The emergence of mathematics

Not all cultures make scientific and technological progress at the same speed. To some extent, many retained the tribal system and ancient customs, according to which one can judge their distant past and obtain information about the era when writing did not yet exist. For example, you can compare the numerical system of the Bakairi tribe in Brazil, in which there are names only for numbers not greater than 6, and the numerical system of the Yoruba tribe in Nigeria, based on a complex subtractive principle and thus understand how the method of naming numbers developed.

European colonialists often could treat such cultures barbarously without respecting their traditions. Many were destroyed, others had to integrate into the existing political and economic system. When scientists gradually realized that such cultures can provide rich material for studying the history of the primitive world, some of them have already disappeared ^{[ neutrality? ]} .

Late twentieth century a branch of science appeared - ethnomathematics studying mathematics as part of traditional culture ^[11] . Studies are beginning to be conducted, during which it becomes known how primitive peoples are believed, shown, called, and written down.

Certain information is provided by archaeological excavations. At the Ishango site in Africa , a bone was found with countable notches , the age of which is estimated from 20 to 40 thousands of years, which provided extensive material for study and conclusions ^[12] . Another artifact - the radial bone of a young wolf with 55 nicks on it - was found in the Upper Paleolithic site of Dolny Westonice (Czech Republic). Mikel Alberti in his book “Mathematical Planet. Traveling Around the World” gives examples of other artifacts ^[13] .

If we systematize the knowledge obtained as a result of ethnomathematical and archaeological research, we can approximately recreate the process of the emergence of mathematics .

The Sense of Number

A series of experiments show that animals in a certain sense can sense the number of objects, not counting them. The English biologist John Lubbock believed that animals already had initial knowledge in the field of arithmetic:

Leroy <...> mentions the case when a person needed to shoot a crow. “To mislead this suspicious bird, it was decided to send two people to its nest, one of which would pass by and the other would remain. But the crow counted them and kept at a distance. The next day the three went, and again she realized that only two left. It turned out that it was necessary to send five or six people to bring her down in the calculations. The crow, thinking that everyone had passed by, wasting no time, returned to the nest. " From this he concludes that a crow can count to four. Lichtenberg talks about the nightingale, who counted to three. Every day he gave him three worms, one at a time. Having finished one, the nightingale returned after another, but after the third he knew that lunch was over <...> There is a funny and suggestive detail in the Tales of a Tropical South African Explorer, Mr. Galton . After describing the weakness of the African tribe of Demara in the long run, he says: “Once, when I watched an African hopelessly trying to count something, I noticed next to Dyne, my spaniel, also puzzled; Dyne was next to half a dozen of her newborn puppies, who constantly moving away from her, she was very worried and trying to find out if they were all here, or someone was missing. She looked at them puzzled, but couldn’t understand anything. She obviously had a vague idea of ​​the score, but here the number was too big for her brain. whether to compare the two of them, a person and a dog, the person is in a disadvantageous position <...> "<...> Thus, we have reason to believe that animals have enough intelligence to distinguish three from four ^[4] .

Original text

Leroy <...> mentions a case in which a man was anxious to shoot a crow. "To deceive this suspicious bird, the plan was hit upon sending two men to the watch house, one of whom passed on, while the other remained; but the crow counted and kept her distance. The next day three went, and again she perceived that only two retired. In fine, it was found necessary to send five or six men to the watch house to put her out in her calculation. The crow, thinking that this number of men had passed by, lost no time in returning. " From this he inferred that crows could count up to four. Lichtenberg mentions a nightingale which was said to count up to three. Every day he gave it three mealworms, one at a time. When it had finished one it returned for another, but after the third it knew that the feast was over <...> There is an amusing and suggestive remark in Mr. Galton's interesting Narrative of an Explorer in Tropical South Africa. After describing the Demara's weakness in calculations, he says: "Once while I watched a Demara floundering hopelessly in a calculation on one side of me, I observed," Dinah, "my spaniel, equally embarrassed on the other; she was overlooking half a dozen of her new-born puppies, which had been removed two or three times from her, and her anxiety was excessive, as she tried to find out if they were all present, or if any were still missing. She kept puzzling and running her eyes over them backwards and forwards, but could not satisfy herself. She evidently had a vague notion of counting, but the figure was too large for her brain. Taking the two as they stood, dog and Demara, the comparison reflected no great honor on the man <...> "According to my bird-nesting recollections, which I have refreshed by more recent experience, if a nest contains four eggs, one may safely be taken; but if two are removed, the bird generally deserts. Here, then, it would seem as if we had some reason for supposing that there is sufficient intelligence to distinguish three from four.

Primitive people inherited this ability. So, according to the recollections of one American missionary, hunters from the wild tribe of Indians, who have names only for the numbers 1, 2 and 3, look before the hunt a large pack of dogs and if at least one is missing, they notice this and start calling it. This phenomenon is known as the “ sense of number ” ^[5] and “ sensory counting ” ^[14] .

Establishing one-to-one correspondence

If there is a set of eight stones and a set of eight shells, you can arrange them so that in front of each stone lay one shell. This is how the process of trade between the two primitive tribes took place. On the contrary, each product from the first tribe put one product from the second tribe and as a result the tribes exchanged the same amount of goods with each other ^[17] .

Such a process, when each element from one set (set) is associated with one element from another set is called in mathematics the establishment of a one-to-one correspondence between two sets ^[18] .

With the establishment of a one-to-one correspondence between the set of counted objects and the set of countable standards, the next stage of the development of the account began.

Of all the counting standards, the most convenient and which is “always with you” are fingers and toes and even other parts of the body ^[15] .

To remember how many animals he killed while hunting, a primitive person just needed to remember on which finger or toe he stopped the count. It could be the second toe of the second foot, the last toe of the first hand and whether all the toes. In some languages, numbers have become so called. Here are some examples:

• The number 18 in the language of one Grendan tribe is called "Three On The Other Leg" ^[19] .
• The same number in the language of one Karaib tribe is called "All my hands, three, my hand" ^[19] .
• In the Zulu language, the word "tatizituna" ("take the thumb") denotes the number 6, and the word "cobmile" (he indicated, that is, the index finger) - 7 ^[20] .

When fingers were not enough, other parts of the body were used, fingers of other people or extension of already bent fingers.

Researcher of New Guinea, N. N. Miklouho-Maclay, suggested that the Papuans count the number of days before the return of the Vityaz corvette by cutting strips of paper for this.

"The first, laying out pieces of paper on his knee, at each trimming repeated“ bang, bang ”(one); the other repeated the word“ bun ”and bent the finger first on one, then on the other hand. Counting to ten and bending the fingers of both hands , he lowered both fists to his knees saying: “two hands,” the third Papuan bent his finger. The same was done with the second ten, and the third Papuan bent his second finger; the same was done for the third ten; the remaining pieces of paper did not constitute the fourth ten and kept on the sidelines. " ^[21]

Often primitive people carried with them special standards of counting - sticks or balls ^[22] .

When the art of counting gradually developed, the concept of number was inseparable from counted objects. The number could not exist by itself. Depending on what was believed, the numbers could be called differently ^[10] . Some tribes to this day there is a division of numerals according to the type of counted objects. For example, in the Tzimshian language, there are seven different types of numerals:

1. For counting flat objects
2. For counting round objects and dividing time
3. For people counting
4. To count long items
5. For canoe counting
6. For measures
7. Indefinite numbers ^{[9] [23]} .

It took a long time for the concept of a number to appear, separated from objects.

Theoretically, you can count any number of items. Their number can be expressed by a number that has never been seen before (for example, 723,945,186 - seven hundred twenty three million nine hundred forty five thousand one hundred eighty six), but nevertheless it will be possible for a person who hears this number to imagine how much this is approximately. The number of items that can be counted is unlimited. For any integer number of objects, there is a well-defined natural number. This phenomenon is called a continuous numerical sequence .

However, the numerical sequence in the language was not always continuous . There are still tribes in whose languages ​​there are only two numerals: one and many . The level of their life does not require any other numerical words. But in connection with technical development, these words become necessary.

The appearance of the word for the number two is a big step in the development of a numerical sequence. After the appearance of the word for the number three, the numerical sequence expands further and further. Gradually, names appear for numbers less than ten .

A few centuries ago, most people did not need to use numbers more than a thousand . For the designation of large numbers, the words "monster", "infinity", "no longer count" were used. So, the prefix "-ter", which means the multiplication of the original unit by 10 ¹² , that is, by a trillion (for example, terabyte), comes from the Roman word "monster", that is, it is cognate with the word "terror". The ancient Russian name for the number 10,000 is darkness . The name million means in old Italian "a big thousand."

In the language of Rwanda, 10,000 is called an elephant, and 20,000 is called two elephants. In Nigeria, the number 160,000 is called "400 meets 400", and the name of the number 10,000,000 can be roughly translated as "There are so many things that their number is immense" ^[24] .

The similarity of numerals among various Indo-European peoples shows that they appeared even when these peoples spoke the same language, i.e., refers to the prehistoric period:

Languages ​​without numerals

There are languages ​​that are completely (or almost completely) devoid of any numerals. In the work of the American mathematician Levy Conent, the languages ​​of the Bolivian tribes Chiquita and Takan are given as examples ^[25] .

Algorithmic and nodal numbers

In science, the numbers that underlie the names of others are given the name " nodal ". Numbers whose names consist of others are given the name " algorithmic " ^[26] . So the numbers three, six, ten, forty, one hundred are nodal, since their names cannot be distinguished by composition. The number sixty is algorithmic, since its name consists of the names of the nodal numbers six and ten. Algorithmic numbers can be formed from nodal in various ways. The following are examples of such formations.

Additive Principle

The first numerical systems used the additive principle. It consists in the fact that the names of the algorithmic numbers are formed from the nodal ones by means of addition , as the name of the number is seven hundred and twenty . The table shows as an example the number system of the Gumulgel tribe living on the Torres Strait Islands and the Bakairi tribe.

As you can see, only 1 and 2 have proper names, the remaining numbers have derived names. For numbers greater than 7, these tribes have only one word, which means a lot.

Subtractive principle

More complex numerical systems also used the subtractive principle. This means that the names of some algorithmic numbers could be formed from nodal ones by subtraction .

The subtractive principle is visible, for example, in the Roman numbering system, where the number 9 is written as IX , that is, as 10-1. Довольно сложной субтрактивной системой счисления с основанием 20 пользовалось африканское племя Йоруба :

Мультипликативный принцип

Мультипликативный принцип заключается в том, что названия некоторых алгоритмических чисел могут образовываться из узловых посредством умножения . Он виден в названиях таких чисел, как "семьдесят", "триста", "четыреста" и т. д.

Для счёта нужно иметь математические модели таких важных событий, как объединение нескольких множеств в одно или, наоборот, отделение части множества. Так появились операции сложения и потом вычитания . Для того случая, когда много раз нужно сложить несколько одинаковых совокупностей, появляется новая операция — умножение .

Другое важное практическое действие — разделение на части — со временем абстрагировалось в четвёртую арифметическую операцию — деление . Свойства арифметических операций открывались постепенно.

Большим "толчком" к использованию арифметических операций послужило развитие измерений . Единицы измерения были связаны прежде всего с частями тела, которыми было легко их (измерения) проводить.

Понятия дроби, как таковой, не было даже после появления письменности. Однако в быту использовались понятия " половина ", " треть ", " четверть ". Такие "дроби" дроби обычно имели знаменателем 2, 3, 4, 8 или 12. Например, у римлян стандартной дробью была унция ( ¹ / ₁₂ ). Средневековые денежные и мерные системы несут на себе явный отпечаток древних недесятичных систем: 1 английский пенни = 1/12 шиллинга , 1 дюйм = 1/12 фута , 1 фут = 1/3 ярда , дюжина = 12 единиц и т. д. Десятичные дроби , удобные в сложных вычислениях получили распространение в Европе только в XVI веке.

В своей практической деятельности человек сталкивался с конкретными геометрическими фигурами и телами. Постепенно происходила их идеализация — люди абстрагировались от дефектов конкретных предметов, создавая идеальные представления. Так появились понятия правильных многоугольников и многогранников, пирамид, призм и тел вращения. Большинство общепринятых названий геометрических фигур являются древнегреческими ^[20] .

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