Chapter 5: Zeno’s Paradoxes
Paradoxes are thought experiments which show that certain ideas we commonly accept as true are logically self-contradictory.
The Greek philosopher Zeno pronounced these classic paradoxes some 2,500 years ago. Since then they have defied a clear solution. Some claim there are mathematical solutions to Zeno's paradoxes and others claim these solutions are unimportant because the paradoxes are fallacious.
I believe they are true paradoxes that have come about because of incorrect ideas on how to interpret motion. All paradoxes are involved with our limited knowledge of how to interpret nature. The aim here is to replace these paradoxes with improved approximations of reality.
The reader can be the judge.
Zeno uses simple words to describe his paradoxes, so let's use simple words to their solution. The failure to explain Zeno's paradoxes is at the heart of the reality crisis in physics. It is remarkable that after 2,500 years they remain unanswered. I will show that these paradoxes follow from our erroneous view of motion.
Zeno's first paradox.
This paradox is about a runner who is ready to start a race. In order to finish the race the runner must first go half the distance to finish line, but to go halfway he must go halfway to the first halfway point and before that he must go another halfway and on and on. There is always another halfway point to reach whatever the distance of the race.
This means there is an infinity of halfway points to reach before the runner can finish the race. The runner spends all his time motionless because he cannot come to the first halfway point.
In reality, the runner does get started and he has forward motion. But what is wrong with our perception of the paradox? Zeno knew his paradoxes did not describe reality. He challenged us, however, to explain why they were paradoxical.
Zeno's second paradox.
This paradox involves a race between the speedy Achilles and the much slower tortoise. The tortoise is given a head start and the challenge is for Achilles to catch and pass the tortoise. Zeno points out that every time Achilles arrives at a point from which the tortoise has departed, the tortoise has moved farther on.
Wherever the tortoise is, by the time Achilles gets there, the tortoise has moved farther on again. Because of this, Zeno argues, Achilles can never catch the tortoise.
We are involved again with an infinity of distances between the departure of the tortoise and the arrival of Achilles. The tortoise is always some distance ahead because Zeno always requires Achilles to arrive at a fixed point from which the tortoise has just departed.
No matter how short the distance, the tortoise will have moved every time Achilles reaches the tortoise's new point of departure. However, in reality Achilles does pass the tortoise. Therefore, where have we gone astray in how we perceive motion?
Zeno's third paradox.
This third paradox involves the motion of the arrow in flight. Zeno insists that if the arrow occupies a specific location at any instant then at any instant the arrow cannot be moving. Anything occupying a place must be at rest.
Since the instant of being somewhere can be any instant along the path of motion then it is presumably clear that the arrow cannot be moving. But the observed arrow does move with respect to the observer.
Has our perception of motion failed to approach the reality of motion?
Zeno exists, insists that objects are either continually moving or they are continually at rest. If an object occupies a specific place then it is at rest. If it is moving it is always out of a specific place. Herein lies the deception in the paradoxes reason tells us that something must be somewhere even while it is in motion, but to provide a place for a something in motion is to identify that place.
The place must be measurable. To measure its location, there must be a specific and changing location for this object in motion. Yet this cannot be, because a specific and changing location indicates a state of rest for the object and hence non-motion.
The points of position that Zeno places on the pathways of motion are the same points of position we place on the pathways of motion to understand motion and where objects in motion will be at any given time.
The Moon and Earth are in continuous motion, and yet it is important to identify their relative positions to know where they will be. For example, due to the influence of the Moon upon tidal changes, it is important to know where the Moon will be at any specific time.
But according to Zeno, if the moon has a specific position at a specific time, then while it occupies a specific location, it cannot be moving. How can we explain this and still end up with a theoretical world with cause and effect and a one-to-one correspondence with reality?
All particles, all objects and all humans use emission and absorption to interact with the world around them. These two separate stages, emission and absorption, depend upon each other.
Stage 1 is departure and the motion of emission. Stage 2 is arrival and the fill of absorption with a cessation of motion.
Emission is the physical of the one using the zero of a pathway for motion, and absorption is the physical of the one entering a zero. Motion is emission used to leave zero and the cessation of motion is absorption where is the entering of a zero.
The cessation of motion means motion has been completed for entering the parameters of the whole. From this phenomenon we can conclude the emission (departure) and absorption (arrival) are opposites which come from the binary zero and one opposites that form the foundation of the universe.
Opposites define, corollary number 9, each other and both stages of emission and absorption are necessary for our awareness of each opposite.
The prior condition of absorption and arrival allows us to gain awareness of emission and emission allows us to gain awareness of absorption. It takes motion to know non-motion and non-motion to know motion as it takes on to know off and off to know on.
This dependency factor between emission and absorption must be a part of the pathways used by Zeno to unravel his paradoxes.
There is no distinction of the pathway 0 unless we have physical 1 borders to define it, and we need borders to define the length of this pathway of motion and this is provided by defining where motion will start and stop.
Every pathway includes a starting place before motion (emission) and a stopping place for absorption and the cessation of motion. Borders of motion are defined by borders of non-motion and borders of non-motion are defined by borders of motion.
By separating motion and non-motion through emission, motion, and absorption, non-motion, there is a straightforward way to solve these paradoxes.
In all of Zeno's paradoxes he provides a path of motion for his moving objects. For the path to be defined, however, we must include the beginning (start) and end (stop) of the path. Before the start and the stop there are non-motion events to define motion. The stop and start give the borders to determine where motion will take place.
It is important to note that in reality motion is always separate from position. Position occurs at the specific locations before the start of motion and at the stop of motion. Position and specific locations are the non-motion and resting states used to define motion's specific pathway. Without knowing the pathway of motion, we cannot define and understand motion.
Non-motion does not mean absolute motionlessness. In reality it appears impossible to reach a state of absolute motionlessness. What appears motionless does have motion at some microcosmic level. The term non-motion names the arrival phase within the hole that is the goal.
In this treatees non-motion is intended to show that the motion necessary for arrival within the boundaries of a particular hole has ceased. Non-motion indicates there is no motion outside the identifiable borders of the hole used for absorption.
An object's motion within a specific hole may continue, but the motion for arrival in this specific hole has ceased. The term motion means the motion necessary for an object's departure from a specific hole and arrival at a specific hole.
Motion and non-motion are relative terms determined by the object's entering and leaving specific holes. Motion is before entering and non-motion is before leaving. This is a relationship between two holes and the motion between them.
The beginning of motion starts from a previous stop of entering a hole and the end of motion stops the motion necessary for entering another hole.
The paradoxes provide a pathway for motion which, like all pathways, is defined by the borders of non-motion. Before the beginning and at the end of motion.
The starting and finishing lines are the specific points necessary for a mission to start from the past absorption, non-motion before the start of motion, and for a mission to end with absorption. As Zenos starts his runners and arrows along his initial pathway, he changes the pathway by placing new points and specific locations of departure and arrival that replace the initial pathway with an infinity of pathways.
In the first paradox, Zeno creates an infinite number of pathways with an infinite number of halfway points with which each require another arrival and another start. The arrival is a stop that depends on a start and conversely.
Arrival is absorption and signals the end of motion. Departure is emission and the beginning of motion starting from the end of motion.
With an infinite number of pathways defined by these new arrival points and points for departure, how can the runner pick only the first one from which to start? He cannot.
The assumption in the beginning is that the runner has only one pathway for motion, but Zeno makes him run on an infinite number of pathways in defiance of reality.
A real running path must always have a single beginning place (start) and a single ending place (finish). Relative to an observer, the runner must either be a part of motion or a part of no motion.
Zeno's halfway points of arrival require infinite processes of motion and non-motion along a path designed for only motion. In reality, the concept of arrival cannot be part of motion. It goes against pure reason and defies direct observation to have both arrival and departure occur simultaneously.
The crux of the problem is that when you start with one path of motion, it can only have one start and one stop and one departure and one arrival. The space in between departure and arrival can only be for the motion of emission.
If you take one path and add more points of arrival then you have changed one path into many separate paths. By definition, every path must include only one start and stop for emission to begin and end. Without these limiting borders, the rate and concept of motion cannot be precisely defined or even understood.
Zeno takes the area between the beginning and end of motion and then adds an infinite number of beginnings and endings. Motion is only a reach for an ending and these arrivals, endings, cannot take place for motion to continue.
If arrival, specific points of locations is placed on the emission phase, this brings motion to an end. To say the runner arrives at specific points while he continues to run contradicts both reason and observation.
Arrival indicates the set location of non-motion. To use arrival during motion is an erroneous use of language and an internal contradiction because you cannot arrive and depart at the same time.
It is observable that opposites occur one after the other and not simultaneously.
The same internal contradiction has been incorporated in the race between Achilles and the tortoise. Zeno requires new arrival points for Achilles where the tortoise begins another departure. He creates his paradox with an infinite series of absorption arrival and emission departure points for both Achilles and the tortoise, and therefore establishes an infinite series of separate paths of motion that originally started out as one path of motion.
Achilles can never pass the tortoise because he is required by Zeno to take an infinite number of separate paths in which he always arrives when the turtle has already departed.
Again, the concept of specific locations and points along the path of motion can never be a part of motion because it is part of non-motion. Motion is one of a pair of opposites. It cannot occur simultaneously with one another of the same pair of opposites, non-motion.
All opposites occur one after the other. The fill follows the empty, or the empty follows the fill. In paradox 3, how can the arrow be in a specific location along the path of motion, when it is in the emptying phase of emission?
While in flight, how can the arrow occupy a specific hole with specific borders at a specific location for its measurement and simultaneously leave its specific location? How can you come to one specific measurement (definition) for the position of an area if the area is continuously changing?
To measure this specific location with a ruler, the geometry of the non-physical between the observed arrow and whatever physical reference point is used for measuring must be stationary and unchanging.
But motion requires the non-physical between the observed and the observer to be in a constant state of change. The ruler becomes an unusable standard of measurement because a stationary frame of reference and a specific distance is unavailable.
To determine that the observed object has a fixed location relative to the observer, the space between them must be fixed. Both the observed and the observer may be moving relative to other objects, but in relationship to each other they cannot move.
As an example, a passenger of an airplane in flight does not have motion with respect to the airplane, but both the airplane and its passenger have motion with respect to the Earth. The concept of motion and non-motion is relative and depends on the relationship between the observer and the observed.
To explain the paradox of the arrow, the relationship between the arrow in flight and the human observer must be considered. The observer may insist on the arrow being in specific locations during its flight because he can mentally imagine it being at one place or another.
The relevant question to ask is, does the arrow really occupy specific places or does the observer's mind use specific locations for perceiving the arrow? Does the mental image of the arrow having fixed locations come from the observer's attempt to substitute that image in place of the arrow's actual motion?
I have concluded that the mind is also an absorber and that knowledge comes from a mental arrival which is the termination of motion. Light emission stimulates our senses with absorption in the brain. If knowledge is an arrival stage in the mind and arrival is a suspension of motion, then how do we identify motion? How can we observe and understand motion if knowledge itself is absorption, the positional requirement of non-motion?
Motion is the means of knowledge, but motion also ends with knowledge acquisition. A solution can be found in comparing the brain to a motion picture camera. After exposing the film to moving objects and processing the negatives, the film consists of series of non-moving pictures that have specific locations.
To create the perception of motion, the film is placed in a projector that will move, transport, the film past the projection lens. The camera is not recording device. The camera is a recording device. The pictures recorded on motion picture film record the act of photon absorption. The photographic film records the light particles that have arrived and reached non-motion through motion.
Each light particle (one) strikes the film and is absorbed recorded at a specific place on the film. Like photographic films, the brain absorbs and records arrivals of information at specific location. But if our brain only records still pictures, how can we perceive motion?
Our sense of motion comes from a rapid series of still images (absorptions) taking place one after the other, similar to the project flashing separate still pictures at a regulated speed. The individual recordings of the mind are accomplished fast enough to also give us this mental sense of motion.
Our perception of motion is created by a rapid series of separate arrival stages. The projector within our mind can play back on command our recorded memory of a series of still mental photographs to create the perception of motion.
The way the mind works helps explain our difficulty in unraveling Xeno’s paradoxes. If the arrow's motion is visualized through a rapid series of specific locations (positions) in the mind, then it is easy to project this visualization upon the actual arrow or any object in motion. We are confusing the mind's perception of an arrow in flight with the arrow's actual flight.
The observer observes a series of non-motion events within the mind, but the arrows flight in motion without non-motion events.
Our eyes and mind absorb the light reflected from the arrow in its motional state, as well as the other objects in non-motion states. But the arrow in flight is in a state of continuous motion. The light particles reflected from the arrow reach non-motion for us and not non-motion for the arrow.
It is through recordings by the process of absorption that we attempt to measure and define the world around us. Absorption occurs whether objects are moving or not moving with respect to us.
When we envision the arrow at a fixed location while in motion, it means this is where the arrow would have been if this motion had stopped. Instead of the arrow actually being at a fixed place, we have assumed that fixed place for the arrow.
An example of this involves the use of a trainer's stopwatch to determine the position of his runner halfway through the race. Before starting, the runner has a specific location with respect to the track and the trainer.
However, at the halfway point, the runner in motion does not have an actual specific location with respect to the trainer or the track. This is because the relationship between the runner in motion and the trainer not in motion is in a continuous state of change.
To take the runner's time at the halfway point, the trainer must assume the runner's position at the halfway point. A return to an actual position relative to the trainer will not occur until the runner finishes his motion at the finish line.
With a stopwatch, the trainer records the runner's actual position at the start by the actual position of the watch at its start. He then records the runner's assumed position at the halfway point by stopping the watch. The runner's motion is surrounded by the actual position of the watch at its start and stop.
At the halfway point, the stopwatch hand has a fixed position, but not a runner in motion who has no fixed position. This is how the runner's rate of motion is measured. This is another example of how opposites define each other.
Let us time the runner at the beginning and end of this motion. We surround the motion of the runner with position (no motion) at the start and stop of the stopwatch to define his motion. We take the position (non motion) of the watch hand at the beginning and at the end of the runner's motion.
The time lapse of motion between the two timing positions on the watch (start and stop) will give us the rate and speed of the runner's motion. We get rates of time for motion whether we assume a position for the object in motion or whether the object actually has position.
In a like manner, to determine the time parameters for position instead of motion, we now surround position with motion. We use the end and the beginning of emission to give us a time frame for absorption and we use the end and beginning of absorption to give us a time frame for emission, motion.
One opposite, motion or position, is used to define the other opposite. Wherever we experience a problem defining something, it is essential to study its opposite to create a definition or enhance its precision. See Corollary number 9.
The above illustration of emission and absorption defining each other explains why they do not occur simultaneously. I have aimed to show that they follow one another. In creating his paradoxes, Zeno placed the stage of absorption specific points and locations within the stage of motion emission.
His paradoxes require departure and arrival at the same time and that objects in motion be placed at a specific location and leave that specific location at the same time. Furthermore, they require that the distance between the observer and the observed object in motion be both changing and not changing at the same time.
Zeno's venerable problems are paradoxical because once their solution is found his paradoxes contradict both reason and observation. Position can be assumed for the observed object in motion, but that object cannot have an actual position relative to the observer.
This does not deny the existence of the observed object, but it does deny that the observed object in motion has an unchanging dimensional relationship with the observer.
In reality, Zeno's runner moves from the start to the finish and Achilles moves past the tortoise because they never arrive at specific points during their motion.
Zeno and his paradoxes, notwithstanding, arrival can never occur during motion and therefore arrival can never be a part of motion. The position of arrival occurs at the end of motion. If knowledge is the positional notations of absorption and if our awareness of motion is an accelerated series of mental photographs that are separate pictures of non-motion, we can reach the conclusion that motion cannot be a direct experience for anyone.
This conclusion is based on observational evidence of how nature works. This is another example that motion is a means to reach observational processes and is not a direct part of observation. Motion is an indirect and prior process that must cease before mental pictures are possible.
This will be an important conclusion from the discussion of quantum physics in the following chapter.
I am not suggesting that because motion and position cannot occur simultaneously, that we must wait for motion to stop before it can be measured.
To be sure, if we waited for physical motion to stop, scientists might as well close their labs and go home. It is important to assume a position for an object in motion even though in actuality it has no position.
This assumption allows assumed positional predictions for the moon and planets of the solar system. By assuming planetary positions, we need not create actual planetary positions during their motion.
It is useful to say whether they will be at a specific time as long as we understand this is where they would have been if they had stopped their motion.
Some may say, so what difference does it make whether position is actual or assumed?
It should be emphasized that the reality crisis in physics is tied to this fundamental question. The mathematics of quantum physics concludes that in the microcosm, position and momentum of subatomic particles are mutually exclusive. Quantum mathematics indicates that when we measure the position of a particle, we cannot measure its momentum and when we can measure momentum, we cannot measure position.
This was noticed in the matrices of Heisenberg's quantum questions. It denies the existence of objects with both specific locations and paths of motion at the same time. However, physicists still insist that position and motion are simultaneous events.
The present conclusion by some is that reality is different in the microcosm than the macrocosm. However, what if the quantum questions are dealing with actual positional processes and are showing that actual position is separate from motion? What if the mathematics indicates that position and motion are not simultaneous?
Through these equations is nature telling us that actual position does not occur during motion? What if in the macrocosm of up there we also cannot have knowledge of actual position and momentum simultaneously as the equation state for down there in the microcosm? Could both the microcosm and the macrocosm be following the same rules? Is the obvious eluding us again?
The importance of solving Zeno's paradoxes is very much involved in the way we interpret the world. The paradoxes invented by Zeno attempt to place the points and specific locations of no motion on the path of the object in motion.
But motion cannot include arrival locations along its path. Arrival can only become part of the end of motion.
To say an object has arrived while still in motion is a false description of motion. To have any meaning, each pathway must have a beginning and an end with motion in between. It goes against reason to inflict an end (arrival at specific locations) to motion while the object continues to move.
Zeno was correct to insist that specific locations for an object and motion for an object cannot occur together. To place actual position on an observed with respect to the observer, you must have non-changing spatial (zero) boundaries between them.
The solution to Zeno’s paradoxes can be found in this conclusion. Actual position and motion (momentum) with respect to observed and observer cannot occur simultaneously.
A simpler way to state this may be that emission and absorption are not simultaneous events. Each requires the prior existence of the other before it can occur. Absorption is the starting point for emission and at the end of emission is the starting point of absorption. Emission is motion that is part of momentum and absorption is the end of motion and the reaching of position.
In creating the paradox of the arrow, Zeno established only the arrow's path of motion by excluding reference to before and after the arrow's motion.
Going beyond Zeno, we start by identifying the arrow at the absorption stage within the drawn bow. This is followed by the arrow's departure, emission and motion, and then on to arrival and absorption at the target.
When Zeno places points (specific locations) on the path of motion, he places the target of arrival on the path of motion when instead it follows after motion.
The bow is also a target for the motion of the arrow to arrive at the drawn position. Position of the arrow in the bow is required for motion to start and position at the target is required for motion to stop.
In the beginning, if we separate both the non-motion, no motion, before the start and the non-motion of the stop with motion in between, then we can define and understand the true pathway for motion.
When we break with Zeno and finally stop changing the parameters of motion with new stops and starts, his classical paradoxes are no longer paradoxical.
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