using measuring sticks and clocks. If we consistently measured velocity
in this way we would cease to think of it as being connected with length
and time and instead think of it as a pure number. This process would
seem quite natural; the measured velocity would after all be independent
both of the distance apart of the mirrors and the time taken to carry out
the measurement. Obviously we would have to associate the concept of
velocity with specified processes but these processes would not involve
concepts of length and time but notions of reflections, events and counting.
We could of course have arrived at the notion that velocity is
dimensionless by showing that each of its composite terms, length
and time, is dimensionless but this would have been an indirect route
involving more steps than is necessary and it was considered more
illuminating to give the direct argument.
The above lines of reasoning lead to the following conclusions.
All physical measurements can be expressed by a ratio which is
the numerical relationship between two similar processes, or in
symbols,
R = Np/Dp
where R stands for the ratio and is dimensionless N stands for
the numerator derived from the process p and D stands for the
denominator derived from the process p
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Generally the denominator will be a standard of some kind
such as the platinum metre for example, and its length will
be taken as one unit. There is no particular reason why it
should be taken as one unit; we could take it as 1000 units
or even 13 units if we so desired. The really important thing
about the denominator is that it is either assumed to be or
defined as constant.
Now normally, when we measure the ratio R and find that
it has altered from some previous value we attribute this
alteration to an alteration in the numerator term Np.
We fail to allow for the possibility that the change in R could
be the result of a change in the denominator term Dp.
For example, if we measure a desk with a foot rule and find
that it is five feet long and then on the next day we measure
it again and find that it is six feet long we would probably say
that our desk had lengthened. However, implicit in this
statement is the assumption, convenient no doubt but logically
unjustifiable, that our ruler is constant rather than our desk.
If we wish to avoid unwarranted assumptions of this kind we
have to make symmetrical statements for the ruler and the
desk and say that our desk has grown longer (implicitly relative
to the ruler) and our ruler has grown shorter (implicitly relative
to the desk) and avoid making statements which are unrelated
to specific physical objects.
The failure to appreciate the importance of the denominator
part of a physical measurement is a crucial error in traditional
physical theories. A similar error occurs in strength of materials
where stresses are thought of in terms of external conditions
only and there is a failure to appreciate that stresses are a
manifestation of differences between external and internal
conditions and can be caused by either an alteration of external
conditions only or by an alteration of internal conditions only
or, and this will be the general case, by an alteration in both
external and internal conditions.
The necessity for appreciating the importance of the
denominator can be illustrated by a further example taken
from an economic context . Suppose a miser has £100 in
bank:.notes which he counts every .night to make sure no
one has stolen any. Every night he goes to bed secure in
the knowledge that he has counted up to 100 and all his
notes are there. The numerator is constant and he imagines
that this is all that matters. However, what he does not
realise is that every day the Government is producing more
notes (as they usually do) and they are thereby stealing his
money by altering the denominator. He does not realise
that the value of his notes is governed not only by the
number he holds, which he can check, but also by the total
number that there are which he cannot check. Another
example of denominator manipulation by the Government
is the use of British Standard Time and no doubt there are
many others. The effect of denominator change appears
to have been appreciated far more in politics than in science
where there has been an uncritical faith in denominator
constancy.
It is true that in one branch of science the importance of the
denominator has been recognized, at least implicitly, and
this is the province of dimensional analysis. This is the reason
why dimensional analysis is so mysteriously powerful and
eminently useful. Its secret of success is that by taking ratios
of concepts of length, time, mass etc. it has cancelled out
the false notions which permeate these concepts; it has taken
away something which was never really there in the first place.
In this it resembles the schoolboy conundrum which involves
taking away the number you first thought of. Dimensional
analysis takes physics back from the erroneous habits of
dimensional thinking in redundant concepts towards a non-
dimensional mode of thinking in terms of number.
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Though the desirability of applying dimensional analysis to a much
wider range of problems than those so far tackled is appreciated
there is no clear understanding of just how this can be accomplished.
Part of the reason for this is that dimensional analysis still uses the
concepts and notation of classical physics which because they are
based on fixed conceptual frames of reference are in some contexts
too simple and in others too complex for efficient application of the
analysis. This leads to the problem of simplicity and complexity of
concepts, structure and mechanism which are considered in the
next section.
THE NATURE OF SIMPLICITY AND COMPLEXITY
Dimensional analysis solves the problem of the hidden
denominator in physical measurements since it always works
in terms of ratios. The solution is unintentional but none the less
real. However, dimensional analysis does not solve the problem
of fixed frames of reference in relation to the concept of simplicity.
In traditional scientific thought certain entities are treated as simple
or fundamental and compounds of these entities are treated as
complex or non-fundamental. There is a failure to realise that
complexity and simplicity are relative terms and that to describe
a term as complex or simple is not in fact a statement about the
entity but a statement about the relation between the entity and
the observer. Alteration in the position of the entity or of the
observer will lead to an alteration of the relationship.
The measurement of physical quantities discussed in the previous
section is a particular case of this with the numerator as the
entity and the denominator as the observer. To forget the
relevance of the denominator term is equivalent to forgetting
the importance of the observer.
Hence, in dimensional analysis, an apparently complex non-
dimensional group is not intrinsically more complex than a
simple non-dimensional group, any more than as has been
shown previously velocity is a more complex concept than
length. Or to express the situation in different terms, the
apparent degree of complexity of an entity is merely a
reflection of how far away in the hierarchical system the
starting point for developing the entity was from the entity
itself. This statement applies to all entities, concrete or
abstract, e.g. physical objects, physical mechanisms,
concepts etc.
The truth of this statement can readily be illustrated for
any hierarchical system. Take volume for example; this
concept is normally thought of as being more complex
than length because we normally start with the concept
of length and proceed via the concept of area to the
concept of volume. However, if we worked the other
way round the concept of length would appear to be
more complex than the concept of volume. This fact is
even reflected in the symbolic mathematical notation
representing the two alternative procedures although
the implications of this fact are not appreciated.
If we start with length denoted by L then volume will
become (L)^3 which is a more complex symbolic notation
than1L. Similarly, if we start with the concept of volume
V then length becomes (V)^(1/3) which is a more complex
symbolic notation than V.
A further illustration of this point is provided by considering the
hierarchical system with which we are most familiar since it is the
system in which we ourselves exist. An (n)th generation of people
can seem a more complex or a less complex entity than an (n + 2)th
generation depending on where we start. If we go man, father,
grandfather then the concept grandfather seems more complex than
the concept man (this increasing complexity shows up in the word
form to some extent. We can make it show up even more if we
amplify the terms slightly to man, manfather and mangrandfather).
However, if we go the other way round,
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i.e. man, son, grandson then the concept grandson seems the most
complex. Of course a man from the west with a western veneration
for youth might maintain that the grandfather, i.e. the (n)th generation
was the most complex because he would always have more generations
of descendants than a person in the (n + 2)th generation.
On the other hand a man from the east with an eastern veneration of
his ancestors would probably hold the opposite view and claim that
the (n + 2)th generation was the more complex concept because a
man in the (n + 2)th generation would have more generations of
ancestors than a man in the (n)th generation. In these circumstances
the most reasonable view to hold would seem to be that the equality
of man embraces both ancestors and descendants and that both
groups are of equal worth. Therefore the net number of generations
is the same for the (n)th and the (n + 2)th generations and of course
every other generation and so each term in the hierarchical series is
equally simple and equally complex.
We can summarize the above discussion by the formal statement:
the complexity or simplicity of an entity is not an intrinsic property
of the entity itself but a statement of the relationship between the
entity and the observer.
The potential simplicity of apparently complex entities has an
important corollary and that is that apparently simple entities are
not as simple as they look but can always be shown to be
compounded of other entities at different places in the hierarchical
system. Failure to appreciate this paradox that the apparently
simple is potentially very complex and the apparently complex is
potentially very simple is a major stumbling block in the path of a
fuller understanding of physical phenomena.
The above discussion draws attention to the need to modify one
of the basic notions of science, i.e. the notion of atomicity.
Though the notion of absolute atomicity in relation to the chemical
atom was destroyed at the beginning of this century by the discovery
of radioactivity, the notion of absolute atomicity per se was not
destroyed. It was merely transferred, first to the proton, electron
and neutron and more recently to the quark. It also crops up in a
somewhat different form as the quantum of energy . The lesson does
not yet seem to have been learnt that there is no absolute atomicity
but only a relative atomicity. This is true not only for concrete
material objects but for abstract immaterial objects such as concepts
also.
At first sight the destruction of the notion of absolute atomicity appears
to be a great loss because it destroys the possibility of a total
explanation. In practice however, we never need a total explanation
and it is a great gain because recognition that all levels are equally
atomic means that we can choose our atom of structure or concept
at a level close to our problem and obtain a simple and sufficient
explanation. This point is best illustrated by an example.
Supposing we drink sweetened tea without realising how it is made
and we find that some days it is sweeter than others. We are not
satisfied with this variation and so we decide to have the tea
analysed to find out what controls the sweetness.
If the analysis is carried out at the chemical element level we will
find that there are many differences between plain and sweet tea.
However the complexity of the differences would defeat our
attempts to control sweetness, especially since differences in milk
and tea content to which we are indifferent might swamp the
differences due to the addition of sugar. If the analysis was
carried out at a much higher level, possibly historically by seeing
how the tea was made, we would soon discover that the degree
of sweetness was governed by the amount of white powder put
in. In the context of our problem it does not matter in the least
that we do not know what the white powder is made of or what
the tea leaves milk and water are made of. Indeed, to know this
redundant information would
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only lead to confusion since the redundant information would hide the
essential information in the same way as noise hides the signal in a
communication channel or the variegated colouring of a chameleon
and its surroundings hides the outline of the chameleon's body.
This section can be summarised by the following statement.
Hierarchical structuring of scientific concepts and material objects
leads to the very important conclusion that terms such as simple and
complex, atomic and compound, fundamental and non-fundamental
are all relative terms which are dependant on the hierarchical frame
of reference from which the structure or concept is viewed.
When confronted with the problem of understanding a particular
entity we require the simplest explanation possible and this requires
a starting point fairly close to the hierarchy in which the entity exists.
THE PROBLEM OF NAMES
Having discussed the nature of space when looked at hierarchically
it is necessary to discuss the allied but much more difficult topic,
the nature of time. However before going onto this it is necessary to
examine the nature of energy and for reasons which will become
apparent this involves the problem of names. A hierarchical system
is extremely difficult to deal with unless one has an adequate notation.
At first sight notation or naming seems to be an arbitrary and
unimportant exercise, 'a rose by any other name ........' but with all
respect to the author of this quotation this is just not true as every
modern advertiser knows. The quotation makes the mistake of
forgetting the denominator term or observer which in this case is
the person smelling the rose. Because names have multiple
associations and conjure up all kinds of images and concepts they
have a great effect on people; surely a rose named Nazi would not
smell as sweet as a rose named Peace.
In mathematics the central importance of notation is illustrated by
the revolution brought about by dropping the Roman system of
notation and introducing the Arabic.
This can be regarded essentially as the substitution of an
inconsistent hierarchical structure by a consistent hierarchical
structure.
In language the importance of notation is illustrated by the greater
ease of learning to read and write in a European notation than in a
Chinese notation. The difficulty of appreciating the relationship of
one element in a hierarchy to another without an adequate system
of names is illustrated by proper names in family structure. If the
Icelandic method of notation is adopted then Peter Thompson's
son becomes David Peterson, David Peterson's son becomes
Francis Davidson and so on. Clearly it becomes very difficult to
appreciate more than immediate relationships in such a system.
The British system of surnames is better since at least this does
indicate a connection through the male line. A statement of the
generation number, B is of the 4th generation, R is of the 9th
generation, would be an improvement and it is not difficult to
define even better numerical schemes.
Besides the very large set of proper names we also have a small
set of relative names, i.e. father, mother, brother, aunt, etc. and
these are invaluable for understanding situations involving local
regions of the hierarchy. They convey the relationship between
the elements without conveying redundant information or noise to
confuse the issue. The relative nature of the various terms in the
hierarchical series is also more obvious than in other notations.
If the hierarchical consistency of physical phenomena is to be
fully recognised then just such a notation needs to be developed,
a notation which can be applied to any hierarchical level.
The terms Solid and Fluid used in this note are an attempt to
develop such a notation for the context of material phenomena.
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At present the various sciences are like various languages that have
grown up in different territorial regions. Originally the elements of
language may have been the same but because of the lack of
intercommunication between the different regions they have grown
away from each other and now contain relatively little trace of their
common origins. The essentially common nature of their problems
and situations is only seen when their problems are put into the
international Esperanto of mathematics (delta2 V = 0 for example)
but since nobody uses Esperanto in their daily affairs or thinks in
Esperanto, the real significance of this similarity is not appreciated.
It is true that some concepts seem to be hierarchical, eg energy,
and it is significant that these concepts are the most powerful.
However, the concept of energy is still unsatisfactory in many
respects. Firstly, it is dimensional; consequently every time we
change hierarchies we have to introduce an arbitrary conversion
coefficient or factor of ignorance to make the equations correct.
Secondly, it is tied in primarily with a conceptual context of
force x distance or strain energy. It would be better if both these
were regarded as derivative concepts of some more fundamental
concept.
A better concept than energy is entropy since here energy is at
least put into a non-dimensional form and thought of in relative
rather than absolute terms.
Unfortunately entropy has unfortunate historical antecedents too.
It is connected primarily with the notion of temperature and is
largely a prisoner in that hierarchy Also it is thought of in the
emotively negative and unattractive terms of increasing disorder
and eventual heat death of the Universe. This is because it is
looked at in a one sided way and the decrease in entropy in a
superior hierarchy which accompanies the increase in entropy
in an inferior hierarchy is neglected.
If, for example, one observed man's food cycle without taking
into account the ordered things it enabled man to do then it might
easily be imagined that the process was merely one of increasing
disorder in which food was degraded into waste products and
heat.
The concept which seems to be ideally suited for a hierarchical
role is the concept of information as used in communication
theory( 12) . This is a numeric concept which has no dimensional
terms associated with it. Its emotional connotations of knowledge,
understanding and meaning are positive and attractive and this fact
is an important one for reasons discussed previously. As will be
shown in the next section, increase in time or increase in length
both lead to increase in information content and thus information
provides an underlying connection between the two which is
readily understandable and therefore can be readily manipulated
in theoretical analysis. This connection between time and length
has been suggested by the theory of relativity, though certainly
for the majority of people and probably for everyone the theory
is unintelligible. This is because length and time have implicitly
been treated as self evident entities and have never been
analysed into simpler elements. One of the reasons preventing
such an analysis is that both concepts have been treated as
continuum concepts and as such are unanalysable.
THE NATURE OF TIME
The nature of time can best be appreciated by developing the
concept of time from Information theory.
Consider a discrete source of information consisting of a single
channel which can take the value 1 or 0. Suppose the maximum
speed with which it can change its state is one change per unit
of time. Suppose its physical dimension is some given direction is
one unit of length.
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In time 10t this source generates 2^10 choices or since Information (I)
equals the logarithm to base 2 of the number of choices
I = lg(2^10) = 10
therefore the source generates 10 units of information in time 10t.
If we increase the size of the information source to 10 channels the
physical dimension is increased to 10 units of length and in one unit of
time the same amount of information is emitted. Thus in terms of
information,time and length are equivalent. But if time and length are
equivalent this means that just as we should be able to increase or
decrease length so we should also be able to increase or decrease time.
In other words increasing length is equivalent to time moving forward
whereas decreasing length is equivalent to time moving backwards.
Now the suggestion that time can move backwards is surely a reductio
ad absurdum as far as the argument is concerned; it seems to contradict
a deeply held psychological experience that time is essentially different
to length in that it is a dimension in which we are always moving forward.
The solution to this conundrum lies in our faculty of memory for it is this
which gives time its arrow for us. In projecting our own subjective
psychological feeling about time into nature we are guilty of an
anthropomorphism every bit as unjustifiable though as excusable as the
geocentric anthropomorphism of the Ptolemaic system. This can be
demonstrated by the following example. Consider an individual moving
forward in time. Certain physical changes are taking place in his body
and certain memory traces are being made in his brain. 'vie can summarize
these processes ~ saying that information is being transferred from the
environment into the object which in this case is the individual concerned.
The individual is conscious of the growth of memory traces because his
consciousness has access to them, i.e. he can recall them at will, he
knows the past. The memory traces which are going to be put in his
brain but which are not yet there, he does not have access to, he does
not know the future. Suppose now that we can control the physical
processes that govern the growth of his body and the input of memory
traces to his brain and suppose we reverse those processes so as to
return both his body and his memory to an earlier state. It is evident
that he would have no knowledge of this reversal of his time because
he would have no access to the information that had been taken out
of his memory and put back into the environment, in other words he
would have no knowledge of his future. It is evident that we can never
be conscious of a negative movement along our own personal time
scale any more than we can turn round in relation to ourselves.
This is because our personal time scale is a conscious time scale and
it is from this time scale that we obtain our impression of moving
forward in time. If we could not recall the past in any respect we
would have no consciousness of time flow. Also if we could call up
knowledge of past and future with equal facility we would have no
consciousness of time flow either.
To return for a moment to the individual who was moved backwards
in his own time scale by reversing his physical and mental processes,
it is clear that during the period that he was moving backwards in his
conscious time scale we were moving forward in ours. We may
therefore think of our time scale as superior to his since his time scale
shows a reversal with respect to ours but our time scale shows no
reversal but a forward jump with respect to his. We would be in a
position where we could predict his future (assuming his environment
to be reasonably constant) and he would no doubt regard us as a
prophet in much the same way as a primitive savage would regard a
man who could predict eclipses of the sun a prophet. As far as we are
concerned we are not prophesying but only saying what for us has
already happened; however, as far as he was concerned, we would be.
Now all prediction in science is based on this ability to know what will
happen in the future because in our superior time scale it has happened
already. To say that a process is periodic or repetitive is to say that its
time scale is inferior to ours. Without periodic or repetitive processes
prediction in science would be impossible.
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It is paradoxical that we think of the pendulum as a time measuring
instrument when the only time it can really measure is that elapsing
between the top and bottom of its swing. In fact of course the
pendulum is only one of the elements we employ in a device to
measure time. The other element is a counting instrument of some
kind, a clock mechanism for example or a heap of stones and a jug
into which we put a stone every time the pendulum swings. The essential
nature of the counting device or memory can be assessed if we consider
what would happen to our measurement of time if every occasion we
dropped a stone into our jug the previous stone dropped through a
hole and was lost.
Any reversible or cyclical process therefore is a process which involves
the ebb and flow of local time. First we may consider a particular
example of this and then a general statement which applies to all cases.
The swing of a pendulum involves an exchange of energy between the
pendulum and the field. During the downward swing energy is flowing
from the field into the pendulum. During the upward swing the reverse
process occurs, the energy flows from the pendulum into the field.
Relative to the pendulum the field energy (which is kinetic energy of
the field particles) is potential. Relative to the field the pendulum kinetic
energy is potential.
Now forward flow of time may be identified with increase in kinetic energy.
Therefore during the downwards swing of the pendulum, time for the
pendulum is moving forwards. During the upwards swing, time is moving
backwards. The reverse situation holds for the field.
The general statement of which the pendulum case above is a particular
example is as follows:
Any cyclical process involves the exchange of information between an
object and its environment. During one part of the cycle information is
flowing from the environment into the object and time is moving forward
for the object but backwards for the environment. During the remaining
part of the cycle information is flowing out of the object into the
environment and time is moving backwards for the object but forwards
for the environment.
Thus time is the change in information dl and since for a change in a
closed system a + b
therefore
dla = -dlb
dla + dlb = a
dta + dtb = a
i.e. in a closed system, time is conserved.
The failure to recognise the complete symmetry between the object
and the environment and the consequent conservation of time is just
another case of the failure to recognise the importance of the
denominator term or observer or environment.
At this point it is useful to return to the pendulum example and
consider the hierarchical time scales which are involved in time
measurement. The time of our pendulum is ebbing and flowing
between zero and N the number of elements in the swing of the
pendulum from its highest to its lowest position, whilst the time
of our counting device is moving forward. Eventually, because
the mechanical device is finite, the jug becomes full with stones,
say, or the hands of the clock point to 12, the time of our
mechanical device returns to zero. In the
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case of the jug which we shall suppose is emptied out and of the clock,
the time jumps back to zero and does not return gradually to zero as the
pendulum time did. If we want a time scale that continues to move
forward we have to introduce a counting device in a superior hierarchy
for which time goes forward and by which we can record the number of
jumps or returns to zero, a calendar for example. This process can be
extended indefinitely. It can be seen therefore that measurement of time
interval does not depend upon one hierarchy but on a whole series of
hierarchies. Furthermore, the hierarchical scales which we use to
determine time do not bear a constant relation to one another;
for example, on earth in 1970 the relationship between the day and the
year is 365. However, this is not so on the moon nor will it be so on
the earth in the year 1970 x 109.
of course it could be argued that although we do use hierarchical scales
to measure time in fact we do not need to do this in principle. We could
just as easily count the most rapid event available and use this count as
our measure of time. However, it can be shown that this notion is just
not true. We do have to use hierarchical scales to measure time and we
can never guarantee the constancy of one hierarchical time scale in
relation to another.
Consider the time interval 9 years, 3 months, 2 weeks, 3 days,
1 hour and 5 seconds. Now as has been pointed out these hierarchical
scales do not bear a constant relation to one another. Surely we can get
over this by eliminating the hierarchical nature of the statement and
expressing the interval all in a single hierarchy in seconds; 293,418,005
seconds in fact. Though we are not used to dealing with time intervals in
figures looking like the national debt, presumably we could soon train
ourselves to. We have surely established that though we do use
hierarchical scales for convenience we do not need to do so.
However, 293,418,005 seconds is expressed in hierarchical scales.
Not scales of seconds, hours, days, weeks, months and years it is true,
but instead scales of seconds, tens of seconds, hundreds of seconds,
thousands of seconds, ten of thousands of seconds etc and these are
just as much hierarchical scales as seconds, hours, days etc.
If we really wanted to express the time interval in some non-hierarchical
way we should have to express it by a number thus 1111111111...
i.e. involving 109 marks on paper. Such a mode of expression would be
completely incomprehensible unless we adopted some implicit
hierarchical structuring such as the number of pages of 1's, the number
of books containing a certain number of pages each etc.
The numbers of items we can comprehend without some implicit
hierarchical structuring of this kind is no more than about 7 and even for
7 and smaller numbers there is probably some hierarchical structuring
taking place at a subconscious level. Suppose then it is accepted that
even when we measure in seconds we do in fact measure time in
hierarchical scales. Surely however, there can be no question of the
scales not bearing a constant relation to one another. After all ten units
are always 10 and ten tens are always 100. In the conceptual numbering
hierarchical system this may be so since we have a complete control of
the system and can make the hierarchical relationship constant by definition.
However, any physically realisable system of measuring time will consist
of some hierarchical arrangement of counting devices and since we do
not have complete control over the present, let alone the future, physical
environments/we cannot guarantee that the relationship between one
hierarchy and another will remain constant.
Suppose we consider a particular example of a physically realisable
system for measuring time, a pendulum say, with a hierarchical counter
resembling a mileometer, i.e. consisting of several toothed cogs with
one turn of the first cog producing 1/10th of a turn of the second cog
and so on. Now over a short period of time the relationship between
the movements of the cogs will be constant just as the relationship
between the day and the year is constant over a few centuries. As the
counting device wears out a point will be reached that the mechanism
slips one tooth say and eleven turns are needed to
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operate the decade counter instead of ten; the hierarchical relationship
has thus altered. In general therefore the relationship is not constant and
so we can conclude that any physically realizable measure of time is
hierarchical and the hierarchies do not bear any fixed relationship to
each other. Precisely the same lines of augments can be set out for
measurements of length to show that measurements of length are
hierarchical and do not bear constant relationships to each other either.
The eminent English philosopher Berkeley pointed out centuries ago
that physically unrealizable systems had no place in physics and one
would have thought that by now his contention would have been
accepted as sine qua non.
In the event it appears to be just as difficult to exorcise magic
from science as it is to exorcise it in other places.
The trouble with moving away from primitive measurements of length
and time such as the barleycorn and the foot, the day and the year,
and going to methods which are somewhat more constant is that we
are deluded into thinking we have solved the problems relating to
hierarchical scales. In fact we do not solve them this way at all but
merely push them into the background where they are more difficult
to see.
If we return to more primitive methods of measuring time and length
we will see that, as could be deduced from the sections on simplicity
and complexity, the notion that time and space are simple entities or
simple concepts is illusory; they are as complex as the world they
describe. Likewise the notion that time and space are one and three
dimensional respectively is also illusory. The dimensionality of any
concept or physical entity is merely a reflection of the hierarchical level
from which we are viewing it. Space has as many dimensions as we wish
to give it and so has time.
CONCLUDING REMARKS
The S-F hierarchical system has opened up so many fresh approaches
to natural phenomena that in writing about it one is faced with an
embarras de richesse.
However a choice must be made and this note has confined itself to
drawing attention to the major aspects of the system. Before closing
it is appropriate to touch briefly on some of the many other consequences
of this theory.
Negative quantities of any kind have no place in physics, i.e. physics
deals with natural numbers and not integers. Negative mass for example
must be the manifestation of an object with a lower mass then the
environment. The masses we measure by weighing or accelerating an
object are the apparent masses not the total masses.
Because the number of hierarchies is unlimited the potential energy in
any lump of material or any volume of space of any size however small
is also unlimited.
The equation E = mc^2 merely refers to the energy conversion factor
at the electromagnetic level. In general the energy available will increase
as the hierarchical level is lowered. It follows that any conservation law
confined to a finite set of hierarchies will always break down if looked
at sufficiently closely. The so-called zeropoint energy of particles is a
Brownian movement caused by the kinetic energy of a lower hierarchy.
There is no absolute indeterminacy about particle motion but only relative
indeterminacy due to our incapacity to discriminate beyond a certain point
with a given set of instrumentation.
There is no conflict between dynamic and static theories; both are aspects
of the same thing viewed from different hierarchical levels.
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Thus for example it is perfectly correct to treat heat as a real fluid and
it is equally correct to treat heat as a form of motion. The two views
are no more contradictory than two different views of a chair. In like
manner there is no conflict between treating light both as a wave and
as a particle. Both these conflicts arose only because of the failure to
realise the possibility of different hierarchical frames of reference and
different levels of discrimination for viewing objects.
It can be seen therefore that existing theories far from being rendered
less true by the S-F system are rendered more true. Boyle's law for
instance is completely true for a gas provided that the level of
discrimination is chosen at the correct level. On the other hand for an
indefinitely high degree of discrimination all laws confined to a finite
set of hierarchies break down.
Because dynamic states can be truly regarded as materials (heat
regarded as a fluid for example) it can be seen that the Gibb's phase
rule equation is a thinly disguised tautology which ultimately reduces to
the number of degrees of freedom = the number of variable concentrations
of different materials
Phase boundaries, using the term in the most general sense possible, are
reflections of environmental discontinuities. By suitably controlling the
environment it is possible to remove these boundaries and move continuously
between any number of phases in precisely the same way that by increasing
the temperature and pressure it is possible to move continuously from a gas
to a liquid without the formation of a surface. It follows that terms such as
solid, liquid and gas for example are parts of a continuum with large pieces
missing in between solid and liquid, liquid and gas because of our particular
environmental circumstances. Thus discontinuities between phases and
components are similar to and as arbitrary as unconformities in a geological
series.
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REFERENCES
1. GRIMER F J 'A laboratory investigation into some of the factors
affecting the strength of soil-cement'
Department of Scientific and Industrial Research,
Road Research Laboratory Note No RN/3288/FJG August 1958
2 GRIMER F J 'A proposed relation between the strength and the
phase volumes of road materials'
Department of Scientific and Industrial Research,
Road Research Laboratory Note No LN/167/FJG August 1962
3. MILLARD D J Discussion on the 'Porosity and strength of brittle
solids' Conference on the mechanical properties of non-metallic
brittle materials Butterworths Scientific Publications 1958
4. WITNEY B D 'Plastics to push out stone and concrete ,
1970 pp 133-134 Engineering 6 Feb 1970 pp.133-134
5. ROAD RESEARCH LABORATORY 'Concrete Roads'
Her Majesty's Stationery Office London 1955
6. SHANNON C E and WEAVER W
'The mathematical theory of communication ,
University of Illinois Press Urbana 1949
7. SCHOFIELD R K
'The pF of the water in soil , Science Oxford 1935
Trans 3rd Int Congr Soil Science Oxford 1935
8. CRONEY D and COLMAN J D Journ Soil Sci 2 (1) p 81
'Soil structure in relation to soil suction'
Journ Soil Sci 5 (1) p 81
9. BRIDGMAN P W 'The compressibility of water'
International Critical Tables of Numerical Data, Physics,
Chemistry and Technology 2 p 40 McGraw Hill New York 1928
10 GRIMER F J AND HEWITT R.E.
'The form of the stress-strain curve of concrete interpreted with a
diphase model of material behaviour'
Intern Conf on Structure, Solid Mechanics and Engineering Design
in Civil Engineering Materials Southampton 1969
11. CAWS P 'The philosophy of science' D Van Nostrand Princeton 1964
12. PIERCE J R
'Symbols signals and noise' Hutchinson London 1962
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