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.
We worship Adam Smith for "his" invisible hand. Oh, how well he describes, with the help of an abstraction, what happens 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.
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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:
a) accuracy, b) ability to throw repeatedly, and c) 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 fight 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 (of Art). 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 P.P.S. April 29 2022 I found one more example (281 * 2693 * 24852697) - (2^5 * 5 * 11 * 17 * 61 * 337 * 30577) = 21 Ivan has a white chessboard with a side of N unit squares, whose diagonal (a1, b2, c3, ...) is painted black. How many are the composite squares (i.e. the squares with a side of C unit squares, where 1<C<N), whose vertex unit squares are all white?
PS Fausto Morales found the following formula (2N^3 - 9N^2 + 10N)/6 - [1 - (-1)^N)]/4 and I'll write something similar to the (virtual) inscription on Carnegie's tombstone, namely Here blogs a man who knows how to attract readers cleverer than himself. "There is no God but Allah and Muhammad is his messenger," cried out the Muslims, waved the green flag and succeeded in capturing the lands from India to France. Even if we have doubts about their faith, we cannot fail to admire its power, i.e. their readiness to die in the name of Allah.
"There is no greater God but Evolution and Darwin is its messenger," cry out today's believers. Even if we doubt the truth of their faith, we cannot fail to admire their pragmatism: as they wave the flags of Evolution, they head en masse to the vaccination points. Faith is dear, but life is dearer. Futurologists are strange creatures. They tell you that 20 years ago it was impossible to predict you'd carry in your pocket a computer more powerful than those of NASA were when you were born. Next minute they tell you that by 2040 the renewable energy will be at least 80% of all energy produced, if the unforeseen developments are ruled out.
There are always unforeseen developments, why should we rule them out? If there were no unforeseen developments, then 20 years ago I and the futurologists would have known that today we'd carry in our pockets computers more powerful than those of NASA were at the time of our birth. Which watch is smarter: the Swiss one (dozens of springs) or the Korean one (thousands of transistors and lines of code)? Which watch owner is smarter: the one with the mechanical watch (which costs thousands of Swiss francs) or the one with the Korean watch (which costs few hundred euros)?
The Swiss watch is simple, you put it on your hand and forget about it. A Korean watch needs time and effort to set up, update, and connect to another smart and much more expensive device that also requires setup and updates. Therefore, a Korean watch wastes a lot of your time. On the one hand, the people with a lot of time seem smarter. On the other hand, if they dedicate their time to their watches, they don't seem so smart. On the one hand, the people with mechanical Swiss watches seem to earn more than enough and are therefore smarter. On the other hand, if they dedicate their money to Swiss watches and similar knick-knacks, they don't seem so smart. I think the smartest watch is the watch that does not say what time it is, but what to do with your time. The same goes for people: the smartest are not those who know what time it is, but those who: a) know what they spend their time on, and b) are happy about it. They definitely don't need watches for that. Perfect numbers like perfect men are very rare. René Descartes
Zumkeller numbers like good men are at least one in a dozen. Yours truly They say that the magic square in Albrecht Dürer's Melancholia is interesting because the painter's goal was to show the year of the sad event (the death of his mother in 1514 AD) that provoked his melancholia. This worthy goal had several unintended consequences. The couples of symmetric quadruplets (first+ultimate and second+penultimate) form identical trapezoids, the second of which is rotated by 180 degrees. There is one more interesting thing: the sums of numbers on the numbered vertices of the two hexagons - blue (3+2+14+15) and red (10+11+7+6) - equal the magic constant of the square.
Oddly enough, combining blankets is not commutative. Blankets A + B can make you warmer than B + A.
Take 7 distinct numbers out of the numbers {1,2,...,9} and use them to fill in every circle so that the products of the numbers on every horizontal and vertical line are equal.
This was the easy part. The hard thing to do is to find several different approaches to the right solution (embodied in the different first questions you may ask when you start thinking about the problem). "People say that dog is man's best friend," said the dog, "but are silent about whether man is dog's best friend."
Once upon a time ... I wrote some program but later found it did lots of needless calculations. I removed the respective part of the program, but instead of running faster it slowed down. When I complained to a friend of mine, who had been programming for decades, he looked down on me the way the Great Incas had been looking down on their subjects and said, "This is what usually happens."
Come to think of it, this is what usually happens not only to our programs but to ourselves. Oh, how often our efforts to remove the needless things from our lives lead us to needless situations. Wishing to remove the needless weight we damage our muscles and tendons and are getting hernias. Wishing to spend the needless money we meet people we do not need to know, do things we do not need to do and waste our time. At the end we learn the hard way that out of all imaginable things time is never needless. I have a friend who does not know I am a synesthete and is amazed by my ability to compare music to colors or Irish whiskey to Bulgarian white wine. He told me more than once that comparing apples to oranges is a forbidden operation.
Not only am I a synesthete, but I am also interested in math. That is why I know one can compare objects of different type. One can even add and multiply them. It’s elementary, my dear Watson, 5 apples plus 3 oranges equals 8 pieces of fruit. 5 jackets times 5 trousers equals 25 different suits. The fact that we think does not necessarily mean we don't have to learn how to think. Same with breathing, once we start practicing sports, yoga, or singing, we learn our "old" breathing doesn't work and we need to learn to breath correctly.
It is better to make your own point than to follow the party line.
... hard to hide and easy to hate.
In an essay called Roberta, from a book called Chronicles of a liquid society, Umberto Eco tries to explain why we are angry when athletes succeed with the help of chemistry, but think it is OK when artists use it for creative purposes. It is elementary, Eco writes, in sports we are interested in the person behind the result, in arts we are interested in the result, not the person. Trying to leave nothing unexplained, Eco gives the following examples:
> if football was firing balls into the goal by a cannon, football would lose all interest; > if a monkey could write The Divine Comedy, then the book would still be miraculous. If we are willing to think a little more, we can reach other conclusions about: Football In football we are interested in the goals, not the persons, which is why we watch football regularly, but read biographies of football stars once every 10 years. Art Think about supply and demand! If monkeys could write books like The Divine Comedy, then (due to the abundant supply) no one would think that their writings are divine. It is comparatively easy to make clever guesses; indeed there are theorems like "Goldbach's Theorem", which have never been proved and which any fool could have guessed.
G.H. Hardy Was G.H. Hardy's guess that it is comparatively easy to make clever guesses a clever guess? I have already shared my doubts. Now, instead of discussing Goldbach's Conjecture I'll turn to Ramanujan. If it was easy to make clever guesses, then why was not Hardy able to instantly guess the mathematical brilliance of Ramanujan, but needed Littlewood to discuss in detail Ramanujan's first letter with? Not only was G.H. Hardy's guess not clever, but also it was not original. It is said someone mentioned Fermat's Last Guess (Theorem) to Gauss and the math wizard replied it was nothing special and he could generate within seconds several similar conjectures. No one mentioned any such conjectures, which makes me think no conjectures were provided (or if they were, they were not particularly interesting to be remembered). Truth is of the highest value, everything else* is in the eye of the beholder. You may argue about the proper temperature of the beer with your friends, but cannot argue about its current temperature with the thermometer.
_____________________________ * propriety, beauty, kindness, importance, usefulness, etc. You seem intelligent only to those whose problems you can solve faster than they can. If you see problems where others do not see them, they think you are peculiar or pessimistic. If you solve your own problems, they think you are an uninteresting person with problems.
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