Chapter II
Where We Stand:
Conceptually
A major factor in the advance of physical science from
its primitive beginnings to its present position has been the availability
of mathematical techniques to aid in the acquisition of knowledge and
to facilitate the systematization and utilization of that knowledge after
it has once been acquired. The practical advantages of having a substantial
portion of the accumulated knowledge in any physical field available in
a form suitable for mathematical manipulation and easy adaptation to the
specific problems at hand are familiar to all concerned and need no particular
comment. Equally important to the investigator is the conceptual freedom,
which is attained by the use of mathematical rather than verbal reasoning.
The answer to a problem of long standing in the physical field usually
involves some significant change in the basic concepts entering into the
phenomena with which the problem is concerned, but the ability of the
human mind to break loose from the shackles of traditional thought and
formulate totally new concepts is severely limited, and finding the solution
to a problem of this kind by direct means is extremely difficult.
If the problem is attacked mathematically, however, the
investigator has much more freedom. He is still circumscribed by the limits
of current thinking with respect to the definitions of his terms and other
general concepts entering into his mathematical expressions, but from
there on he is essentially free of restraint. If a direct function fails
to serve his purpose, he is entirely at liberty to try an inverse function;
if a trigonometric relationship proves to be inapplicable, he is free
to try a logarithmic relation, and so on, even though the conceptual changes
involved in these modifications of the mathematical relationships are
so drastic that they would be essentially inconceivable from the standpoint
of a direct conceptual approach. Because of this freedom of maneuver,
it is often possible to obtain a mathematical solution for the problem
under investigation and to embody this solution in an equation or some
other mathematical expression. With this mathematical knowledge at hand,
the investigator may then be able to go back to the physical meaning of
the mathematical terms and make the conceptual jump that was not possible
without the guideposts established by the mathematical study.
Max Planck’s discovery of the quantum of radiant energy
is a classic example of an investigation that followed such a pattern.
The reason for the observed distribution of spectral frequencies in blackbody
radiation had long been an unsolved problem. Mathematical expressions
formulated by Wienand Rayleigh on the basis of different ideas as to the
mechanism of the radiation were each successful in certain spectral regions
but failed in others. Planck addressed himself to this problem, and after
a long search succeeded in constructing a new expression which correctly
represented the distribution of frequencies through the entire range.
As soon as he obtained this expression–”on the very day when
I formulated this Law,” he tells us–he undertook the “task
of investing it with a true physical meaning,”^{36} and in this way he ultimately conceived
the idea of the quantum. Theoretically, of course, someone could have
hit upon this idea directly, without the benefit of the prior knowledge
of the mathematical relationship, but the concept of discrete units of
energy was so foreign to current scientific thought that scientists were
simply unable to visualize this possibility until Planck was pushed into
making the conceptual adjustment as a matter of sheer mathematical necessity.
Much the same thing took place in this present investigation.
The concept of a reciprocal relation between space and time, the central
idea of the new theoretical system, could have been formulated directly–indeed,
it will be shown in a subsequent chapter that if the question of the basic
structure of the physical universe is examined in a coldblooded, logical
and systematic manner, without making any unsupported assumptions in advance,
the formulation of such a concept is inevitable–but, like the quantum,
it represents such a radical alteration of existing thought that the human
mind was simply unable to make the direct jump. Here again, what could
not be done directly was done indirectly by way of the mathematical approach.
An intensive study of a number of physical phenomena in the first phase
of the project produced some new and more accurate mathematical expressions
for the variability of these phenomena under different conditions. Attention
was then turned to finding a physical explanation for each of these expressions,
and after a long search, the reciprocal postulate finally emerged.
There is no guarantee; however, that the search for a
physical explanation of a mathematical relation will always be as successful
as it was in the two instances that have been discussed. Even though mathematical
information is very helpful, it is still only a clue, not a map or a blueprint,
and the conceptual innovation that is necessary for a complete and correct
explanation may still elude the investigator. In many important studies,
some of which have a bearing on the subject matter of this volume and
will therefore be discussed in the next chapter, the investigations have
produced conceptual answers that are unable to meet the requirements for
proof of their validity, and hence are wrong or, at least, only partially
correct. In many other cases, the problem has been still more recalcitrant,
and the most strenuous efforts have failed to produce any plausible
explanation of the mathematical results.
From a strictly logical point of view it seems rather
obvious that the existence of these correct but unexplained mathematical
expressions of physical phenomena is an indication that the work of the
scientist is still incomplete, and that more time, effort, and ingenuity
will have to be applied to these problems. But this appraisal of the situation
is very distasteful to a profession that, at the moment, is basking in
the sunlight of an impressive record of successes, and in order to avoid
the necessity of admitting failure in these instances, the physicists
are now denying that these problems exist, and ate advancing the curious
contention that the mathematical expressions are complete in themselves
and that further explanation is not only unnecessary, but actually nonexistent.
As expressed by Northrop, they are “trying valiantly to convince
themselves that their present collection of mathematical formulae, which
possess no physical meaning, constitute an ideal state of affairs.”^{37}
Even though this doctrine is devoid of any logical foundation
and is obviously a rationalization of failure that belongs in the “sour
grapes” category, it is tremendously popular because it emancipates
the theorist from the harsh necessity of conforming to physical reality.
The typical presentday contribution to physical theory consists of a
rigorous mathematical “calculus” which is, in the words of Rudolf
Carnap, “constructed floating in the air, so to speak,” and
which deals with terms whose meaning, if any, is vague and indefinite.
“The words may have independent meaning,” says Braithwaite,
“but this is not how the words are used in a treatise on physics.
There they are used as symbols in a calculus which is to be interpreted
as an applied deductive system; they are not understood as having any
meaning apart from their place in such a calculus.”^{38}
The popularity of this method of procedure is easy to
understand. If the theorists were required to make all of their terms
meaningful and to expose their work to comparison with the observed facts
at every step of the way, the pages of the Physical Review and
similar journals would shrink drastically. Genuine additions to theoretical
knowledge are not nearly so easy to produce, as the present volume of
published material would seem to indicate.
It is not the prerogative of the author of this work
to say how anyone else should conduct his investigations, nor what kind
of material should be published in the scientific journals, but the prevalence
of this abstract mathematical approach to physical theory has created
a rather general impression that this is the only proper way to
carry out such activities and that any work which does not follow the
presentday “standard procedure” of mathematical formulation
in terms of tensors, or spinors, or matrix algebra, or some other complex
mathematical device, is automatically devoid of merit. This attitude very
definitely is an appropriate subject for comment, as one of the most important
conclusions reached in the initial phase of this present investigation
was that while mathematical techniques are valuable tools for certain
purposes, as mentioned earlier, the presentday “standard procedure”
utilizing mathematical processes with little or no actual physical meaning
is inherently incapable of remedying the existing deficiencies in physical
theory, and a procedure more definitely tied down to physical reality
and emphasizing the conceptual rather than the mathematical aspects of
the theoretical situation is essential to the attainment of the objectives
of a work of this kind. Development of the Reciprocal System has therefore
followed a pattern altogether different from that of the typical present
day approach.
In this development all terms and concepts are sharply
and explicitly defined, and all of the conclusions that are reached–the
intermediate as well as the final results–are capable of being verified
by comparison with the findings of observation and measurement, to the
extent that observational knowledge is available. There has been no deliberate
attempt to minimize the use of mathematics, but the findings of this work
show that the complex entities and phenomena of the universe are built
up from simple foundations and these simple basic phenomena and relations
do not require complex mathematics for their representation. The correct
mathematical representation of a simple physical relation must itself
be simple. There are many complex phenomena in the universe, to be sure,
but the initial presentation of the Reciprocal System, in this and the
books previously published, does not reach the point in the theoretical
development where any substantial degree of complexity has emerged, and
as a consequence, the mathematical treatment, while entirely adequate
for its purpose, is very simple—so simple, in fact, that to the modern
physicist, accustomed to page after page of mathematical symbolism with
only the bare essentials of a verbal commentary, the work seems to be
wholly nonmathematical.
It is rather ironic that such a judgment should be passed
on the first general physical theory which carries out a complete quantitative
development coincidentally with the qualitative development and which,
for the first time, permits physical magnitudes to be calculated directly
from purely theoretical foundations without the aid of “constants”
obtained by measurement. Nevertheless, this tendency to classify the work
as nonmathematical would be of no particular consequence and would not
call for any special comment, were it not for the further tendency on
the part of the physicists to regard the adjective “nonmathematical”
as synonymous with “defective” or “unacceptable,”
and to adhere to what Northrop calls “the scientific dogma that nothing
is truly scientific which is not mathematical.” ^{39}
The general attitude is clearly expressed by a European
reviewer who repeats the major conclusions of Beyond Newton and
then comments in a caustic manner that the author “arrives at these
revolutionary conclusions without recourse to mathematics, and by the
sole magic of words.” This attitude has been encountered so frequently
that an effort has been made to deal with it in each of the preceding
books in this series, but apparently something more detailed and more
emphatic is needed, and before proceeding with the principal business
of this chapter–a review of our conceptual knowledge of space and
time–it seems advisable to discuss the distinction between conceptual
and mathematical knowledge, and to bring out the point that it is truly
the “magic of words” that invests the results of any
work, mathematical or otherwise, with a meaning.
The findings of this investigation are that the difficulties
which are being experienced by presentday physical science are not due
to any lack of full mathematical treatment–we now have mathematics
running out of our ears–nor to inadequacy of the mathematical tools
available–mathematical methods have reached a peak of sophistication
and complexity far beyond the needs of basic science–nor to the employment
of faulty logic in the development of theory–many of the conclusions
of modern physics are illogical, to be sure, but it is quite obvious that
these are errors of desperation born of long years of frustration; they
are not the cause of the existing difficulties but the result of them.
The root of the present trouble is conceptual. The elaborate and
versatile machinery of modern science has been unable to solve the more
difficult problems of the universe of space, time, matter, electricity,
and so on, simply because all of its efforts have been based on erroneous
assumptions as to the nature of these entities–space, time,
matter, electricity, etc.–with which it is dealing.
In large measure, this is a result of a misconception
on the part of the physicists as to the degree of latitude that they are
privileged to exercise in the construction of theory. Presentday theorists
are under the impression that they are at liberty to define the concepts,
which they use in any way that they see, fit. Herbert Dingle, for instance,
tells us that atoms, as the physicist conceives them, are “creatures
of the imagination, to be formed into the image of our fancies and restricted
by whatever laws we cared to prescribe, provided only that when they behaved
in accordance with those laws they should produce phenomena.”^{40}
Einstein’s attitude toward basic concepts was similar. “The axiomatic
basis of theoretical physics... must be free invention,” ^{41} he insists.
The weakness of this policy is that when conclusions
are reached on the basis of such concepts, they too belong to the world
of fancy, not to the world of reality. If the theoretical physicist entered
into his activities merely as a mental exercise, in the manner of some
of the more abstruse mathematical developments, no exception could be
taken to his procedure, but as matters now stand, the theorist who is
working with “creatures of the imagination” sooner or latershifts
his ground and starts claiming that his conclusions are applicable to
the real world. Thus we find the Copenhagen school of physicists, principal
architects of the presentday “official” atomic theory, asserting
that the particles of which their “atom” is constructed are
not “material particles in space and time” and do not “exist
objectively.”^{42} As a statement about the dream
world of the physicists’ imagination, this may well be correct. But the
Copenhagen theorists are not content to look at it in this light; they
want to apply it to the real physical world and to real physical atoms,
and here it has no relevance at all, because the entire theoretical development
which leads to these strange conclusions has been based on a concept of
atomic structure “formed in the image of their fancies” and
not on a concept of the atom as it actually exists physically.
The Copenhagen model of the atom is essentially the same
kind of a thing as the “billiard ball” model postulated by an
earlier generation of scientists. Both models were devised to represent
certain aspects of the behavior of atoms, ignoring all other aspects,
and both have had a considerable degree of success in these particular
areas. But neither is in any way a picture of the real physical atom,
and whatever conclusions are drawn from either model are conclusions aboutthe
model, not about the physical atom. If we want to arrive at conclusions
applicable to the real physical atom, then we must start with concepts,
which accurately represent the physical atom and its properties;
we have no latitude for error.
It is true that the formulation of such concepts is a
formidable undertaking. The theorists work under a severe handicap because
of the lack of any direct method of ascertaining the true nature and properties
of the physical atom, and they have turned to speculation and assumption
as a matter of necessity, or presumed necessity. But even if speculation
is unavoidable, this does not make the results thereof any less speculative.
If these results can be tested against the facts of observation and pass
this test successfully, then the speculation has paid dividends, but if
they fail in the test or if they are untestable conclusions of such an
absurd character as the Copenhagen contention that physical aggregates
which do exist objectively are composed of parts that do not
exist objectively, it is evident that the speculation has missed its mark
and that the conclusions apply only to the speculative world of fancy,
not to the real world.
In cases such as this the lack of logic in the theorists’
position is obvious, and it is surprising that so few critics have protested
(publicly, at least) the way they are “getting away with murder.”
But there are many other instances in which no one seems to have realized
that the concepts upon which a particular physical theory is based may
not coincide with the actual physical realities, which these concepts
are, intended to represent. The concept of time, for example, is rather
vaguely defined in presentday theory, but all definitions are specific
and in agreement in one respect; that is, the magnitude of a time interval
can be measured by means of a suitable clock. We actually have no assurance,
however, that the quantity thus defined and measured always coincides
with the physical time that enters into such relations as velocity. Modern
theory assumes such a coincidence, but it will be shown in the
subsequent pages that this assumption is frequently incorrect and the
time measured by means of a clock often is not the true physical
time applicable to the phenomenon under consideration.
The consequences of an inadvertent error of this kind
in the definition of a concept are just as serious as those resulting
from a wild speculation that misses its mark, and one of the most essential
tasks of the present investigation has therefore been to examine the basic
concepts of space, time and motion carefully and critically to make certain,
as far as it is possible to do so, that the way in which these concepts
are defined for purposes of the development of theory conforms to the
nature and properties of the physical entities which these concepts are
intended to represent. Our first concern will be to ascertain just how
much actual knowledge about these entities is available. In this
chapter we will examine the available conceptual knowledge.
The term “conceptual knowledge” as used in
this work, refers to any information that has a specific meaning and that
applies specifically to definable physical concepts. These concepts may
be “real” physical objects, in the sense in which this term
is commonly used, or they may be abstractions such as “force,”
the real existence of which is debatable. The essential requirement is
that they be capable of explicit definition, so that we know what we are
talking about. Some items of conceptual knowledge can be expressed in
mathematical terms, and here we have both conceptual and mathematical
knowledge, but it does not necessarily follow that all mathematical expressions
represent conceptual knowledge.
To illustrate this point, let us consider the structure
of some chemical element, sodium, for example. Modern theory tells us
thatthe sodium atom contains 11 extranuclear electrons. If this statement
could be substantiated, it would constitute conceptual knowledge, as herein
defined: authentic information about a specific physical concept, the
sodium atom. But when we examine this alleged information carefully, we
find that what is actually known is that there are certain mathematical
expressions of physical relationships–Moseley’s Law, for instance–in
which each kind of atom has its own characteristic numerical value, and
the value applicable to sodium is 11. The conclusion that this indicates
the existence of 11 units of some kind in the sodium atom is reasonable,
but it is no more than an assumption, and the further conclusion that
these units are electrons has no factual foundation at all. It is purely
an interpretation of the mathematical relations in the light of
current ideas as to the nature of the atomic structure. This closer scrutiny
thus discloses that all we have here is mathematical knowledge;
the currently favored interpretation of the mathematical relation could
very well be wrong (and this present investigation indicates that it is,
indeed, wrong).
This example illustrates the fact that mathematical knowledge
is, in general, incomplete and nonspecific knowledge. In this particular
case, all that the mathematics are able to tell us is that there are 11
units of some kind that are in some way connected with sodium.
The mathematical equations give us no indication as to the nature of the
units nor as to the nature of their connection with the sodium atom. The
number 11 is dimensionless in the equations and it can refer to any kind
of a unit, without restriction. In order to carry knowledge of the mathematical
type to completion we must resort to words; we must find names for the
mathematical terms, which will give these terms their correct physical
meanings.
The need for verbal additions to transform mathematical
knowledge into complete knowledge is not applicable in reverse; that is,
conceptual knowledge expressed verbally can be complete in itself without
any necessity for mathematical addition or elaboration. Qualitative information,
which is by definition outside the scope of mathematical treatment, constitutes
a very important part of the accumulated store of scientific knowledge.
Furthermore, whatever can be expressed in mathematical symbols can also
be expressed in words. The verbal expression may be complicated and awkward,
but if a mathematical expression has a physical meaning, then it must
be possible to state the same thing in words, because it is only through
the medium of words that we give meaning to symbols. For instance, we
can write PV = k. In itself, this means nothing. But if we define these
four symbols in an appropriate manner and add some necessary qualifications
as to the circumstances under which the equation is valid, a “text,”
as Bridgman calls such an explanation, this expression becomes Boyle’s
Law, one of the important relations of physics. When we have thus given
the mathematical symbols definite meanings, it then becomes possible to
reproduce the meaning of the equation in words alone, and the elementary
physics textbooks customarily state Boyle’s Law and similar physical principles
both ways.
The current tendency to magnify the importance of complex
mathematical treatment and to deride and ridicule any development utilizing
purely verbal logic or simple mathematics, “the false worshipful
attitude toward mathematics,”^{43} as Northrop calls it, is a completely
upside down attitude. Mathematics is not essential to thought, nor is
it a substitute for thought. As Freeman J. Dyson warns, “Mathematical
intuition is dangerous, because many situations in science demand for
their understanding not the evasion of thought, but thought.”^{44} The policy that has been followed
throughout this work is to utilize mathematics where and to the extent
that a useful purpose is served, and not otherwise. Where no mathematical
treatment has been required, none has been used. Where arithmetic or simple
algebra are adequate for the tasks at hand, these are the tools that have
been utilized. Where it has been necessary or convenient to call upon
the calculus or other advanced mathematical devices, this has been done.
The mathematical simplicity of the work does not indicate any lack of
mathematical accuracy, nor is it the result of any nonmathematical attitude
on the part of the author. It merely reflects the simplicity of the basic
physical concepts and relations as they emerge from the development of
the consequences of the postulates of the new system.
An additional factor tending to minimize the mathematical
content of this particular volume is that in the normal processes of human
thought the answer to the question “What?” precedes the answers
to the questions “How many?” and “How much?” If we
are asked to explain the operation of an automobile, for example, we first
describe the functions of the various parts in purely qualitative terms,
and if we find it necessary later on to introduce mathematical relationships
such as compression ratio, torque, efficiency, etc., we do so only after
a full qualitative explanation has been given. Even though this present
work is addressed primarily to individuals who are well versed in the
general subject matter of physical science, it is an elementary presentation
of the new theoretical system, comparable to the first explanation of
the principles of automobile operation, and like the latter it is mainly
a qualitative explanation
The primary advantage of utilizing mathematical methods
where they are applicable is the convenience of employing a few symbols
to represent concepts and operations that would require a great many words
for verbal definition. In the process of theory construction there is
a further gain in that once the symbols have been properly defined to
begin with, these definitions can be laid aside and the analysis can proceed
in the symbolic and abstract language of mathematics until the final conclusions
are reached, when the definitions are again called upon as a means of
ascertaining the meaning of the symbols that represent these conclusions.
This procedure not only expedites the intermediate operations very materially,
but also enables these operations to be carried out with the freedom from
the conceptual limitations of human thinking that has already been mentioned
as one of the most important characteristics of the mathematical approach
to physical problems.
Unfortunately freedom, once attained, is often abused,
and so it has been in presentday physics. The justification for ignoring
the meaning of the symbols in all of the intermediate mathematical steps
between the initial premise and the final conclusion is that whatever
meaning is assigned initially remains unchanged throughout the subsequent
manipulation and hence does not require any further consideration until
the final conclusions are ready for interpretation. On this basis every
intermediate step has just as definite and specific a meaning as the initial
and final statements, and the customary practice of handling these intermediate
steps in terms of symbols only is merely a matter of convenience, not
a matter of necessity. However as pointed out by Braithwaite in the statement
previously quoted modern physical science is following an altogether different
procedure, utilizing terms which are never specifically defined
and which have no “independent meaning”; that is, no meaning
aside from the way in which they enter into the mathematical development.
“The possibility of explicit definitions,” says Hesse, “is
not generally one of the considerations which weigh with scientists in
judging a good theory.” ^{45} One of the major virtues of mathematical
treatment in general is the precision with which mathematical statements
can be made, but in presentday physics mathematical methods are deliberately
employed for the opposite purpose: to make theories more “abstract”;
that is, more vague. As Whittaker comments, “the mathematical physicist...
is interested in noncommutative symbolism for a wholly different reason.
He may be said to be, in a certain sense, moving away from precision.”
^{46}
Scientific history shows that physical problems of long
standing are usually the result of errors in the prevailing basic concepts,
and that significant conceptual modifications are a prerequisite for their
solution. But the effect of the new mathematical practices of the theoretical
physicist is to freeze the existing basic concepts and to secure some
sort of agreement with observation by making the mathematical treatment
more complex and its conceptual meaning more vague, rather than locating
and correcting the error in the conceptual foundations upon which the
mathematical treatment is based. Instead of definite answers to our problems,
what we get is a profusion of “mathematical theories which are being
continually formulated at an everaccelerating tempo and in a complexity
and abstractness increasingly formidable.... These have come crowding
on each other’s heels with everincreasing unmannerliness, until the average
physicist, for whom I venture to speak, flounders in bewilderment.”
(P. W. Bridgman). ^{47}
In effect, the modern scientist is taking the stand that
his mathematical techniques are so powerful that they can overcome whatever
handicaps may be imposed by errors in the basic physical concepts. The
present work challenges this assumption, and contends that valid, meaningful,
and physically correct basic concepts are primary requisites for
sound theory, and that a logical development of these concepts is the
essential element in constructing the framework of such a theory. On this
basis, conceptual knowledge is of paramount importance, and it will therefore
be our first concern as we now begin a survey of our present knowledge
of space and time.
It has become increasingly clear in recent years that
the area in which we make direct contact with space and time is only a
very small sector of the universe as a whole. It does not necessarily
follow, therefore, that the properties, which these basic entities possess,
or seem, to possess, in the limited area subject to our direct observation
are properties of space and time in general. Nevertheless, the information,
which we obtain by direct observation, is the cornerstone of any scientific
consideration of the spacetime situation, and it is therefore extremely
important to be certain just what our observations do tell us about
the properties of space and time.
In view of the meticulous–even hairsplitting–attention
that science pays to details in the subsequent stages of development oftheory,
the casual way in which the basic fundamentals are treated in presentday
physical science is a curious phenomenon. Certainly the items that lie
at the very base of the structure of physical theory and therefore have
a material bearing on the validity of every part ofthe whole should have
no less careful and critical scrutiny than the various elements
of the superstructure, but the prevailing opinion appears to be that it
is sufficient to accept, “without examination,” as Tolman^{21} puts it, the superficial impressions
of the lay public as adequate definitions of space and time for scientific
purposes.
There even seems to be an impression that the validity
of the basic concepts is immaterial, and that accuracy can be introduced
later in the development. For instance, R. B. Lindsay tells us that “a
physical theory starts with primitive, undefined concepts, such as the
notions of space and time. It proceeds to the construction of more precisely
defined constructs, for instance, mass and force in mechanics.” ^{48} This casual attitude toward conceptual
foundations is not only highly incongruous in a profession that prides
itself on the “rigor” of its treatment of the subject matter
within its field, but it is also entirely unrealistic. Derived concepts
cannot be more precisely defined than their antecedents. Whatever uncertainties
may exist in the definition of space are carried along undiminished to
the concept of force, since force is defined in terms of mass and space.
Furthermore, there is no such thing as building theories
on “undefined concepts.” If a concept is not explicitly defined,
it is implicitly defined by the way in which it is used. Concepts may
be vaguely defined, as in much of presentday theory, poorly defined,
or erroneously defined, but they cannot be undefined. In the case of space
and time it is merely assumed that the definitions are so well known and
so universally accepted that no further discussion is required. One current
physics textbook, for instance, simply says ”Time intervals are measured
by clocks, with which everyone is familiar,” and it then proceeds
to insert the time terms into all manner of physical relations without
further ado. Newton did essentially the same thing, explaining, “I
do not define time, space, place, and motion, as being well known to all.”
But no structure is any stronger than its foundations,
and one of the most essential tasks of the present investigation has been
to make a detailed study of space and time as they appear under direct
observation, with the objectives of determining first, whether the
conelusions as to their properties that are commonly drawn from these
observations are justified, and second, whether any information that can
legitimately be derived from observation has been overlooked. The remainder
of this chapter will be concerned with the results of that study.
The most conspicuous property of space as we know it
first hand is that it is threedimensional. Of course there is
much imaginative speculation about a fourth dimension, and mathematicians
are fond of constructing hypothetical spaces of n dimensions, but the
sector of the universe which we inhabit very definitely presents a threedimensional
aspect to our observation–no more, no less.
Additionally, space, as we find it, is homogeneous;
that is, so far as we can tell, each unit is exactly like every other
unit, and it is isotropic, that is, its behavior is the same in
all directions. Here again there are many speculations and hypotheses,
which involve directional characteristics or departures from homogeneity,
but there is no direct evidence of anything of this kind, and we are now
considering only the properties of space as they appear under direct observation.
When we have come this far we have exhausted the information
that we can obtain directly. Space is threedimensional, homogeneous and
isotropic in our local environment, and that is all that we can tell from
direct observation. It is frequently claimed that these properties necessarily
call for the existence of certain other properties; for instance, that
“infinity and mathematical continuity (infinite divisibility) follow
directly from its homogeneity.”^{49} But even if there were general
agreement on these points–which there is not–such properties
are not directly observable. If there is a limit to the divisibility of
space, it is below the present observational range, and certainly we cannot
verify the existence of infinite space.
Little as we know about space, our direct knowledge of
time is still more limited. Even those few items that are accepted as
factual are largely assumptions. As Eddington states the case:
We have jumped to certain conclusions about time and
have come to regard them almost as axiomatic, although they are not
really justified by anything in our immediate perception of time. ^{50}
The most conspicuous feature of time as we observe it
is that, in some way, it progresses. In fact, it is only as a progression
that we know it at all; whatever properties we can recognize in time are
simply the characteristics of the progression. We note, for one thing
that the progression is uniform, so far as we can determine. Another
fact that we observe is that in the context of the familiar phenomena
of our everyday life, time is scalar. In the velocity equation
v = s/t for example, the term t is a scalar quantity. We also observe
that time appears to move steadily onward in the same scalar direction
and we have formulated the Second Law of Thermodynamics to give expression
to this empirical observation. Many physicists are therefore inclined
to believe that we know time to be unidirectional, and in the statement
previously mentioned, Tolman lists this as one of the properties of time
which he “assumes without examination.” Other observers, notably
Eddington, have pointed out that there is a serious question as to the
validity of this conclusion because, notwithstanding the assertion contained
in the Second Law, the term t is mathematically reversible in the
equations representing the various physical phenomena. In spite of the
constant direction of “Time’s arrow” in our local region, it
is thus clear that we will have to be cautious about extrapolating the
constancy of direction to the universe as a whole.
So far our reexamination of the observed properties of
space and time has produced no surprises, but we have now arrived at a
place where the lack of a careful and critical study of this kind has
caused physical science to fall into a serious error that has had unfortunate
consequences in many areas of physical theory. As has been mentioned,
time enters into the mathematics of the physical processes with which
we are most intimately concerned as a scalar quantity. From this the physicists
have jumped to the conclusion that time is onedimensional, and this conclusion,
another of those accepted “without examination” by Tolman, is
now, as Eddington put it, regarded “almost as axiomatic.” Capek
explains:
The basic relation in space is juxtaposition; the basic
relation in time is succession. The points of space are beside
one another; the instants of time follow one another.^{51}
Notwithstanding its general and unquestioning acceptance,
this conclusion is entirely unjustified. The point that the physicists
have overlooked is that ”direction” in the context of the physical
processes, which are represented by vectorial equations in presentday
physics always, means “direction in space.” In the equation
v = s/t, for example, the displacement s is a vector quantity because
it has a direction in space. It follows that the velocity v also
has a direction in space, and thus what we have here is a space
velocity equation. In this equation the term t is necessarily scalar
because it has no direction in space.
It is quite true that this result would automatically
follow if time were onedimensional, but the onedimensionality is by
no means a necessary condition. Quite the contrary, time is scalar in
this space velocity equation (and in all of the other familiar vectorial
equations of modern physics: equations that are vectorial because they
involve direction in space) irrespective of its dimensions, because
no matter how many dimensions it may have, time has no direction in
space. If time is multidimensional, then it has a property that corresponds
to the spatial property that we call ”direction.” But whatever
we may call this temporal property, whether we call it ”direction
in time” or give it some altogether different name, it is a temporal
property, not a spatial property, and it does not give time magnitudes
any direction in space. Regardless of its dimensions, time cannot be a
vector quantity in any equation such as those of presentday physics in
which the property, which qualifies a quantity as vectorial, is that of
having a direction in space.
The existing confusion in this area is no doubt due,
at least in part, to the fact that the terms “dimension” and
“dimensional” are currently used with two different meanings.
We speak of space as threedimensional and we also speak of a cube as
threedimensional. In the first expression we mean that space has a certain
property that we designate as dimensionality, and that the magnitude applying
to this property is three. In other words, our statement means that there
are three dimensions of space. But when we say that a cube is threedimensional,
the significance of the statement is quite different. Here we do not mean
that there are three dimensions of “cubism,” or whatever we
may call it; we mean that the cube exists in space and extends into three
dimensions of that space.
There is a rather general tendency to interpret any postulate
of multidimensional time in this latter significance; that is, to take
it as meaning that time extends into n dimensions of space,
or some kind of a quasispace. But this is a concept, which makes little
sense under any conditions, and it certainly is not the meaning of the
term “multidimensional time” as used in this work. When we
here speak of time as threedimensional, as we will later in the discussion,
we will be employing the term in the same significance as when we speak
of space as threedimensional; that is, we mean that time has a property
which we call dimensionality, and the magnitude of this property is three.
Here again we mean that there are three dimensions of the property
in question: three dimensions of time.
There is nothing in the role which time plays in the
equations of motion to indicate specifically that time has more than one
dimension. But a careful consideration along the lines indicated in the
foregoing paragraphs does show that the presentday assumption that we
know time to be onedimensional is completely unfounded, and it
leaves the door wide open to establishing the true dimensions of time
by other means. Errors such as this masquerading as established facts
are among the most serious obstacles to the advance of knowledge, and
unmasking an error of this kind is often the key to solution of a problem
of long standing.
Although the items that have been discussed in the preceding
paragraphs constitute all that we actually know about space and time individually
from direct observation, there is one more source of direct information,
as we have some observational knowledge of the relation between
space and time. What we know is (1) that the relation between space and
time in the sector of the universe accessible to direct observation is
motion, and (2) that in motion space and time are reciprocally related
from a scalar standpoint; that is, moving a greater distance in the same
time has exactly the same effect on the speed, the scalar measure of the
motion, as moving the same distance in less time.
We may now summarize the primary subject matter of this
chapter, the conceptual knowledge of space and time that we have been
able to obtain from direct observation of these entities as they exist
in our local environment:
Space is threedimensional, homogeneous, and
isotropic.
Time progresses uniformly and (perhaps only locally) unidirectionally.
The scalar relation between space and time is reciprocal, and
this relation constitutes motion.
