Home Page


All Posts

Remedial Math

Remedial Physics

Glossary (Vinaire’s Blog)

BOOK: The Disturbance Theory

BOOK: The Mindfulness Approach

BOOK: Education in Information Age

A Critique of Scientology Philosophy


Mathematics & Thinking


Reference: Critical Thinking in Education


The most useful aspect of mathematics is that it provides opportunities:

  1. To think outside the box.

  2. To help learn something new.


Thinking outside the box

There’s a popular story that Gauss, a famous mathematician, had a lazy teacher in his elementary school. The so-called educator wanted to keep the kids busy so he could take a nap; he asked the class to add the numbers 1 to 100.

But Gauss found the answer in less than 10 minutes, and he interrupted the teachers nap with his answer: 5050. So soon? The teacher suspected a cheat, but when he looked at Gauss’s method, he realized that he had a genius in his class.

Here is what Gauss did. He was required to add the first 100 numbers as follows.

1 + 2 + 3 + 4 + … + 98 + 99 + 100


But he split the numbers in two groups (1 to 50 and 51 to 100), and arranged these numbers as follows

1     +   2   +   3  + … + 48 + 49 + 50

100 +  99 +  98 +  … + 53 + 52 + 51


Each row had fifty numbers. He added the corresponding numbers as follows.

1 + 100 = 101

2 + 99 = 101

3 + 98 = 101

48 + 53 = 101

49 + 52 = 101

50 + 51 = 101


Gauss found that the final sum would be

101 + 101 + 101 + … (50 times)    =    101 x 50    =    5050.

This was thinking outside the box. Mathematics provides many such opportunities.


EXERCISE: Add numbers

(a)  1 to 20

(b)  1 to 50

(c)  1 to 33


Helping learn something new

Mathematics also provides many opportunities to help learn something new. For example, the feel for numbers is very important and it helps one learn to add very quickly.

Part of the feel for numbers is to know the gap between a number and the next TEN.

This gap can be seen on a number line at the beginning of this essay, where it helps add 39 + 5 = 44, and 66 + 8 = 74. Here the gap is filled first by the second number and then the rest of the number is added easily to TEN.

A student and his or her study partner can drill these gaps. One of them calls out a number and the other responds with the gap. Such drill is a lot of fun, when the numbers called out are random.

Number           Gap

9                              1

8                              2

7                              3

6                              4

5                              5

4                              6

3                              7

2                              8

1                              9

27                            3

49                            1

54                            6                            etc.

The fundamental aspects of mental math can be learned quite quickly with such drilling. But any such drilling must be followed by proper understanding. For example, the student must first understand that multiplication is “repeated addition” before he or she drills the multiplication tables.


Comments on Einstein’s Last Essay

Disturbed space

Reference: Relativity and the Problem of Space
NOTE: Einstein’s statements are in black italics. My understanding follows in bold color italics.


Einstein wrote this essay in 1952, three years before his death, as a new appendix to his book, “Relativity: The Special and General Theory (1920)“.


It is characteristic of Newtonian physics that it has to ascribe independent and real existence to space and time as well as to matter, for in Newton’s law of motion the idea of acceleration appears. But in this theory, acceleration can only denote “acceleration with respect to space”. Newton’s space must thus be thought of as “at rest”, or at least as “unaccelerated”, in order that one can consider the acceleration, which appears in the law of motion, as being a magnitude with any meaning. Much the same holds with time, which of course likewise enters into the concept of acceleration.

Newtonian physics describes velocity as the rate of change of the distance of a body with respect to another body. Such distances are interpreted in terms of the material dimensions. This leads to the conventional view of space as the “background” in which matter moves. 

Newtonian physics describes acceleration as the rate of change of the velocity of a body. This change is relative to the body itself, and no distance is involved. It involves change in an inherent characteristic of the body called inertia. This leads to the view of inertia as the “background” in which the very matter is formed. 

The background is confused with space only, though it gives meaning to both space and inertia. However, this background is neither space nor inertia. Space and inertia are dimensions that are abstracted from the fact of matter.

Newton himself and his most critical contemporaries felt it to be disturbing that one had to ascribe physical reality both to space itself as well as to its state of motion; but there was at that time no other alternative, if one wished to ascribe to mechanics a clear meaning.

The reality of space exists as the abstraction of material dimensions. The reality of inertia (state of motion) exists as the abstraction of material (mass). Here the physical reality of the background is limited to being the reference point of zero dimensions and zero inertia.

There seems to be a confusion between the objectivity of the physical reality and the subjectivity of abstraction. It is very difficult to conceive the physical reality of “zero”. So we incorrectly assign physical reality to space and inertia, which are truly abstractions.

It is indeed an exacting requirement to have to ascribe physical reality to space in general, and especially to empty space. Time and again since remotest times philosophers have resisted such a presumption. Descartes argued somewhat on these lines: space is identical with extension, but extension is connected with bodies; thus there is no space without bodies and hence no empty space. The weakness of this argument lies primarily in what follows. It is certainly true that the concept extension owes its origin to our experiences of laying out or bringing into contact solid bodies. But from this it cannot be concluded that the concept of extension may not be justified in cases which have not themselves given rise to the formation of this concept. Such an enlargement of concepts can be justified indirectly by its value for the comprehension of empirical results.

The physical reality applies to material dimensions and not to space, which is an abstraction. Therefore, there is no “empty space” with physical extensions. The idea of “empty space” is totally subjective. Einstein’s argument hides the fact that there can be a physical reality of zero extensions. This makes Decartes’ argument that there is no “empty space”, closer to reality.  

The assertion that extension is confined to bodies is therefore of itself certainly unfounded. We shall see later, however, that the general theory of relativity confirms Descartes’ conception in a roundabout way.

What brought Descartes to his remarkably attractive view was certainly the feeling that, without compelling necessity, one ought not to ascribe reality to a thing like space, which is not capable of being “directly experienced”.

Our reality consists of physical perceptions. It is then supported by mental abstractions. Absence of physical perceptions amounts to no physical reality. The background is, therefore, a complete absence of physical dimensions. The idea of background as space in which objects move is purely subjective.

The psychological origin of the idea of space, or of the necessity for it, is far from being so obvious as it may appear to be on the basis of our customary habit of thought. The old geometers deal with conceptual objects (straight line, point, surface), but not really with space as such, as was done later in analytical geometry. The idea of space, however, is suggested by certain primitive experiences. 

We try to measure the gap between objects by estimating the sum of extensions of material objects that will fill that gap. Thus the gap is measured in terms of material extensions. This suggests an idea of space having extensions of its own. When this becomes customary to our thought we have abstraction of space and locations as points, lines, surfaces and volumes of geometry. Analytical geometry provides further abstraction of space.

Suppose that a box has been constructed. Objects can be arranged in a certain way inside the box, so that it becomes full. The possibility of such arrangements is a property of the material object “box”, something that is given with the box, the “space enclosed” by the box. This is something which is different for different boxes, something that is thought quite naturally as being independent of whether or not, at any moment, there are any objects at all in the box. When there are no objects in the box, its space appears to be “empty”.

The space within a box may be measured by arranging objects in a way so that it becomes full. This would define the space enclosed by the box. This space is there whether the box is full or empty This space is the property of the box. It depends on the extensions of that box.

So far, our concept of space has been associated with the box. It turns out, however, that the storage possibilities that make up the box-space are independent of the thickness of the walls of the box. Cannot this thickness be reduced to zero, without the “space” being lost as a result? The naturalness of such a limiting process is obvious, and now there remains for our thought the space without the box, a self-evident thing, yet it appears to be so unreal if we forget the origin of this concept. One can understand that it was repugnant to Descartes to consider space as independent of material objects, a thing that might exist without matter.  (At the same time, this does not prevent him from treating space as a fundamental concept in his analytical geometry.) The drawing of attention to the vacuum in a mercury barometer has certainly disarmed the last of the Cartesians. But it is not to be denied that, even at this primitive stage, something unsatisfactory clings to the concept of space, or to space thought of as an independent real thing.

By reducing the thickness of the walls of the box to zero we can make the box disappear. We are then left with impressions of the extensions of the box being projected on a background of zero extensions. The background simply acts like a “blank canvas” on which impressions of the box are “drawn”. Einstein is calling it “physical space without the box”. But all we have is a non-physical mental impression of the box, which we are holding up as space. It doesn’t matter if there is anything filling up that space or not.

The ways in which bodies can be packed into space (e.g. the box) are the subject of three-dimensional Euclidean geometry, whose axiomatic structure readily deceives us into forgetting that it refers to realisable situations.

The axiomatic structure of Euclidean geometry basically applies to the three-dimensional impressions left by solid objects, or created by the mind. These mental impressions follow the rules of material extensions from which they are derived.

If now the concept of space is formed in the manner outlined above, and following on from experience about the “filling” of the box, then this space is primarily a bounded space. This limitation does not appear to be essential, however, for apparently a larger box can always be introduced to enclose the smaller one. In this way space appears as something unbounded.

“Bounded space” is the three-dimensional impression of material objects projected on a null background. By increasing the extension of these impressions we may arrive at the impression of unbounded space. But this “unbounded space” is simply an impresion of a box of infinite dimensions.

I shall not consider here how the concepts of the three-dimensional and the Euclidean nature of space can be traced back to relatively primitive experiences.

Rather, I shall consider first of all from other points of view the rôle of the concept of space in the development of physical thought.

It has to be understood that this whole process of defining space is subjective. This concept of space is derived from our experience with material extensions. The rules followed by this concept are those of material extensions. These rules have embedded themselves in our thought. This is mental conditioning.

When a smaller box s is situated, relatively at rest, inside the hollow space of a larger box S, then the hollow space of s is a part of the hollow space of S, and the same “space”, which contains both of them, belongs to each of the boxes. When s is in motion with respect to S, however, the concept is less simple. One is then inclined to think that s encloses always the same space, but a variable part of the space S. It then becomes necessary to apportion to each box its particular space, not thought of as bounded, and to assume that these two spaces are in motion with respect to each other.

Einstein is pointing out this mental conditioning that forms the background of our understanding of the physical universe. But he is ignoring the real background, which is the “blank canvas” of zero dimensions, on which such mental impressions are “drawn”. 

Before one has become aware of this complication, space appears as an unbounded medium or container in which material objects swim around. But it must now be remembered that there is an infinite number of spaces, which are in motion with respect to each other.

The concept of space as something existing objectively and independent of things belongs to pre-scientific thought, but not so the idea of the existence of an infinite number of spaces in motion relatively to each other.

This latter idea is indeed logically unavoidable, but is far from having played a considerable rôle even in scientific thought.

We are conditioned to think of space as an unbounded medium or container in which material objects swim around. Einstein considers each object as having its own space, and explores the idea of an infinte number of spaces, which are in motion with respect to each other. Such spaces in motion, however, are still mental impressions, same as before.

But what about the psychological origin of the concept of time? This concept is undoubtedly associated with the fact of “calling to mind”, as well as with the differentiation between sense experiences and the recollection of these. Of itself it is doubtful whether the differentiation between sense experience and recollection (or simple re-presentation) is something psychologically directly given to us. Everyone has experienced that he has been in doubt whether he has actually experienced something with his senses or has simply dreamt about it. Probably the ability to discriminate between these alternatives first comes about as the result of an activity of the mind creating order.

Motion exists due to sequential changes in material extensions. Such changes, or motion, provides the impression of time.

Just as space is the impression of “material extensions”; time is the impression of “material changes”. Thus time is also subjective. Because of this subjectivity, what one has experienced is confused with what one has projected.

The objective physical reality lies in actual material extensions and material changes. This makes space and time deterministic when compared to a background of “zero” extension and “zero” change.  The theory of relativity does not consider such a background.

An experience is associated with a “recollection”, and it is considered as being “earlier” in comparison with present “experiences”. This is a conceptual ordering principle for recollected experiences, and the possibility of its accomplishment gives rise to the subjective concept of time, i.e. that concept of time which refers to the arrangement of the experiences of the individual.

Our experience is made up of perceptual elements that may be arranged in a matrix-like fashion. The logical elements of continuity, harmony and consistency, make the matrix arrangement quite deterministic. Different paths among the perceptual elements lead to different logical sequences. Each logical sequence represents an experience of time. This is different from our conditioning to a subjective time due to rotation of the earth.

What do we mean by rendering objective the concept of time? Let us consider an example. A person A (“I”) has the experience “it is lightning”. At the same time the person A also experiences such a behaviour of the person B as brings the behaviour of B into relation with his own experience “it is lightning”. Thus it comes about that A associates with B the experience “it is lightning”. For the person A the idea arises that other persons also participate in the experience “it is lightning”. “It is lightning” is now no longer interpreted as an exclusively personal experience, but as an experience of other persons (or eventually only as a “potential experience”). In this way arises the interpretation that “it is lightning”, which originally entered into the consciousness as an “experience”, is now also interpreted as an (objective) “event”. It is just the sum total of all events that we mean when we speak of the “real external world”.

When we talk about many people reacting to the same event we consider that event to be “objective”, because it is entering our consciousness from “outside”. This provides a very limited view of consciousness.

Consciousness is not limited to us. We are the simply the most evolved part of the universe that is looking back at the universe. This universe, as a whole, is conscious. That consciousness simply peaks in us.

Thus, the objectivity of time has to do with actual sequence of material changes and not with some general abstraction of those changes that gets interpreted relative to the rotation of earth.

We have seen that we feel ourselves impelled to ascribe a temporal arrangement to our experiences, somewhat as follows. If b is later than a and c later than b then c is also later than a (“sequence of experiences”).

Now what is the position in this respect with the “events” which we have associated with the experiences? At first sight it seems obvious to assume that a temporal arrangement of events exists which agrees with the temporal arrangement of the experiences. In general, and unconsciously this was done, until sceptical doubts made themselves felt.  In order to arrive at the idea of an objective world, an additional constructive concept still is necessary: the event is localised not only in time, but also in space.

The “temporal sequence” of events is rather subjective. Objectivity lies with a “logical sequence” of events that have the characteristics of continuity, harmony and consistency. An event may appear to be localized in space and time but it must be continuous, harmonious and consistent with rest of the space and time.  

In the previous paragraphs we have attempted to describe how the concepts space, time and event can be put psychologically into relation with experiences. Considered logically, they are free creations of the human intelligence, tools of thought, which are to serve the purpose of bringing experiences into relation with each other, so that in this way they can be better surveyed.

Einstein’s use of psychology to describe space and time is subjective. Physical perceptions are what they are. Our psychological experiences needn’t be any different if we maintain continuity, harmony and consistency in observation, and do not introduce assumptions.

The attempt to become conscious of the empirical sources of these fundamental concepts should show to what extent we are actually bound to these concepts. In this way we become aware of our freedom, of which, in case of necessity, it is always a difficult matter to make sensible use.

Empirically, we are part of the phenomena that we are perceiving. Objectivity is determined by continuity, harmony and consistency between the phenomena observed and the observer. 

Here we are going deeper than the frame of reference used by Einstein and Kant. Here we are using the frame of reference used in the Vedas.

We still have something essential to add to this sketch concerning the psychological origin of the concepts space-time-event (we will call them more briefly “space-like”, in contrast to concepts from the psychological sphere). We have linked up the concept of space with experiences using boxes and the arrangement of material objects in them. Thus this formation of concepts already presupposes the concept of material objects (e.g. ”boxes”). In the same way persons, who had to be introduced for the formation of an objective concept of time, also play the rôle of material objects in this connection. It appears to me, therefore, that the formation of the concept of the material object must precede our concepts of time and space.

It is true that the formation of material object precedes our subjectively conditioned concepts of space and time as elucidated by Einstein. We still need to investigate how material objects are formed.

All these space-like concepts already belong to pre-scientific thought, along with concepts like pain, goal, purpose, etc. from the field of psychology. Now it is characteristic of thought in physics, as of thought in natural science generally, that it endeavours in principle to make do with “space-like” concepts alone, and strives to express with their aid all relations having the form of laws. The physicist seeks to reduce colours and tones to vibrations, the physiologist thought and pain to nerve processes, in such a way that the psychical element as such is eliminated from the causal nexus of existence, and thus nowhere occurs as an independent link in the causal associations. It is no doubt this attitude, which considers the comprehension of all relations by the exclusive use of only space-like concepts as being possible in principle, that is at the present time understood by the term “materialism” (since “matter” has lost its rôle as a fundamental concept).

Matter somehow emerges from the background of zero dimensions and zero change. We then abstract from matter the space-time-event and other psychological concepts. Therefore it appears as a materialistic approach to explain the universe. But there seems to be a spiritual element underlying the very formation of matter in the first place.

Why is it necessary to drag down from the Olympian fields of Plato the fundamental ideas of thought in natural science, and to attempt to reveal their earthly lineage? Answer: in order to free these ideas from the taboo attached to them, and thus to achieve greater freedom in the formation of ideas or concepts. It is to the immortal credit of D. Hume and E. Mach that they, above all others, introduced this critical conception.

When an idea is not clear and cannot be explained fully it is important to examine it more closely no matter how sancrosanct it is considered to be.

Science has taken over from pre-scientific thought the concepts space, time, and material object (with the important special case “solid body”) and has modified them and rendered them more precise. Its first significant accomplishment was the development of Euclidean geometry, whose axiomatic formulation must not be allowed to blind us to its empirical origin (the possibilities of laying out or juxtaposing solid bodies). In particular, the three-dimensional nature of space as well as its Euclidean character are of empirical origin (it can be wholly filled by like constituted “cubes”).

The subtlety of the concept of space was enhanced by the discovery that there exist no completely rigid bodies.

The Euclidean geometry derives its concepts of space and time from the rigidity of the dimensions of solid material objects. These concepts are now enhanced by the discovery that there exist no completely rigid bodies.

All bodies are elastically deformable and alter in volume with change in temperature. The structures, whose possible congruences are to be described by Euclidean geometry, cannot therefore be represented apart from physical concepts. But since physics after all must make use of geometry in the establishment of its concepts, the empirical content of geometry can be stated and tested only in the framework of the whole of physics.

The concepts of space and time must take into account the changes in material extensions due to temperature and other physical phenomena. The Euclidean geometry must considered within the framework of the whole of physics.

In this connection atomistics must also be borne in mind, and its conception of finite divisibility; for spaces of sub-atomic extension cannot be measured up.

Atomistics also compels us to give up, in principle, the idea of sharply and statically defined bounding surfaces of solid bodies. Strictly speaking, there are no precise laws, even in the macro-region, for the possible configurations of solid bodies touching each other.

In physics, the atoms are not uniformly solid. They are made of fields of frequency gradients from zero (space) to very high frequency (the nucleus of atom). The dense frequencies appear as solid.

Thus the spaces of sub-atomic extension cannot be measured up. The material objects cannot be considered to be bound by sharply defined surfaces. They do not exactly touch each other. This modifies the concept of space and time.

In spite of this, no one thought of giving up the concept of space, for it appeared indispensable in the eminently satisfactory whole system of natural science.

Mach, in the nineteenth century, was the only one who thought seriously of an elimination of the concept of space, in that he sought to replace it by the notion of the totality of the instantaneous distances between all material points. (He made this attempt in order to arrive at a satisfactory understanding of inertia).

Such minutiae in the concepts of space-time-event become important when working with the concept of Inertia.


The Field

In Newtonian mechanics, space and time play a dual rôle. First, they play the part of carrier or frame for things that happen in physics, in reference to which events are described by the space co-ordinates and the time. In principle, matter is thought of as consisting of “material points”, the motions of which constitute physical happening. When matter is thought of as being continuous, this is done as it were provisionally in those cases where one does not wish to or cannot describe the discrete structure. In this case small parts (elements of volume) of the matter are treated similarly to material points, at least in so far as we are concerned merely with motions and not with occurrences which, at the moment, it is not possible or serves no useful purpose to attribute to motions (e.g. temperature changes, chemical processes).

In Newtonian Mechanics space and time act as the frame of reference in which extensions, and changes in such extensions, are displayed. These extensions and changes are treated the same way for both space and matter. This should be obvious because space is treated as abstraction of material extensions.

Changes in extensions constitute physical motion. Changes, such as, chemical and temperature, are not described in this frame of reference because they do not constitute physical motion.

Both space and matter are also treated as continuous and rigid. Space has location; whereas matter has”material points”. Many location in space may be combined as a point on a map. Similarly, the material points of an object may be combined as a “center of mass”.

The second rôle of space and time was that of being an “inertial system”. From all conceivable systems of reference, inertial systems were considered to be advantageous in that, with respect to them, the law of inertia claimed validity.

A material point is differentiated from surrounding space locations by its inertia. Locations in space and time have no inertia. The material point, therefore, follows certain laws of motion. Furthermore, because of this inertia, matter can have structure; whereas, surrounding space and time have no structure.  

In this, the essential thing is that “physical reality”, thought of as being independent of the subjects experiencing it, was conceived as consisting, at least in principle, of space and time on one hand, and of permanently existing material points, moving with respect to space and time, on the other. The idea of the independent existence of space and time can be expressed drastically in this way: If matter were to disappear, space and time alone would remain behind (as a kind of stage for physical happening).

Thus, the essential difference between space and matter is inertia. This difference is seen to be so sharp that space and time on one hand, and matter on the other, are thought to be completely separate and independent of each other. Matter is seen as existing in space; and, if all matter were to disappear, it is thought that space and time would still exist.

The surmounting of this standpoint resulted from a development which, in the first place, appeared to have nothing to do with the problem of space-time, namely, the appearance of the concept of field and its final claim to replace, in principle, the idea of a particle (material point). In the framework of classical physics, the concept of field appeared as an auxiliary concept, in cases in which matter was treated as a continuum. For example, in the consideration of the heat conduction in a solid body, the state of the body is described by giving the temperature at every point of the body for every definite time. Mathematically, this means that the temperature T is represented as a mathematical expression (function) of the space co-ordinates and the time t (Temperature field). 

The law of heat conduction is represented as a local relation (differential equation), which embraces all special cases of the conduction of heat. The temperature is here a simple example of the concept of field. This is a quantity (or a complex of quantities), which is a function of the co-ordinates and the time. Another example is the description of the motion of a liquid. At every point there exists at any time a velocity, which is quantitatively described by its three “components” with respect to the axes of a co-ordinate system (vector). The components of the velocity at a point (field components), here also, are functions of the co-ordinates (x, y, z) and the time (t).

The Newtonian standpoint came to be surmounted by the concept of field, which was first developed in the subjects of thermodynamics and fluid dynamics. In these disciplines, matter was treated as a continuum, and its states, such as, temperature and velocity, were described at every point of the body for every definite time.

In other words, a field describes the state of matter continuously in space and time co-ordinates. It takes differential equations to completely describe the complexity of such states. In principle, this concept of a continuous field claimed to replace the idea of a particle (material point).

It is characteristic of the fields mentioned that they occur only within a ponderable mass; they serve only to describe a state of this matter. In accordance with the historical development of the field concept, where no matter was available there could also exist no field. But in the first quarter of the nineteenth century it was shown that the phenomena of the interference and motion of light could be explained with astonishing clearness when light was regarded as a wave-field, completely analogous to the mechanical vibration field in an elastic solid body. It was thus felt necessary to introduce a field, that could also exist in “empty space” in the absence of ponderable matter.

In the classical sense, the field described a state of matter, and, therefore, it could occur only within a ponderable mass. Thus, it was assumed that where no matter was available there could also exist no field. 

But in the first quarter of the nineteenth century it was shown that the phenomena of the interference and motion of light could be explained with astonishing clearness when light was regarded as a wave-field, completely analogous to the mechanical vibration field in an elastic solid body. It was thus felt necessary to introduce a field, that could also exist in “empty space” in the absence of ponderable matter.

This state of affairs created a paradoxical situation, because, in accordance with its origin, the field concept appeared to be restricted to the description of states in the inside of a ponderable body. This seemed to be all the more certain, inasmuch as the conviction was held that every field is to be regarded as a state capable of mechanical interpretation, and this presupposed the presence of matter. One thus felt compelled, even in the space which had hitherto been regarded as empty, to assume everywhere the existence of a form of matter, which was called “aether”.

Since space was viewed as an abstraction of material extensions, it was assumed that the “empty space” must have mechanical properties. This was the idea underlying the concept of “aether” which was considered essential as the medium for the propagation of light. The field concept was assumed to have a mechanical carrier.

The emancipation of the field concept from the assumption of its association with a mechanical carrier finds a place among the psychologically most interesting events in the development of physical thought. During the second half of the nineteenth century, in connection with the researches of Faraday and Maxwell it became more and more clear that the description of electromagnetic processes in terms of field was vastly superior to a treatment on the basis of the mechanical concepts of material points. By the introduction of the field concept in electrodynamics, Maxwell succeeded in predicting the existence of electromagnetic waves, the essential identity of which with light waves could not be doubted because of the equality of their velocity of propagation. As a result of this, optics was, in principle, absorbed by electrodynamics. One psychological effect of this immense success was that the field concept, as opposed to the mechanistic framework of classical physics, gradually won greater independence.

During the second half of the nineteenth century, in connection with the researches of Faraday and Maxwell it became more and more clear that the description of electromagnetic processes in terms of field was vastly superior to a treatment on the basis of the mechanical concepts of material points.

By the introduction of the field concept in electrodynamics, Maxwell succeeded in predicting the existence of electromagnetic waves, the essential identity of which with light waves could not be doubted because of the equality of their velocity of propagation.

There were no material points of mechanical properties that acted as the medium of electromagnetic waves. The wave-field had its own extension and changes. It simply existed in a background of zero dimension and zero change. 

Nevertheless, it was at first taken for granted that electromagnetic fields had to be interpreted as states of the aether, and it was zealously sought to explain these states as mechanical ones. But as these efforts always met with frustration, science gradually became accustomed to the idea of renouncing such a mechanical interpretation. Nevertheless, the conviction still remained that electromagnetic fields must be states of the aether, and this was the position at the turn of the century.

Howver, the electromagnetic waves were intensely investigated as being states of a mechanical aether.

The aether-theory brought with it the question: How does the aether behave from the mechanical point of view with respect to ponderable bodies? Does it take part in the motions of the bodies, or do its parts remain at rest relatively to each other? Many ingenious experiments were undertaken to decide this question. The following important facts should be mentioned in this connection: the “aberration” of the fixed stars in consequence of the annual motion of the earth, and the “Doppler effect”, i.e. the influence of the relative motion of the fixed stars on the frequency of the light reaching us from them, for known frequencies of emission. The results of all these facts and experiments, except for one, the Michelson-Morley experiment, were explained by H. A. Lorentz on the assumption that the aether does not take part in the motions of ponderable bodies, and that the parts of the aether have no relative motions at all with respect to each other. Thus the aether appeared, as it were, as the embodiment of a space absolutely at rest. But the investigation of Lorentz accomplished still more. It explained all the electromagnetic and optical processes within ponderable bodies known at that time, on the assumption that the influence of ponderable matter on the electric field – and conversely – is due solely to the fact that the constituent particles of matter carry electrical charges, which share the motion of the particles. Concerning the experiment of Michelson and Morley, H. A. Lorentz showed that the result obtained at least does not contradict the theory of an aether at rest.

The questions asked were: How does the aether behave from the mechanical point of view with respect to ponderable bodies? Does it take part in the motions of the bodies, or do its parts remain at rest relatively to each other?

Observations related to the “aberration” of the fixed stars, and the “Doppler effect” for known frequencies of emission, were explained by H. A. Lorentz on the assumption that the aether does not take part in the motions of ponderable bodies, and that the parts of the aether have no relative motions at all with respect to each other. Thus the aether appeared, as it were, as the embodiment of a space absolutely at rest.

Investigations by Lorentz also favored the assumption that the constituent particles of matter carry electrical charges, which share the motion of the particles.

In spite of all these beautiful successes the state of the theory was not yet wholly satisfactory, and for the following reasons. Classical mechanics, of which it could not be doubted that it holds with a close degree of approximation, teaches the equivalence of all inertial systems or inertial “spaces” for the formulation of natural laws, i.e. the invariance of natural laws with respect to the transition from one inertial system to another. Electromagnetic and optical experiments taught the same thing with considerable accuracy. But the foundation of electromagnetic theory taught that a particular inertial system must be given preference, namely that of the luminiferous aether at rest. This view of the theoretical foundation was much too unsatisfactory. Was there no modification that, like classical mechanics, would uphold the equivalence of inertial systems (special principle of relativity)?

According to Classical mechanics, there is invariance of natural laws with respect to the transition from one inertial system to another. It, therefore, appears unsatisfactory to regard aether of mechanical nature to be at rest in two different inertial systems while not also interacting with matter.

The answer to this question is the special theory of relativity. This takes over from the theory of Maxwell-Lorentz the assumption of the constancy of the velocity of light in empty space. In order to bring this into harmony with the equivalence of inertial systems (special principle of relativity), the idea of the absolute character of simultaneity must be given up; in addition, the Lorentz transformations for the time and the space co-ordinates follow for the transition from one inertial system to another. The whole content of the special theory of relativity is included in the postulate: The laws of Nature are invariant with respect to the Lorentz transformations. The important thing of this requirement lies in the fact that it limits the possible natural laws in a definite manner.

Einstein, therefore, rejected the idea of a mechanical ether and replaced it by his special theory of relativity. The bridge from one inertial system to another was now provided by the nature of wave-field, instead of the nature of matter. This bridge was represented by the constant ratio of wavelength to period of the wave. This invariant was referred to as the “speed” of light.

But this again is unsatisfactory because it assumes that an electromagnetic wave, such as light, has zero inertia and it does not interact with matter.

What is the position of the special theory of relativity in regard to the problem of space? In the first place we must guard against the opinion that the four-dimensionality of reality has been newly introduced for the first time by this theory. Even in classical physics the event is localised by four numbers, three spatial co-ordinates and a time co-ordinate; the totality of physical “events” is thus thought of as being embedded in a four-dimensional continuous manifold. But on the basis of classical mechanics this four-dimensional continuum breaks up objectively into the one-dimensional time and into three-dimensional spatial sections, only the latter of which contain simultaneous events. This resolution is the same for all inertial systems. The simultaneity of two definite events with reference to one inertial system involves the simultaneity of these events in reference to all inertial systems. This is what is meant when we say that the time of classical mechanics is absolute. According to the special theory of relativity it is otherwise.

The problem of space is that it is an abstraction. Time is basically the “change” aspect of space. The three spatial coordinates apply to the extension of the bodies. The time coordinate applies to the changes in that extension. These dimensions do not exist in the absence of the bodies.

The beginning of twentieth century brought about the consideration that in the absence of bodies there is electromagnetic field. Space and time of the theory of relativity represent the dimesions of the field, which are wavelength and period respectively. The invariant ratio ‘c’ of wavelength to period simply establishes the continuity of space with time, or changes in space.

Thus, the theory of relativity assumes space and time to be the abstractions of field extensions instead of material extensions. From Newton to Einstein the difference is only in the abstraction of space and time. But the field, like matter, still must exist in a background of zero dimensions and zero change.

The sum total of events which are simultaneous with a selected event exist, it is true, in relation to a particular inertial system, but no longer independently of the choice of the inertial system. The four-dimensional continuum is now no longer resolvable objectively into sections, all of which contain simultaneous events; “now” loses for the spatiaIly extended world its objective meaning. It is because of this that space and time must be regarded as a four-dimensional continuum that is objectively unresolvable, if it is desired to express the purport of objective relations without unnecessary conventional arbitrariness.

In the Newtonian mass-based system, the sum total of events which are simultaneous with a selected event exist in all inertial systems. This is because, on the broad scale, the inertial characteristics of mass do not change appreciably from one inertial frame to another.

But in a field-based system, the inertial characteristics of field do change appreciably from one inertial frame to another with change in frequency. Therefore, the sum total of events which are simultaneous with a selected event does not exist in all inertial systems of the field.

Since the special theory of relativity revealed the physical equivalence of all inertial systems, it proved the untenability of the hypothesis of an aether at rest. It was therefore necessary to renounce the idea that the electromagnetic field is to be regarded as a state of a material carrier. The field thus becomes an irreducible element of physical description, irreducible in the same sense as the concept of matter in the theory of Newton.

It is the invariance of the inertial characteristics of the mass-system that determine mechanical properties. Since this invariance can no longer be supported for field-systems, the electromagnetic fields cannot be regarded as having a mechanical carrier. Thus there is no fundamental mechanical basis for space as “aether”.

The field thus becomes an irreducible element of physical description, irreducible in the same sense as the concept of matter in the theory of Newton.

Up to now we have directed our attention to finding in what respect the concepts of space and time were modified by the special theory of relativity. Let us now focus our attention on those elements which this theory has taken over from classical mechanics. Here also, natural laws claim validity only when an inertial system is taken as the basis of space-time description. The principle of inertia and the principle of the constancy of the velocity of light are valid only with respect to an inertial system. The field-laws also can claim to have a meaning and validity only in regard to inertial systems.

Einstein says, “Natural laws claim validity only when an inertial system is taken as the basis of space-time description.” From mass to field, the inertial characteristics are changing. The inertial characteristics at the frequency of light would probably be non-existent because that is the reference one is using for inertia at the level of mass. Therefore, the space-time description shall also be “non-existent” at the frequency of light.

This conclusion is highly unsatisfactory. Evidently, there is inertia at the frequency of light. The special theory of relativity does not account for inertia.

Thus, as in classical mechanics, space is here also an independent component in the representation of physical reality. If we imagine matter and field to be removed, inertial-space or, more accurately, this space together with the associated time remains behind. The four-dimensional structure (Minkowski-space) is thought of as being the carrier of matter and of the field. Inertial spaces, with their associated times, are only privileged four-dimensional co-ordinate systems, that are linked together by the linear Lorentz transformations. Since there exist in this four-dimensional structure no longer any sections which represent “now” objectively, the concepts of happening and becoming are indeed not completely suspended, but yet complicated. It appears therefore more natural to think of physical reality as a four-dimensional existence, instead of, as hitherto, the evolution of a three-dimensional existence.

It is untrue that if we imagine matter and field to be removed, the inertial-space (space together with the associated time) remains behind. This view of Einstein is just an abstraction, and a subjective interpretation of reality.

Objectively,  if we imagine matter and field to be removed, only the background of zero inertia, zero dimensions and zero change would remain behind. The Minkowski space is a mathematical abstraction of matter and field. It is not an actuality that can be sensed physically. Only matter and field are sensed physically.

Einstein’s theory correctly acknowledges the reality of the field, which was experimentally determined by Faraday, and mathematically confirmed by Maxwell. He also see correctly the physical reality of inertia, extensions and changes, which was worked out by Newton. He does a wonderful job of bringing a better alignment to the works of Newton, Faraday and Maxwell.

But Einstein fails to see that the reference point of the physical reality of inertia, extensions and changes would be the background of zero inertia, zero extensions and zero change. Therefore, Einstein deviates from the accurate desription of physical reality as it exists, or from an objective sense of “now”.

This rigid four-dimensional space of the special theory of relativity is to some extent a four-dimensional analogue of H. A. Lorentz’s rigid three-dimensional aether. For this theory also the following statement is valid: The description of physical states postulates space as being initially given and as existing independently. Thus even this theory does not dispel Descartes’ uneasiness concerning the independent, or indeed, the a priori existence of “empty space”. The real aim of the elementary discussion given here is to show to what extent these doubts are overcome by the general theory of relativity.

The special theory of relativity is a four dimensional analogue of the three-dimensional aether theory. It does not dispel Descartes’ uneasiness concerning the independent, or indeed, the a priori existence of “empty space”. (This is per Einstein himself.)

The “empty space” is empty of matter and field, but it has not been empty of their abstractions. The truly empty space shall be empty not only of matter and field, but also of their abstractions.

The truly empty space is the background of zero inertia, zero extensions and zero change. This would definitely align with the reality postulated by Descartes.


The Concept of Space in the General Theory of Relativity

This theory arose primarily from the endeavour to understand the equality of inertial and gravitational mass. We start out from an inertial system S1, whose space is, from the physical point of view, empty. In other words, there exists in the part of space contemplated neither matter (in the usual sense) nor a field (in the sense of the special theory of relativity). With reference to S1 let there be a second system of reference S2 in uniform acceleration. Then S2 is thus not an inertial system. With respect to S2 every test mass would move with an acceleration, which is independent of its physical and chemical nature. Relative to S2, therefore, there exists a state which, at least to a first approximation, cannot be distinguished from a gravitational field. The following concept is thus compatible with the observable facts: S2 is also equivalent to an “inertial system”; but with respect to S2 a (homogeneous) gravitational field is present (about the origin of which one does not worry in this connection). Thus when the gravitational field is included in the framework of the consideration, the inertial system loses its objective significance, assuming that this “principle of equivalence” can be extended to any relative motion whatsoever of the systems of reference. If it is possible to base a consistent theory on these fundamental ideas, it will satisfy of itself the fact of the equality of inertial and gravitational mass, which is strongly confirmed empirically.

When there is neither matter nor field, there is no inertia either. It will be the pure background of zero extensions, zero changes and zero inertia. This background cannot be “accelerated” as such. To bring about acceleration one must introduce some kind of inertia as the basis of extension and change.

An inertial system begins with constant frequency in case of a field; and with constant velocity in case of matter. An inertial system is without acceleration, which means that neither frequency nor velocity is changing.

An inertial system has constant inertia. An “accelerating system” is no longer an inertial system. Acceleration requires addition of inertia, therefore, an accelerating system shall consist of a gradient of inertia. This would be the case with a gravitational system, which consists of acceleration.

Therefore, to have gravitation there must be a gradient of inertia. Two different points in a gravitational field shall have different inertia. Due to this difference there shall be a flow of gravity  from the point of higher inertia towards the point of lower inertia. This is similar to a flow from a point of high pressure towards a point of low pressure.

Inertia and gravitation are closely tied together.

Considered four-dimensionally, a non-linear transformation of the four co-ordinates corresponds to the transition from S1 to S2. The question now arises: What kind of non-linear transformations are to be permitted, or, how is the Lorentz transformation to be generalised? In order to answer this question, the following consideration is decisive.

Einstein is working with pure abstraction of space and time. His approach is completely mathematical. Here we need to understand the meaning of acceleration and gravity.

Constant frequency represents a field of constant wavelength and period, and of static inertia. Acceleration would then be a uniformly increasing frequency and inertia. Gravity is essentially a manifestation of changing frequency and inertia. The gravitational field shall consist of uniformly increasing frequency and inertia, and uniformly decreasing wavelength and period.

We ascribe to the inertial system of the earlier theory this property: Differences in co-ordinates are measured by stationary “rigid” measuring rods, and differences in time by clocks at rest. The first assumption is supplemented by another, namely, that for the relative laying out and fitting together of measuring rods at rest, the theorems on “lengths” in Euclidean geometry hold. From the results of the special theory of relativity it is then concluded, by elementary considerations, that this direct physical interpretation of the co-ordinates is lost for systems of reference (S2) accelerated relatively to inertial systems (S1). But if this is the case, the co-ordinates now express only the order or rank of the “contiguity” and hence also the dimensional grade of the space, but do not express any of its metrical properties. We are thus led to extend the transformations to arbitrary continuous transformations.  This implies the general principle of relativity: Natural laws must be covariant with respect to arbitrary continuous transformations of the co-ordinates. This requirement (combined with that of the greatest possible logical simplicity of the laws) limits the natural laws concerned incomparably more strongly than the special principle of relativity.

In Newton’s inertial system the space is rigid, and the time is mechanical, because they are abstractions of matter. This is the case when the system is not accelerating. But, according to Einstein, when this system is accelerating, the space and time remain contiguous but their dimensional grade changes. With acceleration, there occurs a transformation in the characteristics of field and matter.

We observe the characteristics of field and matter existing together in the region of atom, which are currently addressed by quantum mechanics. Within the atom, the “extremely dense” field appears as matter, and the less dense field appears as space between sub-atomic particles. With acceleration, the whole atomic structure seems to shift higher on the frequency spectrum. With this transformation the wavelength and period decrease on average, changing the dimensional grade of space and time. A contiguity is maintained through a constant ratio between wavelength and period. This ratio is referred to as the universal constant ‘c’. It is called the “speed of light”.  

This is an attempt to express Einstein’s mathematics in terms of physical reality.

This train of ideas is based essentially on the field as an independent concept. For the conditions prevailing with respect to S2 are interpreted as a gravitational field, without the question of the existence of masses which produce this field being raised. By virtue of this train of ideas it can also be grasped why the laws of the pure gravitational field are more directly linked with the idea of general relativity than the laws for fields of a general kind (when, for instance, an electromagnetic field is present). We have, namely, good ground for the assumption that the “field-free” Minkowski-space represents a special case possible in natural law, in fact, the simplest conceivable special case. With respect to its metrical character, such a space is characterised by the fact that dx1² + dx2² + dx3² is the square of the spatial separation, measured with a unit gauge, of two infinitesimally neighbouring points of a three-dimensional “space-like” cross section (Pythagorean theorem), whereas dx4 is the temporal separation, measured with a suitable time gauge, of two events with common (x1, x2, x3). All this simply means that an objective metrical significance is attached to the quantity

ds² = dx1² + dx2² + dx3² – dx4²    (1)

as is readily shown with the aid of the Lorentz transformations. Mathematically, this fact corresponds to the condition that ds² is invariant with respect to Lorentz transformations.

Einstein is assuming the concept of field to be independent of matter. His mathematical concept of gravity comes from the acceleration of space devoid of matter. He calls it “pure gravitational field” linked with the idea of general relativity. Einsteins differentiates this “pure gravitational field” from the laws for fields of a general kind (when, for instance, an electromagnetic field is present).

Einstein’s assumes that the “field-free” Minkowski-space is the simplest conceivable special case of the general laws of fields. He attaches a metrical significance to it based on Lorentz transformation of space and time. Thus he assumes the following (see Validity of Lorentz Transformation).

(1) Light has zero inertia (its speed is the same in all inertial systems).

(2) The gamma “fudge” factor is the same in different inertial systems.

These assumptions are not totally satisfactory.

The Minkowski-space is an abstraction of the electromagnetic field. The space element (ds) is represented by the “three-dimensional wavelength” of the electromagnetic field. The time element (dt) is represented by the period of the electromagnetic field. The ratio (ds/dt) is maintained as the constant ‘c’, that maintains continuity among the frequencies of the electromagnetic field. The gravitational field is represented by the gradients of frequencies of the electromagnetic field. This mathematics does not make subjective assumptions as those made for Lorentz transformations.

Thus, an electromagnetic field is a field of constant frequency. The inertial or gravitational field is manifested within the electromagnetic field where the frequency increasing on a uniform gradient. The background simply provides a reference point of zero frequency.

It is the uniform gradient of increasing frequency that leads to the formation of stable particles such as electrons, protons and neutrons. Therefore, the assumption that the concept of field is independent of matter becomes questionable.

If now, in the sense of the general principle of relativity, this space (cf. eq. (1) ) is subjected to an arbitrary continuous transformation of the co-ordinates, then the objectively significant quantity ds is expressed in the new system of co-ordinates by the relation

ds² = gik dxi dxk    (1a)

which has to be summed up over the indices i and k for all combinations 11, 12, . . . up to 44 . The terms gik now are not constants, but functions of the co-ordinates, which are determined by the arbitrarily chosen transformation. Nevertheless, the terms gik are not arbitrary functions of the new co-ordinates, but just functions of such a kind that the form (1a) can be transformed back again into the form (1) by a continuous transformation of the four co-ordinates. In order that this may be possible, the functions gik must satisfy certain general covariant equations of condition, which were derived by B. Riemann more than half a century before the formulation of the general theory of relativity (“Riemann condition”). According to the principle of equivalence, (1a) describes in general covariant form a gravitational field of a special kind, when the functions gik satisfy the Riemann condition.

It follows that the law for the pure gravitational field of a general kind must be satisfied when the Riemann condition is satisfied; but it must be weaker or less restricting than the Riemann condition. In this way the field law of pure gravitation is practically completely determined, a result which will not be justified in greater detail here.

Space and time are related in the sense that time represents a change in space that maintains continuity through the ratio ‘c’. Therefore, this ratio should appear in the description of space element ‘ds’ in (1) given by Einstein. This shall reduce the arbitrariness of Einstein’s assumptions. The ‘Reimann condition’ in mathematics simply assures the continuity of the physical reality of field and matter.

ds² = dx1² + dx2² + dx3² – c²ds²

or, (1 + c²) ds² = dx1² + dx2² + dx3²                (1)

In short, when the frequency of the electromagnetic field is subjected to continuous uniform increase, a gravitational field is produced. The continuous change occurs under the constraints imposed by the ratio ‘c’, and the boundary condition of zero frequency and inertia of the background. This results in a curvature forming the mass particle. The details of the gravitational field can be worked out from the above constraint and boundary condition.

We are now in a position to see how far the transition to the general theory of relativity modifies the concept of space. In accordance with classical mechanics and according to the special theory of relativity, space (space-time) has an existence independent of matter or field. In order to be able to describe at all that which fills up space and is dependent on the co-ordinates, space-time or the inertial system with its metrical properties must be thought of at once as existing, for otherwise the description of “that which fills up space” would have no meaning.  On the basis of the general theory of relativity, on the other hand, space as opposed to “what fills space”, which is dependent on the co-ordinates, has no separate existence. Thus a pure gravitational field might have been described in terms of the gik (as functions of the co-ordinates), by solution of the gravitational equations. If we imagine the gravitational field, i.e. the functions gik, to be removed, there does not remain a space of the type (1), but absolutely nothing, and also no “topological space”. For the functions gik describe not only the field, but at the same time also the topological and metrical structural properties of the manifold. A space of the type (1), judged from the standpoint of the general theory of relativity, is not a space without field, but a special case of the gik field, for which – for the co-ordinate system used, which in itself has no objective significance – the functions gik have values that do not depend on the co-ordinates. There is no such thing as an empty space, i.e. a space without field. Space-time does not claim existence on its own, but only as a structural quality of the field.

Space-time coordinates exist only in the presence of field and matter. Otherwise, they are abstractions by definition. Such abstractions are subjective and not objective. So, the conceptualization of space in classical mechanics is subjective.

In classical mechanics, space exists objectively only when it is coincident with matter that fills it. In the absence of matter, there is no space objectively.

In special theory of relativity, space exists objectively only when it is coincident with matter and/or field that fill it. In the absence of matter and/or field, there is no space objectively.

In general theory of relativity, it is mathematics that brings the abstract space in relationship with what fills it. This indicates that space is the same as what fills it. This is an objective view.

The gravitational field is thus an extension of matter. When there is no matter and field, there is no space either. There is no such thing as “empty space”.

Thus Descartes was not so far from the truth when he believed he must exclude the existence of an empty space. The notion indeed appears absurd, as long as physical reality is seen exclusively in ponderable bodies. It requires the idea of the field as the representative of reality, in combination with the general principle of relativity, to show the true kernel of Descartes’ idea; there exists no space ’empty of field’.

Thus Descartes was not so far from the truth when he believed he must exclude the existence of an empty space.


Generalized Theory of Gravitation

The theory of the pure gravitational field on the basis of the general theory of relativity is therefore readily obtainable, because we may be confident that the “field-free” Minkowski space with its metric in conformity with (1) must satisfy the general laws of field. From this special case the law of gravitation follows by a generalisation which is practically free from arbitrariness.

Einstein’s theory of the pure gravitational field is based on the acceleration of Minkowski space. Acceleration is a condition relative to the accelerating entity itself.

Minkowski space is an abstraction of an electromagnetic field of constant frequency. Acceleration in this field means a gradient of uniformly increasing frequency. This condition manifests as a gravitational field.

The further development of the theory is not so unequivocally determined by the general principle of relativity; it has been attempted in various directions during the last few decades. It is common to all these attempts, to conceive physical reality as a field, and moreover, one which is a generalisation of the gravitational field, and in which the field law is a generalisation of the law for the pure gravitational field. After long probing I believe that I have now found  the most natural form for this generalisation, but I have not yet been able to find out whether this generalised law can stand up against the facts of experience.

The general principle of relativity does not lead to a precise theory because it disregards the Boundary condition of the background of zero extension, zero change and zero inertia. The gravitational field results from the gradients in the electromagnetic field, so we are talking about two different but related fields.

The question of the particular field law is secondary in the preceding general considerations. At the present time, the main question is whether a field theory of the kind here contemplated can lead to the goal at all. By this is meant a theory which describes exhaustively physical reality, including four-dimensional space, by a field. The present-day generation of physicists is inclined to answer this question in the negative. In conformity with the present form of the quantum theory, it believes that the state of a system cannot be specified directly, but only in an indirect way by a statement of the statistics of the results of measurement attainable on the system. The conviction prevails that the experimentally assured duality of nature (corpuscular and wave structure) can be realised only by such a weakening of the concept of reality. I think that such a far-reaching theoretical renunciation is not for the present justified by our actual knowledge, and that one should not desist from pursuing to the end the path of the relativistic field theory.

The existence of a variety of field laws means that a fundamental reference point is missing. If frequency is looked upon as the basis of the electromagnetic field, then the fundamental reference point shall be a “field” of zero frequency. This we identify as the background SPACE.

The field theory comprises of electromagnetic and gravitational fields. The electromagnetic field consists of constant frequency. The gravitational field consists of uniform frequency gradients. Einstein’s observations do lead toward this form of field theory.

The physical reality then consists of “disturbances” that consist of frequencies and their uniform gradients. An electromagnetic field of constant frequency provides a three-dimensional space. The fourth dimension adds to it the gravitational field.

The quantum theory takes a statistical approach because there are too many moving parts to reality without a reference point. With the reference point of a background SPACE of zero dimensions and zero inertia, it now become possible to directly specify the state of a system.

The physical reality of matter has not weakened with the discovery of the field. Both matter and field have frequency as their basis. Both the theory of relativity and quantum mechanics suffer from a lack of reference point. That reference point is now provided by a background SPACE of zero dimensions and zero inertia.

CONCLUSION #1: Our concept of space is based on material extensions. Space may be considered independent of matter only as an abstraction and not as something real. That means the physical reality of space is non-existent in the absence of matter.

CONCLUSION #2: Our concept of time is based on changes in material extensions. Time may be considered independent of matter only as an abstraction and not as something real. That means the physical reality of time is non-existent in the absence of matter.

CONCLUSION #3: Space and time as abstractions are useful the same way that mathematics is useful. But any abstraction is only as useful as it is consistent with the physical reality.

CONCLUSION #4: The abstract concepts of space and time are preceded by matter. Therefore, matter is more basic to both space and time. Matter is best understood by the concept of inertia. Einstein does not delve deeply into explaining the phenomenon of inertia. Instead he covers it by mathematics based on Lorenz Transformations. The validity of Lorentz transformations is questionable where the explanation of inertia is concerned.

Validity of Lorentz Transformation


The Educational Approach


Reference: Critical Thinking in Education


There are two distinct educational approaches.

  1. The Greek Academy System: This educational approach believes in the student learning to think rationally on his own.

  2. The Scholastic Model: This educational approach believes in forcefully impressing data.

The scholastic model uses an examination system to forcefully impress data. It raises the student’s anxieties of what might happen if he does not “pass” an exam. The student becomes confused and unable to think rationally. He resorts to memorizing data without understanding. The system passes him with good grades if he can regurgitate data verbatim.

Under the scholastic system, a good grade is supposed to be synonymous with a bright mind. However, it is no more than the ability to memorize and recall data impressed by others. Such forcefully impressed data conditions the mind. It reduces the ability to understand and analyze data rationally.

Education must avoid becoming a mode of conditioning if it is to produce effective human beings. The first vital principle in teaching is to do everything possible to keep the student alert and aware of the subject on a rational plane.

The alert mind is extroverted and analytical. Its essential mode is self-learning. It thrives best when it is least “molded.”


The Approach Needed

The approach needed in education today is to let the mind become alert, extroverted and able to self-learn. This is accomplished by resolving the existing confusions in the mind on major subjects.

In teaching a subject one should first check the key points of understanding, and clean up the confusion surrounding those points. For example, in mathematics, the key points of understanding in sequence are: (1) The purpose of learning mathematics, (2) reading and writing large numbers, (3) the operation of division, and (4) the use of fractions.

Besides mathematics, the other major subject is language and grammar.

The resolving of confusions in major subjects helps students become self-learners.

The students must learn to think rationally on their own.


Study Materials & Supervision


Reference: Critical Thinking in Education


The purpose of a Tutorial Class is to encourage students to self-learn directly from materials, and to strengthen that learning by students assisting other students.



The text material for a tutorial class must be written in a language that is easy to understand. It should be supported by dictionaries that consist of easy-to-understand definitions and pictures.

The tutorial class material must present how a subject came about, and the reason why one should study it. The materials should then provide an overview of the subject before diving into the details. The details should be presented starting from the earliest concepts on which that subject is based. The materials then gradually build up the subject on a gradient such that no gaps are created in the student’s understanding.



Given proper study material, the students should be able to self-learn. If a student is unable to focus, it is because the assigned materials do not address his basic confusions in that subject.

The job of the supervisor is to quickly isolate the student’s confusion and give him the right materials to study. Sometimes it may require a bit of troubleshooting in getting the student set up properly.

Here is an actual example of a troubleshooting session.

SUPERVISOR:     “Is there something in math you don’t feel quite comfortable with?”

STUDENT:            “Yes… multiplication.”

SUPERVISOR:     “Alright.  What does the word MULTIPLY mean?”

STUDENT:            “Umm…”

(The SUPERVISOR explained the process of multiplication as “repeated addition.”)

SUPERVISOR:     “I am going to check you out on the multiplication of two single-digit numbers.  What is ‘three times two’?”

STUDENT:            “Six.”

SUPERVISOR:     “What is ‘four times three’?”

STUDENT:            “Twelve.”

SUPERVISOR:     “What is ‘six times six’?”

STUDENT:            “Oh, that’s a big number.”

(The student could multiply with very small numbers, but got nervous when larger numbers were asked.)

SUPERVISOR:     “Six times six would be adding six to itself six times.  Can you do this addition and tell me the sum?”

STUDENT:            (Pause) “Oh! I don’t like adding either.”

(The SUPERVISOR then demonstrated the process of addition as “counting together.”)

SUPERVISOR:     “Adding is counting numbers together. Are you comfortable with counting?”

STUDENT:            “Yes, I can count.  One, two, three, …”

SUPERVISOR:     (Stops her at the count of twenty) “Very good.  Now count for me starting from eight hundred ninety five.”

STUDENT:            (Taken aback) “Oh! That is a big number… (thinking) eight hundred ninety-six, eight hundred ninety-seven, eight hundred ninety-eight, eight hundred ninety-nine (long pause) two hundred, two hundred one…”

The student did not know what number followed eight hundred ninety-nine.  By this time it was evident that the student was shaky in her understanding of the numbering system itself.  The student was then assigned appropriate materials to study. She was then able to focus and make rapid progress.



In the normal course the supervisor applies the principle of gradient to help the student overcome his difficulties. Here is an actual example of assisting a young child write numbers.

SUPERVISOR:     “Is it ok if I ask you to write some numbers for me?”

STUDENT:            “Yes.”

SUPERVISOR:     “Alright.  Can you write six thousand, seven hundred eighty-three?”

STUDENT:            “Umm…”

SUPERVISOR:     “That’s ok.  See if you can write seven hundred eighty-three?”

(The student thinks for a moment and writes “700 83”.  The SUPERVISOR noticed that she could write eighty-three correctly.)

SUPERVISOR:     “Ok.  Can you write eighty-three for me?”

(The student smiles and writes “83”.)

SUPERVISOR:     “Excellent.  Can you write one hundred?”

(The student writes “100” correctly.)

SUPERVISOR:     “Very good.  Now, can you write one hundred one?”

(The student writes “101” correctly.  The SUPERVISOR then asked the student to write “one hundred nine” and “one hundred ten”.  The student wrote them correctly.)

SUPERVISOR:     “Excellent.  Can you write one hundred eighty-three?”

(The student pauses then writes “183” correctly.)

SUPERVISOR:     “That is correct.  Now write seven hundred eighty-three for me?”

(The student feeling more confident writes “783”.)

And so on…

The general supervision is basically devoted to helping the students develop better study habits. The supervisor encourages the student not to go past any word he does not understand the meaning of. He must look up such words in a dictionary.

Usually a dictionary has many definitions for a word. The student selects the definition that fits the context. If the student cannot find the right definition then he must seek the help from the supervisor. But very soon he develops the skill of finding the right definition by himself.

Sometimes the student cannot understand a sentence even after he has looked up the words in that sentence for their definitions. In this case the student should make examples of the meaning of that sentence–how something is that way, or it is not that way—which then resolves the problem. Sometimes it is the wrong definition used for small simple words that causes the problem.



In a tutorial class, the student studies correct materials under proper supervision. He then practices to become a self-learner.

The students may be studying different lessons in the same tutorial class. However, a student must understand a lesson fully before moving on to the next lesson. The supervisor may go around quizzing the students verbally on the sections they have completed to make sure they are not going by concepts that they do not understand.

The supervisor may ask a student, who has completed a section on a lesson, to help another student who is still studying that section. The effort is to help the whole class move forward together as much as possible. When the whole class has completed a lesson, it is followed by a Q&A (Question & Answer) period in which questions from the students are answered by the supervisor. A diagnostic test may then follow to pick up anything that is still not fully understood.

The supervisor also teaches the students on how to help each other.


The Emotional Curve


Reference: Mindfulness Approach


The weighted center of the mental matrix is perceived as the “I” of the person. It directs attention at parts of the matrix to activate them. The activation of the matrix forms thoughts. The cumulative effect of thoughts appears as emotional charge. This charge acts on the endocrine system of the body, which makes the body exert effort.

Thus, emotion is the direct index of the state of the mental matrix or “I” that is also reflected in the state of the body. The effort manifested in the body is then perceived back by the mental matrix. Accomplishment of intended effort discharges the emotional charge.

When there is a sudden drop in the emotional state of a person, we have an emotional curve. It comes about with the realization of failure or inadequacy. A person has at least one emotional curve in their past. Contemplation on the emotional curve leads straight to the incident, which gave the person his computation.

In this exercise the student focuses on a time when he was happy and suddenly was made sad. He contemplates on it until the actual emotional charge comes up . He fully experiences the emotions and contemplates on the incident over and over until the emotional charge is fully gone. One should be extremely thorough about discharging the emotional curve.


EXERCISE: The Emotional Curve

PURPOSE: To discharge the sudden emotional downturns in one’s life.

PREREQUISITE:  In case of extreme discomfort, return to the exercise Accessibility of Memory.

GUIDING PRINCIPLE: The Discipline of Mindfulness


  1. Focus your attention on a time when you were happy and suddenly were made sad. Let the data come up and freely associate. Continue with this contemplation until a memory of sudden emotional downturn comes up with full force.

  2. Experience those emotions fully without holding anything back. You may get into grief discharges, fear discharges or anger discharges. Experience them over and over until the charge is fully gone. Be extremely thorough about discharging the emotional curve.

  3. Make sure you let the mind carry out its associations naturally without being directed. In case there is increasing mental stress, return to the exercise Accessibility of Memory

  4. Repeat the above steps to discharge other emotional curves from your life. The exercise is completed when no more emotional curves are coming up.

  5. Continue with this exercise in, at least, 20 minutes long sessions until it is completed.