Zeno’s Motion Paradoxes—Essay #2
Paradoxes cannot exist; and therefore, all paradoxes can either be solved, tucked away in iSpace where they become meaningless, or both.
Today’s philosophers recognize four motion paradoxes created by Zeno. The first two, “The Dichotomy Paradox” and “Achilles and the Tortoise Paradox,” were discussed and solved in the essay, “Solving Zeno’s Paradoxes”; but the second two, “The Arrow” and “The Stadium” were not discussed. I feel it necessary to solve the second two so as to have addressed all four.
But first, I have decided that more of an explanation needs to be added to the first two.
When I solved the first two of Zeno’s motion paradoxes I stated that infinite anything, that is, bits of matter, quantities of energy, or numbers cannot exist between two points in three dimensional reality. There are two reasons why. First, imagine two bits of matter, A and B, speeding away from each other. Regardless of what the time frame is, and no matter how long they speed away from each other, there will never be an infinite measurement nor an infinite amount of energy between the two points. And it only makes sense if the two points are speeding toward each other, there cannot be an infinite distance between them. And second, numbers are a manmade tool, which means numbers do not exist in Nature; therefore, ½ of ½ of ½ ad infinitum cannot describe the distance between two objects. In Nature, in three dimensional reality, there is simply a distance between points A and B. The distance between A and B can be compared to the distance between A and C without using numbers. A and B can be shorter or longer or equal to the distance between A and C It’s not difficult to understand that the Earth is closer to the sun than Jupiter. Numbers can be added to the distance as a convenience of comparison, or as a working tool, but the numbers do not exist in reality.
Even though these two explanations are true, a statement cannot be made without empirical or logical proof. More of an explanation must be laid in the foundation in order to make it even less of a theory and more of a fact.
Further explanation will now be added.
Relativity gives every bit of matter throughout infinite space a unique address. Scientists agree that two or more bits of matter cannot take up the same space at the same time, and since matter is infinite, then every bit of matter is related to all other bits of matter with regard to their distances from all other bits of matter. Note: some might argue that there are alternate worlds that exist in the same space as another world. The problem with this argument is they are now talking fourth dimensional reality. The solution of Zeno’s motion paradoxes exists in the third dimension.
Some would like to argue, and perhaps it is true, that there are Universes out there that are identical to ours with identical people and with identical lives. Somewhere in the reaches of infinity there is another you living an identical life to yours. But these Universes are not and can never be identical in addresses, simply because of the relation these Universes have to all other matter.
Since all matter has a unique address, then all matter has to have a relationship to all matter. The Earth has a relationship to the sun. And even though the distance between the sun and the Earth continually changes as the Earth rotates around the sun, both the sun and the Earth continue to have a unique relationship to each other in space. To further expand this idea, the Earth and the sun have a unique address with every other bit of matter throughout infinite space. Jupiter, which has a unique address with the Earth, is further from the sun than the Earth is from the sun. The next nearest star is further from our sun than Jupiter, and so on.
All addresses of matter have distances between them, because they are finite in relation to each other. But regardless of how far one address is from another there will never be an infinite distance between them.
With this information I have added unique addresses to the solution of Zeno’s first two motion paradoxes. And it has been shown that these addresses can have only finite distances between them.
A side note: when Sir Isaac Newton discussed absolute time, he made it clear that numbers and motion cannot be assigned to it. In essence he was stating that absolute time is “existence.” Something similar can be seen with infinity. Numbers cannot be assigned to infinity, and infinity is not a distance. Infinity is a “state of being.”
3. The Arrow Paradox.
This paradox was created using discrete units of time. It states that at any given in time an arrow is not moving, but rather stationary; therefore, how can the arrow go into motion and move to the next instant? It’s similar to taking a photograph of an arrow in flight. The arrow in the picture is stationary.
Dictionary. Com describes “instant” as “an infinitesimal or very short space of time; . . . .” It goes on to state that an instant is the same as a “moment.” In essays discussing Zeno’s motion paradoxes both “instant” and “moment” are used.
Since time is a major part of the paradox it will be used to solve the paradox. In order to create time, motion of one object has to be in relation to another object. On Earth, time represents the motion between two objects; and more specifically, the distance travelled by the Earth around the sun. One complete revolution is one year, and from there we coin other words to compare time to one year. We use eons, ages, centuries, months, weeks, hours, minutes, seconds, microseconds, instants, and others. All of these words represent a certain distance that the Earth has travelled. So, when the word “instant” is being used, it actually means a very small amount of distance. In order for an arrow to travel an instant, it must be in motion. Even though the amount of distance travelled by the arrow in an instant is infinitesimal, it is still a distance and it is still in motion. Since an instant is defined as time, it cannot be a stationary place in space. Time and distance are synonymous; therefore, the arrow is always in motion during its flight.
If the arrow appears to be stationary in a photo, it is because cameras are so fast they can fool the human eye by reducing the blur of motion to a state of invisibility. When lightening is captured in a photograph, it may appear to be stationary, but we know it is moving very fast. The image is being captured in an instant.
So the solution to this paradox lies in the fact that time is a manmade tool. And it represents the Earth traveling in a continuous motion around the sun. There are no stops and starts in time. In fact, an hour represents the Earth travelling in a continuous motion for approximately 67,000 miles. It makes no difference if time is presented in discrete units, such as a clock ticking off seconds. This is just another manmade view of time.
4. The Stadium Paradox
This paradox is often portrayed as follows. There are 3 sets of equal objects. The objects in the middle of the stadium, the “A’s,” are stationary. The objects racing from the left are the “B’s.” And the ones racing from the right are the “C’s.” As an example each of the A’s represent one second in relation to the speed of the B’s and C’s. For the sake of simplicity each letter will be one millimeter (mm) in width.
Both the B’s and the C’s are traveling the same speed at one second per mm as they approach the A’s.
When the B’s and C’s start to pass each other we are confronted with the paradox.
It has taken both the B’s and C’s two seconds to get to the third A, but it has only taken one second for the B’s to get to the second C, and one second for the C’s to get to the second B. This means the C’s and the B’s are travelling 2mm/sec toward each other, but only 1 mm/sec when traveling toward the A’s. And thus, the paradox.
A simpler way to consider it:
There are three men: Herman, Henry, and Arvid. Herman is standing in the middle of the stadium. Henry is fifty meters to the left of Herman, and Arvid is fifty meters to the right of Herman. Henry and Arvid are Olympic runners. They start running all out at twenty miles per hour toward Herman. Together they are running forty miles an hour toward each other. But that can’t be, because each man is running twenty miles per hour toward Herman.
He [Zeno] says, in effect, that if someone is running toward me from the west at the maximum possible speed, and someone else is approaching me from the east at the maximum possible speed, then they are approaching each other at twice the maximum possible speed…which is a contradiction.1
This is an interesting paradox because it is another example of “time dilation.”
In the essay “Is, Ts, Infinity” I have proven that time dilation does not exist in three dimensional reality. It exists only in the imagination, or iSpace.
The following is how the logic is involved. If a man is standing on the sideline watching the event, he sees Henry and Arvid running toward Herman at twenty miles per hour, and toward each other at forty miles an hour.
The man standing on the sideline is in what I call a “secondary frame of reference,” as he observes the 40 miles per hour; and in regards to time dilation, a secondary frame of reference can have no consequence in three dimensional reality. Whether or not a frame of reference, either primary or secondary, can be a part of reality, has to do with Cause and effect. Reality can only exist in a primary frame of reference. Combined, Henry and Arvid are running forty miles per hour, but since that is being viewed from a secondary frame of reference it can have no effect on either of the runners.
Henry is in a primary frame of reference, and he is traveling twenty miles per hour. He can pass Arvid in front of Herman, keep running, run off the track, and run into a brick wall at twenty miles per hour, but not forty miles per hour. There can only be cause and effect from the primary frame of reference. The forty miles per hour is meaningless, because nothing can happen with the forty miles per hour. A person standing on the sideline is in his own primary frame of reference, and he can observe other primary frames of reference, and the paradox proves that he can also observe secondary frames of reference. Even if Arvid is the name of a racecar, and Henry and the racecar are approaching each other at 120 miles per hour, in this scenario, from a secondary frame of reference, this speed is once again meaningless and can have no cause and effect, which means it is not a part of three dimensional reality. Henry cannot run into a brick wall at 120 miles per hour. Another example: three cars are travelling down a four lane highway. On the right side a car (A) is travelling 60 miles per hour. On the left side is a car (B) moving 40 miles per hour. It is in the slow lane and in front of a car © speeding 80 miles per hour in the fast lane. Car C passes car B. We have no need to add their speeds, but if we did, the 120 miles per hour, being in a secondary frame of reference, would have no meaning. Car A is approaching cars B and C. Once again to add their speeds would have no meaning.
As a note: only when two or more primary frames of reference interact can they produce one meaningful, primary frame of reference. So, the man on the sideline watches as Arvid and Henry, instead of passing each other, they collide head on. Now, they have created a single, primary frame of reference involving both Arvid and Henry, and the cause—colliding at forty miles per hour, can have disastrous effects.
So, the solution of this paradox lies in the fact that Zeno created, along with the primary frames of reference, a secondary frame of reference. Henry running toward Herman at 20 miles per hour has nothing to do with running toward Arvid with a combined speed of 40 miles per hour. Without an interaction between Arvid and Henry, the 40 miles per hour will effect nothing. The secondary frame of reference cannot exist in three dimensional reality because it has no purpose and no meaning. It will never interact with another secondary frame of reference, nor with a primary frame of reference.
This paradox is solved and will be tucked away in iSpace.
1 written by Kevin Brown