1.0 Direction belongs exclusively to Physics
The first quantitative (mathematical) concept discovered by civilization was probably the positive integer.
  At some point in Paleolithic time, Man began to count apples and oranges and to represent the
results with tally marks.  Eventually, man assigned symbols known as numerals to differentiate these
Merchants and traders later developed negative numbers, partly from the necessity to record debt in
“ For purposes of addition and subtraction, one can think of negative numbers
as debts. Adding a negative number is the same as subtracting the corres-
ponding positive number”  
In other words, the minus sign (-) is not really a mathematical concept such as a number, but a symbol
that represents an operation. Negative simply meant that you owe the amount to me.
In retrospect, it was predictable that the mathematicians would sooner or later extend the notion of
'negative' to direction:
“ vector: A mathematical entity that has both magnitude (which can be zero)
and direction.” 
“ vector: In physics and in vector calculus, a spatial vector, or simply vector,
is a concept characterized by a magnitude and a direction.” 
“ If the scalar is negative, it also changes the direction of the vector by 180º.” 
In the context of vectors, ‘negative’ means contrary motion. A minus sign prefixed to a vector quantity
indicates that the motion is in a direction contrary to a reference. Therefore, if you travel 3 kilometers
southwards and retrace your steps 1 kilometer northwards, you traveled a net distance of 2 kilometers
southwards from your point of origin (Fig. 1).
The mathematicians got into the habit of symbolizing vectors with little arrows. The arrow is supposed to
represent an itinerary, a bunch of locations. In the foregoing example, the arrow pointing south is thrice
as long as the arrow pointing north.
There are, however, fatal problems with this symbolism:
measured in units such as meter per second or miles per hour.
• The arrowhead represents direction, a static and qualitative concept. Direction
should not and does not involve Mathematics.
single piece.’ We can chop and hack kilometers and kilometers per hour into
discrete little pieces. We cannot divide direction even in half.
Hence, it is ludicrous to attempt to synthesize distance-traveled and direction with a single line as the
mathematicians do routinely.
But the insanity of Mathematics doesn’t stop here. The truly amusing part of the Math show comes up
when you travel 3 kilometers south and retrace the same 3 kilometers north. The idiots of Mathematics
'illustrate' such operations with null and position vectors, which from a physical point of view can only
mean that the object moved and returned to origin (dynamic) or that it never left in the first place (static):
“ In physics, moving position vectors are used in mechanics and dynamics
to keep track of the positions of particles, point masses, or rigid objects.” 
[Moving positions? Oh man, this is getting scary!]
The Wolfram site speaks volumes regarding the insurmountable troubles the mathematicians run into
when they attempt to illustrate null and position vectors. The page that defines the word vector depicts an
arrow that symbolizes magnitude and direction.  The vector is labeled with the words head and tail.
The page that defines the null vector shows absolutely nothing.  The null vector has neither head nor
tail, poor fella! The null vector is… how can I say this… an arrow-less arrow (Fig. 2). If ever a mathema-
tician defines a vector as having magnitude and direction and illustrates it with an arrow, ask him to draw
for you a null vector. Insist on seeing the arrowhead so you know in which direction you are not moving.
Copyright © by Nila Gaede 2008
What this shows, again, is that the magnitude of a vector is exclusively a mathematical concept. The
direction half of a vector is exclusively a qualitative physical interpretation of the operation. Indeed, the
same is true of every operation in Mathematics. Whether we add, subtract, multiply, divide, raise a
number to a power, or take a square root, the amounts belong exclusively to Mathematics. The
operations and their physical interpretations belong exclusively to Physics. If you have 3 shoes and the
thief takes away 2 of them, we still have three shoes. They are just apportioned differently. They are
merely in different locations. Destroying the two is not the same thing as 'taking them away.' Subtract
means to withdraw, not to annihilate. Likewise, adding refers to existing shoes, not to the act of creating
them. You must have shoes before you add or subtract them.
We have to conclude that direction has absolutely nothing to do with Mathematics, which means that
the mathematician is statutorily barred from defining this strategic word. Mathematics has no use for
qualitative concepts such as direction any more than it has use for concepts such as love or justice. In a
scientific context, direction is a qualitative physical interpretation of a mathematical operation which the
mathematicians in their immense ignorance symbolize with a negative sign.
2.0 ...yet no one in 'Physics' defines the word direction
Having said this, it doesn't come as a shock when we fail to find a formal definition of the word direction
in specialized dictionaries or encyclopedias of Physics where it belongs. You will not find the word
direction at the Wolfram Physics site  or at the Infoplease Encyclopedia of Physics.  You will not
find it at the Wikipedia under Physics.  Where you will find 'examples' of direction is in a dictionary
“ signed number: a number preceded by a plus (+) or minus (-) sign, indicating
a positive number (+) or negative number (-). Signed numbers are also called
directed numbers since the sign can indicate position (or direction) of a
number, relative to zero on a number line.” 
“ The direction from an object A to another object B can be specified as a
vector v = AB with tail at A and head at B. However, since this vector has
length equal to the distance between the objects in addition to encoding
the direction from the first to the second, it is natural to instead consider
the unit vector v (sometimes called the direction vector), which decouples
the distance from the direction... The direction from an object A to another
object B can be specified as a vector v= AB with tail at A and head at B.” 
“ A direction is determined by an ordered pair of two numbers (a,b), not both
zero, called direction numbers. The direction corresponds to all lines parallel
to the line through the origin (0,0) and the point (a,b).” 
Actually, the famous direction the idiots of Mathematics are talking about has nothing to do with
direction. It has to do with the particular way in which numbers increase or decrease on a number line.
In Mathematics, the word negative simply means less than zero.
“ Keep in mind that a positive y direction is up, a negative y direction is down,
a positive x direction is right, and a negative x direction is left.” 
[These are just conventions. The number line of Math can be placed diagonally
if I wish. The alleged 'direction' in which the arrowhead points merely indicates
that numbers increase.]
But what location or position does zero have? How would the mathematician know that the negative
direction is towards the left if he has not yet defined the word direction? The words left and right may
qualify as examples of directions, but not as definitions. Before the mathematician can use the word
direction in a sentence he must define it. Indeed, it is the mathematicians’ insistence on regarding
direction as a mathematical concept that explains why this word remains undefined. The definitions of
Mathematics are not definitions, but rather proofs. The mathematicians have never defined a single
word in their entire vocabulary. Mathematics is not founded upon definitions, but upon axioms. 
3.0 The scientific definition of direction
The task before us, then, consists of defining the word direction for the purposes of Physics. We should
expect a wholly qualitative definition which we are going to use to provide a physical interpretation to a
phenomenon. Since mathematical physicists rely on non-scientific notions that English majors have
developed over the years, this is a good place to start.
The most popular meanings of the word direction include:
1. The course along which something moves, aims or points.
2. A line leading to a place or point, or along which something lies, aims or points.
3. To aim or move along a line.
4. A position to which motion or another position is referred.
5. The alignment of two points in space, or their mutual angular position, or orientation.
Unfortunately, these definitions are too vague and have mutually conflicting elements to be used
consistently. Some emphasize a target, but lack an itinerary. Others have it the other way around. The
last one is a tautology: direction = alignment.
The absence of rigor creates communication problems during the physical interpretations of mathema-
tical theories. For example, what can Weyl mean when he states that ‘a curve preserves its direction
unchanged’ ( p. 115).  Or what is the significance of Heidemann's statement that we can go around
the Earth always in the same direction (p. 71).  Evidently, Weyl and Heidmann are not referring to a
rectilinear displacement towards a stationary target. They implicitly equate direction with forward,
casually relying on the ‘course along which something moves’ or ‘to set or hold a course’ definitions.
Specifically, the examples offered by Weyl and Heidmann fail to meet the ‘straight’ requirement of the
definition of direction. Therefore, we definitely must come up with a definition of the word direction that
can be used consistently.
But is this really a problem? Doesn't it all boil down to the prosecutor's choice of definitions?
It becomes a problem in Physics when we run up against misguided conclusions founded on such
statements. Hawking claims that time points or flows in the ‘forward’ direction and depicts this
parameter with a straight axis capped by an arrowhead that he says points towards the ‘future.’ If the
future is the ‘direction’ in which an object moves when it moves forward, do flanking motions to the left
and to the right qualify as directions towards the present in his model? Is backward motion a direction
into the past? Is my future direction your past direction when we meet head on? If a balloon expands
radially, which part of the surface is moving in the direction of the future?
For instance, we can rephrase Heidemann’s description and say that we move around the Earth by
walking ‘forward.’ Or we can rephrase and say that Weyl meant that the last point on a curve is ahead of
the one preceding it. What these statements indicate is that you are transferring your eyes to this next-
to-the-last point and looking ‘forward’ at the back of the other. The jury would get a vague idea that, by
'direction,' the presenter actually means ‘forward.’ But what can the jury understand when the
prosecutor clarifies that time flows or points ‘forward’ or in the direction of the future. We have a
concept – time – facing in the direction of another concept – the future. So where is this physical object
called 'future' located? If direction belongs to Physics, there absolutely must be a physical object
underlying the concept direction and this object must have a location. Time by itself has neither
direction nor location because this word does not depict a physical object. Nevertheless, isn’t it
incumbent upon the prosecutor to define the term direction if it is such a fundamental concept of
Physics? Or, to see things in light most favorable to relativists, shouldn't we define direction if, as they
say, it is an essential component of their mathematical vector?
The unscientific usages of direction that I just alluded to are not a complete loss, however. They
nevertheless serve as a basis to determine what attributes a direction should or must have. A direction:
• should embody a location but make no provision for motion
• must point parallel to a straight line, specifically, to the line of sight
• must make a provision for a target. (i.e., involves two objects.)
• must be objective rather than subjective, defined rather than proven
(i.e., it shouldn't involve an observer).
• must make a provision for aiming, pointing, facing and tilt.
Let's analyze each of these attributes and see if we can incorporate them into a definition of direction
that we can use consistently in Science.
Direction is conceptually static and indivisible. We cannot divide direction in half, chop it into little
pieces, or construct it with discrete components. When we invoke motion, we imply a minimum of two
locations, meaning two frames in the cosmic film. A mathematician may get away with saying that he
can place a straight line between two static points. His argument is guaranteed to fail if he claims that
two successive locations of one object constitute a straight line or rectilinear itinerary. With respect to
what? With respect to Jupiter? The only way we can depict direction is with a continuous straight line.
This cannot involve more than one frame in the movie.
In Physics, the word straight roughly means:
“ the quality or state of extending in one direction without turns, bends or
curves; or being without influence or interruption.” 
“ A curve is a line no part of which is straight.” (p. 112) 
These are not perfect definitions, but they capture the essence of what most people understand by the
word straight. Therefore, by definition, a curve does not have direction – no portion of it lies entirely
along the line of sight.
Of course, the mathematical physicists couldn’t use this definition. It stood in the way of their
supernatural physical interpretations. So they got around it by incorporating straight under the
definition of curve:
“ In mathematics, the concept of a curve tries to capture the intuitive idea of a
geometrical one-dimensional and continuous object. A simple example is the
circle. In everyday use of the term "curve", a straight line is not curved, but in
mathematical parlance curves include straight lines and line segments.” 
“ A line can be described as an ideal zero-width, infinitely long, perfectly straight
curve (the term curve in mathematics includes 'straight curves')” 
No one but an idiot of Mathematics believes that a straight line is also a curve. What meaning could
such nonsense have? In the rational world of Physics, curved and straight are antonyms.
The mathematical definition of the word curve nevertheless reveals the true intentions of the mathema-
ticians. They are not talking about the 1-D, about continuous, about an object, about a curve, or about
straight. They are talking about itineraries that deviate from a rectilinear path. They are not talking about
curved, but about curvilinear. A ball is curved. When the batter strikes it, the ball moves curvilinearly.
The straight line of Mathematics is purportedly a trajectory, a movie of a dot supposedly traveling in a
‘straight line’ from one frame in the cosmic movie to another.
However, in the real world, not a single object travels rectilinearly, for I ask, with respect to what? In
Physics, it is surrealistic to speak of rectilinear itineraries. The only context in which we can say that the
Moon moves in the direction of the Andromeda Galaxy or that the gun is pointed in the direction of the
target is in a static context. We must visualize the Moon and the gun standing still and facing the target
straight ahead along the line of sight perpendicular to the selected plane. Again there must be two
objects and the scene must be static. If the mathematician invokes a second frame, the movie is
guaranteed not to have 'direction.'
The problem with the word forward is that the object uses itself as a reference. A lonely object in the
Universe may be conceived as pointing forward – just choose a face – but we still know nothing about
direction. The word direction implicitly invokes a second object. A line tangent to a sphere may have
direction because we reference one end against the other, but what is the direction of the lonely ‘point’
it allegedly touches if we consider it all by itself? In what direction does any point on the surface of a
sphere point or face? Until we have a second object that serves as a reference, we still do not have
direction. Words such as forward, future, and up are neither objects nor locations. They stealthily use a
preferential tilt or orientation of the object at the center of attention as a reference. They tell us nothing
about either the itinerary or the destination of this object. In Physics, there are no directions such as
love or happiness. Qualifiers such as northward or eastward stealthily embody extrinsic objects as a
reference. On Earth, northward means toward the North Pole. In this context, the mathematician treat
the North Pole as a physical object. But the object that is pointed towards the North Pole tacitly does it
through the body of the Earth, not along its surface. A geodesic may consist of forward motion, but it
does not have direction. Its direction changes in every frame of the movie.
The word forward is also a poor criterion to define direction because it has subjective connotations. It
depends on an observer. Your forward direction is not the same as mine. Therefore, the jurors have to
second-guess the prosecutor and fill in the blanks on their own. Direction, on the other hand, should
not be an issue that needs to be decided with an experiment. The words aim, point, and face are
objective. The barrel of the gun is either physically tilted towards the target or it isn’t. Opinions have no
bearing on the reality of whether it is or not. Likewise, the Milky Way either moves or doesn't in the
direction of the Andromeda Galaxy. We don't need a conscious observer to measure whether it does or
not. The definition of direction cannot have us second-guess whether the rocket intends to curve
towards the North Pole when it travels tangentially to the Earth. Only in each individual frame of this
movie can we talk about direction. We can say, for instance, that in a given frame the rocket is traveling
in the direction of Mars. Hence, the static and objective word straight can be used to define direction and
not the subjective word forward.
Aim, point, face, and tilt
The word face derives from the Latin facies meaning the surface at the front of the head. The verb ‘to
face’ derives from this noun and refers to the alignment of the body with the plane of the face. We have
extrapolated this notion to objects. We say that a cube faces the wall, meaning that we momentarily
substitute our face – more specifically our eyes – for the plane in question and look out from it. The verb
to face reduces to offering an imaginary plane perpendicular to the line of sight.
For instance, the cue of a game of billiards has what we refer to as a ‘point’ at the tapered end. When
seen head on, this tip is not typically characterized as facing, but as pointing. Upon closer inspection,
however, the tip of the cue is a surface, indeed, much larger than the surface of the tip of a pin. But then,
even the tip of a pin shows up as a surface under the microscope. And so does the surface of a
hydrogen atom! There is not a single real object in the Universe that will not exhibit a surface under the
right magnification. Hence, for the purposes of Physics, the words facing and pointing are synonyms.
We are treating the surface in question as a plane, substituting our eyes for this plane, and looking
outwards along our line of sight. Face and point are not verbs, but static attributes like tilt and
orientation. The object does not occupy more than one location when it faces or points. Facing differs
from direction in that it implies self-reference. Direction requires an extrinsic reference. If facing means
'to look,' direction means 'to see.' We can replace direction with a straight line, but we cannot do the
same with facing or pointing. The word tilt, in comparison, is the position of the object from another
object's perspective. Here the observer places his eyes on the target rather than on the source. It is the
merging of face and tilt that imbues direction with reciprocity.
Let's put it all together now.
facing or pointing: Imaginary substitution of a specific plane of an object with
the observer's eyes. (v.) To look outwards.
tilt: The face an object presents from a particular perspective.
direction: Line of sight. The mutual facing of two objects; a conceptual linking
of two locations. Direction is a property of straight lines and edges.
Direction and facing emphasize the source. We are placing our eyes on the source object. Orientation
and tilt emphasize the target. Our eyes are on our face again staring at the target from afar.
The following statement synthesizes the meaning of direction:
“ The book is tilted in such a way with respect to the ceiling that if I place my
eyes on that specific face of the book, I would see a spec in the ceiling and
Dr. Math asks rhetorically whether it is possible to define the word direction without using the words
point and line:
“ Your definition would require us to first define 'ray' and 'direction.' Can you
do that without reference to 'point,' 'line,' and 'plane'?” (p. 5) 
Here I showed Dr. Math that we can. Therefore, students don't need to take Math 101 to understand
what a direction is. It suffices that they take English 101. The reason relativists have never formally
defined direction despite that they use the word in so many of their writings is that they have not
realized the importance of language to the scientific method. They give lip service to the word rigor and
use definitions inconsistently anyways. Without proper definitions, the prosecutor is talking about a
rock and the jury is visualizing a tree.
What is it that you don't
understand, Bill? This is
positive direction, this is
negative direction, and this
is zero direction.
|The mathematicians depict displacement
with arrows they call vectors. Therefore,
if you subtract 1 mile north from 3 miles
south in the religion of Mathematical
Physics, you end up with 2 miles south.
That’s the net magnitude and direction of
|The problem is that direction is conceptually static and made of a single piece
(i.e., a photograph). We cannot chop direction into pieces. Magnitudes and
displacements, on the other hand, are divisible. A magnitude is the result of
counting (i.e., a movie). Displacement involves several frames of the
Universal Film. Therefore, it is ludicrous to attempt to synthesize the
magnitude of a displacement and direction with a single line. The magnitude
of a mathematical vector belongs exclusively to Mathematics. The direction
part of the mathematical vector belongs exclusively to Physics. Never the
twain shall meet.
So now for the amusing part...
|Fig. 2 Null and position vectors
1. This page: The mathematicians invented negative direction
|If you travel 3 miles south and then 3
miles north and arrive at your starting
point, the mathematician says you
never left. He depicts these non-motion
type of operations with null or position
vectors, which is no vector at all. The
null or position arrow of Math doesn't
even have an arrowhead, for in which
way is 'position' moving or pointing?
So the next time you visit your local
mathematician, ask him to illustrate a
nulll vector for you. Let's see what he