Bill's Blog
A line is not the shortest distance between two points
I.  'What' is a noun!

The word point is regularly used as a hypothetical entity in the course of scientific discussion. Relativists define an event as 'a point in spacetime' and a black hole as 'a point of infinite curvature'.
(1) But what do we mean by point?

Euclid defined a point as 'that which has no parts' and, miraculously, this resilient axiom has survived unscathed to our days. Relativists continue to regard a point as a dimension-less, structure-less entity that has position as its only attribute.
(2) Others define point as the smallest particle you can imagine, and yet others, as the  intersection between two lines. (3) In principle, however, these definitions appear to fall short of the mark. 'Something' that is dimension-less and without extension is commonly referred to as 'nothing'. Structure-less is synonymous with void. And for those who just finished visualizing the smallest particle they can think of, as homework, magnify it and then divide it in half.

The problem is that for the last 2500 years mathematicians have believed that point has something to do with size. If you have trouble seeing my point, please feel free to bring your personal scanning electron microscope (SEM); you  no longer have to strain your eyes. Make my point as big as you want and imagine it the shape of a circle, a square, or a pyramid. If point wants to join the exclusive Noun Club of Physics it will have to submit a portrait for its ID like the rest of the members. A point is nothing more than a hypothetical subset of a noun, of which the minimum requirement is that it consist of two dimensions. If you can imagine it, if you can see it, if you can draw it, whether point, line, triangle, or sphere, it has at least width and height! The set of figures that is one-dimensional is an empty set. For the purposes of Physics, we define noun and point respectively as follows:

        Noun: that which is amenable to visualization in a single frame,
                   image, or position.
(4)

        Point: an imaginary constituent of a noun used as a hypothetical
                   entity.
Fig. 1     A point is a noun 
Essentially, a noun is portrayable in a still picture because its sine qua non property is form, and a point is merely a subsystem up to and including the noun itself.
Now for those who define a point as a cross between two straight lines, we must first define what we mean by straight and what we mean by line.  
II.   I walk the line

Euclid defined line as (a) 'endless breadth'. From him, relativists developed mantras which they sing over and over, seemingly, with little skepticism:
(b) 'a line is the shortest distance between two points' and (c) 'two points in space constitute a straight line'. (We note that despite the constraining points, relativists subsequently clarify that their line may extend infinitely in both directions.) Most people, for their part, have this vague notion that a line is (d) 'one-dimensional'. Mathematically speaking, a line is defined as
(e) 'a set of points that satisfies ax + by + c = 0, where both a and b cannot be zero'.
(5) From a physical point of view, this suggests that an ideal line is
(f) 'a homogeneous, continuous set of points', since we can replace the variables in this equation with any fraction on the number line.
(6) Let's test the integrity of these definitions to determine if we can live with any of them.

(a) Euclid's axiom fails because he is depriving us of the benefit of imagining the shape of his noun. Euclid's line extends forever and as a result we are eternally unable to visualize WHAT he is talking about. Let's magnify Euclid's line under the scanning electron microscope to make this absolutely clear.
Fig. 2     Euclid's endless line
Upon closer inspection, Euclid's line consists of
two parallel lines that
extend beyond our field
of view and which he
defines to have no ends
(Field of view # 1). If we
magnify one of these
lines, lo and behold, we
find that it also consists
of parallel lines without
endpoints (FOV # 2) and
so on ad infinitum (FOV
# 3).
Euclid's line is perpetually under construction in all four cardinal directions, which means that he is not yet ready to offer a final version of his axiom. Meanwhile, don't let Euclid persuade you that the magnified lines are really the edges of a continuous geometric figure extending from top to bottom. What geometric figure, Euclid? All we see before us is two parallel lines. Prove to us that you placed a wooden board under the microscope!

More disappointing yet with Euclid's line is that there is actually very little to see. A breadth is the invisible gap between two points, the absence of something, a synonym of distance, displacement, linear space, nothingness. Therefore, if by breadth Euclid meant separation, endless breadth is an oxymoron. And, were we to be less generous and remove the constraining points -- which we don't need any longer once we have the breadth Euclid claims is his line -- voila! Euclid's famous line just vanished!

(b) The relativistic rosaries run into similar troubles. If a line is alleged to be synonymous with distance, it stretches the imagination how it can also consist of points.  Like Euclid's breadth, the relativistic line is pure emptiness, vacuum; we should never be able to draw such a line. Either a line is a shapely figure that we can visualize and illustrate or it is a spiritual gap beyond conceptualization. The succession of points relativists trace with their pens on a sheet of paper bares scant resemblance to the ghostly 'distance' embodied in their definition.

Nevertheless, this misconceived definition (the shortest distance between two points) dares the skeptic to perform a measurement to verify this claim (i.e., that this is in fact the shortest distance between two points). This does not appear to be the definition of a noun. It is at best the definition of an adverb. The skeptic is required to go through a tedious thought experiment where he walks over and over from one point to the other through different paths and then compares all these measurements. Or the mathematician must slide beads on his electronic abacus and measure angles to demonstrate that a line constituting a given figure is the briefest of segments. This leads us to believe that the relativistic line is a proof derived from experience rather than an axiom taken at face value. 'Shortest' implies that we have established certain relations and are now publishing our results. 

But let's throw a final bucket of cold water on the relativist proposal: a  distance IS the shortest gap between two points, a fact that reduces the relativistic definition to a tautology. Relativists seem to be confusing displacement with distance. Displacement is synonymous with rectilinear motion of a single point while distance is a static separation between two points. The trouble is that the word distance is also used to refer to the set of non-linear routes taken by a point.  This motion-related distance is conceptually distinct from the definition of distance as a gap. The former belongs exclusively to Mathematics; the latter, exclusively to Physics. The distance that relativists identify as a geometrical line is not alleged to grow from one point to the other while a clock ticks, but spans a frozen separation at time = 0. We are at a loss to compare the relativistic distance against another intrinsic chasm in this system because, in Physics, there is only one 'distance' between any two points.

(c) In the event, however, that distance were synonymous with trajectory, this would merely falsify the second relativistic mantra. The definition of a structure self-destructs if its mere existence is made contingent upon  motion. If relativists so much as suggest that two points constitute a straight line because you can draw a straight line or walk in a rectilinear path between them, they are not characterizing a noun. They are describing a verb. Drawing a line of necessity involves more than one frame or image. We are not asked to  look at a still picture of a structure. We are asked to view a movie of a line being erected between two points. We may never find out what a straight line looks like if the second point is imagined to be at the other side of the universe, more so if the trace zigzagged during the trip!
(7)

Less convincing yet, this relativistic definition is self-defeating. A definition of an object that requires us to draw the object that we are attempting to define implies that two points do not automatically constitute a straight line (i.e., we are asked to construct the line 'subsequent' to the existence of the two lonely points!). With 'two points in space' we are still one step away from conceptualizing a straight line

                                              
SUMMARY

A. Euclid's endless breadth


                             
o < ------ breadth ------ > o

Problems:

  
(1) 'endless breadth' is an oxymoron and, hence, beyond
        conceptualization.
   (2) a line can be illustrated; a breadth cannot (separation is not a
        noun for the purposes of Physics).


 
B.  Relativistic 'shortest distance' (also purported to be infinite)

                              
o < ----- distance ----- > o

Problems:

  
(1) if infinite, the line is not the 'shortest', let alone a 'distance'
   (2) 'shortest' is an adverb (qualitative comparison)
   (3) without the endpoints, there is nothing to see (cannot be
        illustrated)
   (4) a distance is a unique gap between two points, not one of
        many possible trajectories of one point.



C.  Relativistic 'distance' traveled between 'two points' in space

                               
o >>> displacement >>> o

  Problems:

   
(1) not an axiom, but a proof
    (2) this line is a description of a verb (must 'walk' between the
          two points to comprehend the 2 or 3 dimensional figure)
    (3) without points, what remains is structureless motion
    (4) displacement is the rectilinear itinerary of 1 point rather
          than the gap between 2 points.
    (5) rectilinear motion is beyond conceptualization in a universe
         where all matter is in perpetual motion 
    (6) self-defeating 'axiom' (two points do not automatically
         constitute a line if  we are required to draw the line
         subsequent to the existence of the two points)


(d) The suggestion that a line is one-dimensional is very disconcerting. Many regard dimensionality to be a property of space, a notion extrapolated from the mathematical idea that a volume is the measure of extent in space. (8) However, a simple thought experiment should suffice to test whether this claim is true. For example, let's remove all matter from the universe and leave you alone with a ruler in empty space. The trick is for you to establish the width or height of space. Where will you  anchor your ruler to determine this property of the void? Clearly, dimension is a property of that which has form, for why else would relativists resort to the object 'straight line' as an example of one-dimension
Fig. 3   Relativistics 1D line under the SEM 
The line segment AB appears
at first glance to satisfy the requirements of what we typically understand by 1D. Ideally, the line has a single dimension -- length -- and exhibits no angles greater than 0 degrees. An object
that forms a 45 degree angle, on the other hand, has at least length and height, and is regarded to be two-dimensional.  
But as we said earlier, the reason you can see this line is that it consists of width and height. If we so much as imagine a point or a line, we cannot be visualizing the first of the dimensions; we are at least in the second. (Magnify any line under your personal SEM to verify this assertion). The fact that the ends A and B form a greater than 0 degree angle with our line of sight further contradicts our initial hypothesis. We don't have the luxury of visualizing a solitary dimension from a lateral perspective because we would be attempting to cheat structure of one of its fundamental pillars. When we imagine slicing a 3D chunk of matter, first in one direction until we reach a plane, and then in another direction until we reach a 'dimension', we do not end up with structure, length, width, line, or point. We end up with the coordinate of longitude, i.e., line of sight or visualization! There is no object in the 1D.

These misconceptions have developed from a combination of bad habits and poor definitions. In General Relativity, an arrow depicts a magnitude and a direction (i.e., the terms dimension, coordinate, and vector are synonymous). However, a magnitude is a quantitative relation of Math while direction is a qualitative property of Physics. A magnitude requires the observer to establish a standard and to perform a  measurement. If we are to represent such comparisons with symbolical arrows, we need at least two of them: one for our standard and another for the test object. These arrows are not continuous dimensions, coordinates, or vectors, but segmented number lines. The arrowheads indicate that numbers are increasing, not that an object is pointing or traveling in a given direction. For example, the 'second' is a quantitative relation between two itineraries. One is the displacement of a cesium wave (or photon) and the other, the motion of the hand of a clock. These itineraries are not perpendicular. Hence, a solitary arrow is woefully inadequate to represent scalars such as time, mass, and energy, more so if the axis is alleged to embody direction.
(9)

The 'length' inherent in the physical notion '3-dimensional', in contrast, is conceptually distinct from extrinsic magnitude or measurement. Three-dimensional is a property that objects have of facing or pointing in three mutually-orthogonal directions. The dimension we label as 'length' is a continuous, numberless, qualitative concept that is restricted to one of these directions. Length does not arise as a result of a comparison, but by definition. Hence, 'length' may be depicted by a single arrow or axis only with the tacit understanding that this arrow points perpendicular to intrinsic width and height. Similarly, the coordinate of longitude is by definition positioned perpendicular to latitude and altitude, and the vector of depth runs  perpendicular to breadth and elevation. In Physics, dimension has to do with structure, coordinate with position, and vector with motion. They are not to be confused under any circumstances.
(1
Fig. 4     Dimension, coordinate, and vector
(e) The equation also fails to satisfy the definition of a line because, like in the previous case, it is predicated on motion. We are urged to replace variables with numbers in order to find out what a line is all about. An extra-terrestrial is deprived of visualizing one of the simplest geometrical figures known to man pending ENIAC's number-crunching. There is no way to represent architecture  with numbers, variables, or other symbols.
Fig. 5     Equations and functions 
Equations never represent structures. Dynamic that they are, equations symbolically represent itineraries, not roads. We construct an ongoing trajectory when we replace variables with a series of numbers. Each number that we plot on a coordinate system represents a location, not a structure. (11)
(f) If the definition invoking continuity holds, it reinforces our earlier argument that two points in space do not automatically constitute a straight line. Our isolated points do not yet meet the 'continuous' requirement, more so if the position of each point can be mathematically represented by a discrete fraction on the number line. A point is an island, not a peninsula, which implies that space or another medium delineates its entire perimeter. Hence, two points can never constitute a line regardless of how they are aligned or distributed in space if we are to be consistent with our hypothesis. We simply cannot construct a continuous wall using discrete bricks. If ever a line bends during your gedanken experiment, or slackens, or becomes malleable, look for a discrete stowaway at the elbow. He's got no business hiding there. To be straight, a line must be conceived to be an indivisible, continuous entity from the start. In Physics, the terms continuous and straight are synonymous.

   
Straight (adj.): continuous; a term used to describe the property
     of a hypothetical noun made of a single piece.


    
Line (n): an imaginary, continuous, two-dimensional figure
     pertinent exclusively to Physics, assumed to have the shape
     of a rectangle; typically used to depict qualitative structure,
     position, or motion-related concepts such as dimensions,
     inclination, orientation, and direction. 


    
Curve (n): an imaginary, directionless trace consisting of
     discrete points; typically used to depict segmented or
     quantifiable, motion-related concepts such as trajectories,
     itineraries, orbits, and number lines.



III. Conclusion

The foregoing arguments expose weaknesses in definitions taken for granted by the relativistic community. 'Occur' and 'event', as well as the related adverbs 'here' and 'now', entail a minimum of two images and for time to be greater than zero. The shape of a point, on the other hand, can be conceptualized in a single frame. Therefore, a point in space at time zero is not the definition of an event, but the characterization of location.

Similarly, a black hole cannot be synonymous with the noun point if it is defined in terms of the effect the singularity has on a clock that happens to drift into or a photon that attempts to escape out of its structure-less event horizon. None of the mathematical scalars typically used to describe the influence exercised by a singularity -- time, force, field, energy, and mass -- can be visualized in a single frame, image, or position. Like all artifices of relativity and quantum, a black hole is all verb and no noun!
(12) If, instead, by 'a point of infinite curvature' relativists mean 'a location of infinite curvature', they again have failed to define a structure. (13)
Fig. 6   Event, black hole, and poi
A line is essentially what has heretofore been known as a line segment. It is characterized by two ends rather than by two endpoints. A line is a two-dimensional, geometric figure that consists of a recognizable shape. If in addition the line is alleged to be straight, it is made of a single piece. We propose the word 'curve' for segmented concepts (i.e., number lines) and suggest that, for clarity of exposition, they be depicted with dashed lin
References

1.  

2. See for example the definition of black hole.
Copyright (c) 2001- 2006 by Bill Gaede 
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