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In recent decades, there has been a growing stream of studies that cast doubt on the reliability of many statements of historical science. Behind its quite decent façade, there is a darkness of fantasies, fables and simply outright forgeries. This also applies to the history of mathematics.
Consider closely and biasedly the figures of Pacioli and Archimedes, Luke and Leonardo, Roman numerals and the Egyptian triangle 3-4-5, Ars Metric and Rechenhaftigkeit and much, much more …
When did people learn to count?
We can safely say that this happened to their distant ancestors, long before they became homo sapiens. Arithmetic penetrates into all aspects of life, even animals. For example, it was found thata crow can count to eight.If a crow has seven chicks and one is removed, then she will immediately start looking for the missing and count her offspring. And after eight, she doesn't notice the loss. For her, this is some kind of infinity. That is, every creature has some kind of numerical limit.
It also exists among people who do not know mathematics. This was reflected in various languages, in particular in Russian.
Only six to seven centuries ago, the troops of the most formidable and victorious Asian conquerors were clearly divided into divisions only up to a thousand people… They were headed by commanders who were called foremen, centurions and thousanders. Larger military units were called "darkness" and they were headed by "temniki". In other words, they were denoted by a word meaning "so many that it is impossible to count." Therefore, when we meet large numbers in the Old Testament or in the “ancient” chronicles, for example, 600 thousand men whom Moses brought out of Egypt, this is a clear sign that the number appeared, by historical standards, quite recently.
The real science of mathematics began somewhere in the 17th century. Its founder was Francis Bacon, English philosopher, historian, politician, empiricist (1561-1626). He introduced what is called experiential knowledge. Science differs from scholasticism in that in it any statement, any knowledge is subject to verification and reproduction. Before Bacon, science was speculative, at the level of some logical constructions, guesses, hypotheses and theories were expressed, but they were never tested. So physics and chemistry as sciences until the 17th century did not exist in the modern sense… The same Galileo Galilei (1564-1642), the founder of experimental physics, climbed onto the Leaning Tower of Pisa and threw stones from there, and only then did he find out that Aristotle was wrong when he said that bodies move in a straight line and evenly. It turned out that the stones are moving with acceleration.
Aristotle argued so not because he was lazy to check, but because even the simplest experimental scientific methods had not yet been born. We emphasize again: no verification - no reliable knowledge.
One example, not known to everyone. The first work on physics in China was published in 1920. The Chinese explain this by the fact that for centuries they did without it, because they were guided by the teachings of Confucius (556-479 BC). And he sat down and contemplated and drew everything, like Aristotle, from the air. Checking Confucius is just a waste of time, the Chinese believe. This is highly suspicious in light of claims that they were the first to invent paper, gunpowder, compass and a bunch of other inventions. Where did all this come from if they had no science?
Thus, the very first attempts to believe when and how certain scientific, including mathematical results appeared, show that there are a lot of myths in the history of scienceespecially when it comes to time before the invention of printing, which made it possible to consolidate the history of certain studies on paper. One of these fables, wandering from book to book, is the myth of the Egyptian triangle, that is, a right-angled triangle with sides corresponding to 3: 4: 5. Everyone knows that this is a myth, but it is stubbornly repeated by various authors. He talks about a rope with 12 knots. A triangle is folded from such a rope: three knots at the bottom, 4 on the side and five knots on the hypotenuse.
Why is such a triangle so wonderful? The fact that it satisfies the requirements of the Pythagorean theorem, that is:
3.2 + 4.2 = 5.2
If this is so, then the angle at the base between the legs is right. Thus, without having any other tools, neither squares nor rulers, you can depict a right angle quite accurately.
The most amazing thing is that in no source, in no study there is any mention of the Egyptian Triangle.It was invented by the popularizers of the 19th century, who supplied ancient history with some facts of mathematical life. Meanwhile, only two manuscripts remained from the ancient Egipt, in which there is at least some kind of mathematics. This is the Ahmes Papyrus, a study guide to arithmetic and geometry from the Middle Kingdom period. It is also called the Rind papyrus by the name of its first owner (1858) and the Moscow metematic papyrus, or the papyrus of V. Golenishchev, one of the founders of Russian Egyptology.
Another example - "Occam's razor", a methodological principle named for the English monk and nominalist philosopher William Ockham (1285-1349). In a simplified form, it reads: "You should not multiply things unnecessarily." It is believed that Occamah laid the foundation for the principle of modern science:it is impossible to explain some new phenomena by introducing new entities, if they can be explained with the help of what is already known… This is logical. But Occam has nothing to do with this principle. This principle was attributed to him. Nevertheless, the myth is very persistent. It is used in all philosophical encyclopedias.
Another fable - about the golden ratio- dividing a continuous quantity into two parts in such a ratio in which the smaller part relates to the larger one, as the larger one relates to the whole quantity. This proportion is present in the five-pointed star. If you write it in a circle, then it is called a pentagram. And it is considered a devilish sign, a symbol of Satan. Or the sign of Baphomet. But nobody says thatthe term "golden ratio" was coined in 1885by the German mathematician Adolph Zeising and was first used by the American mathematician Mark Barr, and not by Leonardo da Vinci, as they say everywhere. This is what is called a "classic of the genre", a classic example of the description of the past in modern concepts. Because here an irrational algebraic number is used, a positive solution to a quadratic equation - x.2 –x-1 = 0
There were no irrational numbers either in the era of Euclid, or in the era of da Vinci and Newton
Was there a golden ratio before? Certainly. But she called divina, that is, divine proportion, or devilish, according to others. All Renaissance warlocks were called devils. There was no question of any golden ratio as a term.
Another myth is Fibonacci numbers… We are talking about a series of numbers, each term in which is the sum of the previous two. It is known as the Fibonacci series, and the numbers themselves are Fibonacci numbers, after the name of the medieval mathematician who created them (1170-1250).
But it turns out that the great Johannes Kepler, the German mathematician, astronomer, optician and astrologer, never mentions these numbers. The complete impression that not a single mathematician of the 17th century knows what it is, despite the fact that Fibonacci's work "The Book of Abacus" (1202) was considered very popular in the Middle Ages and in the Renaissance and was the main one for all mathematicians of that era … What's the matter?
There is a very simple explanation. At the end of the 19th century, in 1886, Edouard Luc's wonderful four-volume book "Entertaining Mathematics" for schoolchildren was published in France. There are many excellent examples and problems in it, in particular, the famous puzzle about a wolf, a goat and a cabbage, which must be transported across the river, but so that no one eats anyone. It was invented by Luca.He also invented the Fibonacci numbers. He is one of the creators of modern mathematical myths that have become very firmly established in circulation. Luke's myth-making was continued in Russia by the popularizer Yakov Perelman, who published a whole series of such books on mathematics, physics, etc. In fact, these are free and at times literal translations of Luke's books.
It must be said that there is no possibility to check the mathematical calculations of the times of antiquity.Arabic numerals, (the traditional name for a set of ten characters: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9; now used in most countries to write numbers in decimal notation), appear very late, at the turn of the 15-16 centuries.Before that, there were so-called Roman numerals that cannot be used to calculate anything.
Here are some examples. The numbers were written like this:
888- DCCCLXXXV111, 3999-MMMCMXCIX
With such a record, no calculations can be made. They were never produced. But in ancient Rome, which existed, according to modern history, one and a half thousand years, huge amounts of money were circulating. How were they counted? There was no banking system, no receipts, no texts related to mathematical calculations exist. Neither from ancient Rome nor from the early Middle Ages. And it's clear why: there was no way to write mathematically.
As an example, I will give how the numbers were written in Byzantium. The discovery, according to legend, belongs to Raphael Bombelli, an Italian mathematician and hydraulic engineer. His real name is Matsolli (1526-1572). Once he went to the library, found a mathematical book with these notes and immediately published it. By the way, Fermat wrote his famous theorem on its margins, since he could not find another paper. But this is by the way.
So, the writing of the equation looks like this, (There are no corresponding icons on the cybord, so I wrote it down on a separate piece of paper)
This method of mathematical notation cannot be used in calculations.
In Russia, the first book in which there was some kind of mathematics was published only in 1629. It was called "The Book of Soshny Letter" and was devoted to how to measure and describe urban and rural land holdings (including land and industries) for the purpose of state taxation (conventional tax unit - plowThat is, not only for tax officials, but also for land surveyors.
And what turns out? The concept of a right angle did not exist yet… That was the level of science.
Another misconception. The great Pythagoras invented his theorem. This opinion is based on the information of Apollodorus the calculator (the person is not identified) and on the lines of poetry (the source of the verses is not known):
He raised a glorious sacrifice for him by bulls."
But he did not study geometry at all. He studied occult sciences. He had a mystical school, in which, in particular, occult significance was attached to numbers. The two was considered female, the three was male, the number five meant "family." The unit was not considered a number. It was defended by the Dutch mathematician Simon Stevin (1548-1620). He wrote the book "The Tenth" and in it he proved that one is a number, and introduced the concept of decimal fractions.
What were the numbers?
We discover Euclid (about 300 BC), his essay on the foundations of mathematics "Beginnings". And we find that mathematics was then called "ARS METRIC" - "The Art of Measurement". There all mathematics is reduced to measuring segments, prime numbers are used, there is no option for division, multiplication… There were no funds to carry them out. There is not a single work of that era where there would be calculations. Count on the counting board abacus.
But how were bridges, palaces, castles, bell towers calculated? No way. All the main structures that we know appeared after the 17th century.
As you know, St. Petersburg in Russia was founded in 1703. Only three buildings have survived since then. Under Peter 1, no stone buildings were erected, mainly mud huts made of clay and straw. Peter issued a decree, which spoke specifically about the huts. Stone buildings were built, in fact, only in the era of Catherine II.Why did the Russian people go to Europe on the orders of the tsar? To learn fortification, construction, the ability to make mathematical calculations of buildings and structures.
We recently carried out calculations for Paris. All major buildings were built in the 18th and 19th centuries.One of the first stone buildings in this city is the Saint Chapel - Saint Chanel. You cannot look at it without tears: crooked walls, crooked stones, no right angles, a cave structure, the oldest in Paris from the 13th century. Versailles was built in the 18th century. Then, on the site of the Champs Elysees, there was a Goat's Swamp.
Take Cologne Cathedral, which began to be built in the Middle Ages. It was completed in the 20th century! It was completed using modern methods. The same story with the Sacre Coeur, the Basilica of the Sacred Heart. This cathedral was allegedly badly damaged during the Great French Revolution: statues, stained-glass windows and so on were smashed. Everything is restored but this was done in the 19th and even in the 20th century. All French ancient buildings have been restored using modern methods. AND we see not the buildings that were once, but those that look the way modern restorers imagine.
The same applies to Peter and Paul Fortress In Petersburg. It is made of glass and concrete and looks very nice. And if you go inside, there are rooms that have been preserved since the time of Peter 1. Terribly wretched rooms, with walls made of cobblestones, fastened with clay and straw, are practically shapeless. And this is the 18th century.
The history of the Intercession Cathedral in the Moscow Kremlin, also called St. Basil's Cathedral, is well known. It collapsed during construction, since there were no calculations and methods for this calculation. This is reflected in the written sources. Therefore, Italian builders were invited, and they began to build both the Kremlin and all the other buildings. And they built one to one in the style of Italian cathedrals and palaces. The Italians had something that made a revolution not only in construction, but throughout civilization. They were proficient in the methods of mathematical calculation.
Arithmetic clearly suggests that without knowledge of these methods, nothing worthwhile will be built. Bridges are complex technical structures, unthinkable without preliminary calculations. And until such mathematical calculations were developed, there were no stone bridges in Europe. There were wooden, water-type pontoons. 1st stone bridge in Europe - Charles Bridge in Prague. Either the 14th or the 15th century. It fell apart more than once, because the stone has an expiration date, and because the calculations were improved. The first and last stone bridge in Moscow was built in the middle of the 19th century. It stood for 50 years and fell apart for the same reasons.
Born, mathematics gave rise to not only modern science. The invention of Arabic numerals and the positional numbering system, positional numbering, when the value of each numerical sign (digit) in the number recording depends on its position (digit), made it possible to perform calculations that we still do today: addition - subtraction, multiplication - division. The system was very quickly adopted by merchants, and the result was a surge in the financial system. And when we are told that this system was invented by the Knights Templar in the 13th century, this is not true. Because there were no such ways to manage it.
But mathematics gave birth to much more, as always happens with the greatest achievements of mankind. She turned the 16th century into a dark and sinister era. The heyday of obscurantism, witchcraft, witch hunts. In 1492 - the establishment of the Inquisition in Spain, in 1555 - the establishment of the Inquisition in Rome. Meanwhile, historians are trying to convince us that the Inquisition is a product of the 13-15 centuries. Nothing like this. Why did all this come about? How did it start? With a mania to calculate everything. They even counted how many devils fit on the end of the needle. And witches were determined by weight: if a woman weighed less than 48 kg, she was considered a witch, since, according to the inquisitors, she could fly. This is the 16th century. There even appeared the term "computation-Reckenhaftigheit."
As a curiosity, it is worth noting that that century gave us something else. For example, the words "Computer, printer, scanner"… Computers were called those who were engaged in calculations, that is, calculators. A printer is a person who is busy with book printing, and a scanner is a proofreader. These meanings have been lost, and words have revived in our time with new meanings.
Simultaneously, in 1532, science chronology appears… And this is natural: while there were no ways to count, there were no chronological calculations. At the same time, astrology begins to develop, also based on calculations.… It is necessary to mention and numerology… They begin to see magic in numbers. In numerology, certain properties, concepts and images are assigned to each single-digit number. Numerology was used in the analysis of a person's personality in order to determine character, natural talents, strengths and weaknesses, predict the future, choose the best place to live, determine the most appropriate time for making decisions and for actions. Some with her help chose partners for themselves - in business, marriage. One of the largest numerologists was Jean Boden (1529-1594), politician, philosopher, economist. Appears and Joseph Just Scaliger (1540-1609), philologist, historian, one of the founders of modern historical chronology. Together with the theologian and monk Dionysius Petavius they calculated retroactively a number of historical dates in past history and digitized the facts and events that were known to them.
The example of Russia shows how hard and difficult it was to introduce arithmetization into the consciousness of society.
1703 can be considered the year of the beginning of this process in the country. Then Leonty Magnitsky's book "Arithmetic" was published. The very figure of the author is fictional. This is just a translation of Western manuals. On the basis of this textbook, Peter the Great organized schools for naval officers and navigators.
One of the summer cottages of the book - problem number 33 - is still used today in some educational institutions.
It reads like this: “They asked a certain teacher how many students he had, since they wanted to give his son to him as a teaching. The teacher replied: "If as many disciples come to me as I have, and half as many and a quarter as many, and your son, then I will have one hundred disciples." How many students did he have?"
Now this problem is solved simply: x + x + 1 / 2x + 1 / 4x + 1 = 100.
Magnitsky does not write anything like this, because back in the 18th century 1/2 and ¼ were not perceived as numbers. He solves the problem in four stages, trying to guess the answer according to the so-called "False Rule".
All mathematics in Europe was at this level. The book "Mathematical Ingenuity" by B. Kordemsky says that the mathematical book of Leonardo of Pisa became widespread and for more than two centuries was the most authoritative source of knowledge in the field of numbers (13-16 centuries). And the story is given of how the high reputation of Fibonacci brought the emperor of the Roman Empire Frederick II to Pisa in 1225 with a group of mathematicians who wanted to publicly test Leonardo. He was given the task: "Find the most complete square that remains a complete square after increasing it or decreasing it by five."
A / 2 + 5 = B / 2, A / 2 - 5 = C / 2
This is a very difficult task, but Leonardo allegedly solved it in a few seconds.
Back in the 18th century, they did not know how to work with ½ plus ¼, but Leponardo and the audience work great with them. But fractions as numbers were not recognized until the late 18th century.
Only then did Joseph Louis Lagrange do it. What's the matter? Frederick II and the whole story were invented by the same Luke in his book "Entertaining Mathematics".
Euclid is credited with discoveries in mathematics made many centuries later. For instance, squaring the triangle.
But in the 16th century, the Hungarian engineer and architect Johann Certe wrote to the great Albrecht Durer: “I am sending you a theorem about a triangle with three unequal angles. I found a wonderful solution … But making a square of the same area out of a triangle is an art. I suppose you understand that very well."
This means that in the 16th century Cherte invented the quadrature of a triangle, which, it would seem, was solved by Euclid many centuries ago, and everyone, it would seem, knows how to look for the area of a triangle.
It all boils down to what the 16th century mathematicians did under ancient names. There were so-called Euclid commentators, and they are now said to have perfected him. In fact, they worked under the name of Euclid, under the name of the trade mark. And this is not the only case.
Back in the 18th century, a certain Greek Pelamed was declared the inventor of everything. He invented numbers, chess, checkers, dice and many other things. It was only at the end of the 19th century that it was believed that chess was invented in India.
Some works that enjoyed authority and popularity in ancient times and did not survive or came down in the form of separate fragments, attracted the attention of falsifiers because of the author's surname or the subjects described in them. Sometimes it was about a whole series of sequential forgeries of any composition, not always clearly connected with each other. An example is the various works of Cicero, the many forgeries of which gave rise to heated debates in England in the late 17th and early 18th centuries about the very possibility of falsifying the primary sources of real historical knowledge. The writings of Ovid in the early Middle Ages were used to include the miraculous stories they contained in the biographies of Christian saints. In the 13th century, a whole work was attributed to Ovid himself. The German humanist Prolucius in the 16th century added a seventh chapter to Ovid's "Calendar". The goal was to prove to opponents that, contrary to the testimony of the poet himself, this work of his contained not six, but seven chapters.
Most of the forgeries in question were a kind of reflection of the peculiarities of not only the political struggle, but also the prevailing atmosphere of the hoax boom. At least such an example allows one to judge its scale. According to researchers, in France between 1822 and 1835 more than 12,000 manuscripts, letters and autographs of famous people were sold, in 1836-1840 11,000 were put up for sale at auction, in 1841-1845 - about 15,000, in 1846-1859 - 32,000 Some of them were stolen from public and private libraries and collections, but the bulk were forgeries. An increase in demand gave rise to an increase in supply, and the production of forgeries was ahead of the improvement in methods of detecting them at this time. The successes of the natural sciences, especially chemistry, which made it possible, in particular, to determine the age of the document in question, new, as yet imperfect methods of exposing hoaxes were used rather as an exception.
As soon as new methods appear, new challenges appear. There is a kind of race going on. As already mentioned, they began to calculate everything, up to the size of the planet. Columbus considered the Earth to be three times smaller than it really is. An amazing fact. After all, it was believed that the Greek mathematician and astronomer Erastophenes of Cyrene (276-194 BC) accurately calculated the diameter of the planet. Why didn't Columbus know this? Because Erastofen was part of the 16th century project. These were the people who took the ancient names.
One of the greatest philosophers of the twentieth century O. Spengler put forward the thesis that Greek and modern mathematics have nothing in common, that they are, in essence, two different mathematicians, different ways of thinking. It is the difference in the ways of thinking that is revealed at the turn of the 16th and 17th centuries.
To understand the meaning of changes in science, life, in human consciousness generated by modern mathematics, K. Marx's characterization of technologies as a general social phenomenon helps: “Technology reveals the active relationship of man to nature - the direct process of the production of his life, and at the same time his social conditions of life and the spiritual ideas that flow from them. " Almost a hundred years later, one of the classics of civilizational methodology, A. J. Toynbee, defines technology as a "bag of tools."
Mathematics became the reason for the unprecedented improvement of these "tools" and changed the course of civilization.