When the pharaoh asked whether geometry could not be made easier, Euclid replied: "There is no royal road to geometry". He was badly mistaken. Not only is there a royal road to geometry (and mathematics as a whole), but there is also a road for ordinary noblemen. Marquess de l'Hôpital had to pay annual fee to Johann Bernoulli in order to drive on it.
"Everything you can imagine is real," Picasso used to say. He had never heard of the imaginary numbers, poor fellow.
"Всичко въображаемо е реално", казвал Пикасо. Не бил чувал за имагинерните числа, горкият.
Armstrong and Aldrin left a plaque on the Moon, which reads as follows: "We came in peace for all mankind." Then they stuck the American flag next to it. A century earlier, as if especially for this occasion, Kozma Prutkov warned people that when they see the elephant's cage labeled "Buffalo", they must not believe their eyes.
A picture is worth a thousand words, Henrik Ibsen thought. He was badly mistaken. If Nike's swoosh could say it all, their slogan "Just do it." would be superfluous.
Sometimes even colorful people see the world in black and white.
Artist X said about artist Z that he was a white crow, which was why he was never accepted by the people of black&white thinking. Doesn't that sound stupid? Dividing people into people with colorful and people with black&white thinking is a manifestation of binary, i.e. black&white, thinking.
Ambiguity is not necessarily a bad thing, sometimes you profit from it. When a woman asks you if the dress makes her look fat, it's wise to be vague about the answer. This is what a writer with a lot of life experience wrote.
My life experience makes me think otherwise: in the above situation, the clear and honest answer is preferable. You answer NO and you are always right:
a) When a woman is skinny, a dress cannot make her fat.
b) When she is fat, the fault is not in the dress.
We are so enlightened we can see how the Universe will die of cold (Heat death). We are so blind we can't see how we make the Earth sweat (Global warming).
"I hated war, but I loved the warrior spirit. I hated the sword, but loved the samurai." is what Phil Knight wrote in his autobiography. I smell something fishy there. Without the samurai the samurai sword is a work of art. Without his swords the samurai is nothing.
Let A be the set of your friends and B be the union of C (the set of your friends' friends) and D (the set of your enemies' enemies). Politics is an effective nonmathematical instrument that can easily disprove your naive faith that B is a subset of A.
Giants dislike people standing on their shoulders, it is them who enjoy standing on the shoulders of others.
Inventors devote their lives to Simplicity. Artists - to Beauty. Lawyers - to Justice. Mathematicians - to Truth. There is, however, one essential difference: Simplicity, Beauty, and Justice are subjective, Truth is not. Many mathematicians would admit that Euclid's Theorem on the infinitude of primes is more beautiful than the Pythagorean Theorem, but not one would say it is truer.
Apple's market capitalization has exceeded 2 trillion dollars. Let me remind you that
Market capitalization = number of shares outstanding (i.e. all shares minus those held by the company itself) * share's current market price
Let me remind you one more thing you've probably forgotten. The number of shares outstanding is an objective quantity, and the current market price is an intersubjective quantity (i.e. a non-objective quantity, which is not entirely subjective because it is shared by different subjects).
Mathematics teaches us that Pi and e are irrational and transcendental numbers, but no one knows whether Pi*e is irrational or transcendental. Similarly, no one knows whether the product of objective and intersubjective quantities is objective quantity. In short, people may believe that market capitalization is something real, but they do so at their own peril.
Science is mighty when it deals with small things and weak when it deals with big things. It can accelerate elementary particles, but cannot slow down global warming.
"If the climate were a bank, the US would have already saved it," former Venezuelan president Hugo Chavez once said. I have something to say too: "The climate may not be a bank, but Mother Nature is. How can we live if we do not borrow from the deposits of clean air and water somebody else made?"
The magical world of numbers is much better than the real world of ours. In the former, the sum of two irrationals is sometimes rational. For example:
Unfortunately, that does not work with irrational people.
"Man only plays when he is in the fullest sense of the word a human being, and he is fully a human being only when he plays," German poet Schiller wrote. Had he been a man in the fullest sense of the word, he would have played with his name and change it to Spieler (meaning player in German).
There was another German-speaker who was not afraid to play with his name and change it from Alois Schicklgruber to Alois Hitler. You guessed right, I am talking about Adolf Hitler's father. Had Alois chosen to play with himself rather than with Klara Poelzl, he would have done a great service to Humanity. Therefore, my dear Schillers and Phil Knights of the world, just playing is not enough, it greatly matters what you play and who you play with.
Type-1 people say that the world moves forward (evolution). Type-2 people say it goes to the left (to the fastest lane). Type-3 people say it goes down (to hell). Type-4 people say it goes back in time (to barbarism).
The world does not go anywhere, people do. They rush in all directions and think the world obediently follows their lead.
All happy families are alike ... No, I am not Leo Tolstoy, let me write about something else.
All healthy people are alike, but all sick people are sick in their own way. You can not find equally sick people. Treated by different combinations of drugs two similar patients, i.e. patients with similar sicknesses, are moving on diverging roads. This is due to their inability to realize that the drugs they take interact not only with them but also with each other. If you take a lot of different medications, you might get cured of something, but you will definitely get something else.
How many drugs are on the market? There are different answers on the net, averaging around 12000. Sounds reasonable, as there are about 12000 diseases in ICD-9 (International Classification of Diseases of the World Health Organization)*. Combinatorics says that the number of different interactions between 12,000 drugs is 2^12000 (a number with 3613 digits, which is greater than the number of atoms in the visible universe). If you believe that there exists some supernatural being able to study all these drug interactions (and find that all of them are health-improving), then you definitely need medication.
Now we know why every sick person is sick in their own way (they use different combinations of drugs to reach unique complaints). And why are the healthy people alike? Because they do not take any medications.
* this is a translation of an old text of mine, now we have ICD-11
Some people simplify their lives by producing candles at home instead of working somewhere (and for someone) else. Other people simplify their lives by buying home-made candles instead of making them themselves. Buy as many candles as you want dears, but please understand that your homes and lives would be simpler without any candles.
"Stupidity spreads at the speed of light," Nicolás Gómez Dávila wrote. I don't know what he meant, but immediately find several examples confirming his claim (light pollution and television signals).
... can take you on their different wings to the same endpoint. This is what Economics Nobelist Thomas Schelling wrote:
How satisfactorily each individual adapts to his social environment is not he same thing as how satisfactory the social environment thus formed is.
Jidu Krishnamurti needed fewer words to say it:
Being well adapted to a sick society is not a sign of good health.
They say we are imperfect copies of the Almighty. I am sure it is so. Imagine how awful the Almighty would be if we were his perfect copies. Imagine how awful the world would be if we were the templates.
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