For the new school year
Most readers were somewhere on vacation - whether in our beautiful country, in neighboring countries, or maybe even overseas. Let's take advantage of this while the borders are open for us ... What was the most frequent sign in our short and long journeys? This is an arrow pointing towards the exit from the motorway, the continuation of the mountain path, the entrance to the museum, the entrance to the beach, and so on and so forth. What's so interesting about all this? Mathematically, not so much. But let's think: this sign is obvious to everyone ... representatives of a civilization in which archery was once shot. True, it is impossible to prove this. We don't know any other civilization. However, the regular pentagon and its star-shaped version, the pentagram, are more mathematically interesting.
We don't need any education to find these figures intriguing and interesting. If, Reader, you have been drinking five-star cognac in a five-star hotel on the Place des Stars in Paris, then maybe… you were born under a lucky star. When someone asks us to draw a star, we will draw a five-pointed one without hesitation, and when the interlocutor is surprised: “This is a symbol of the former USSR!”, We can answer: Stables!”.
The pentagram, or five-pointed star, a regular pentagon, has been mastered by all mankind. At least a quarter of countries, including the US and the former USSR, have included it in their emblems. As children, we learned to draw a five-pointed star without lifting the pencil from the page. In adulthood, she becomes our guiding star, unchanging, distant, a symbol of hope and destiny, an oracle. Let's look at it from the side.
What are the stars telling us?
Historians agree that until the XNUMXth century BC, the intellectual heritage of the peoples of Europe remained in the shadow of the cultures of Babylon, Egypt and Phoenicia. And suddenly the sixth century brings a renaissance and such a rapid development of culture and science that some journalists (for example, Daniken) claim - it is difficult to say whether they themselves believe in this - that this would not have been possible without the intervention of the prisoners. from space.
When it comes to Greece, the case has a rational explanation: as a result of the migration of peoples, the inhabitants of the Peloponnesian peninsula learn more about the culture of neighboring countries (for example, the Phoenician letters penetrate into Greece and improve the alphabet), and they themselves begin to colonize the Mediterranean basin. These are always very favorable conditions for the development of science: independence combined with contacts with the world. Without independence, we doom ourselves to the fate of the banana republics of Central America; without contacts, to North Korea.
Numbers Matter
The XNUMXth century BC was a special century in the history of mankind. Without knowing or perhaps not hearing of each other, the three great thinkers taught: Buddha, Confucius i Pythagoras. The first two created religions and philosophies that are still alive today. Is the role of the third of them limited to the discovery of one or another property of a particular triangle?
At the turn of the 624th and 546th centuries (c. XNUMX - c. XNUMX BC) in Miletus in modern Asia Minor lived Such. Some sources say that he was a scientist, others that he was a wealthy merchant, and still others call him an entrepreneur (apparently, in one year he bought all the oil presses, and then borrowed them for a usurious payment). Some, according to the current fashion and the model of doing science, see him, in turn, as a patron: apparently, he invited the wise men, fed them and treated them, and then said: “Well, work for the glory of me and all Science.” However, many serious sources are inclined to assert that Thales, flesh and blood, did not exist at all, and his name only served as the personification of specific ideas. As it was, so it was, and we probably will never know. The historian of mathematics E. D. Smith wrote that if there were no Thales, there would be no Pythagoras, and no one like Pythagoras, and without Pythagoras there would be neither Plato nor anyone like Plato. More likely. Let us leave aside, however, what would have happened if.
Pythagoras (c. 572 - c. 497 BC) taught at Crotone in southern Italy, and it was there that the intellectual movement named after the master was born: Pythagoreanism. It was an ethical-religious movement and association based, as we would call it today, on secrets and secret teachings, considering the study of science as one of the means of purifying the soul. During the life of one or two generations, Pythagoreanism went through the usual stages of development of ideas: initial growth and expansion, crisis and decline. Truly great ideas don't end their lives there and never die forever. The intellectual teaching of Pythagoras (he himself coined a term he called himself: philosopher, or friend of wisdom) and his disciples dominated all antiquity, then returned to the Renaissance (under the name of pantheism), and we are actually under his influence. today. The principles of Pythagoreanism are so ingrained in culture (at least in Europe) that we hardly realize that we could think otherwise. We are surprised no less than Molière's Monsieur Jourdain, who was surprised to learn that he had been speaking prose all his life.
The main idea of Pythagoreanism was the belief that the world is organized according to a strict plan and harmony, and that the vocation of man is to know this harmony. And it is the reflection on the harmony of the world that constitutes the teaching of Pythagoreanism. The Pythagoreans were certainly both mystics and mathematicians, although it is only today that it is easy to classify them so casually. They paved the way. They began their studies of the harmony of the world, first studying music, astronomy, arithmetic, etc.
Although mankind succumbed to magic "forever", only the Pythagorean school elevated it to a generally applicable law. "Numbers rule the world" – this slogan was the best characteristic of the school. Numbers have a soul. Each one meant something, each one symbolized something, each one reflected a particle of this harmony of the Universe, i.e. space. The word itself means "order, order" (readers know that cosmetics smooth the face and enhance beauty).
Different sources give different meanings that the Pythagoreans gave to each number. One way or another, the same number could symbolize several concepts. The most important were six (perfect number) i ten - the sum of consecutive numbers 1 + 2 + 3 + 4, made up of other numbers, the symbolism of which has survived to this day.
So, Pythagoras taught that numbers are the beginning and source of everything, that - if you imagine - they "mix" with each other, and we see only the results of what they do. Created, or rather developed by Pythagoras, the mysticism of numbers does not have a “good print” today, and even serious authors see here a mixture of “pathos and absurdity” or “science, mysticism and pure exaggeration.” It is difficult to understand how the famous historian Alexander Kravchuk could write that Pythagoras and his students filled philosophy with visions, myths, superstitions - as if he did not understand anything. Because it only looks like this from the point of view of our XNUMXth century. The Pythagoreans did not strain anything, they created their theories in perfect conscience. Maybe in a few centuries someone will write that the whole theory of relativity was also absurd, pretentious and forced. And the numerical symbolism, which separated us from Pythagoras for a quarter of a million years, deeply penetrated into culture and became a part of it, like Greek and German myths, medieval knightly epics, Russian folk tales about Kost or the vision of Juliusz Slovak the Slavic Pope.
Mysterious irrationality
In geometry, the Pythagoreans were amazed figurami-podobnymi. And it was in the analysis of the Thales theorem, the basic law of the rules of similarity, that a catastrophe occurred. Incommensurable sections were discovered, and hence irrational numbers. Episodes that cannot be measured by any general measure. Numbers that are not proportions. And it was found in one of the simplest forms: a square.
Today, in school science, we bypass this fact, almost not noticing it. The diagonal of a square is √2? Great, how much can that be? We press two buttons on the calculator: 1,4142 ... Well, we already know what the square root of two is. Which? Is it irrational? Perhaps it is because we use such a strange sign, but after all actually it's 1,4142. After all, the calculator does not lie.
If the reader thinks that I am exaggerating, then ... very well. Apparently, Polish schools are not as bad as, for example, in British ones, where everything is immeasurability somewhere between fairy tales.
In Polish, the word "irrational" is not as scary as its counterpart in other European languages. Rational numbers there are rational, rationnel, rational, i.e.
Consider the reasoning that √2 it's an irrational number, that is, it is not any fraction of p/q, where p and q are integers. In modern terms, it looks like this ... Suppose that √2 = p / q and that this fraction can no longer be shortened. In particular, both p and q are odd. Let's square: 2q2=p2. The number p cannot be odd, since then p2 would also be, and the left side of the equality is a multiple of 2. Hence, p is even, i.e., p = 2r, hence p2= 4r2. We reduce the equation 2q2= 4r2. we get d2= 2r2 and we see that q must also be even, which we assumed is not so. Received contradiction the proof ends - you can find this formula now and then in every mathematical book. This circumstantial proof is a favorite trick of the sophists.
I emphasize, however, that this is modern reasoning - the Pythagoreans did not have such a developed algebraic apparatus. They were looking for a common measure of the side of a square and its diagonal, which led them to the idea that there could be no such common measure. The assumption of its existence leads to a contradiction. The hard ground slipped from under my feet. Everything should be able to be described by numbers, and the diagonal of a square, which anyone can draw with a stick on the sand, has no length (that is, it is measurable, because there are no other numbers). “Our faith was in vain,” the Pythagoreans would say. What to do?
Attempts were made to save themselves by sectarian methods. Anyone who dares to discover the existence of irrational numbers will be put to death, and, apparently, the master himself - contrary to the commandment of meekness - carries out the first sentence. Then everything becomes a curtain. According to one version, the Pythagoreans were killed (somewhat saved and thanks to them the whole idea was not taken to the grave), according to another, the disciples themselves, so obedient, expel the adored master and he somewhere ends his life in exile. The sect ceases to exist.
We all know Winston Churchill's saying: "Never in the history of human conflict have so many people owed so much to so few." It was about the pilots who defended England from German aircraft in 1940. If we replace “human conflicts” with “human thoughts”, then the saying applies to the handful of Pythagoreans who escaped (so little) from the pogrom at the end of the XNUMXs. XNUMXth century BC.
So "the thought passed unscathed." What's next? The golden age is coming. The Greeks defeat the Persians (Marathon - 490 BC, Payment - 479). Democracy is getting stronger. New centers of philosophical thought and new schools are emerging. Followers of Pythagoreanism are faced with the problem of irrational numbers. Some say: “We will not comprehend this mystery; we can only contemplate it and admire Uncharted." The latter are more pragmatic and do not respect the Mystery: “If something is wrong with these figures, let's leave them alone, after some 2500 years everything will become known. Maybe numbers don't rule the world? Let's start with geometry. It is no longer the numbers that are important, but their proportions and ratios.
Supporters of the first direction are known to historians of mathematics as acousticsThey lived for a few more centuries and that's it. The latter called themselves mathematics (from the Greek mathein = to know, to learn). We do not need to explain to anyone that this approach has won: it has lived for twenty-five centuries and succeeds.
The victory of mathematicians over auzmatics was expressed, in particular, in the appearance of a new symbol of the Pythagoreans: from now on it was a pentagram (pentás = five, gramma = letter, inscription) - a regular pentagon in the shape of a star. Its branches intersect extremely proportionally: the whole always refers to the greater part, and the greater part to the lesser part. He called divine proportion, then secularized to gold. The ancient Greeks (and behind them the whole Eurocentric world) believed that this proportion was the most pleasing to the human eye, and met it almost everywhere.
(Cyprian Camille Norvid, Prometidion)
I will finish with one more passage, this time from the poem "Faust" (translated by Vladislav August Kostelsky). Well, the pentagram is also an image of the five senses and the famous "sorcerer's foot". In Goethe's poem, Dr. Faust wanted to protect himself from the devil by drawing this symbol on the threshold of his house. He did it casually, and this is what happened:
Faust
M epistopheles
Faust
And this is all about the usual pentagon at the beginning of the new school year.
