Whereof one cannot speak, thereof one ... must outsource to Asia.
Това, за което не може да се говори ... се аутсорсва в Азия.
Това, за което не може да се говори ... се аутсорсва в Азия.
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To get your teeth into them they must be ripe.
... it is not important how much you have, but what you do with it.
Happiness is multiplied by division. Patricie Holečková
Better two lovers of fifty than one of a hundred. Hervé Le Tellier In 1827 Norwegian mathematician Abel completed his only European mathematical journey and returned to Norway. Critics (and Abel himself) thought his journey was a failure. His modest government funding has been cut off and in order to support himself and fight tuberculosis Abel had to take out a loan. Since he died in poverty that very same year, he left no legacy and his heirs were unable to repay the loan.
History is a strange beast. Our contemporaries, unlike Abel's, consider his European journey a great success. Our contemporaries, unlike Abel's, believe he left a huge legacy. That is why in 2001 Norwegian Academy of Sciences began awarding an annual Abel Prize in Mathematics of about a million US dollars. There is only one problem, the prize is given to mathematicians who have already been recognized in the mathematical community. On behalf of Abel who spent his life in obscurity and poverty, a huge award is given to well-known and financially successful mathematicians. If he learned that young, poor and unknown mathematicians were still being ignored and his name was used as an excuse, Abel would turn in his grave. Abel is not alone, he is in good company. How uncomfortable would God, Son, Motherland, Democracy and Humanity feel if they knew what is being done in their name? Creativity seems to have no limits. Ten thousand years ago A invented counting. He was a shepherd and carried a bag of pebbles, one for each of his sheep. I am sure you know how the story (about one-to-one correspondence) goes.
Creativity seems to have no limits. Centuries later B decided to stop carrying pebbles and invented an easier way of counting: one sheep, two sheep, three sheep. Later, B decided to omit the repetitions (sheep, sheep, sheep). That is how B invented the numbers: one, two, three. Creativity seems to have no limits. Centuries later C decided to arrange the numbers visually and the number line was born. Several hundred years before Christ, a Chinese (D) started arranging the numbers in two dimensions and the magic square was born. And people started playing with it. Creativity seems to have no limits. 2 millennia passed. K decided to arrange the numbers of the magic square in three dimensions. Each cell in the square was given a height equal to the number in it. The magic squares started looking like projects in New York. K did not stop there. He saw that if it rained, in some projects the water would run off. Other projects would retain water. The concept of water retention was born. Now the question was, "How much water will project X retain?" Creativity seems to have no limits and we can expect arrangements of numbers in 4, 5 and more dimensions, as well as inventions of other operations with numbers. Before those limitless possibilities start making our heads spinning, let us forget about number operations and think about water operations. We perform the same operations with water that A did tens of thousands of years ago: we look for it and find it, drink it and wash with it. On the one hand, we can keep it better than A could (in dams and bottles), but on the other, we spend more than A did. And most importantly, just like A we are unable to produce water (although we seem to know how). Creativity, no matter what we think of it, has its limits. If time was money, we would have already spent it.
Real people are like real numbers. They do not imagine they are mostly irrational. Corollary (by Fausto Morales): Most leaders will tell you the current health crisis is complex. And so it is, by virtue of their own contributions: when virologists and healthcare professionals provide those leaders with REAL facts and forecasts, the latter flee into the IMAGINARY axis right away. From that point on, the only meeting point of both groups is ... ZERO. A drug addict with a syringe in his hand and longing in his eyes ...
A suicide attempter with a knife in his hand ... A drunkard at the steering wheel ... A pedophile approaching a kindergarten ... A general with his finger over the nuclear button ... Just do it? The less we waste adjectives, the harder it is to lie. Nicolás Gómez Dávila
An in-depth study of Chinese social mobility between 1371 and 1904, conducted by researchers at Hong Kong University (cited here), found that the state exams for senior administrative positions had been producing social mobility by employing many lowborn people by the government. At the same time the Chinese society degraded precisely because it followed the "religion" of eternal values, which was contrary to the idea of mobility. Those state exams exemplified the "religion" too, since the candidates were required to memorize what Confucius had written two millennia earlier. The mobility was supposed to be social, but the Chinese society was demobilized. It is no coincidence that in the 19th century the British Lord Elgin wrote: "Uninvited, and not by the most appropriate methods, we have destroyed the barriers with which those ancient nations tried to hide from the world, at least in the case of China, the rotten remains of their disappearing civilization." Which shows that by social mobility we have mistakenly chosen to denote the personal rise of some lowborn Chinese, but not the demobilization of the Chinese society. Now you know why I started with this particular quote from Dávila. ... arguing about whether humans are fundamentally good or evil makes about as much sense as arguing about whether humans are fundamentally fat or thin. David Graeber and David Wengrow, The Dawn of Everything
Why are there fundamental theorems of arithmetic, algebra and calculus, but no fundamental theorems of geometry and combinatorics? How can geometry function without a fundamental theorem but arithmetic cannot? Do you think that if aliens organized the Miss Universe pageants, the winners would be females from the Homo sapiens species?
Speaking of Homo sapiens (i.e. Wise man), do you think that if aliens made the taxonomic classification of organisms, man would be named Homo sapiens? You have many double-six sets of dominoes. Find formulas for the minimum/maximum number of pips (dots) in any square of 2*n^2 tiles, having in mind that tiles from a new domino set are allowed only after the tiles of the current set have been exhausted.
PS 8 December 2021 Fausto Morales kindly provided the following formulas: With k(0) = 34 , k(1) = 34, k(2) = 25 , k(3) = 19, k(4) = 32, k(5) = 29, k(6) = 11 and k(7) = 29, Minimal case: P(n) = 12*n^2 - k{abs[(n mod 12) - 7]} Maximal case: M(n) = 12*n^2 + k{abs[(n mod 12) - 7]} " ... intelligence is not the same thing as rationality, since being good at computing something is no guarantee that a person will try to compute the right things," Steven Pinker wrote in his latest book. Seven years earlier I wrote something similar in a blog-post called "Brains":
Having been mathematical geniuses could not prevent Georg Friedrich Bernhard Riemann, Kurt Gödel and John Forbes Nash from acting nonsensically. The first one tried to prove the truth of the Bible mathematically, the second one (out of fear of possible poisoning by unknown villains) committed suicide by starvation, and the third one was convinced that aliens were communicating with him. Although he was not a mathematical genius, Rüdiger Gamm was genius enough to mentally calculate 81 to the 100th power. Scientists were so impressed that they regularly subjected him to PET scans while thinking. Rüdiger's brain was so busy calculating, that he was unable to do the elementary calculation that each scan exposed him to radiation 700 times stronger than an ordinary chest X-ray. Kiriakoula the Numerologist is trying to persuade Spiros the Historian that history is predetermined. "There is nothing coincidental in the fact that the famous chess player Bobby Fischer was born under the sign of Pisces*," she says.
"What about the fact that Donna Summer was born on 31st of December?," Spiros asks. "And what about you? Your name is Kiriakoula** but you were born on Wednedsay." ________________________ * Fish in Latin ** Sunday in Greek You have many double-six sets of dominoes. Find a formula for the minimum number of pips (dots) in any square of 2^(2n - 1) tiles, having in mind that tiles from a new domino set are allowed only after the tiles of the current set have been exhausted.
PS 17 November 2021 I was able to find recurrences for the minimum number of pips p(n) = 3*p(n-1) + 3*p(n-2) + 4*p(n-3) + 246, with p(1) = 1, p(2) = 19, p(3) = 173 and p(4) = 734, and for the maximum number of pips m(n) = 3*m(n-1) + 3*m(n-2) + 4*m(n-3) - 246, with m(1) = 23, m(2) = 77, m(3) = 211 and m(4) = 802. Fausto Morales showed much better thinking (as always) and found closed formulas for the minimum number of pips p(n) = 3*(2^2n) - 5*(2^3 - 2^((n-1) mod 3)) + 1, with p(1) = 1, and for the maximum number of pips m(n) = 3*(2^2n) + 5*(2^3 - 2^((n-1) mod 3)) - 1, with m(1) = 23. Eagles plane on the wings of the wind and, similarly, intelligence uses the wings of imagination. And as the wind is mightier than the eagle, the imagination is mightier than the intelligence. Proof: When your intelligence is not so great, your imagination might persuade you that it is.
The contrary, i.e. the claim that when your imagination is not so great, your intelligence might persuade you that it is, is not true. That is why some very intelligent, but not very imaginative, people prefer to busy themselves with writing 300-page-long treatises on the philosophy of Thomas Aquinas or with learning the first 22000 digits of Pi (which someone more imaginative has already found how to calculate). What a piece of work is a man! He can't define what hardcore pornography is, but he can recognize it when he sees it (in the words of the world-famous American judge Potter Stewart).
With prime numbers, the situation is quite different - we can easily define them, but we cannot recognize them when we see them. Pierre de Fermat thought that 2^32 + 1 = 4294967297 is a prime number, but much later Leonhard Euler discovered it is not. We worship Adam Smith for "his" invisible hand. Oh, how well he described, with the help of an abstraction, what is happening in the economy, we say to ourselves with admiration.
It is an abstraction, yes, but not quite. On the one hand, it is an abstraction as it is a hand unattached to a body. On the other hand it has many fingers – producers and consumers, and therefore it resembles the human hand. A true abstraction would be the bodiless and fingerless hand. Even geniuses can hardly envisage such a hand. For example, Euclid could see how the invisible hand of prime numbers, with its countless fingers, is spinning the Machine of mathematics, but he was unable to imagine the number zero – the number of fingers of a fingerless hand. They tell you progress is self-evident and give you justifications: thanks to physics (new ways of javelin-throwing) and chemistry (new javelin materials) a modern record-breaker can throw farther than the ancient Achilles. Even so, today's javelin throwing is "half-throwing". In the time of Achilles, in addition to distance the following things were sought:
> accuracy, > ability to throw repeatedly, > ability to throw while keeping your life and health, i.e. ability to throw wearing uncomfortable armor and helmet. Oh, how I pity the modern record-holder who would face Achilles. There is an old philosophical problem that can be compressed into just one sentence: If a tree fals in the forest and no one is around to hear it, does it make a sound? The subjectivists argue that, since sound is subjective experience, no subject means no sound. The objectivists argue that, since sound is vibrations in the air, the falling tree makes a sound. The followers of the middle path, some of them otolaryngologists and psychiatrists, think that in order to have a sound there must be objective reasons leading to a subjective experience; otherwise we have deafness (reasons without experience) or hallucination (experience without reasons).
Many people think they are intelligent and spiritual themselves, because they have a subjective experience of their potential. No, in the "human forest" subjective experience is not enough. To pass as intelligent and spiritual you have to demonstrate objective results. Otherwise people might think your intelligence and spirituality are your hallucinations. A pair (a, b) such that sigma(a) = sigma(b) = a+b (where a<b and sigma(n) is the sum of the positive integer divisors of n) is called amicable pair. The number a (OEIS A002025) is always abundant and the number b (OEIS A002046) is always deficient. Is a+b (OEIS A180164) always abundant?
They say a picture is worth a thousand words. In fact, it was Henrik Ibsen who first said it. I wonder why he preferred to say it instead of paint it.
P.S. August 27, 2021 Maybe the answer is The grass is always greener on the other side. Let c(x) be the x-th composite number, c(y) be the least composite number greater than c(x) that is co-prime to c(x), and d = c(y) - c(x). Is there an x, such that the respective d is neither prime, nor perfect square?
P.S. March 30 2021 Wow! Fausto Morales found that (75083 * 250480717) - (2^2 * 3^2 * 11 * 13 * 17 * 19 * 23 * 29 * 31 * 547) = 35 |
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