Like true love, a paradox cannot be explained by logic alone. Simply put, a paradox is a statement that contradicts itself. Any idea, situation, puzzle, statement, or question that challenges your ability to reason and leads you to an unexpected and seemingly illogical conclusion can be considered a paradox.
The classic example of a paradox is the so-called grandfather’s paradox. Imagine a psychotic time traveler who goes back in time and kills his grandfather before his father is conceived. This means that the traveler would not have been conceived, and if he had not been conceived, then who returned to kill his grandfather?
The answer to this theoretical time travel mystery is still unclear, as it is with many other interesting paradoxes. In the information age, logic helps us understand what we know, but a paradox reminds us of what else we need to know. Let’s dive into it.
How to define a paradox?
A paradox is a thought that can seem reasonable and illogical at the same time. The Cambridge Dictionary defines paradox as a situation that might be true but is impossible to understand due to its contrary characteristics. In Greek, ‘para’ translates to ‘abnormal’, ‘distinct or’ contrary ‘and’ dox ‘means’ idea’ or ‘opinion’. Therefore, according to some Greek philosophers, a paradox is a abnormal belief or contrary to oneself or an idea which ultimately leads to an insoluble contradiction.
You don’t have to time travel to create a crazy paradox. For example, in the famous crocodile paradox (of which there are many variations), a magical crocodile steals a child and promises to return it only if the father can correctly guess what the crocodile is going to do. If the father says “The child will not be returned,” what can the crocodile do? If he does not return the child, it means that the father’s assumption was true, so he should have returned the child. If he returns it, then the father’s guess was wrong, so he shouldn’t have. It’s a paradox, nothing the crocodile does can satisfy the situation.
It is believed that this paradox originated centuries ago in ancient Greece, but there are hundreds of different paradoxes that are also found in literature, mathematics, philosophy, science, and various other fields. Although a true paradox may seem both true and false, logic tends to suggest most paradoxes as invalid statements.
There are four main types of paradoxes:
- Falsidic paradox: A paradox that leads to a false conclusion resulting from a misconception or a false belief. For example, Achilles and Zeno’s turtle.
- True paradox: When a situation or statement tells us about an outcome that seems absurd but is in fact valid by logic. Shrodinger’s cat is a famous example of a true paradox.
- Paradox of antinomy: A question, riddle, or statement that does not lead to a solution or conclusion is called an antimony paradox (also known as a self-referential paradox). One of its examples is Barber’s paradox (discussed below).
- Dialetheia: When the opposite of a situation and the original situation coexist together, such a paradox is called a dialetheia. No concrete examples are known but some real-life situations can be considered dialetehia (for example when you are standing at the kitchen door, and a family member asks you if you are in the kitchen? answer yes or no.
Why paradoxes matter
Paradoxes are important because they make us think. They force us to re-evaluate what we thought we knew and to think about things from unusual angles. A paradoxical state of mind, in which we adopt contradictory (or seemingly contradictory) ideas is the key to success, some studies showed. It was found that the main thinkers spent a considerable amount of time developing ideas and counter-ideas at the same time, what is called the Janusian process.
The study of paradoxes is also important, especially for mathematicians. Mathematicians like to break everything into small pieces and define things carefully, and they do so with paradoxes. For example, consider a simple paradox called the temperature paradox, which states:
“If the temperature is 90 and the temperature is increasing, that would seem to mean 90 is increasing.”
Obviously 90 is not increasing, it is a fixed number, it cannot increase. We know it intuitively, but how do we prove it? American mathematician and philosopher Richard Montague addressed this paradox (and many others) and explained that the paradox emerges from linguistic vagueness, which can be resolved by mathematical clarity. The linguistic formalization of the paradox would look like this:
- The temperature is rising.
- The temperature is ninety.
- Therefore, ninety is up. (invalid conclusion)
But the mathematical formalization implies that point 1. marks the evolution of temperature over time, while point 2. makes an assertion about the temperature at a given moment. Therefore, we cannot draw conclusions based on this point in time alone.
This type of paradox, which emerges from problems of language and ambiguity is not often important, but other paradoxes, especially those which cannot be resolved by normal means, matter because they help us find better definitions of objects and relationships. A good example of this is Curry’s paradox.
Now that we know the types of paradoxes and why they are important, let’s take a look at the most popular and craziest paradoxes of all time:
Examples of paradox
“This sentence is wrong”
This so-called liar paradox is the canonical example of a self-referential paradox. Other classic examples are “Is the answer to this question ‘no’?” “And” I lie “.
Mathematicians have tried to dissect and analyze this paradox in detail because it may have some significance for define the limits inherent in mathematical axioms. The Liar Paradox was used in 1931 by a mathematician named Kurt Gödel to define mathematical axioms, but the paradox itself dates back to at least 600 BC. “
The barber’s paradox
Proposed by the British mathematician Bertrand Russell, this paradox states that if a barber is defined as the person who shaves only individuals who do not shave alone, then who shaves the barber? In this case, the barber would shave – but then, according to the definition, he is no longer the barber as he cannot shave a person shaving on their own.
Now, if he doesn’t shave on his own, then he’s one of those supposed to be shaved by the barber. In this case too, the barber should shave. Therefore, the barber paradox suggests that no such barber can ever exist who calls himself a barber because he only shaves people who do not shave themselves. Well, what is a barber?
If there is a pile of sand which has a million grains, and one by one, the grains are removed from the pile so that at the end of the process only one grain remains, would it still be considered like a bunch? If not, when does the pile of sand become a non-pile? Sounds crazy, right? But this is the Sorites paradox given by Eubuilde de Milet around the fourth century BC, and until today no genius of mathematics has been able to give a logical solution to this problem.
Another similar type of puzzle is the so-called Theseus ship. The mythological hero Theseus continues his adventures, and at some point one of the parts of the ship must be replaced. It’s still the same ship, right? Only one part was replaced. But part after part, every component of the ship is replaced. Is it still the same ship? If not, when did it stop being the same ship?
Zeno’s Achilles and the Turtle
In this paradox developed by the ancient Greek philosopher Zeno, there is a race between the great Greek warrior Achilles and a turtle. The turtle has a lead of 100 meters. Achilles runs faster than the turtle to catch up with it. But here’s how Zeno looked at it:
- Stage 1: Achilles runs to the starting point of the turtle as the turtle moves forward.
- Stage 2: Achilles runs to where the turtle was at the end of stage 1, while the turtle goes a little further.
- Stage 3: Achilles runs to where the turtle was at the end of stage 2 while the turtle goes even further.
- … Etc.
The gaps get smaller and smaller each time, but there is an infinity of these stages, so how can Achilles overcome an infinite number of gaps and catch up with the turtle? How does something catch up with anything, for that matter? Obviously things are catching up with other things, so what’s going on here?
The ancient Greeks did not have the mathematical tools to resolve this paradox, but nowadays we know better. There can be an infinite number of steps, but they are also infinitely small. It’s kind of like how 1/2 + 1/4 + 1/8 + 1/16 +… infinity adds up to 1. It’s an infinite number of steps, but the steps become infinitely small, and in the end, they add up to something tangible.
Animalia Paradoxa – The Classification of Magical Creatures
This is in fact not a paradox but a biological classification of beasts and magical creatures which are also mentioned in ancient story books. In versions of Systema Naturae which arrived before its sixth edition, author Carl Linnaeus (father of modern taxonomy) listed creatures like Hydra (seven-faced serpent), Draco (a dragon with bat wings and the ability to spit fire), Unicorn (beautiful horse), Lamia (half human, half animal), etc.
From a scientific point of view, these creatures do not exist, so why did a genius like Carl Linnaeus mention such creatures in his greatest scientific work? It seems paradoxical that the man who defined our classification of biological creatures introduces unreal creatures; we can say that it is a little paradoxical.
Paradoxes have a unique appeal because they appeal to human curiosity and mystery. They seem to arouse the curiosity of the human mind for thousands of years and will likely continue to do so for many years to come.