Albert Einstein always loved paradoxes. In his view, it is precisely the study of them that allows a person to understand the surrounding reality more deeply. Actually, his famous theory of relativity arose from the contradiction in physics at the end of the 19th century between Newtonian mechanics and the assertion of the finiteness of the speed of light. So, what are the most interesting ancient Greek paradoxes in history, the knowledge of which can help us see our world differently?
Zeno’s Paradoxes
The most interesting paradoxes are the well-known ancient Greek paradoxes.
The ancient Greek philosopher Zeno formulated many paradoxes, the most famous of which tells the story of the tortoise and Achilles. If this hero tries to catch up with the tortoise, by the time he reaches its initial position, the tortoise will have moved forward. Then Achilles will run to a new point, but the tortoise will again have moved forward a bit. Thus, the hero will never catch up with the slow tortoise!
Despite the obvious fallacy of this conclusion, it is very difficult to explain right away where the catch is. Studies of the paradox pushed mathematicians to create the theory of infinite series. On the other hand, the paradox suggested to the ancient Greeks that space and time cannot be infinitely divided into smaller parts. Modern quantum physics, which emerged in the 20th century, confirms this.
The Petersburg Paradox
Let’s say a person is offered to play a game. He pays an entry fee, after which a coin is flipped. If it lands heads, the player receives 1 ruble; if not, the coin is flipped again. If heads appear on the second attempt, the winnings will be twice as much (2 rubles); otherwise, the coin is flipped again. Each time, the amount of winnings doubles (4 rubles, 8 rubles, 16 rubles). The question is: what is the fair cost of participating in the game?
Usually, mathematicians determine the benefit of participating in a game through the mathematical expectation, but in this case, it equals infinity! That is, from the point of view of probability theory, the game is profitable at any cost of participation, even a billion rubles. But obviously, no one will spend all their money on such a game.
This paradox illustrates the law of diminishing returns. A second car is not as necessary for a person as the first. Similarly, with money. Ten thousand dollars is a huge sum for a beggar, but a billionaire can give it to a waiter for tea. Therefore, if we evaluate the winnings in the game not by the amount of money, but by their utility, then the mathematical expectation will be finite.
The Barber Paradox
One of the most famous and simple paradoxes. In the city lives a barber who shaves everyone who does not shave themselves, but he refuses to shave anyone else. Does he shave himself? There are similar paradoxes, such as the priest paradox, the catalog paradox, and the reflexive adjective paradox.
Such riddles sometimes seem like amusing jokes, but in reality, they have had a serious impact on the development of mathematics. All of them can be reduced to Russell’s Paradox, discovered in 1901. Let a “simple” set be defined as a set that is not an element of itself. Will the set of all “simple” sets be “simple”? The resulting contradiction destroyed the entire naive set theory, which was unquestioned by leading mathematicians of the 19th century.
So does the barber shave himself? The answer is very simple—such a barber cannot exist.
Time Paradox
First described in the book “The Time Traveler” published in 1943. If a person travels back in time and kills one of their ancestors (grandfather, grandmother, father, mother), they themselves cannot be born, and therefore cannot commit the murder. The simplest solution to the paradox is that time travel is impossible.
However, there are other versions. For example, when traveling into the past, an alternative universe is created, unrelated to the one from which the traveler came. Another theory states that it is possible to travel to the past, but it cannot be changed. In any case, all discussions about time travel remain purely theoretical for now. However, modern physics does not rule out the possibility of time travel.
The Omnipotence Paradox
Most monotheistic religions claim that God is omnipotent. The question arises—can God create a stone so heavy that he himself cannot lift it? Can God construct a triangle on a plane whose sum of angles is not equal to 180°? Dozens of similar statements can be devised, the essence of which leads to an ambiguous interpretation of the term “omnipotence.”
The simplest conclusion from the paradox is that an omnipotent being cannot exist. However, for obvious reasons, such an answer does not satisfy theologians and believers, as it undermines the foundations of all world religions. Therefore, theologians sometimes refine the concept of “omnipotence” as the ability to do what is possible in principle.
In other words, God cannot create a stone that he cannot lift because such a stone cannot exist in the first place. Moreover, some philosophers point out that God cannot hate, lie, sin, and perform many other actions that are perfectly accessible to ordinary people. However, no religion has formulated a definitive answer to the paradox.
The Chicken or the Egg?
The question of which came first, the egg or the chicken, is very popular. Biologists point out that the chicken as a species evolved from another species of birds that laid eggs, so eggs have existed longer. Even before birds, dinosaurs and reptiles laid eggs. But there was a time when there wasn’t a single species on earth that laid eggs!
That’s true, however, during the time when life had not yet emerged from water, some marine creatures began to reproduce by laying eggs. Gradually, the eggs became larger, and their shells hardened. The transition process was very slow, so it is impossible to accurately determine when the first egg appeared. Similarly, it is impossible to pinpoint the first human on Earth, as the transition from apes to Homo sapiens was also very slow.