If an irresistible force was to act on an immovable object, what would happen?
A mathematician named Mike Alder decided to approach this philosophical paradox from a scientific perspective. He proposed a simple answer to the paradox – it is not worth discussing.
Alder argued that for an object to be immovable, all known forces must be acted upon it with no effect. Similarly, an irresistible force can only be called that if literally no object could ignore its effects. Therefore, the two cannot possibly exist in the same universe, meaning that the paradox is pointless. As Alder would put it – “Language is bigger than the universe”, as it allows us to formulate impossible scenarios that ignore the rules of science.
The implication of this line of thought is that if you cannot tangibly test an idea, then there is no point in arguing it as it would not add to scientific knowledge. This is a purist view of the fundamental principle of science that is falsifiability.
Sir Isaac Newton was one of the earliest pioneers of this philosophy. He wrote: “hypotheses non fingo”, or “I do not engage in untestable speculation”. Newton challenged the classical school of philosophy, where one would challenge and develop an idea through thought, discussion and argument. When faced with philosophical questions such as whether animals had rights, he would ask: “What set of observations do you consider would establish the truth of your claim?”.
Alder named his principle – that one should only discuss matters that can be tested and verified – Newton’s Flaming Laser Sword (as he believed all good principles should have sexy names). This is a play on Occam’s razor, the philosophical principle that once you shave away the complexities, the simplest truth remains. Alder believed that Newton’s Flaming Laser Sword was a much sharper and more dangerous tool than Occam’s Razor, meaning that as useful as it is, it should be used with care.
Of course, this is an extreme school of philosophy that is only upheld by a group of philosophers who we now call “scientists”. There are still many intangible issues that could only be solved through thinking, such as ethics. Thus, the battle between scientists and philosophers continue.
One of the more humorous sides to numbers is mathematicians’ attempts to categorise numbers as “interesting” or “dull”. For example, 1 is interesting because it is the first positive integer. 73 is interesting because it is the 21st prime number and 21 is a multiple of 7 and 3. The number 1729 is a good example of how a number can seem dull but later found to be interesting. When the British mathematician G. H. Hardy visited Indian mathematician Srinivasa Ramanujan, he commented that the number of the taxicab he rode in on was 1729 – a number he found to be rather dull. Ramanujan objected and stated it is very interesting as it is the smallest number expressible as the sum of two cubes in two different ways (1729 = 1³ + 12³ = 9³ + 10³). Such numbers are now referred to as taxicab numbers and 1729 is called the Hardy-Ramanujan number.
A way to discover the smallest most uninteresting number is through the Online Encyclopaedia of Integer Sequences, which documents every integer worth noting as it is in some sort of arithmetic sequence. The smallest integer that does not appear in this encyclopaedia as part of a sequence could be considered as objectively the smallest “uninteresting” number. In 2009, this number was 11630, but has since changed to 12407, then 13794 and now 14228 (22 April 2014).
But paradoxically, the smallest uninteresting number is interesting in itself by being the smallest most uninteresting number. This is known as the interesting number paradox. By this paradox, every natural number is unique and ergo, “interesting”.
(Image source: http://www.xkcd.com/899/, and here’s an explanation of some of the numbers on it)
Mirrors are perhaps one of the most useful yet underrated inventions that we use every day. From shaving in the morning to fixing make-up during lunch, the modern man or woman will use a mirror (or some other reflective surface) at least once a day. Mirrors show us an accurate reflection of the world that we cannot see. We can only look forwards and need a mirror to reflect light going the opposite way to see behind us or – more importantly – ourselves. To do this, a mirror must directly reflect every photon (particles that make up light) at exactly the right angle so the image is not distorted. If the mirror is not completely flat or perfectly polished, light will not be reflected at the exact angle and we will see a distorted image – much like looking into a mirror at the circus. Therefore, one could say that a perfectly flat, clean mirror is absolutely honest, as it will reflect exactly as it sees.
However, this statement is not entirely true as what you see in the mirror is a mirror image of reality. This may seem trivial, but it has significant consequences. This is most obvious when you hold a book up to a mirror. Without training, it is very difficult to read something that is mirrored. This is why Leonardo da Vinci wrote his notes in mirror image. This phenomenon of something becoming completely different is also seen in chemistry. Because of the way molecules are arranged, it is possible to have a property called chirality – where two molecules with the same elemental composition are built in the mirror image of one another. Essentially, it is as if the molecule can be either left- or right-handed. It turns out that even if the composition is the same, two molecules of different chirality (called enantiomers) can act completely differently. This effect may be as simple as changing the way a liquid polarises light to making a drug completely inert or even toxic. For example, the amino acid carvone that gives the spearmint taste only tastes like spearmint if it is L-carvone (“left-handed”), whereas D-carvone (“right-handed”) is tasteless despite having the same molecular formula.
Since the topic of chirality is rather technical and hard to understand, let us move on to the field of literature. One of the best examples of how mirrors can completely change something is seen in Lewis Carroll’s novel Through the Looking-Glass and What Alice Found There. Lewis Carroll understood the significance of mirror images in chemistry and wrote this novel to portray how quirky and strange a “mirror world” may be. Through the Looking Glass is a sequel to the famous book Alice in Wonderland and describes a world that is the mirror image of Wonderland. Carroll cleverly wrote the first book so that it would be the opposite of the first book. The first book starts outdoors, is set in the summer, uses changes in size as a plot device and focuses on the theme of trump cards. The second book starts indoors, is set in the winter, uses changes in direction as a plot device and draws on the theme of chess. There are even characters such as Hatta and Haigha who are the mirror images of the Mad Hatter and the March Hare from the previous book. Although they are very similar, they are just not the same and hence Alice does not recognise them. Perhaps the line that best shows Carroll’s understanding of the dangers of mirror worlds is this: “Perhaps Looking-glass milk isn’t good to drink”.
The field of psychology is also heavily interested in mirrors. It is a well-known fact that our brains recognise the purpose of mirrors. If you put a mirror in front of someone, you know that the person will examine themselves, groom themselves or simply make funny poses. A simple experiment shows how used to mirrors we are. If you angle two mirrors at right angles and fit a transparent sheet of glass in front of the two to make a prism shape, the image you see through the glass is a reflection that is not mirrored. Because it is not mirrored, you can hold up a book to it and still read it fine. This is known as a non-reflecting mirror. An interesting experiment shows that if you make people use this kind of mirror, they become incredibly confused as they are too used to using a mirror image to see themselves. Even though the reflection they see is a “truer” image, because their brain automatically flips the mirrored image, they become uncoordinated and keep moving their hands in the opposite direction.
As mentioned at the start, mirrors are a human invention. Although reflection occurs in nature, such as on a clear surface of water, animals generally are incapable of using mirrors. This is such a universal fact that animal psychologists use a mirror test to determine whether a specie of animal is self-aware or not. The test is done by showing an animal a mirror. Most animals will see their reflection and automatically believe that it is another animal, as they are incapable of thinking that it is a reflection of themselves. Hence, they will try to threaten, attack or flee from the image they see. But if you show a higher-order animal such as an ape or dolphin a mirror, they will start to groom themselves as they realize that the mirror is simply showing themselves.
This is what sets us apart from animals. Not only are we capable of recognizing ourselves in a mirror, but we have the ability to go one step further and reflect on ourselves using the mirror of our minds. Some people may take a look at the person in this mirror and be content with who they are. But some will gaze into the mirror and, much like the animals in the mirror test experiments, see a completely different person they do not recognise. This may cause disappointment, frustration or even disgust as we realize that we are not who we think we are or aspire to be. Then again, sometimes you will gaze into the mirror and see a person that has strengths such as courage – a person you could be if you realized your true potential. The most frightening realisation would be to discover that there is no one in the mirror.
Lastly, we could consider the mirror of behaviour. Goethe said that “behaviour is a mirror in which everyone displays his own image”. The corollary to this is that human beings read behaviour to try and interpret another person’s character. One can use this to greatly improve the relationship and connection with another person. Mirroring is the act of subtly copying the other person’s behaviour to build rapport– where an empathic bridge is constructed between two people. Rapport is particularly useful in jobs that involve earning the trust of strangers in a short time, such as in healthcare or business. By matching the other person’s body language, such as posture or actions like taking a sip of water, the other person will open up more easily to you. The same applies to verbal and emotional mirroring where you subtly reuse the words the other person spoke and reflect their emotions such as excitement. Obviously, one must be subtle with mirroring as a direct imitation will appear mocking and strange. If you are able to subtly copy their behaviour, the other person’s subconscious mind will be tricked into thinking that you are similar in character and trust you more. This skill is extremely useful in improving your interpersonal and social skills.
A mirror is a paradoxical object that is absolutely honest yet relatively deceitful. Reflections in the mirror are true yet completely different. If you take a peek into the mirror of your mind, perhaps you will see the person you think you are now or the person you could be in a mirror world. If you are happy with what you see, then cherish that and be proud of who you are. Otherwise, you can always do what Alice did and jump through the looking-glass to find an alternate you – the best you that you can be.
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 = ½ + ¼ + 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?
Most people believe that as an incorrect option (goat) is ruled out, their odds of winning the car go up from 1/3 to ½ even by staying on the same Door 1 and there is no benefit to switching. However, it is better to switch doors as this will double your odds of winning the car. To illustrate this point, the following three scenarios (with the car being behind Door 1, 2 or 3) can be imagined, using the above rules of the game:
In Scenario 1, you have already chosen the car (Door 1) so Monty Hall will randomly open Door 2 or 3. Switching will obviously lead you to losing the car. The chance of you losing after switching, therefore, is 1/6 + 1/6 = 1/3 (as either Door 2 or 3 could be opened)
In Scenario 2 and 3, because you chose the wrong door (goat) and Monty Hall will open the door with the goat behind it, switching will lead you to choosing the car (no other choices). As the odd of either scenario happening is 1/3 each, your odds of winning after a switch is 2/3 – double the odds of winning after not switching (1/3, the odd of your first guess being right).
Of course, this is only under the assumption that the rules of the game were followed and that Monty Hall will always open a door with a goat behind it. This problem and the answer suggested was extremely controversial as tens of thousands of readers refused to believe that switching could be a better choice. However, as the above illustration shows, the Monty Hall problem is a veridical paradox – a problem with a solution that appears ludicrous but is actually proven true by induction.
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.