Chapter 2
FUNDAMENTAL LAWS AND POSTULATES
The basic fact underlying all physical phenomena is that the entities which make up the physical world are non-coincident. If each of these entities coincided in every respect with all others there would be no way of distinguishing between them and so far as could be determined the entire universe would be one single entity.
This may not be a very profound observation, yet we will find as we look deeper into the workings of the physical universe that the most important advances in knowledge are not the result of the discovery of complicated truths or esoteric doctrines but rather materialize through a more complete understanding of the true significance of essentially simple facts. This becomes apparent when we pass on to a corollary of the initial observation. Since non-coincidence is the basis of all physical phenomena it follows that the magnitude of the non-coincidence and the changes therein are representative of the magnitudes of the phenomena themselves.
We must also note that our ability to express these phenomena in quantitative terms is due to the fact that the non-coincidence is definite and can itself be evaluated mathematically through some process of measurement. So far as we are aware there is no basic reason why this should necessarily be so. It is by no means impossible to conceive of a universe in which magnitudes are fluctuating rather than fixed and measurement are not reproducible. But we find nevertheless that our actual universe obeys this principle wherever we have the ability to apply a test.
The type of non-coincidence which naturally occurs to us most readily by reason of its familiarity is non-coincidence in place. The magnitude of this non-coincidence we call distance and we measure it in the first instance by utilizing a standard, on a specially prepared tape, making successive applications of this standard to determine the ration between it and the distance being measured.
Absolute distance is a scalar quantity, having magnitude only. But when a reference point is introduced distance becomes directional. We may have distance toward the reference point or away from it, positive or negative distance, we may say. Two reference points establish a reference line and give us a two-directional distance, or as we usually call it, a distance in two dimensions. The distance may be along the reference line or at any angle to it. A third reference point not in the same straight line establishes a reference plane and defines distance in three dimensions. Here the distance may be in reference to the plane itself or at any angle to it.
In a three-dimensional universe the true distance must necessarily be the distance measured in three dimensions, but under various circumstances we may encounter the projection of such a distance on a two-dimensional plane or a one-dimensional line. Here the magnitude of the distance as measured in one or two dimensions will vary with the position of the reference line or plane. So far as the actual distance itself is concerned, however, the findings of this work are that distance is absolute in magnitude and independent of the reference system.
Like many another foundation stone of our physical knowledge, this statement is of such a nature that it cannot be proved by any direct means. In fact the physical theories currently in vogue definitely contradict it and explicitly deny the existence of absolute distance, contending that the magnitude of any distance varies with the situation of the observer. But there is no basis on which we can argue this question on its own merits; we can only follow each line of thought to its logical conclusions and see which set of results is the more complete and consistent. The explanation of the physical universe which will be presented in these pages requires that the existence of absolute distance be postulated. The justification for this step must lie in the fruits which it brings forth.
The second kind of non-coincidence which we recognize is non-coincidence in time. Two distinct objects may be coincident in three-dimensional distance; that is, they may occupy the same location in space, as long as they do so at different times. Here distance is zero but there is a measurable difference in time.
To the human race time has always seemed much more mysterious than distance. The latter appears concrete and tangible: something that one can readily grasp and understand, whereas time has a shadowy and elusive character that sets it apart from the more familiar landmarks of the physical world. It is commonly regarded as a one-dimensional unidirectional continuum moving past us at a uniform velocity. Newton considered it to be absolute in magnitude and independent of distance, and it is so treated in Newtonian mechanics. Einstein’s relativity theory makes it one dimensions of a four-dimensional space-time system, the other three dimensions being in space.
In the course of the present study it was found that neither of these hypotheses furnishes an adequate explanation of the most fundamental physical relations and before an accurate mathematical correlation of basic physical phenomena could be developed it was necessary to formulate a new concept of the nature of time. In order to lay the foundation for the introduction of this concept let us first look at the question of the measurement of time and distance.
We have no means of measuring either time or distance without in some way involving the other, as the two are inextricably mixed in ordinary life. We can, however, measure distance in a manner which eliminates the effect of time and hence gives us a measurement which is independent of time, even though time was involved in the measuring process. This we accomplish by utilizing a rod or measuring stick of some kind, laying it down over successive intervals of the distance from A to B and thereby arriving at a measurement of AB in terms of our standard measuring unit.
This does not mean that time does not enter into the picture at all. The measurement cannot be made instantaneously. From its very nature some time must elapse between the beginning and the end of the measuring process and we cannot rule out a priori grounds the possibility that this passage of time has modified distance in some manner. Furthermore the reference frame in which AB is located may be in motion (another time effect) and we must consider the possibility that this motion may alter the magnitude of AB. Einstein and his disciples assert that this is actually true. By repeated measurements of the same distance at different times and under different conditions, however, we have determined that we can arrive at results which are reproducible independently of time effects. This justifies the assumption that we are measuring a definite entity which is independent of time, at least to the extent that time enters into the phenomena with we ordinarily deal.
As distance can be measured independently of time (even though time enters into the process of measurement in an indeterminate manner) so we can measure time independently of distance. Here again distance will enter into the measurement but only in such a manner that its magnitude is immaterial. We may, for instance, rotate some three-dimensional body around one axis and use a circumferential point as a datum. The interval between cycles is a definite quantity of time which we may use as a standard of measurement just as we use a yardstick for measuring distance. It is true that we cannot make this measurement without bringing distance into the picture, as our reference point must be at a finite radial distance from the axis of rotation in order that we may observe it, but the essential point is that the magnitude of this distance is not material to our measurement. We do not even need to know it. Furthermore, we may select locations at different radial distances as reference points without altering our measurement of time.
In addition to measuring time and distance separately we are also able to measure a relation between the two in motion. We observe that certain objects move through a definite distance in a definite length of time. By measuring distance traveled and time elapsed we arrive at velocity, the ratio of distance to time.
Now let us look more closely at the concept of motion and its two components. As a simple example let us consider a point C in motion along a line from A to B. One step toward determination of the velocity of C obviously is to measure the distance either from A to B or between two other reference points A’ and B’ on the line of motion. Since we want the relation between distance and time it might logically be assumed that the next step would be to measure the time difference between the reference points. But no, if we are proceeding along Newtonian lines we take the stand that there is no difference in time between these points; that A, B and C are always coincident in time. We way that time progresses but that it progresses uniformly and simultaneously at all locations and that at any stage of the motion the time at A is the same as the time at B. Einstein gives us a somewhat different explanation but one that still accepts the idea of a continuous unidirectional progression of time.
Obviously our ideas as to the basic nature of time are pure assumption. We have no means of determining whether time actually possesses these characteristics with which we have endowed it. Still these assumptions must be justified if they constitute the only reasonable explanation which we can find for the observed phenomena. The basic facts of the universe are not necessarily susceptible to proof. But it takes only a minimum of consideration to demonstrate that the accepted ideas about the nature of time do not constitute the only possible explanation of the phenomenon that we are examining. We can arrive at exactly the same result by attributing to distance those properties, which we now ascribe to time. On this hypothesis A,B and C would always be at the same location but “here” would be in constant movement just as “now” is ordinarily assumed to progress. The motion in this case would be entirely in time, whereas in the usual interpretation, where time is assumed to be the same at all locations, the motion is entirely in distance.
Under the assumption or a uniform progression in distance the non-coincidence of A, B and C would be measured in time and the velocity of C (which would remain the same under both hypotheses) unsteady of being expressed as
difference in distance/elapsed time at uniform rate of progression
would be expressed as
elapsed distance at uniform rate of progression/difference in time
On examination of this hypothesis we can see that it accounts for the observed phenomenon equally as well as the one-dimensional uniform velocity hypothesis of time. In fact, the two are absolutely parallel. It becomes evident that when we are considering this phenomenon by itself we are in the position of the passengers on a train who observe another train on a parallel track. If the second train is viewed without reference to the adjacent terrain we will probably get the impression that it is moving and that we are standing still. By exerting an effort however, we can picture the second train as stationary and the result will be that our own train will seem to be in motion. Without some outside system of reference we cannot choose between these alternatives. Furthermore we know that there is still a third possibility: both trains may be moving.
So it is in this case. We may postulate a full system of distance, complete in three dimensions, coexisting with a bobtailed sort of time restricted to one dimension, or we may reverse the assumption and visualize a three-dimensional time coexisting with a one-dimensional distance. Or we may adopt the third alternative and consider both time and distance as three-dimensional. Since the basic assumptions with respect to the nature of time and distance are thus found to be readily reversible in application to simple motions it follows that both should have the same dimensions; that is, a three-dimensional entity can exchange positions with another three-dimensional entity or a one-dimensional entity with another one-dimensional entity, but a three-dimensional entity cannot enter into a reversible exchange of position with a one-dimensional entity. It is also quite obvious that time and distance cannot both be one-dimensional since we clearly have three dimensions of something.
The logical conclusion is that both time and distance are three-dimensional. We see time as a uniform unidirectional progression only because we have taken this one dimension of time as a reference datum just as one train appears stationary only because we take it as the reference datum from which to appraise the movement of the other. It will be brought out in the subsequent discussion that there is a very good reason why the characteristics of time appear to be altogether different from those of distance but irrespective of the limitations imposed upon our powers of observation by reason of our own peculiar position in the universe we can free ourselves from the restrictions of this artificial one-dimensional reference datum through mental exertion just as we can visualize the adjoining train as stationary and our own in motion if we make an effort to do so. Let us therefore postulate three dimensions of time as well as three dimensions of distance and see where this leads us.
Since the three-dimensional nature of distance is commonly recognized there is an ample terminology by which dimensional distinctions can be clearly drawn. We have the term “distance,” which is normally used in a one-dimensional sense. “Area” refers to two dimensions and “volume” to three dimensions. The word “space” in its narrowest connotation also refers to three dimensions although in common practice it has a wider range of usage than volume and is rather indiscriminately applied to any number of dimensions. Thus we speak of the space between one object and another, meaning distance, and the wide open spaces, meaning area. Because of the lack of recognition of more than one dimension of time, however, we have no dimensional terms for time analogous to those referring to distance. To preserve dimensional clarity it will therefore be necessary either to coin some new terms for two and three-dimensional time or to specify the number of dimensions to which the word applies whenever this is not apparent from the context. The latter appears to be the preferable alternative, as unfamiliar terms are usually rather confusing and in the subsequent discussion “space” and “time” will be regarded as general terms applying to any dimension. Where some particular number of dimensions is involved the expressions “two dimensional time,” “one-dimensional space,” etc. will be used. “Distance,” “area” and “volume” will be used where appropriate interchangeably with one, two and three-dimensional space.
When we postulate three-dimensional time we have endowed time with all of the properties that we recognize in space. The question then naturally arises as to what distinction can be drawn between the two. Whenever the two entities are associated, as in motion, more time is the equivalent of less space and vice versa. An increase in time per unit space is equivalent to a decrease in space per unit time. From this we arrive at the conclusion that Time is the reciprocal of space.
The two entities with we we started have now become one. Instead of space and time we have have space-time, a single entity with reciprocal forms. Having arrived at this tentative conclusion, we are next confronted with the question as to where such things as mass, electrical phenomena, etc., fit into the picture. Are they in the space-time universe or part of that universe? Here we have no guideposts as yet to point out the truth. One conclusion seems as likely as the other, so far as any direct evidence is concerned. But the mathematical considerations which entered into the original approach to this problem described in Chapter I demand a one-component universe and it is therefore necessary to postulate that space-time is the sole component.
This leads to another fundamental issue. Is space-time continuous, or is it composed of discrete units? Until comparatively recently such a question would never have received serious consideration. Observation seemed to indicate that all of the basic physical magnitudes, with the possible exception of mass, were continuous and infinitely divisible and this was accepted without challenge. The classic work of Planck on the subject of blackbody radiation opened up an entirely new line of thought. Energy, one of those fundamental physical quantities whose continuity has always been taken for granted, was now shown to be composed of separate indivisible units. A major step toward a better understanding of the structure of the universe had been taken.
Much effort has been expended on the elaboration and extension of this unit or quantum theory of energy since it was first formulated, but thus far the property of existing only in discrete units has been accepted as merely a peculiarity of some physical phenomena such as mass and energy. Having postulated the universe as consisting entirely of a single all-embracing component we are now in a position to step forward to a bolder and more comprehensive hypothesis. If energy, which we postulate is one of the various manifestations of space-time, exists only in discrete units, it is highly probable that space-time the antecedent of energy, itself exists only in discrete units. We will so postulate:
On the basis of the considerations outlined in the preceding paragraphs the First Fundamental Postulate has been formulated as follows:
The physical universe is composed entirely of a single entity: space-time, existing in three dimensions, in discrete units, and in two reciprocal forms: space and time.
Obviously there are no means by which a fundamental postulate can be proved to be true. Any proof must necessarily rest upon something else and that something then becomes fundamental instead of the postulate we are trying to prove. Neither can we disprove a fundamental postulate by any method that is rigorously valid. When a proposition is advanced as fundamental it is by the same token presented as being more authoritative than anything else which may possibly be in conflict with it, and any conflict reacts against the less authoritative proposition, not against the more authoritative. This is the impregnable basis on which a diversity of religious doctrines exist. They cannot be shaken in the least by contradictory facts for the mere existence of a conflict with accepted fundamental truth automatically stamps these alleged facts as untrue, no matter how strong the supporting evidence may be.
Furthermore, the conflict between fundamental postulates and observed facts may be only apparent and not real. Many theories that were faced with seemingly insurmountable difficulties at one stage or another have triumphed in the long run when the alleged discrepancies have been resolved through a more complete understanding of the factors involved. But from a practical rather than a philosophical standpoint we are actually more interested in a consistent theory than a correct theory. Essentially we want something that will give us the right answers; not just something that is correct in itself. From this point of view the matter of conflict with observational data becomes extremely significant. The physical theory that meets our requirements most adequately is the one that is the most consistent with our observations of the physical world in which we live.
This does not imply that being consistent with observed facts is incompatible with being correct. On the contrary, we are on solid ground in contending that those fundamental assumptions that produce consistent explanations for the greatest number and variety of physical phenomena and which conflict the least with observed facts are the most probably correct. But this leaves us open to endless argument for it puts us in a place where we cannot prove our point. We are on a much more secure footing if we define the aim of science as the formulation of general principles consistent with observed facts and designate the degree of this correlation as the measure of approach to our goal.
It is true that this might be regarded as setting up a concept of scientific truth as distinguished from truth in general. But at any rate it puts the fundamentals of science on a scientific basis and excludes those metaphysical and philosophical speculations that merely confuse the fundamental issues without contributing towards the attainment of the specific goal of science. Of course, the question as to whether scientific truth, as thus defined, is the absolute truth still remains unsettled but it becomes an external rather than an internal question from the viewpoint of science. It is an issue that cannot be resolved by scientific means and hence it is proper that we leave it outside the boundaries of science.
The fundamental postulates of this work have been formulated on this basis. They have not resulted from a search for the absolute truth, whatever that may be, but from an effort to establish a working basis by which the ordinary phenomena of everyday life could be explained qualitatively and quantitatively. It will be demonstrated in the subsequent pages that the mere existence of a universe such as that defined by the First and Second Fundamental Postulates must necessarily result in phenomena identical with those which we actually observe in the physical world about us and in physical magnitudes that are identical, within the limits of accuracy of our measurements, with the results that we obtain from our measuring devices. To the extent of this correspondence the postulates and the relations derived therefrom represent scientific truth as herein defined.
According to the First Postulate time and space are reciprocal quantities. Two units of space are therefore equivalent to one-half unit of time; ten units of time are equivalent to one-tenth unit of space and so on. Furthermore a single unit of space is the equivalent of a single unit of time. But when we postulate space-time as the sole component of the physical universe to follows that there can be no phenomena within the universe except those resulting from inequality between space and time; when the two are equal everything is uniformity and there are no physical phenomena of any kind. A space-time ratio of unity, indicating inequality between space and time, is therefore the condition of rest in the space-time universe, the datum level from which all action starts.
It is apparent from these same considerations that the significant measurement of space-time is not the total magnitude but the excess above unity. This quantity, which we will call the space or time displacement, is a measure of the divergence fro the neutral unit level and hence a measure of effectiveness in physical phenomena. The occurrence of physical events depends entirely on the presence of these space and time displacements. It is clear, however, that the First Postulate merely permits such displacements and does not require their existence.
Of course, the postulate could have been expressed differently, but close scrutiny of the actual physical universe fails to reveal any mechanism whereby a space or time displacement can originate. Displacements already in existence can be transferred or altered in form but so far as we are able to determine there is no internal means of creating additional displacements or of extinguishing those already in existence. The First Postulate has been so expressed that it conforms to this observed property of the actual physical universe. From it we will be able to derive certain relationships which will tell us that if a certain event takes place then a definite consequence will follow. There is nothing in the system, however, which requires the initial event to occur. On the contrary, the postulate is entirely consistent with a static universe in which nothing ever happens.
This in no way affects the validity of the laws and principles applying to possible events. As an analogy let us consider a pendulum. From our knowledge of the principles of mechanics we can work out a complete system of relationships expressing its motion in detail. We can calculate the period, the displacement for a specific force, the velocity at any point, and so on. But we cannot determine from any of these calculations or from any of the laws of the pendulum whether it will ever move at all. So far as we can tell from any information available within the system, it may remain motionless forever. The impetus that puts the pendulum in motion must come from outside the system.
Similarly, the impetus that set our physical universe into motion and created a dynamic organism out of a static uniformity must have come from the outside, for we find nothing within the system that could originate action any more than the pendulum could start moving of its own volition. Where that impetus came fro we do not know. So far no one has been able to improve upon the simple explanation set forth in the first chapter of Genesis: “In the beginning God created the Heavens and the Earth.” Perhaps the act of creation was the push that set the pendulum in motion.
But for present purposes it is not necessary that we know why or how the universe was activated, it suffices to known that these space and time displacements actually exist and that we are justified in assuming that the events dependent on the existence of displacements will actually occur.
One more point remains to be considered before the consequences of the First Postulate can be evaluated; the quantitative behavior of the universe. The question as to whether physical magnitudes are absolute or merely relative has already been mentioned. Another closely related issue is the degree to which the space-time universe conforms to ordinary mathematical relations; that is, whether two and two equal four, the product ab equals the product ba, multiplication is the inverse of division, and so on.
Until recently comparatively few persons would have challenged these relations but now we have mathematical systems based on assumptions of a different nature, sot that two and two do not always equal four, and even though we are unable to discover any physical reality corresponding to these unorthodox forms of mathematics they at least serve the purpose of calling our attention to the fact that the basic tenets of ordinary mathematics are only assumptions, not self-evident truths. We have no warrant for contending that the physical universe must necessarily follow the normal mathematical relations. Even though it clearly does so in the limited region of which we have direct observational knowledge, there is no certainty that we are correct when we extrapolate this conclusion applicable to the accessible region into regions where direct observation is not possible. Consequently if we want to use mathematical processes in the development of theory we must first postulate that the physical universe conforms with the mathematical relations which we intend to use.
The situation with respect to the geometry of space-time is much the same. We are accustomed to thinking of the straight-line postulate and the Euclidean system of geometry founded on it as expressions of fixed and unchangeable natural law, but as soon as it was demonstrated that self-consistent geometrical systems can be developed on the basis of contrary assumptions it became clear that this is not necessarily true. In order to make use of Euclidean geometry in developing the theoretical framework we must therefore postulate that this is the actual geometry of the universe.
Putting these items together we arrive at the Second Fundamental Postulate:
Space-time conforms to the relations of ordinary mathematics, its magnitudes are absolute and its geometry is Euclidean.
Essentially this postulate is merely a statement that the universe in its entirety follows the same rules of quantitative behavior as that portion of the whole which is accessible to our observation.
Implicit in the fundamental postulates are certain relations which are of such a general nature and so far-reaching in their application that we are justified in classifying them as basic natural laws. From the standpoint of their utility and their broad field of coverage these principles might even be regarded as taking precedence over the postulates themselves, but since they can be derived from the postulates as logical and necessary consequences of the latter they are not quite in the same category as the postulates, which stand on the farthest boundary of our knowledge and beyond which we cannot go. These general principles derived from the postulates by mathematical and logical processes will be called laws to distinguish them from the postulates upon which they are erected.
The first of these laws is a consequence of the external origin of the initial space-time displacements. Since there is nothing within the system defined by the postulates which can create or extinguish these displacements it follows that any event which alters the relationship between space and time in any one physical phenomenon must be accompanied by an event of the same magnitude and the opposite space-time direction in some other phenomenon. Thus if event A results in an increase in the ratio of space to time in some physical magnitude it must be accompanied by an event B which will result in a decrease of the ratio of space to time in some other magnitude. We will call this the General Law of Reaction and will express it as follows:
Every physical event is accompanied by a reciprocal event equal in magnitude and opposite in space-time direction.
The expression “space-time direction” in the statement of this law refers to the direction of the displacement from the neutral unit space-time ratio. Thus an increase in time relative to space is a progression in the direction away from space and towards time.
An application of this general law in one of the many fields to which it applies is expressed by Newton’s Third Law of Motion, which states that “for every action there is an equal and opposite reaction.”
It should be noted that no contention is being made that sufficient observational data are available to provide an adequate verification of the General Law of Reaction. This law and all of the other laws and principles that will be set forth in the subsequent pages are derived from the fundamental postulates as logical and necessary consequences of the postulates. It is imperative, of course, that all of the derived principles should be in harmony with observation and measurements of actual physical phenomena where these are available, otherwise the whole structure is imperiled, but, as pointed out in the introductory chapter, it does not follow that experimental verification of every link in the chain is essential. If an integrated theoretical structure of this nature meets the acid test wherever tests can be applied we are justified in taking the stand that it is equally sound elsewhere, even though there are locations inaccessible for testing.
Of course, broad generalizations of this kind are always subject to disproof from the standpoint of the concept of scientific truth as defined earlier in this chapter. If even a single case can be found in which the law definitely does not hold good it collapses completely. At first glance it appears that the General Law of Reaction could actually be overthrown in this manner but it is soon found that there are extraordinary difficulties in the way of so doing. There are, it is true, a number of physical processes which do not seem to be accompanied by any inverse phenomena as would be required by the Reaction Law, but this does not prove that these phenomena do not exist, it may merely mean that we are unable to recognize them as matters now stand. As these processes are subjected to further analysis in the subsequent chapters the missing reactions will in many cases be identified. Now let us look at the same physical event from a slightly different standpoint. In some manner, the exact details of which are immaterial for present purposes, a displacement of space relative to time has taken place in a physical magnitude through the action of an outside agency. For the example, the pendulum has been displaced from the neutral position, thereby causing the system to acquire potential energy while kinetic energy remains zero.
Now let us see what happens if a change takes place in the displacement through the action of the system itself; that is without the intervention of any outside agency such as that which cause the original displacement. From the same considerations which gave us the General Law of Reaction we find that any such change in the displacement of the original magnitude must be accompanied by an equal and opposite change in the reciprocal magnitude, leaving the total displacement unchanged. When the pendulum is released it starts down toward the neutral point and as it progresses it loses potential energy, but this loss of potential energy is balanced by a gain in the reciprocal phenomenon: kinetic energy, and the total energy, which represents the space-time displacement in this case, remains unchanged. It is apparent that once a displacement has occurred the total amount of displacement must remain constant until and unless the system is again acted upon by an outside force. In other words it requires an outside agency to alter the amount of space-time displacement in either directions. We thus arrive at another basic physical law, the General Law of Conservation:
The total mount of space-time displacement cannot be altered by any process within the physical universe.
This does not rule out the possibility that there may be outside forces acting upon the physical universe either intermittently or continuously. We know that the pendulum, which theoretically should continue in motion forever if isolated from external forces, actually comes to a stop sooner or later under the influence of friction, air resistance and other agencies. It is not inconceivable that some agency outside the physical universe may provide the equivalent of friction and the other forces acting to increase or decrease the motion of the pendulum. But if any such agency exists its point of application is hidden from our view as our measurements confirm the validity of the Conservation Law in all of its various subsidiary forms to the degree of accuracy attainable with our present measuring facilities.
The third general law which we should consider at this time is a purely mathematical relation. In many of the physical events which together constitute the activity of the universe the ultimate result is not necessarily fixed and may assume any one of a number of possible forms, all of which are consistent with the fundamental laws governing the particular class of event. In these cases we are not able to predict the outcome of any one specific action as between these various alternatives but where a substantial number of similar events are occurring simultaneously or successively the results for the group as a whole can be evaluated by means of the mathematics of probability, since we have postulated that the universe will follow ordinary mathematical relations. This leads to the General Probability Law:
Where a physical event may have more than one possible result the proportionate number of each alternative resulting from a number of evens of this kind is equal to the mathematical probability.
This work is not an appropriate place to enter into any extended discussion of the mathematical theory of probability. It has been postulated that the physical universe conforms to the relations of ordinary mathematics, in which we include probability mathematics, and for our present purposes these mathematics can be taken just as they come from the hands of the mathematicians. However, the theory of probability is not as far advanced as most of the other common branches of mathematics and there is not the substantially unanimous agreement on all major points which characterizes most mathematical theory. For this reason it is advisable to comment briefly on some points which have a bearing on the subsequent discussion.
One important fact which is not always fully appreciated, even among mathematical authorities, is that the probability relations do not create something out of nothing; that is, they do not give us any new information. They simply enable us to add up fragmentary information which we already possess and to put it in usable form. The primary basis of probability mathematics is that our knowledge is never quite complete and our ignorance is never quite total. Through its processes we are able to take the available bits of information, which are of little or no value in determining the situation of individuals, and accumulate them to the point where we have substantially accurate knowledge of the situation of the group. In essence we may regard the application of probability mathematics as a process of magnification. In optics we magnify distance at the expense of field of vision; in mechanics we magnify force at the expense of distance; in probability mathematics we magnify accuracy of information at the expense of applicability to individuals or small groups.
This demonstrates the error underlying the contentions of those mathematicians who argue that if nothing whatever is known as to the truth or falsity of a statement the probabilities are equal. This doctrine seems very plausible if the basic principles of probability are not examined closely and it has gained widespread acceptance. It is responsible for many of the so-called “paradoxes” of probability wherein the application of probability mathematics leads to conclusions that are manifestly absurd. In reality, if nothing is known as to the truth or falsity of a statement the probability relations cannot be applied at all, since no matter how many times we multiply our knowledge by the probability method the result is still zero.
Another point brought out by the concept of probability mathematics as a process of magnification is the relation of the size of the group to the accuracy of the results. Even though the probable error of an individual observation may be great it is often possible to obtain very accurate results by making a large number of observations. Of course, this accuracy is affected by the randomness of the errors; that is, a systematic error repeated in every observations will be introduced into the final result regards of the size of the group. In the usual case, however, an increase in the total number of observations also increases the probability that the errors will have a random distribution. The variability of these factors of size of group, randomness of distribution, and extent of the knowledge appertaining to the situation is undoubtedly responsible for the lack of agreement as to the exact nature of the probability relations. At the present stage of development of theory there are a number of probability functions in common use, each of which seems to have its advantages for certain applications. One of the striking features of present-day practice is in the use of probability mathematics is the necessity for an arbitrary selection from among those accepted mathematical functions.
A determination of the exact probability function applicable to each of the physical phenomena discussed herein is beyond the scope of this work. In all cases where probability mathematics have been used in the subsequent discussion sufficient accuracy for present purposes has been attained by the use of the so-called “normal probability function” which can be expressed as:
In this connection it might be stated that there are grounds for believing that this normal function may actually be the accurate probability expression applicable to all physical phenomena, rather than just an approximation close enough for our purposes. It is not unlikely that there is a true probability function expressing the results of pure chance unaffected by other considerations. Such a function would be strictly applicable only where the units are all exactly alike, the distribution is perfectly random, the units are infinitely small, the variability is continuous and the size of the group is infinitely large. The ordinary classes of events around which most of our present probability theory has been constructed, such as coin and dice experiments, obviously fail to meet these requirements by a wide margin. Coins, for instance, are not continuously variable with an infinite number of possible states; they have only two states: heads and tails. This means that a major item of uncertainty has become almost a certainty and the shape of the distribution curve has been altered accordingly. Strictly speaking, it is no longer a true probability curve but a combination curve of probability and knowledge. If the two elements could be separated and expressed individually in all such cases the existing multiplicity of probability functions would likely be eliminated.
Students of probability have commented that the normal probability function appears to be a sort of a limiting function which all of the others tend to approach under ideal conditions. If any one of the commonly used functions is the true probability function, this is the most likely candidate. On the other side of the picture, the basic physical phenomena conform closely to the requirements for the validity of the laws of pure chance. The units are nearly uniform, the distribution is random, the variability is continuous or nearly continuous, and the size of the group, although not infinite, is extremely large. While these facts are both not conclusive they are very suggestive, particularly when considered in connection with the good correlation with experimental data where comparisons have been made.
A familiar expression of the General Probability Law in application to the flow of energy is found in the Second Law of Thermodynamics. This law has been stated in many different ways, all of which mean essentially that the naturally occurring flow of energy is always in one direction: from the state of less thermodynamic probability to the state of greater thermodynamic probability. In the realm of thermodynamics where the law was originally formulated it is strictly in accord with the General Probability Law as set forth in this work. It has become common practice, however, to extend the application of this law to a much wider field and this extension has led to some conclusions with which this present study does not agree. The best known item of this kind is the contention that the Second Law of Thermodynamics indicates that the universe is continually running down and that it will ultimately reach a dead level of uniformity in which there will be no activity at all, since there will be no energy differential to cause action. From the standpoint of the General Probability Law, however, it is apparent that this extends the Second Law into an area in which it is no longer valid. It is quite true that the flow of energy is always in the direction of the state of greater probability but it is not necessarily true that this is always the direction of thermodynamic probability.
The true situation is an irreconcilable conflict between individual probability and group probability. The most probable state for the individual is the average. The most probable state for the group is a condition in which there are individual deviations from the average. Instead of pointing towards the ultimate uniformity envisioned by those who are guided solely by the Second Law of Thermodynamics the General Probability Law requires a never-ending conflict in which individuals tend to approach the average but are continually driven back away from the average by the tendency of the group to approach a normal distribution.
By way of illustration let us imagine a number of gas molecules enclosed in a totally inert container and let us assume that these molecules have the ability to transfer energy from one to another by some process of radiation involving a very large number of units of energy in proportion to the number of molecules. If we now observe the movement of this radian energy we will find that the net flow is always from the molecules of higher energy to those of lower energy and we might well deduce a “Second Law of Thermodynamics” to account for this situation. Viewing the system as a whole, however, we can see that the transfer of radiant energy can never result in an energy equality and hence can never cease, for the probability principles tell us that the collisions which will take place between the molecules will tend towards normal distribution, not towards equality, and they will continually restore the inequality which the radiation is endeavoring to eliminate.
There is every indication that this is just the situation which actually prevails in the physical universe. Instead of a universe that is running down and will ultimately lapse into stagnation for want of a driving force, a more thorough consideration of the probability principles leads to the conclusion that we are as near the most probable state now as we can ever get and that the present state of affairs in the universe will go on indefinitely, barring external action of the kind responsible for putting the mechanism into motion in the first place.