Because of the way negative numbers work, this solution is equally feasible. Ergo, both 0 and 1 are acceptable answers.
How can one series possibly have two different answers? Grandi used the fact that both 0 and 1 are possible from his series as proof that God exists, as something (1) can be made from nothing (0).
Grandi’s series becomes even stranger when a more advanced technique is applied.
Let us say that Grandi’s series is denoted by S (S = 1 – 1 + 1 – 1…). We can then break down the series as 1 – (1 + 1 -1 + 1…), because the plus and minus signs can be inverted together. Ergo, S = 1 – S → 2S = 1 → S = ½
Now we have three answers to Grandi’s question: 0, 1 and ½. For over 150 years, mathematicians fiercely debated the answer to Grandi’s question. By the 19th century, mathematics had evolved and mathematicians had figured out better ways to solve infinite series.
The classic example is the solution to the series: 1 + ½ + ¼ + ⅛… To solve this, you can add the partial sums, where you add each number to the sum of the previous numbers to see what number you are approaching (the limit).
1 → 1.5 → 1.75 → 1.875 → 1.9375… until we infinitely approach 2 (or 1.9999999…)
If we apply this method to Grandi’s series, we do not approach a single number because we keep swinging between 0 and 1. (1 → 0 → 1 → 0 → 1…)
So we can apply another method, where we average the partial sums as we go instead of adding.
e.g. 1 → ½(1 + 1.5) = 1.25 → ⅓(1 + 1.5 + 1.75) = 1.416 → ¼(1 + 1.5 + 1.75 + 1.875) = 1.531… until we approach 2.
Eventually, the series appears to converge on ½, showing that the answer to Grandi’s series seems to be ½.
The problem with this method is that Grandi’s series does not actually have a limit, but we are applying a solution as if it has a limit. This is similar to using a divide by 0 trick to prove that 1 + 1 = 3. In mathematics, when rules are bent, we end up with weird, paradoxical results.
To show this empirically, consider the thought experiment of Thomson’s Lamp:
Imagine a lamp that is turned on after 1 minute, turned off after ½ minute, turned on again after ¼ minute ad infinitum. This incorporates both infinite series discussed above. Ergo, we know that the sum of time is 2 minutes. So, at the end of 2 minutes, is the lamp on or off? If Grandi’s series solves to 0, the light is off; if it is 1, the light is on. Then what does it mean if Grandi’s series solves to ½? Is the light on or off?
Imagine that there are twelve coins in front of you. They are exactly the same size and shape, but one is either lighter or heavier than the other 11. To determine which coin is the odd one out, you are allowed to use a scale exactly three times. How do you find the unique coin while figuring out whether it is heavier or lighter than the other coins?
If I have three gloves, there must be at least either two left gloves or two right gloves. It is impossible to have one left glove, one right glove and a third glove that is neither left nor right (usually). This logic is called the pigeonhole principle. It is named because of the logic that if you have n pigeons and m pigeonholes where n > m (e.g. 10 pigeons in 9 holes), then at least one pigeonhole must contain more than one pigeon. This is because the biggest spread of the pigeons is putting at least one in each box, but as n > m, there is a pigeon left over and it must go in a box with another pigeon. The pigeonhole principle seems like a basic counting principle, but its implications are quite interesting.
For example, let’s say that your sock drawer is very unorganised and has a mix of black and white socks. What is the minimum number of socks you need to pick out before you get two of the same colour? The pigeonhole principle dictates that when n > m, each “slot” must be filled with more than one item. Here, the slot is colour. As there are two colours (m = 2), you only have to pick three socks out to have a matching pair (n = 3, 3 > 2).
The pigeonhole principle allows us to make seemingly impossible conjectures, such as the fact that a person living in London will have the exact number of hairs on their head as at least one other person living in London. An average human head has about 150,000 hairs and it would be a safe assumption to say that no one would have more than a million hairs on their head (m = 1,000,000). The population of London far exceeds a million (n > 1,000,000), therefore, there must at least two people living in London with the exact same amount of hair on their head. Similarly, if you are in a room with 366 other people, you are guaranteed to share a birthday with at least one person.
In the 17th century, French philosopher Blaise Pascal made the following argument for believing in a god:
There is a god or there is not.
You can choose to believe in a god or not (the wager).
If there is a god, you will be rewarded eternally in the afterlife for your faith, but be punished eternally if you do not believe.
If there is no god, you lose a finite amount of your time and maybe some material wealth for believing in a god.
Ergo: As the rewards and punishments that follow in the case of god existing is infinite, it is better to bet that there is a god, no matter how infinitesimal the odds may be.
Pascal’s wager does not deal with the possibility of whether gods exist or not; that is irrelevant to the wager. He merely suggests that the odds suggest that you should believe. But is this really the case?
To begin with, what Pascal promotes through this wager is not true belief or faith, but a rational choice to believe – something that is not really possible. Believing is not a product of reasoning but more of an alternative. Furthermore, if there really is an omniscient god, would he not easily see the impure motives behind your “faith”?
Secondly, how do we know that the god you believe in is the true god? There have been thousands and thousands of religions throughout history. Who is to say that the deity that you will face in the afterlife will not be Hades, Odin or Yama? If that is the case, then you will have lined up behind the wrong god and you will be punished for your “idol worship”. This argument nullifies the mathematical advantage of infinite rewards that Pascal suggests.
Lastly, one cannot rule out the possibility should a god exist, there is no way of knowing whether that god is benevolent or malevolent. Pascal’s wager only deals with the two possibilities of a benevolent god and the absence of god, but if a malevolent, wrathful god exists, then what is the gain from worshipping him? When you kill an insect, do you judge whether that insect has faith in you then reward or punish it accordingly? It is likely that in this scenario, worshipping such a god will be a waste of time and you will be relatively better off not believing in god.
In 1990, an American philosopher named Michael Martin presented a counter-wager to Pascal’s wager – the so-called atheist’s wager. He argued that if a benevolent god existed, then he should reward good deeds regardless of your faith. If a god does not exist, then your good deeds will leave a good legacy and the world will (hopefully) be a slightly better place to live in after you pass away.
Ergo, the wager we should be making is not whether a god exists or not, but that we should be good.
(If you are interested in this, you should read The God Delusion by Richard Dawkins, he explains this very elegantly)
You are walking along the road when you reach a fork, where you see two guards chatting with each other. You approach them to ask which way you should go, but you notice a peculiar sign that says: “One of us always tells the truth, one of us always lies”. There is no way for you to know who tells the truth and who tells lies. If you could only ask one question to help you walk down the right path, what would you ask the guards?
The solution is simple: all you have to do is ask “If I asked the other guard which road is the right one, which road would he tell me to take?” then go to the opposite road. The reason being, no matter who you ask – truth-teller or liar – they will tell you the wrong answer. If you asked the truth-teller, he would honestly reply with what the liar would say, which is the wrong answer. If you asked the liar, he would tell you the opposite of what the truth-teller would say, which would ultimately be the wrong answer.
Author Bernard Werber (the inspiration for this Encyclopaedia) posited the following theory: if we could see the future, would we not actively build towards a better future? Imagine a tree soaring high into the sky, stretching countless branches in all directions. The many branches of the tree branch off into smaller branches, which branch into even more smaller branches. At the end of each branch, there hangs a leaf. This tree is not a normal tree; it is a Tree of Possibilities that represents the flow of time from the beginning of the universe to the distant future. Each split in a branch represents the creation of two different futures due to a choice or a change, while a leaf represents the final future created from the cumulative effects of these changes. Thus, the Tree of Possibilities is the ultimate crystal ball showing all the pasts that could have been and all the futures that can happen.
Of course the Tree of Possibilities is a fictional model created in our imaginations. But what if we could actually make this tree? First, we would create an organisation of the greatest scientists, mathematicians, sociologists, psychologists, historians, philosophers, science fiction writers etcetera that represent the many fields of knowledge. These people are gathered in a location far from the reaches of governments and the media, where they can discuss without any interference. These specialists will debate over all sorts of topics, amalgamating their knowledge and intuition to generate a tree diagram as mentioned above. This is a diagram free from ethics, morals, laws, optimism, pessimism and individualism – the ultimate objective view of all possible futures that humanity and the Earth may face. The experts may agree with each other at times and disagree at times. There is ample possibility that their postulations are wrong. But none of these matter. The important point is not that the Tree is “accurate” or not, but that it is an extensive scenario database of all the paths humanity can walk on towards the future.
The Tree of Possibilities will have various conjectures such as: What if nuclear war broke out? What if artificial intelligence is perfected? What if chimpanzees reach the intelligence levels of human beings? What if we build cities on the Moon? However, the future is altered much more easily that you would think. Thus, there will also be branches representing much more trivial and ordinary (even bizarre) postulations as well: What if smoking is banned? What if the average age women gave birth is older? What if rhinoceroses were domesticated pets? What if pianos do not exist?
On analysing these numerous postulations, a branch bearing the leaf with the ideal future will be found. Ergo, we can choose to follow a path of least resistance, where all the choices we make will ultimately lead to that ideal future. Essentially, the Tree of Possibilities is a tool that is used to predict the future. However, it is not “fortune telling” as it is based on logic rather than magic and divinity to see into the future. The future the Tree tells is not a set “destiny”, but rather one “possibility”. Thus, instead of fearing the future like we do with fortunes, we would instead feel excitement over the potential of finding the ideal future. If the path we are currently on is fated to an unhappy ending, then we can simply jump onto a different path with the guidance of the Tree. Unlike fortune telling, which destroys all uncertainty and any other possibilities in the future, the Tree of Possibilities provides humanity with the greatest gift: dreams of a better future.
As you could imagine, the possibilities of the future are infinite so a drawn-out diagram of the Tree of Possibilities would take up extensive amounts of space. Ergo, the ideal form of the Tree of Possibilities would be a computer program. As computer programs only need sufficient storage space, it provides a perfect environment in which the Tree may grow. The program would generate a Tree based on the information provided by the scholars, drawing out each branch and leaf, while also calculating the effects of any action on each of the possible futures. If we further applied the engine used in chess programs to predict the next few moves, then we may be able to create a program that can calculate the ideal future and the path of least resistance for humanity.
My ideal future is this. There is an isolated island, far from any interference, with a large building. At the centre of this building, there lies a supercomputer running The Tree of Possibilities. The computer is surrounded by lecture theatres, conference rooms and residential areas. Thus, specialists of each field may come to stay and use their knowledge to water the Tree and foster it. This island will provide humanity with hopes and dreams, leading them towards the best possible future based on logic and imagination.
The Tree of Possibilities will radically change our day-to-day lives. One of the greatest weaknesses of human beings is the inability to see the long-term happiness and sacrificing it for short-term gain. However, if we were able to see precisely how our actions will affect the future, then would we not act differently? Armed with insight and foresight, people will understand what is best for the future, and instead of the current near-sighted attitude of only seeing the gain right before our eyes, they will act in the best interests of their children and grandchildren. Politicians will see how useless bickering over trifling issues is and instead focus on policies that take a while to show the effects (yet nonetheless important), such as environmental conservation. The Tree of Possibilities will help us make rational decisions to create a world that the future generation will be happy living in, without being swayed by emotions and selfish greed. And so, we will build towards a utopia.
The greatest weapon a person has is imagination that can build the future.
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.
The Kasiski examination can be used to attack polyalphabetic substitution ciphers such as the Vigenère cipher, revealing the keyword that was used to encrypt the message. Before this method was devised by Friedrick Kasiski in 1863, the Vigenère cipher was considered “indecipherable” as there was no simple way to figure out the encryption unless the keyword was known. But with the Kasiski examination, even the Vigenère cipher is not safe anymore.
The Kasiski examination is based on the fact that assuming the number of letters of the keyword is n, every nth column is encoded in the same shift as each other. Simply put, every nth column can be treated as a single monoalphabetic substitution cipher that can be broken with frequency analysis. Ergo, all the cryptanalyst needs to do to convert the Vigenère cipher into a Caesar cipher is know the length of the keyword.
To find the length of the keyword, look for a string of repeated text in the ciphertext (make sure it is longer than three letters). The distance between two equal repeated strings is likely to be a multiple of the length of the keyword. The distance is defined as the number of characters starting from the last letter of the first set of strings to the last letter of the second set of strings (e.g. “abcdefxyzxyzxyzabcdef” -> “abcdef” is repeated” -> distance is “xyzxyzxyzabcdef” which is 15 letters). The reason this works is that if there is a repeated string in the plaintext and the distance between these strings is a multiple of the keyword length, the keyword letters will line up and there will be repeated strings in the ciphertext also. If the distance is not a multiple of the keyword length, even if there is a repeated string of letters in the plaintext, the ciphertext will be completely different as the keyword would not match up and be different.
It is useful recording the distance between each set of repeated strings to find the greatest common factor. The number that factors the most into all of these distances (e.g. 6 is a factor of 6, 12, 18…) is most likely the length of the keyword. Once the length of the keyword is found, then every nth letter must have been encrypted using the same letter of the keyword. Thus, by recording every nth letter in one string, you can obtain what is essentially a Caesar cipher. The Caesar cipher is then attacked using frequency analysis. Once a few of these strings (of different positions on the ciphertext) are solved, the keyword can be revealed by checking the shift key against a tabula recta (e.g. if a certain string of nth letters is found to have been shifted 3 letters each, then the corresponding letter in the keyword must be “D”, which shifts every plaintext letter by 3 in the Vigenère cipher). When the keyword is deduced, every message encrypted using that keyword can now easily be decoded by you.
Although the Kasiski examination appears to be complex, attempting to try it reveals how simple the process is. Thus, it is useful to try encrypting a message using the Vigenère cipher then trying to work out the keyword using the Kasiski examination. Much like the frequency analysis, it is an extremely useful tool in the case of needing to break a secret code.
A cipher is a message that has been encoded using a certain key. The most common and basic type of ciphers are encrypted using letter substitution, where each letter represents a different, respective letter. For example, the message may be encoded in a way so that each letter represents a letter three values before it on the alphabet (e.g. if a=0, b=1… “a” becomes “d”, “b” becomes “e” etc.). This creates a jumble of letters that appears to be indecipherable.
However, the characteristics of substitution ciphers make them the most decipherable type of encryptions. As each letter can only represent one other letter, as long as the key is cracked (i.e. what letter is what), the message and any future messages can be cracked. The most important tool in decrypting substitution ciphers is pattern recognition and frequency analysis.
Frequency analysis relies on the fact that every language has certain letters that are more used than others. In the English language, the letters that are most used, in order, are: E, T, A, O, I, N, S, H, R, D, L, U (realistically, only E, T, A, O are significant and the rest are neither reliable nor useful in frequency analysis).
For example, if Eve intercepted a long, encrypted message that she suspects to be a simple substitution cipher, she will first analyse the text for the most common letter, bigram (two letter sequence) and trigram. If she found that I is the most common single letter, XL the most common bigram and XLI the most common trigram, she can ascertain with considerable accuracy that I=e, X=t and L=h (“th” and “the” are the most common bigram and trigram respectively). Once she substitutes these letters into the cipher, she will soon discover that certain patterns arise. Eve may notice words such as “thCt” and deduce that C=a, or find familiar words and fill in the blanks in the key. The discovery of each letter leads to more patterns and the vicious cycle easily breaks the code.
Frequency analysis is extremely useful as it can be used to attack any simple substitution ciphers, even if they do not use letters. For example, in Sir Arthur Conan Doyle’s Sherlock Holmes tale The Adventure of the Dancing Men, Sherlock Holmes uses frequency analysis to interpret a cryptogram showing a string of hieroglyphs depicting dancing men.
To reinforce this weakness in substitution ciphers, many cryptographers have devised better encryption methods such as polyalphabetic substitution, where several alphabets are used (e.g. a grid of two alphabets – also called a tabula recta).