The Comprehensive Solution of Zeno’s Arrow Paradox 1

1. Introduction

Zeno’s Arrow Paradox is a very famous paradox. Aristotle described in Physics ⅵ “everything when it occupies an equal space is at rest, and if that which is in locomotion is always occupying such a space at any moment, the flying arrow is therefore motionless.” Surely, Zeno is right because the arrow is still at any instant. However, Aristotle said, “This is false, for time is not composed of indivisible moments any more than any other magnitude is composed of indivisibles.” If Aristotle’s statement is right, Zeno’s statement is only a matter of course. If time does not consist of moments,  there is no problem even if the arrow is still at any moment. First, Aristotle’s countermeasure is considered in this post. However, if an arrow is still at a moment, can we postulate that the speed of the arrow at the moment is zero? So, instantaneous velocity is the next problem.

2. Motion Is Created by Our Brain

Even though time has never stopped until now, we have the concept of a moment. Where did the concept of the moment come from? In fact, the concept of a moment comes from the feature of our eyes. Our eyes resemble the digital camera. Hence, our eyes only capture still images.

Figure 1

 

Figure 1 shows that a digital camera captures photons reflected by a ball. Even though the ball is moving, we can take only a still image of it using a digital camera. Because the flying ball is sufficiently slower than a photon, we can regard it standing still.  So, when I take a picture of a flying ball with a high-speed camera, I get only a still image of it. This discussion can be generalized. Ordinarily, the moving object moves enough slower than a photon. So, we consider it as motionless. Hence, we can know only the position of the object as far as we detect a bounced photon.

Theoretically, we can take only a still image using our eyes. Since there are about 6.5 million pyramidal cells in one eye, our eyes roughly correspond to a digital camera with 6.5 million pixels. Each pyramidal cell contains opsins, which are photosensitive molecules. The photoreceptor of each opsin is retinal.

Figure 2

Figure 2 shows the isomerization of the retinal. In the beginning, 11-cis-retinal captures a photon. Immediately, it is isomerized to all-trans-retinal. The photoisomerization of the retinal is the necessary reaction of the visual perception. That is, our eyes capture photons with opsins. Because digital cameras and eyes capture reflected photons, both of them can obtain only still images.　Why can we see motions? We cannot see the movement directly by eyes. Donald Hoffman says that your brain creates the motion. He shows many examples in Visual Intelligence. 

Figure 3

Additionally, I represent one simple example. Figure 3-1 shows that there is a black circle on the left side. Figure 3-2 shows that there is a black circle on the right side. Animation 1 shows these two figures by turns ten times. As a result, you may see the reciprocal movement of the black circle. However, there are only two still images. Furthermore, there is no motion of the black circle. In this example, the motion consists of two stages. Firstly, Two still pictures at two instants are constructed by our brains. Secondly, our brains create the motion between them. This example is the basic form of the creation of the motion by our brain.

 

Animation 1

 

 

4. Time Does Not Consist of Instants

As mentioned above, the movement we perceive is what the brain created from still images. Hence, Zeno’s arrow paradox is the paradox of the motion created by our brains because Zeno used the word, moment. Because time has not stopped, we do not know whether the moment exists actually. The moment may be only in our brains. Thus, we shall consider the motion created by our brains.

Figure 4

Because the motion created by our brains is continuous, space and time are assumed to be continuous. Figure 4 shows the motion of an arrow based on the assumption. At the time A, B, C, D, E, the tip of the arrow is at point P, X, Y, Z, Q, respectively.

Originally, the brain complements between still images with continuous motion. Reversely, we think that we can extract still images at every moment from continuous motion. This is impossible in reality. No matter what high-speed camera you use, the blur will always remain. Because the shutter speed is not zero at all, blurring is always a problem in photographs. However, if we pause a moving picture, we always get a still image. A moving picture is composed of still images called frames., and regular moving pictures are 30 frames per second. The frame rate is sufficient for normal movement, but in the case of the arrow, a very fast frame rate is required to obtain accurate still images. Ideally, an infinite frame rate is required for the arrow.

Figure 5

Assume the ideal videotape that recorded the movement of the arrow. If we assume that the ideal videotape has infinite frames, we can get an accurate still image at any moment. The length of the videotape corresponds to the time. The upper line of figure 5 corresponds to the videotape. Lower line segments of figure 5 correspond playback time of the videotape and intervals of line segments are correspond to pausing times. No matter how many pauses, the videotape playback time is constant. That is, even if the arrow is stopped at any moment, the arrow fly without problems. At this point, Zeno’s arrow paradox has been solved by half. Furthermore, we shall consider Aristotle’s statement “time is not composed of indivisible moments.”

Figure 6

As a prerequisite, assume that space and time are continuous and time corresponds to the length of line segments. Furthermore, assume that Euclid Elements is true.

Book 1, Definition 3: The ends of a line are points.

Consider a radius of a circle. Ends of the radius are the center of the circle and any point on the circumference. There is one to one correspondence between the center of the circle and a point on the circumference about each radius. We can draw any number of radii. If we extract radii, then the center point is divided into ends of radii. Figure 1-1 shows four radii: AO, BO, CO, DO. When we extract these radii, figure 1-2 shows AO₁, BO₂, CO₃, DO₄. No matter how many a point is divided, the total magnitude of points equals zero.

Therefore, no matter how many points on the circumference are gathered, the total length is zero. The center point can potentially correspond to all points on the circumference.

Since the magnitude of the point is zero, an independent point is undetectable. Furthermore, because the center of the circle is ends of all radii, it can be regarded as a gathering of many points. Therefore it is reasonable to consider a point on a line as a possibility. Here we rethink about Figure 5. When one line segment is cut at four points, it will be cut into four line segments. Concurrently, four points, which existed as a possibility, are actualized as eight points. In this way, you can realize points by cutting the line. However, you can only realize a finite number of points with this method. The reason is as follows. There is always a distance between two points because if there is no distance between two points, they are the same point. Assume that the minimum distance between two points is ε and the length of the line is l. According to Archimedes’ axiom, there is a natural number n such that:

If we accept mathematical induction, any natural number is finite.

1. 1 is a finite number.
2. If n is a finite number, then n+1 is a finite number.
3. Any natural number is finite.

Because any large natural number is finite, the number of realized points is finite.

Next, we shall reconsider the ideal videotape. If you want to show all still images in order, the ideal videotape should be played. However, there is no still image next to the first still image. If there is no time between the first still image and the next still image, they are the same still image. If there is time between the first still image and the next still image, there are potentially infinite still images between them. As a result, the above-mentioned logic can be applied to the ideal videotape. No matter how you play the ideal videotape, you can only do a finite number of poses until you finish playing the ideal videotape. Therefore, time does not consist of instants.

Next, we shall consider the real motion of an arrow, not videotape. Assume an ideal camera that can take a picture of a moment. Can we use it to break down the arrow motion into still images? Firstly, we can take a picture of the arrow of the first moment. However, we cannot take the next still image of the arrow. Consider the first still image of the arrow and another still image of the arrow. If there is no time between them, they are the same still image. If there is time between them, the later still image of the arrow is not the next still image of the arrow because there are potentially infinite still images of the arrow between them. As in the case of the ideal videotape, we can only capture a finite number of still images of the arrow with the ideal camera. In conclusion, we cannot decompose the arrow motion into still images. With similar logic, we cannot compose time from moments. If there is no time between two moments, they are the same moment. Hence, time is necessary between two different moments. That is, The gap between two different moments cannot be filled. Therefore, Aristotle’s objection is legitimate.