by Kalman Klim Brattman
Give me the simplest form of matter and motion,
and I will construct, out of them, the world of Nature.
"Give me matter, and I will construct a world out of it."
Immanuel Kant, Kant's Cosmology
("Universal Natural History and Theory Of Heavens")
3. On Transphysical Distances and Frames of Refernces Concepts
with emphasis on
Space and Time Concepts VS. the Nonsensical Spacetime Concept 

The study of motion, plays a pivotal role in TRUTON --as motion indeed is an integral and independent component of the formation of Nature to be studied in the next page.

The study of motion cannot be separated from the concept of distance. There are two, and only two, kind of distances that the Mind can conceive to exist when studying Nature:
  • one, a distance in extent called Space; and
  • the other, a distance in duration called Time.

Those two concepts, Space and Time, were already inherited and implanted into Mind's existence as the nervous system of all locomotive living organisms (organisms that are able to move), without exception, were endowed with a recognition of their surrounding environment that was affected by a Time (e.g., day or night, the beginning or ending of a season, etc.) and by a Space (e.g., distance to a safer place, to water, to a tree, to a warmer place, etc.). That universal primeval recognition of a Space and a Time common to all locomotive livings, while quite different for each species, had one thing in common: that it all developed from one singular interest that was the interest of survival. That survival interest was born out of these two simple but basic concerns: WHEN to move (e.g., day or night) or WHERE to move (e.g., places that are safer than other places, places near water, etc.) in order to stay alive.

That simple recognition of the survival interest existing in all locomotive living creatures of Nature is remarkable indeed, because it implies that the nervous systems of all moving living organisms, regardless how primitive they were, all, by virtue of evolution, were able to "communicate" with and recognize their surrounding Nature via their own mechanism of perceptions in identifying their surrounding Space and Time. Biological body-clocks of all livings are the result of their Time-perception relationship with Nature, while the recognition and the marking of their habitat is the result of their Space-perception relationship with Nature.

The Space and Time concepts devised by the human Mind evolved out of that survival interest of the living and eventually become unique because of the Mind's unique ability to engage in abstract thinking. And by UMCOT, there is no limit in how far that ability can develop and expand. The outer Space or the Cosmic Space is one example of such an expansion. Mathematics, through its ever increase development, was able to create --through its ever deeper logical abstract inferences-- even deeper abstract mental objects and concepts that led to a considerable expansion of the Space and Time concepts coined by the Mind.

With respect to the Space concept, the development of Mathematics was channeled into two major directions that were able to generate new and quite complex abstract spaces:

i) one development was towards mathematical structures giving rise to a number of new type of abstract spaces such as the non-Euclidean spaces, group and field spaces, topological spaces, etc.;
ii) the other development was related to the dimension of mathematical spaces, generating new concepts such as mathematical fields, manifolds, n-spaces, etc.

The concept of Time was also affected by the advancement of Mathematics. The gradual introduction and acceptance of negative numbers (once called and regarded, in the 15th century France, as "absurd numbers") was able to pave the incorporation of the "negative time" into the Time concept. Thus, with this new abstract addition, we were able to associate past events with corresponding to a negative time.



It is important to stress that our formed Space and Time concepts incorporated into the study of Nature are creations of the Mind and, as such, they reside only in our Mind, and nowhere else, as they do not have a separate material existence in Nature. They are just
mental frames of reference created by the Mind and nothing else. To them we cannot associate, in a meaningful way, physical characteristics such as temperature, density, elasticity, etc., because they are not physical entities.

In TRUTON, an absolute demarcation is always made between abstract mental objects --called mentalons and the real physical objects of Nature called realtons. Let us stress again that mentalons, as oppose to realtons, do not possess physical properties because they are objects of the Mind and not of Nature. What mentalons can possess is a close affinity to Mathematics as outlined further below.

Mentalons --the exclusive products of the Mind, could be of different kinds. Some mentalons are images created by the Mind of real or imaginary objects and,
  • called imagelons, when representing imaginary objects of fantasies and dreams, and
  • called imagetrons, when representing real objects observed from Nature.

Mentalons that are not images, are thoughts that
  • could be vested in rational deductive reasoning called rationalities (with Mathematics being a prime example of being made entirely of rationalities); or
  • could be vested in fantasies, dreams, and irrationalities (aka, irrational thinking) forming the Mind's Playground whose constituents, collectively, are called playlons.

Mentalons can be used to study the Mind itself or to study the real material Nature. Mentalons that are "transported" to study the Mind are called transmentals, while when "transported" to study Nature are called transphysicals. The study of transmentals is the object of Psychology that is the academic study of mental functions and behaviors, while the study of transphysicals is done in Physics.

There are seven (7) major types of transmentals capabilities and processes of the Mind:

• one, with respect to the Mind's rationalities --the Logic functionality;

• other, with respect to the Mind's capability to store information --the Memory;

• other, with respect to human Mind's capability of self-reflection, self-examination, and self-learning from experience --the Introspection;

• other, with respect to human Mind's capability of holding a particular conjecture or premise to be true outside experience or its sphere of accumulated knowledge --the Belief;

• other, with respect Mind's capability of directing and committing to a particular course action --the Volition;

• other, with respect to the Mind's Playground vested in its playlons --the Imagination; and finally,

• other, with respect to Mind's states involving perception, cognition, creativity, motivation and emotion -- the Mental States.

Transphysicals, on the other hand, are the Reference concepts of Space and Time that are indispensable for the study of Motion in Nature. In the remaining part of this page, we take a closer look at the two identified transpysicals, Space and Time, through the lens of Mathematics.

 Time of Nature Through the Lens of Mathematics 

Before we "transport" our transphysicals into the material world of Nature, it is useful to discern their characteristics. For instance, we recognize that the concept of time evolved from the
'linear distance' in duration. But a linear distance in Mathematics is represented through a line (straight curve). Therefore, the 'time' concept has incorporated all the mathematical properties of the 'line' concept. We can see all this from the comparative table below:

Formalization of 'line'
from Mathematics
Borrowed formalization
of the 'time' concept

1. A line has no beginning nor end. Between any two distinct points there is one, and only one, line that passes through them both. Because of this,



It is meaningless to talk about a "condensed" or a "stretched" mathematical line. A Mathematical line is of one, and only one, kind.

1'. Between any two distinct events, called EventPoints or TimePoints, there is one, and only one, straight line that passes through both of them called the EventLine or TimeLine of Nature (TILON) with no beginning and no end. Because of this,



It is nonsensical to talk about a condensed or stretched TimeLine/EventLine.

2. Between any two points in a line, there is a point between them.

2'. Between any two EventPoints/TimePoints of the EventLine/TimeLine, we always can have another EventPoint/TimePoint.

3. To a line L, we can associate a one-to-one function f=f(x) such that for any two points P and Q on L we have |PQ|=|f(P)-f(Q)| that is called the distance between P and Q. The number |PQ| is called the length of the line segment PQ and the points P and Q are called endpoints.

3'. A segment of the EventLine or TimeLine, called an EventSegment/TimeSegment (or a time episode), is a part or a portion of the EventLine/TimeLine that is bounded by two EventPoints/TimePoints, say P and Q, called end-EventPoints/TimePoints and contains every EventPoint/TimePoint on the EventSegment/TimeSegment between its end-EventPoints/TimePoints.
The duration of a PQ EventSegment/TimeSegment or a PQ episode is the number |PQ|=|f(P)-f(Q)| where f is called the time function, f=f(t), that is a one-to-one function associated with the EventLine/TimeLine. The variable t is called the time variable or simply 'time.'



Remark 1:
Any two endpoints whose distance between them is 0 (zero) are said to be one and the same.



Remark 1':
Any two EventPoints/TimePoints that have a zero EventSegment/TimeSegment between them are said to be simultaneous.



Remark 2:
The variable x of function f(x) is not a discrete but a continuous variable on L. The continuous variable x is said to be mapping the line L, and it goes without saying that x
  • is a non-stretchable and a non-compressional entity;
    and that x
  • has no beginning nor end.



Remark 2':
The continuous time variable t (or time) of the time function f=f(t) is said to be mapping the TimeLine of Nature (TILON). Since TILON has no beginning or end, it follows that
  • that 'time' cannot be "compressed" or "stretched."; nor
  • that time (the time variable t) has an absolute beginning or an absolute end.

4. The coordinatization of the line:
The mapping f---->f(x) represents the coordinatization of the line. That is, it defines a coordinate system on the line. There is a point that corresponds to 0 (zero) and, there are points that correspond to the positive real numbers and points that correspond to the negative real numbers.

4'. The arrangement of the EventLine/TimeLine:
.There is an EventPoint/TimePoint called zero-EventPoint/TimePoint or the BirthTime that corresponds to the birth of a particular formation. The time referenced to the formation is called the relative time or the intrinsic time of that formation.
.There is an EventPoint/TimePoint called the now-EventPoint/TimePoint or the present-EventPoint/TimePoint that corresponds to a given number called present.
.There are EventPoints/TimePoints called bygone-EventPoints/TimePoints or past-EventPoints/TimePoints preceding the present-EventPoint/TimePoint that are corresponding to negative real numbers. We say that those points represent points of the past.
.There are EventPoints/TimePoints called EventPoints/TimePoints-to-be or future-EventPoints/TimePoints that are corresponding to positive real numbers. We say that those points represent points from an imaginary or virtual future.

5. A half-line or a ray-line is defined trough the concept of "order" of points of a line. When we have coordinatization, we say that the line is directed.
A half-line is part of a line which is finite in one direction, but infinite in the other.

It can be defined by two points, the initial point, A, and one other, B. The half-line is all the points in the line segment between A and B together with all points, C, on the line through A and B such that the points appear on the line in the order A, B, C.

If A is a point on a line such that the set of points C satisfy the relation f(A)f(C), then the set of points C are said to form a ray with A as its origin. If the set of points C satisfy the relation f(A)f(C), then the set of points C are said to form the opposite ray.

5'. The EventLine/TimeLine has two halves. A half-EventLine/TimeLine is defined through the concept of "order" of its EventPoints/TimePoints. When the EventPoints/TimePoints are arranged in the order of their successive occurrence, we say that the EventLine/TimeLine is directed. A half-EventLine/TimeLine is a part of the EventLine/TimeLine that is finite in one direction, but infinite in the other.
.We call the bygone-EventLine/TimeLine or past-EventLine/TimeLine, the half-EventLine/TimeLine whose origin is the present-EventPoint/TimePoint and whose all subsequent EventPoints/TimePoints are the bygone-EventPoints/TimePoints. We call that half-EventLine/TimeLine the entire past-line of Nature (EPLON).
.We call the EventLine/TimeLine-to-be, the half-EventLine/TimeLine whose origin is the present-EventPoint/TimePoint and whose all subsequent EventPoints/TimePoints are the future-EventPoints/TimePoints. We call that half-EventLine/TimeLine a virtual futuristic-line.


Internal biological mechanisms for measuring the duration of a time episode (as defined in point 3' hereinabove) have existed and perfected from the beginning of life. Natural time-markers, as we have noted, are an essential element of life. The evolved human species added a new dimension of keeping track of time not through its existing biological clocks but through external clocks creating timekeeping devices called timepieces capable of functioning outside the body of human species. The evolution of those timekeeping devises from sundials and water clocks to pendulum clocks and culminating with quarts and atomic clocks has been indeed a most remarkable journey.


 Space of Nature Through the Lens of Mathematics 
And  On the Nonsensical Spacetime Concept 

The arena of existence in extent and the arena of existence in duration in Nature are expressed by two fundamental concepts, respectively --that of space and that of time. A location is represented or expressed through the concept of a point in space while an event in Nature is represented or expressed through the concept of a point in time.

Since in Nature, the location of a point in space can be identified exactly through three and only three independent like-parameters, called space parameters, noted say as [x, y, z], we say that the Space of Nature is a 3-dimensional continuum. The mathematical "structure" of that continuum is defined formally (in the language of Mathematics) through a so-called metric, called the metric of space.

Unlike a point in space that can travel randomly in a 3-dimensional volume, a point in time can travel only linearly in the unidimensional space of the line called the timeline. Because of that characteristic, a time point can be identified completely through one, and only one, single parameter, called the timeline parameter and usually denoted with t.

Thus, for a particular point of Nature, its space parameters are function of its sole timeline parameter and, we can write this as x=x(t), y=y(t), and z=z(t). So to reflect this time-parameter, the space [x, y, z] can now be written as [x(t), y(t), z(t)].



It is important to note, right here and right now, that unlike the parameters x, y, and z that are of the same nature being therefore able to be interchanged one with another, the timeline parameter t is of an entirely different nature. As such, it is meaningless indeed to unite the timeline parameter with any other kind of parameters as such a union cannot be combined in a meaningful matter. In a meaningful combination all of parameters forming a space must be of the same kind or nature. Thus, the timeline parameter t can never ever --in a meaningful way-- be added as a forth spatial dimension to represent a space much less the Space of Nature. The Space of Nature can be represented only as [x(t), y(t), z(t)] but never ever, in a meaningful way, as [x, y, z, t]. Nevertheless, such an absurdly, introduced first by Albert Einstein, continue to this day, to be employed without any notable reservation.Kant

The Space of Nature cannot have a physical structure because it is a mental object --a mentalon in the sense envisioned by Immanuel Kant in his seminal work "Critique of Pure Reason" where both 'space' and 'time' are being viewed as mental frameworks that humans have created to study Nature.

Physical structures can be associated only with matter, but not with Space or Time that are mentalons. Space and Time and nothing else but mental receptacles that, by their very nature, cannot be associated with having any physical properties. "Bending" of Space or the "dilation" or "contraction" of Time are nonsensical concepts as they confuse mentalons with realtons.

Thus, the Space of Nature is the Space of Immanuel Kant and Isaac Newton, being a mental container devoid of any implanted distorsions, and being thus, par excellence, the Euclidean Space. Because the Space of Nature is governed by the Euclidean Geometry, and only by Euclidean Geometry, we can see from the diagram below how any point source (S) of Nature will spread its influence equally, in all directions, obeying the Inverse-Square Law Of Nature (ISLON): that the intensity of the influence at any given radius r is the source strength divided by the area of the sphere. Being strictly of geometric origin whose Space is Euclidean, the ISLON applies to all unperturbed source phenomena of propagation. Point sources of gravitational force (Newton's Law of Universal Gravitation), electrostatic force (Coulomb's Law), light source, sound, or radiation sources --are all obeying ISLON.


Let us stress again: ISLON, by being a geometrical and not a physical property, exists in Nature only because the structure of the Space of Nature is Euclidean. ISLON will not exist in a Space of a different mathematical structure. And that recognition is most important indeed as, through it, we are able to recognize at once the validity of the Euclidean Geometry as the geometrical structure of the Space of Nature.

As already stated, the current adopted Space structure of Nature, of the so-called spacetime continuum structure introduced by Einstein's Theory of Relativity, combining Space and Time into one single substratum is an utterly absurd concept because such an union can never ever generate a continuum in the mathematical sense nor in any other coherent sense. That is because, as noted, the parameter 'time" can never ever combine, in a meaningful way, with a parameter that is of a different nature. To have combination of parameters forming a continuum, all parameters must be of the same nature.

That truly absurd spacetime continuum concept that forms the backbone of Einstein's General Theory of Relativity (GTR) purporting to represent the reality of Nature's Space, "explains" gravity not as being a force [sic!!], but as being a derivative springing from the Space's purported curvature!

Finally, we note here that because the motion of an object in Nature can be referenced only through Space and Time, those two concepts --Space and Time-- are also called the base anchoring concepts of Nature (BACON). Motion is the underlying concept of BACON creating therefore the bridge of communication (BOC) between the Space and Time concepts. Motion therefore is an indispensable primary given characteristic of Nature that need to be incorporated right from the beginning of our study. All these ideas will considerably be magnified as we advance with TRUTON. The general blueprint that we have sketched herein will further be expanded, magnified, and enhanced as we go along. So, without further ado, let's begin with our journey that we have set to travel.



Kalman Klim Brattman