The science of mathematics develops from the mental aspect in the abstract part of brain function. In part of the development period of a human, the mind that has the ability to understand abstraction and imagine concepts that do not have a physical presence, emerges and is completed and forms intellectual maturity. Concepts like infinity are the result of this abstract thinking that has occupied many thinkers in the history of science. In this article from Arman Computer magazine, we want to address a paradox in the abstract space, the solution of which lies in a practical and physical space. This opposition of abstraction and the real world has created a beautiful dichotomous paradox.
Dichotomous Paradox
The essence of this paradox lies in infinity and infinite divisions. This paradox can be rewritten as follows:
Something in motion must reach the halfway point before reaching the goal.
Therefore, whenever a movement object travels a path, it must have already traveled half of the path before the path is completed. In this way, each time the remaining path is divided into an infinite number of smaller paths that are the result of dividing the path into two parts. If we consider the time to travel each of these routes in proportion to the length of the route (which is considered a rational condition), we have an infinite amount of time that we must pass through to reach the destination. In the same way, an infinite distance must be traveled for the entire path to be completed by the object. Since in the real world an infinite distance cannot be covered in a limited time, it seems that according to this paradox, even a path of one meter, for example, will never be completely covered.
What does the Dichotomous Paradox say?
The paradoxical conclusion would be that travel along any finite path can neither be completed nor begun, and therefore all movement we observe in the real world must be an illusion. Perhaps this opinion is more compatible with a world like Plato, who states that everything we see is an image of another world. As a result, infinity there has a separate concept from our world.
A perspective on Dichotomous Paradox
In a family meeting, I brought up the paradox of two parts and trying to introduce my daughter’s mind to the concept of infinity. Although she is only 10 years old, she understood the problem well and was even able to provide a perspective on the two-part paradox. I raised the issue as if the rabbit and the tortoise had a race. But the tortoise has moved one meter ahead of the rabbit. The rabbit must travel half the distance to reach the tortoise. On the other hand, to cover this half distance, it must have already covered half of this distance. Therefore, since these divisions can be made in a very large number, practically the rabbit will never reach the tortoise. I remind you that the concept of division is taught in the third and fourth grade of primary school.
Problem with finite space and solution in infinite space
We were all very curious to solve the problem of this contradiction. But a little later, my daughter replied that this problem happened because we did not determine the size of the rabbit’s steps. This paradox cannot occur if the rabbit’s steps are not too small. It was quite clear that this contradiction was a confrontation between the discrete world (rabbit’s steps) and the continuous space (division infinity). Therefore, this contradiction shows that solving a problem in a finite space with the help of a solution in an infinite space creates a problem and may lead the mind astray.
Geometric series and dichotomous paradox
From the point of view of mathematics, traveling a distance in times proportional to the length of the distance can be considered as a geometric progression with a ratio of 0.5 or $\frac{1}{2}$.
$$\{\frac{1}{2}, \frac{1}{4}, \frac{1}{16},…\}$$
Using the geometric series, the result of the following sentences will be equal to 1. Therefore, it is clear that the infinite product of time fractions will be equal to one. In this way, the desired distance will be covered in finite time because the value of the ratio is less than 1, and the first term is also equal to the value of the ratio, i.e. $\frac{1}{2}$. Therefore:
In this relation $a = \frac{1}{2}$ and $r = \frac{1}{2}$ which is calculated as follows.
Result
According to these views on the two-part (dichotomy) paradox, it can be said that doing this, contrary to the two-part contradiction or the dichotomy paradox, is not only possible but also certain, and our movement will definitely reach its end in finite time.
