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 cold-blooded, 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 non-existent. 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 present-day 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 present-day “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 present-day “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 non-mathematical.

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 non-mathematical 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 “non-mathematical” 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 present-day 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. Present-day 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 later shifts 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 present-day “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 about the 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 present-day 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 that the sodium atom contains 11 extra-nuclear 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 non-specific 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 non-mathematical 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 present-day 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 present-day 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 non-commutative 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 ever-accelerating tempo and in a complexity and abstractness increasingly formidable…. These have come crowding on each other’s heels with ever-increasing 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 space-time 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 hair-splitting—attention that science pays to details in the subsequent stages of development of theory, the casual way in which the basic fundamentals are treated in present-day 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 of the 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 present-day 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 conclusions 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 *three-dimensional.* 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 three-dimensional 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 three-dimensional, 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 one-dimensional, 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

besideone another; the instants of timefollowone 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 present-day 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 one-dimensional, but the one-dimensionality 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 multi-dimensional, 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 present-day 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 three-dimensional and we also speak of a cube as three-dimensional. 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 three-dimensional, 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 multi-dimensional 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 quasi-space. But this is a concept, which makes little sense under any conditions, and it certainly is not the meaning of the term “multi-dimensional time” as used in this work. When we here speak of time as three-dimensional, as we will later in the discussion, we will be employing the term in the same significance as when we speak of space as three-dimensional; 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 present-day assumption that we *know* time to be one-dimensional 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*three-dimensional, 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.*

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