This is a painting named The Treachery of Images, painted by the famous surrealist artist René Magritte in 1929. The picture shows a pipe and below it, the words Ceci n’est pas une pipe, which is French for “This is not a pipe”. However, the painting clearly shows a plain pipe. Is Magritte implying that he did not actually draw a pipe? Is the object actually some other clever invention?
What Magritte is saying is that this is not a pipe, but an image of a pipe. The painting is only a realistic representation of a pipe, but it is not real. No matter how hard you try, you will not be able to stuff the pipe and smoke it. Ergo, if Magritte had written “This is a pipe” under the image, he would have been lying.
Magritte was a master of painting realistic pictures and then changing something subtly (or sometimes obviously) to completely change the context, making the picture very surreal. He knew for a fact that his painting of the pipe and the “paradoxical” subtext would rub people the wrong way because people are predictable in some ways. Without the explanation that it is an image of a pipe, many people will experience cognitive dissonance as they see a pipe, yet something is telling them it is not a pipe. This makes people wonder about what Magritte means, until they either figure it out, ask someone about it, or become angry and insult the painting because they have no idea what it means.
In 450 BC, a Greek philosopher named Zeno thought of the following paradox. Let us imagine that Achilles and a tortoise were to have a footrace. Achilles, obvious being faster than the tortoise, allows the tortoise to have a head start of 100 metres. Once the race starts, Achilles will quickly catch up to the tortoise. However, within the time he took to cover the distance, the tortoise would have travelled some distance as well (say 10 metres). When Achilles runs the 10m to catch up again, the tortoise has once again toddled on another metre. Thus, whenever Achilles reaches somewhere the tortoise has been, he still has farther to go. Because there are an infinite number of points Achilles must reach where the tortoise has already been, theoretically the tortoise will be ahead of Achilles for eternity.
According to this thought experiment, motion is paradoxical and theoretically impossible. However, we know for a fact that motion happens. So how can we break Zeno’s paradox?
The main flaw of Zeno’s paradox is that he uses the concept of “eternity”. If we record the story mathematically, the time taken for Achilles to run the footrace is (if it took him 10 seconds to run 100m): 10 + 1 + 0.1 + 0.01 + 0.001… = 11.111… Ergo, the tortoise is only ahead of Achilles for less than 11.2 seconds (rounded). After 11.2 seconds pass, the time passed exceeds the sum of the infinite series and the paradox no longer applies.
Although it is a flawed paradox, the story of Achilles and the tortoise teaches the concept of geometric series - that something finite can be divided an infinite amount of times. For example, 1 = 1/2 + 1/4 + 1/8 + 1/16… ad infinitum. This principle is a crucial part of mathematics and has significant implications in the field of economics. For example, it can be used to calculate the value of money in the future, which is necessary for working out mortgage payments and investment returns. Perhaps it is because of this mathematical principle that it seemingly takes an infinite amount of time to pay off a mortgage.
Zeno’s paradox teaches us that one should not take the concept of infinity for granted.
On a hot afternoon visiting in Coleman, Texas, the family is comfortably playing dominoes on a porch, until the father-in-law suggests that they take a trip to Abilene (a city 53 miles north of Coleman) for dinner. The wife says, “Sounds like a great idea.” The husband, despite having reservations because the drive is long and hot, thinks that his preferences must be out-of-step with the group and says, “Sounds good to me. I just hope your mother wants to go.” The mother-in-law then says, “Of course I want to go. I haven’t been to Abilene in a long time.”
The drive is hot, dusty, and long. When they arrive at the cafeteria, the food is as bad as the drive. They arrive back home four hours later, exhausted. One of them dishonestly says, “It was a great trip, wasn’t it?” The mother-in-law says that, actually, she would rather have stayed home, but went along since the other three were so enthusiastic. The husband says, “I wasn’t delighted to be doing what we were doing. I only went to satisfy the rest of you.” The wife says, “I just went along to keep you happy. I would have had to be crazy to want to go out in the heat like that.” The father-in-law then says that he only suggested it because he thought the others might be bored.
The group sits back, perplexed that they together decided to take a trip which none of them wanted. They each would have preferred to sit comfortably, but did not admit to it when they still had time to enjoy the afternoon.
This anecdote was written by management expert Jerry B. Harvey to elucidate a paradox found in human nature, where a group of people collectively decide on a course of action that is against the best wishes of any individual in the group. Essentially, the group agrees to do something that would not benefit any one, or the group as a whole. This is the Abilene paradox, colloquially known to us through the idiom: “do not rock the boat”.
As seen in the anecdote, there is a breakdown of communication where each member assumes that the majority of the group will decide to follow the action, pushing them towards conformity. There is a mutual mistaken belief that everyone wants the action when no one does, leading to no one raising objections. This is a type of phenomenon called groupthink (coined by George Orwell in his dystopian novel, Nineteen Eighty-Four), where people do not present alternatives or objections, or even voicing their opinions simply because they believe that will ruin the harmony of the group. They are also under peer-pressure, believing that by being the one voice against the unanimous decision they will become ostracised.
The Abilene paradox explains why poor decisions are made by businesses, especially in committees. Because no one objects to a bad idea (falsely believing that that is what the group wants), even bad ideas are accepted unanimously. This is particularly dangerous when combined with cognitive dissonance, where the group will believe that they chose that decision because it was rational and logical. To prevent this paradox from destroying individual creativity in the group, one should always ask other members if they actually agree with the decision or are merely the victims of groupthink.
A long time ago in ancient China, there was a merchant who sold weapons. He would pick up a spear and advertise it as a spear that can pierce any shield. Then, he would pick up a shield and proclaim that it can block any spear. A wise man who was walking past the merchant questioned: “So what would happen if you took your ultimate spear and threw it at your ultimate shield?” The merchant could not answer.
That is why the word for contradiction, or something that does not make logical sense and cannot co-exist, in Korean, Chinese and Japanese is 모순(矛盾), meaning “spear and shield”.
Is time travel possible? In 1943, a science fiction writer called René Barjavel posited the following paradox.
A man travels back to the past and kills his biological grandfather before he meets his grandmother. Thus, his grandparents would not have sired a son (the man’s father) or daughter (mother), which then suggests the man could not have been conceived. If so, who killed the grandfather? As there was no one to kill the grandfather, he would have had a child and the man would ultimately be born, travelling back to the past and killing his grandfather. This paradox suggests that time travel is impossible.
Some people use the parallel universe theory to argue against the paradox. They suggest that as soon as the man travels to the past to kill his grandfather, an alternate universe is created where the grandmother meets a different man and the course of time is changed. This is a valid theory but the grandfather paradox still holds strong in disproving time travel. However, the grandfather paradox only states that travelling back in time is impossible; it says nothing about time travelling to the future.
Imagine that you are on a game show and you are given the choice of three doors, where you will win what is behind the chosen door. Behind one door is a car; behind the others are goats, which you do not want. The car and the goats were placed randomly behind the doors before the show.
The rules of the game show are as follows:
- After you have chosen a door, the door remains closed for the time being.
- The game show host, Monty Hall, who knows what is behind the doors, opens one of the two remaining doors and the door he opens must have a goat behind it.
- If both remaining doors have goats behind them, he chooses one at random.
- After Monty Hall opens a door with a goat, he will ask you to decide whether you want to stay with your first choice or to switch to the last remaining door.
Imagine that you chose Door 1 and the host opens Door 3, which has a goat. He then asks you: “Do you want to switch to Door 2?”
Is it to your advantage to change your choice?
>>> Read more <<<
Cats always fall on their feet. Buttered toast always seems to fall buttered side down. So what would happen if we tied a buttered toast on a cat’s back and then dropped the cat? Would the cat land on its feet or would the toast land on its buttered side?
Or would we achieve perpetual motion and anti-gravity simultaneously as they cancel each other and never touch the ground?
Although the paradox is obviously a humorous thought experiment, there is some truth to the separate adages.
Cats have a natural righting reflex that allows them to twist their upper body so that they land on their feet. This gracious manoeuvre is developed as a kitten and actually involves quite complex physics where the cat is able to turn around without changing their net angular momentum. Since cats have a small body and very light body weight, their terminal velocity (100km/h compared to a human’s 210km/h) when falling is much less and allows them to absorb the shock easily when landing. Furthermore, when falling cats naturally spread their limbs out to slow their fall as much as possible. All these factors let a cat land safely on its feet even if dropped from a high place. Ironically, the lower they are dropped from, the more likely that the cat would fall on its back.
The other side of the paradox is slightly more complicated. The adage that toast falls buttered side first is actually an example of how if something bad can happen, it will happen. However, physicists have discovered that toast is more likely to fall on its buttered side.
When toast falls off a plate, it is highly likely to tip as it hits the edge. This causes it to rotate as it begins to fall. There are two explanations on why the buttered side is more likely to be facing down. Firstly, butter adds weight to one side and heavier objects fall faster in the face of gravity. Secondly, using experimental data it has been found that toast only rotates about 180 degrees by the time it falls the height of the table or person from where it was dropped from.
Despite it only being a tongue-in-cheek thought, one can only wonder how many scientists have made some toast, buttered it, tied it to a cat and dropped the cat off a ladder.
A Cretan named Epimenides once said: “All Cretans are liars.”
So if Epimenides is a Cretan and he is a liar, then the statement is false. But that means that Cretans tell the truth, and Cretans are in fact liars. So what is the truth? This paradox continues ad infinitum due to the self-referencing nature of the statement.
This is a well-known example of a logical fallacy, or a flaw in a logic. It is also referred to as the Liar Paradox, seen in: “This sentence is false”.
The power of a paradox is best portrayed in the following parable.
A wise woman who worked as a fortune teller was tried for being a witch. In her trial, the king demanded she tell a fortune. If the fortune was correct, she would be drowned. If the fortune was wrong, she would be burnt at the stake. The woman smiled, and replied: “I will be burnt at the stake”.
A few years ago, a theoretical physicist studied population growth in cities to find the mechanism of how cities operate. What he found was an astonishing law.
Wherever the city, as the population doubled in size, the average income, number of patents, number of educational and research facilities and other important numbers all increased around 15 percent. Although it is normal for such statistics to increase as a city grows, it is interesting to see that almost all of them increasing at a similar rate, despite being so different sometimes.
More fascinating is the fact that not only do the above “good” statistics increase equally, but so do crime rates, pollution, smog occurrence, stomach flu and AIDS prevalence all increase approximately 15 percent.
Therefore, a city can be seen as a double-edge sword that is both the source of fast growth, wealth and ideas, but also waste, pollution, stress and disease.
Biologically speaking, an organism has a tendency to have slower growth and pace of life as it gets larger. For instance, an elephant’s heart beats slower than a mouse, and its cells do less work on average too. However, a city exhibits a snowball effect where it grows faster as it gets larger. To achieve this extremely high rate of growth, it must consume an immense amount of resources, which ultimately ends up as large quantities of waste and pollution. Also, as people get busier, the overall “quality” of the society falls, leading to increased stress and disease prevalence.
If so, should we abandon our current productivity and live a slow, village life and ignore our potential as a species? Or should we continue our exponential growth at the cost of using up nature’s well-maintained resources like no tomorrow?
A piece of paper has two sides. However, a Möbius strip has only one side. Ergo, if you walk on a Möbius strip, you walk on both sides and end up on the opposite side on the same location you started at in one trip. Because it has one side, it also has one boundary. This means that if you cut a Möbius strip along its length, you end up with not two rings, but one thinner, longer loop with an extra twist.
A similar structure is the Klein bottle. This structure is a self-paradoxical, single curvature, as its opening meets with its base, making the inside and outside indistinguishable. The entry is the exit, the inside is the outside, and the top is the bottom.
Our universe might be such a space where there is no distinction between the beginning and the end.