# Zeno Paradoxes and the Elea School

June 26, 2021Zeno of Elea (495 – 430 BC) was an ancient Greek philosopher.

In Plato’s dialogues, Parmenides and Zeno’s BC. He tells that they came to Athens in 450 BC, Parmenides was quite old at that time and Zeno was around 40 years old. Zeno knew his teacher Parmenides mostly from his works. In his works, the important parts of which have reached us, Zenon preferred to make a critical comment instead of explaining nature. Zeno’s work is a very sharp approach to logic. In his work, Zenon developed his thoughts and reached very definite results.

Zeno first researched the concept of “infinite”, which is very prone to contradiction. The concept of the infinite is first encountered in Anaximandros. Anaximander thought of the infinite and unlimited “Apeiron” as the basic principle of the universe. Likewise, the Pythagoreans, who studied mathematics, were introduced to the concept of the infinite. Pythagoreans especially; They were interested in the infinitely small and the infinite divisibility of a line or a plane. Here, Zeno criticized the concepts of infinitely small and infinite divisibility in the Pythagoreans. He tried to show the difficulties observed in these concepts. Zenon was the first to find the “aporie” and “antonomie” that the concept of infinity contains. Since that day, the history of philosophy has not lost anything of its relevance to these concepts.

Mathematics was able to make these calculations without conflicting with the concept of the infinitesimal. However, despite this, the difficulties in these concepts have not been resolved. The difficulties in these concepts were first discovered by Zeno of Elea, who used them to confirm the ideas of his teacher, Parmenides. He suggested that concepts such as infinite division, change, movement, and multiplicity can be explained with the help of the concepts of place and time, which also contain apories. Space and time are made up of parts that are inseparable from one another. That is why infinite division is a contradictory concept. As a matter of fact, the concepts of movement and change can only be considered in space and time. Because movement is a change in place in time. Change can only happen over time. Since multiplicity is also a dispersion within the place, it must necessarily be related to the place. Because the concepts of place and time themselves carry insoluble difficulties; It is natural that there are some contradictions in the concepts such as infinite division, change, movement and multiplicity that depend on them. Because of these contradictions, all of these concepts are appearances, not reality.

In the explanations he made to confirm his teacher Parmenides, Zeno put forward the following proofs: Suppose an object is made up of parts, and these parts are made up of other parts, which are also divided into parts again. Let this continue. Let us deduce from this the following conclusion: Every object occupying space is infinitely divisible. As such, there are two possibilities: These infinitely dividing little pieces take up space in space, or they don’t occupy space in space. If these coins don’t take up space, no matter how many of them I put together, nothing will happen. Indeed, if the volume of the parts is zero, there is no result from adding the zeros. If these pieces take up space in space, something infinitely large will form, no matter how small their volume. So, when we consider an object to be composed of infinitely divisible parts, this object will be either zero or infinitely large in terms of occupying space. In both cases, there is a contradiction.

Achilles Paradox

Zeno’s criticism of the reality of the movement, the example he gave for this is important and very famous: Achilles (Arschylos), the fastest runner of his time, was in charge of the Greek army. Achilles competes with a tortoise. However, Achilles gives some advance to the tortoise and the race begins. First, Achilles needs time to run the distance given to the turtle as an advance, that is, to catch up with the tortoise. But while Achilles was running, the tortoise did not stop, he also walked a certain way. Now Achilles needs time again to run the distance left behind by the tortoise. But meanwhile the tortoise has advanced again. I can extend this as long as I want. I realize that Achilles will never catch up with the tortoise. It is counterintuitive for Achilles to actually cross the tortoise. With this proof, Zenon wanted to display the contradictions of the thesis that space and time can be infinitely divided, in an entertaining way.

All of Zeno’s arguments have one purpose: to show that the concepts of change, motion, and multiplicity contradict each other. For this reason, Zeno is the first thinker who discovered the aporie and antinomies that the concept of infinite contains. The logical difficulties in these concepts continued to be of interest afterwards. As a matter of fact, Kant also dealt with them. In fact, this interest continues even today. However, modern mathematics has gained the opportunity to do some mathematical operations with the concept of the infinitesimal. In other words, he overcame the difficulties in the concept of infinity technically. For example, modern m