# Jan Popken's inaugural lecture

Jan Popken gave his inaugural lecture De jeugdperikelen van het getal following his appointment as professor at the University of Utrecht on 20 October 1947. The lecture was printed in Euclides 23 (2) (1947-48), 80-97. We give below an English version of the lecture.

Primitive peoples' difficulties with number

by

Jan Popken

In the interior of Brazil live the Botocudos Indians, who are at an exceptionally low stage of development. Agriculture is an unknown concept for them and they do not have canoes, no pottery, no clothing.

The mathematical knowledge of this people can only impress us moderately; they only count to two. Their only counting words are: "pogik" and "kripo", literally translated: "a finger" and "double finger"; all numbers above two are invariably called "uruhu", i.e. "many".

There is also a people in Indonesia who only have the numerals 1 and 2, the so-called Tapiro dwarfs from New Guinea. Moreover, these people also ruffle other people's feathers, because according to the English researcher Wollaston, these numerals being identical to the corresponding ones in the neighbouring Papuan tribes.

The Botocudos and the Tapiro dwarfs are no exceptions; it is a rule that primitive peoples know very few numerals and also have an extremely inadequate number concept. Francis Galton, the same person who is best known for his statistical studies in the field of heredity, describes the difficulties he had in trading with the Damaras, a South African Negro tribe. He describes how sheep were exchanged for tobacco, with the Damara receiving two rolls of tobacco for each sheep, but this transaction had to be carried out for each sheep individually: he had sometimes tried as a test to take two sheep at the same time and put down four rolls of tobacco for them. Then the black trader was completely upset. In such a case, the man took two of the four roles aside, and at the same time cast a glance at one of the two sheep to ensure that it was at least paid fairly. To his surprise, exactly two roles remained as a purchase price for the other sheep. But then suddenly he started to doubt again; he took the first pair of tobacco rolls in his hands again and then looked from one sheep to the other; in short, he was completely lost. The end of the story was that he cancelled the purchase and was not satisfied before the first sheep was paid, and, after it was driven away, the second.

A heifer was bought from the Damaras for ten rolls of tobacco. To calculate the correct number, the Negro spread his large hands on the floor and a roll was placed on each finger. If one wanted to buy a second heifer, the entire process was repeated again, but sometimes, one then successfully applied the trick of laying down half instead of whole rolls. The man was so absorbed by the counting process that he didn't even notice this deception.

In many cases the natural person will avoid the use of abstract numerals and help himself in some other way. The German ethnologist Thurnwald cites as an example how his native servant announced the size of a company that had come to visit: "A man with a big nose, an old man, a child, someone with a skin disease and a little man are waiting."

This inadequate number concept is connected with the fact that in many cases the time calculation is missing. If one asks a native about his age, he answers, for example: "I was so and so big when the sun was sick" and the sick sun means the last eclipse of the sun. The story of that venerable old man from the tribe of the Sakei in Malacca is also well-known. When asked how old he was, he said, "Sir, I am three years old." Three had the meaning of "a lot" for him.

In fact, we should not be surprised about such things. A Crow-Indian once said to a linguist: "Decent people have no opportunity to use numbers above ten." Indeed, a decent man who did not commit robbery usually only had to mention very modest numbers; for example, how many wild game he had caught that day, how many trees felled, perhaps also how many scalps he had conquered in his life, or possibly how many women the "Lonely Wolf" chief kept.

The arithmetical arts of the most primitive peoples are not at all developed to high standards. These people hardly know possessions. I do not know if you have ever admired the interior of a hut in the peat area of Drenthe or one of the most horrible slums in a slum district of a big city, where the typical smell of rotting dirt predicts a dorado for the rat world. These are often palaces full of precious treasures compared to living quarters with very primitive peoples. In the Aryan language of the Bos-Wedda's in Ceylon, the word for "dwelling" is identical to "hollow tree" and in such a place of residence one can only expect riches in fairy tales. Such primitive people live from day to day by hunting, fishing and on wild fruits. Stocks are practically not formed and therefore trade is almost completely absent.

Numbers above a certain limit usually leave the natural person completely cold. His interest and knowledge is limited solely to the small and isolated area in which his tribe lives. He is often astounded with all the details of it, but he cannot imagine the possibilities that are not realized therein and therefore leaves him indifferent. When Thurnwald once took counting tests with a native in the following way: one pig, two pigs, and so on, the man refused to continue when they had come close to sixty to eighty pigs with the justification that more pigs did not exist. A number of a thousand pigs, for example, was as unacceptable to him as √-1 pigs are to us.

Also when a natural person does have possessions, for example, in the form of cattle, then he will not count them in many cases, where we would certainly do so under such circumstances. Some writers mention how the owner checks his large herd of cattle - sometimes hundreds of them - on arrival and how he immediately notices that there is one animal missing, simply because he doesn't see a familiar face. We too possess such a capacity, but this faculty is much more developed among the natural peoples than ours. This is related to the fact that memory plays an extremely large role in primitive man, especially when it concerns the memory of concrete things. One must, for example, marvel at the multitude of detailed points that surround primitive man in his stories; how he can remember after years exactly what A said, how B answered, and so on. The natural person knows all the details of his environment; he knows exactly where water can be found and he immediately recognizes the tracks of each species, even the different footprints of his tribesmen. This has to do with an organization of the mind that differs from ours. Primitive man uses his brain primarily for memorization. We, on the other hand, work much more with abstract concepts, while writing relieves us from remembering a confusing amount of detailed points.

From the aforementioned, however, one should not, above all, conclude that in all natural peoples the counting words are just as poor and the numerical concepts just as flawed as in the aforementioned cases. On the contrary, many have taken it further in this regard. As an example, I mention the residents of the Tonga archipelago in Polynesia. In the present day these people have largely taken over the white civilization, but I am speaking here about a time when they had barely come into contact with our culture: Even in those days, this trading people had their own numerical system and counted up to one hundred thousand. The French researcher Labillardière even suspected that the system was going even further and he bothered the population until they told him about new numerals, which showed that their system would even go on until 1015. This was a wonderful result, which was published triumphantly. It was painful, however, when a further investigation revealed that the alleged numerals consisted in part of meaningless constructions and, worse, formed a small vocabulary of unwelcome words.

A mistake that is often made is that one identifies the number concept of a person with his knowledge of numerals. Someone may very well be in possession of the concept of number without knowing even one numeral word. For example, suppose that someone who cannot count and who has never heard of numbers has to be taught a number concept in the shortest possible time. We then put a pile of apples and a second quantity of pears for our student. Now we teach him how to take an apple from one heap and a pear from another and let him repeat this operation until it stops. If both piles are exhausted at that time, we say that there were as many apples as pears. If, on the other hand, there are still apples, while the batch of pears has disappeared, we say that there were more apples than pears; in the opposite case, however, it is said that there were fewer apples than pears. The student now knows what "the same amount", "more" and "less" is, but he cannot yet name any number.

This is not merely theory, but many natural peoples count according to the principle just mentioned. To this end, they have a helping collection, for example, a pile of stones. One of the stones is then placed next to each object of the quantity to be counted. The latter are then collected and the pile of stones obtained represents the number of quantities counted in the aforementioned manner. Such a pile of stones is sometimes a suitable means of checking whether the neighbours are honest; for this purpose, over time a piece of stone is laid next to each object.

The head of one of the Fidzhian Islands once showed a missionary his father's accounting, which consisted of a cairn of 872 pieces. The missionary was greatly upset when he learned that each stone represented one person who had died.

Many natives also use sticks, shells, cereal grains or other small objects instead of stones. Once Gessi, a lieutenant of Gordon, held a trial in the Southern part of Sudan and on that occasion the people showed him, by means of a bundle of straws and twigs, how many women, children and cows had been taken away by the slave drivers. The longest straws naturally had to represent the most important possession, namely the cows.

The way in which numbers are indicated here fits in perfectly with the very concrete way of thinking of primitive people. Sometimes they knows higher numerals, while they cannot imagine the corresponding numbers; first the concrete representation by a pile of stones, or some other things, tells them something. In their standard work on the Toradjas of Central Celebes, Adriani and Kruyt tell how a head man came back from a trial and said with peace of mind that a fine of twenty buffalo had been imposed on him. When they told him that the fine was extremely heavy, the man asked: "How big a heap is that?" and immediately he counted twenty pieces of areca nut. When, however, he saw those pieces lying, one for each buffalo, he suddenly got such a shock that he said: "Good Heavens, could you pay that?"

In very many cases, numbers are also recorded by means of knots in a cord or in a leaf. This means is, for example, widely used in Indonesia and also among the Indians. It is said that the Huichol Indians tribes use this process to register their successive lovers.

Especially one very specific application is extremely popular. If a major event, a party, a raid on a neighbouring tribe or something like that will happen after a certain period of time, cords will be handed out in which knots indicate how many nights must pass before the time arrives. Such a cord is hung up by the hangers and every morning one knot is loosened when getting up. It is then automatically noticed when the big day has arrived.

We also know this method from history. Herodotus tells us how Darius, when crossing the Ister, i.e. the Danube, left his auxiliary troops with the task of guarding the built bridge for sixty days and for that purpose he gave them a belt in which sixty knots were made and of which one knot had to be released every day. The cord with knots was also used in China in ancient times, but this system was given the highest development in the Inca empire in the form of the famous quipu.

Another well-known means of recording numbers is the notch stick. The savage who makes a notch in the handle of his club, every time he has successfully used this instrument, already employs this method. Also listen to Multatuli in his story about Saïdjah and Adinda. In it the girl asks: "But, Saïdjah, how can I know when to go and wait for you at the ketapan?"

Saïdjah thought about this for a moment, and said:- "Count the moons. I will be away three times twelve moons ... this moon does not count. See, Adinda, cut a line in your rice pounder with every new moon. If you have incised three times twelve stripes, I will arrive the day that follows that, at the ketapan. Do you promise to be there?"

However, it is not necessary at all to undertake an exploration of the jungle to discover more of such examples. After all, in our own national life the notch once occupied a prominent place; this was generally used by small suppliers.

In the financial world, the notch stick used to be highly regarded. People usually used a stick here, which consisted of two parts that fit together exactly. The notches were then incised, while the two parts were held together, so that the amount indicated on both pieces was the same. The creditor and debtor each retained one of the two parts. Someone who lent money to the "Bank of England" was therefore called a "stockholder" and he owned a "bankstock". The English treasury has also been doing bookkeeping for centuries with the help of notch sticks and only in 1826 was this slightly out-dated method abolished. They did not know what to do with the stock of useless notch sticks that filled an entire room in the parliament building. In 1834 it was finally decided to burn them, for which purpose a stove in the upper house was used. Apparently, the stokers, in their enthusiasm, did not take sufficient account of how flammable these rotten pieces of wood were. The stove overheated, a fire started and the parliament buildings went up in flames. So it can be seen that the notch stick already has something on its notch stick.

Let us not laugh too much at these conservative Englishmen: few know that a double notch today still constitutes legal proof of claim in our country: Art. 1924 of our current B.W. [Burgerlijk Wetboek, the Civil Code of the Netherlands] reads as follows:
"Notch sticks, sticks with their double matching, prove faith between those who are used to supplying, which they do in retail, or receiving, to prove in this manner."

In the examples we cited above, the counting was always done by means of a helping set. The principle was always that each of the objects to be counted corresponded with just one thing from the auxiliary set. In our own counting we actually do nothing else, but our helping collection does not consist of stones, sticks or similar objects, but words, i.e. the numerals "one", "two", "three", and so on, which are permanently etched in our memory in this order. We count, for example, four spots on a dice, then let those spots, one after the other, correspond with the numerals one, two, three and four. The last used counting word always allows us to state the number. As can be seen, this method of indicating numbers is already rather abstract and we must therefore consider the numeral system as a relatively late acquired spiritual good of humanity.

It is assumed that spoken language was not the original form of expression of man, but that a gesture is more primary than a word. This is not the place to go into this problem in detail; I only point to the extremely important role that a gesture plays in the life of primitive man. Characteristic of early thinking is the plastic and concrete forms of presentations and the most natural form of expression is the gesture.

For a natural person, counting is usually accompanied by fixed gestures. Let's listen to what a missionary has to tell us about it: "There is no Indian in America who speaks a number without giving it with fingers, hands or feet. When the Indian says: give me one fruit, he immediately raises one finger; he will never say five without showing the hand, never ten without extending both hands, and never twenty without pointing the fingers of both hands towards the toes." In some cases, the numeral word is even a direct translation of the gesture. Thus, in the Zulu language, six is expressed by "tatisitupa," which literally means "taking the thumb," because six counts down one hand and passes to the thumb of the other. The number seven is indicated by the verb "komba", i.e. "to point", because this number is expressed by the index finger. When a Zulu says "amahashi akombile" or "the horses have pointed", he means that there were seven horses. The Negro tribe of the Subiyas on the upper reaches of the Zambezi expresses the concept of "six men" by the sentence: "men who have bent one finger", because when describing the number six they bend one finger of the right hand.

The fact that in counting gestures are the original and in numerals the secondary formations, is clear from this, that in some peoples the number gestures exist but not the corresponding numerals. For example, the Bushmen only know a few counting words; above a certain limit they give all numbers by the word "a lot" but in the meantime they often express that "multitude" very precisely with a gesture. More such phenomena are known, for example, with the Bakai and Papuans. A strange habit exists with some Negro tribes, such as the Nkosi. Here the speaker does not mention any number in the course of his story, but each number is indicated by a gesture. The listener, on the other hand, is expected to always pronounce the numeral.

In many cases, for primitive peoples, the numeral for 5 is associated with "hand", for 10 it is with "two hands", "both hands" or something similar, while 20 is often indicated by the word "human" because one then also counts toes. A Caribbean tribe speaks poetically of "the children of the hands" to indicate the number 10. Peculiar numerical word formation is also found in some Melanesian languages in New Guinea, in which 5 is pronounced as, for example, "my hand is dying" or "my hand is dead". People imagined the fingers; which first stand upright, but when counting piece by piece, "die", i.e. fall down. The number 99 is in one ancient language: "four people die, two hands come to an end, one foot comes to an end and four". Indeed, 99 = 4 × 20 (4 people) + 10 (2 hands) + 5 (one foot) + 4.

The idea of repeating the count periodically, every time the fingers have been counted down, we find very obvious, but it must have taken a great deal of effort for humanity to reach this step. It is said of the Kamchadads, the descendants of the original inhabitants of Kamchatka, that after the countdown of all their fingers and toes they did not know how to proceed, and something similar is reported from the Negro tribe of the Bergdamas. More genius is shown by the majority of Papua tribes, who include all sorts of other parts of the body in addition to the fingers and toes. A Sulka on New Britain, when he does not have enough of his own fingers and toes, calls on the help of a tribesman on whose fingers and toes he counts further. He doesn't need more than one assistant, because a Sulka doesn't overload his brain with numbers over forty.

The way in which the Greenland Eskimo used to count is also interesting. A number such as, for example, 53 is expressed by the sentence: "on the third man on the first foot three", which apparently means that in order to reach 53, one must first count two men, yielding 40, then the fingers of a third, so that one to 50, and finally three toes of his foot. Counting must have been a cold job in the polar winter.

Such examples can further be given and they clearly illustrate how counting could be made according to a base twenty system. There is no doubt that our own dozens of systems originated in such a way in counting on the fingers, a theory that Aristotle already brings up in his "Problems".

The schoolboy, who struggles with those new and complex number representations in the first class, experiences the same difficulties as primitive man. He also likes to call on the help of his fingers. Wise pedagogues, however, have banned this means, and therefore our boy will preferably hide his clandestine calculator cunningly under the couch. Yet the last word in this area has not yet been spoken and there are some pedagogues, such as for example Meumann, who have defended counting; they call this the best and simplest way to illustrate the numbers.

Yet this counting on the fingers, in which one finger after the other turns, has not been a starting point. To see this, we would do well to go a bit deeper into the number concept of the most primitive tribes; we will then see that theirs essentially differs from ours. The abstract number concept that we have made our own is completely strange to nature. We already saw, from a few examples, how this conception is tied to certain very concrete representations. In some parts of New Guinea the number ten is indicated by the word "crocodile" because, in the spirit of the native, this number is inextricably linked to the image of the track of a crocodile, which consists of ten dashes. As a final example, I mention how in the African tribe of Mandigos the number forty is called a "mat" because the fingers and toes of a man and a woman, when they are lying on their mat, represent precisely this number.

The phenomenon of the so-called counting words classes is also related to this. For some peoples the counting words used vary according to the nature of the objects to be counted. In such cases, e.g. a separate set of numerals for people, others for canoes, etc. The language of the Nasioi on Bougainville may well set the record in this regard with more than forty classes. However, we still find remains of such systems with us, e.g. when, when counting days, we speak of a week, a month, or a year, and when counting soldiers of a company, a regiment, or a division.

The heterogeneous way in which numbers are interpreted by many primitive peoples is also very typical. We just talked about a language in which ten was indicated by the word "crocodile". In the same language, five is referred to as "hand", six as "three and three" and twenty as "human." What a difference with the system in which we think the numbers are placed! With us, every number is in the sequence 1, 2, 3, 4, ... a link from a homogeneous system; we imagine the numbers generated from the unit and this is how we count: one, two, three, etc. by adding one to each number to get the following. Natural man does not have such a logically structured system at all; he only knows individual numbers, where one number has a much greater meaning for him than the other.

The number two in particular has always made a special impression on primitive man. He saw it tangible in the breaking of a stick and in the symmetry of the human body. He encountered it in fundamental contradictions, such as the one between man and woman, right and left, day and night, heaven and earth, etc. It is striking how many natural peoples still live in a world of such contradictions. In the Papuans, for example, society is divided into two groups, that of the "land side" and that of the "sea side", and these two groups complement each other. For example, during marriage a man from the land side is dependent on a woman from the sea side and vice versa. Also here and there in Indonesia, e.g. on Ambon, one finds this same phenomenon.

This "dualism" is an essence of the primitive state of mind. It manifests itself, for example, in the appearance of human figurines, one half of which is painted red, the other black, and also in the well-known double masks.

The ancestors of the now civilized peoples must also have been greatly under the spell of dualism. This is the only way to explain the phenomenon of dual forms in linguistics. I also refer to the categories table of the Pythagoreans and the dualistic philosophy of the ancient book "Yih-King" of the Chinese.

But modern Western man is also clearly influenced by this way of thinking. I recall the distinction between left and right in politics. It is generally claimed that the difference in the thinking of the cultivated and that of the primitive man lies largely in the fact that the first is logical and the second is what is called prelogic. But this highly praised logic also contains very clear dualistic principles. Think e.g. of our two-fold concept of truth: a meaningful statement is necessarily true or a lie; a third possibility does not exist. Also think about the logical rule "tertium non datur". In this connection I recall that modern science also has logical systems with a multiple concept of truth, in which dualism is rejected, while the so-called Intuitionism of L E J Brouwer directs its most violent attacks against the unlimited application of the "tertium non datur" in mathematics.

Forming pairs, linking two things together to form a higher unity, is perhaps the most fundamental principle - I would call it primeval intuition - of human mental activity. We have already seen how it also forms the basis of our counting, as it is clearly expressed in the savage, who puts a stone next to every object to be counted. But otherwise there are many examples from mathematics: I need only remind you of the concept of function.

However, the synthesis also includes analysis; in this case the division of a quantity into two parts. In the ancient measuring systems of the Egyptians for fields and for corn, the unit measure was divided into two equal parts, the new measure again into two parts, etc. Halving and doubling were formerly fundamental operations in arithmetic, on which, for example, multiplication was based. Russian farmers probably still count in this way, while our bankers stubbornly show a great preference for fractions such as $\large\frac{1}{2}\normalsize$, $\large\frac{1}{4}\normalsize$, $\large\frac{1}{8}\normalsize$, $\large\frac{1}{16}\normalsize$, etc .; just check a stock list to check this.

In some of the most primitive peoples - I name three from three continents: the Australians, the Bushmen, and the Bakarian from Brazil - the number two also plays a predominant role in counting. These natives only have their own counting words for "one" and "two", while all other numerals are compiled from these according to the schedule:
3 = 2 + 1

4 = 2 + 2

5 = 2 + 2 + 1

6 = 2 + 2 + 2.

Furthermore, the system then usually does not run. To take an example: For the Bakarian, the unit is "tokále" and "two" is called "aháge"; therefore the number five is pronounced "aháge-aháge-tokále". The famous explorer Karl von den Steinen has done interesting tests about the primitive counting art of this people. He explained, for example, that he put down three corn kernels and he asked about the number, then the Bakarian pushed two together with the right hand and left the third grain aside. The pair was subsequently touched again, often even investigated. Then the little finger and the ring finger of the left hand were taken with the right hand, after which "aháge" was pronounced. Only then was the third grain touched, the middle finger of the left hand shifted to the little finger and ring finger, "tokále" was said and finally it was proclaimed "aháge-tokále". To relate in a story that the ancestor Kamuschini felled five trees, the narrator needed three sentences; the Bakai said, "He fell two Piki trees. He felled two more. He felled one." After six, the Bakai no longer has numerals; he then still counts on the fingers and toes; but then always say "méra", i.e. "this". This counting, however, causes little enthusiasm for him and he immediately complains about "kinaráchu iwáno", which means both "brain labour" and "headache". If he has also counted the toes and has not reached the end, he grabs his hair and pulls it apart in all directions.

With slightly higher developed numeral systems one finds, apart from one and two, also a numeral for three or for both three and four. For example, in his book "To the Snowy Mountains of New Guinea", Pulle says that the Pesechems had numerals for one, two and three, while the other numbers are indicated by combinations; e.g. four is "two-two", five is "two-three" and six: "three-three". In very many cases one has individual numerals for the first four numbers. The number six is then made up of two triplets, eight of two quarters, five of three and two or of two pairs plus one, seven of four and three or of two triplets plus one, etc.

Such numeral formations can be found with the primitive peoples everywhere on earth. They are very common with Negro tribes, but also in Canada, California, Mexico, South America, New Guinea and South Asia. They are even found in England, that is to say only among the Gypsies. Seven is "dui trins ta jek" or "two triplets and one" and eight is "dui stors", i.e., "two quarters."

The Gypsies speak an Indo-German language and therefore there is an indication in the foregoing that our ancestors also counted in such a primitive manner. I suspect that we are dealing here with the remains of an ancient Indo-German counting method, in which only individual numerals existed for the numbers from one to four. It is true that the occurrence of these numerical wordings does not mean much to the English Gypsies, but there are a few other facts that support our hypothesis: The numeral for eight is clearly a dual in various Indo-German languages. I need only remind you of the Indian and Gothic languages in which languages this numeral "astau" resp. "ahtau" occurs. These words for eight are apparently expressions for a couple. A few of what? It is obvious that a pair of foursomes is meant here.

Another important fact is that in the oldest Indo-German languages only the numerals of one to four are inflected, the next one is not.

Finally, we point to the shape of the signs that occur in a certain number script. The script referred to here was found in the Indies and probably dates from the first century BC, so that it may be of Indo-German origin. The numbers one, two and three are respectively indicated by one, two and three vertical stripes, the number four on the other hand by a cross, the widespread symbol for the four points of the compass. Well, in this script the sign for eight consists of two crosses, that for five of a cross and a vertical line and that for six of a cross and two lines.

However primitive the systems discussed here may seem, there is nevertheless a kinship with our own system. Is, after all, for example, the numeral "twenty-five" also not compiled?

For modern man, the significance of the number lies largely in the role it plays in trade, in technology, and in natural science. For natural people, the number usually has a completely different meaning. When we talked about dualism, we already saw the meaning that the number two can have for the imaginary life of primitive man. Two is not alone in this respect. As an example, I cite how in some higher developed natural peoples the social structure of society is partly governed by certain numbers. To this end, I refer to the investigations that were made in 1929 by the then inspector H J Jansen in the Ambon Moluccas and in which the significance of the numbers 3, 5, 7, 9 for the Native population was highlighted. This is the curious concept of "oeli", which is probably best described as an organic one: an autarkic group of families or villages. These oelis are almost always composed of 3, 5, 7 or 9 parts. The oeli with the simplest structure consists of three parts, one of which, the middle, has social dominance over the other two and also fulfils a representative function. Two such systems can form a new unit, with the two centres merging. In this way an oeli is formed consisting of the middle and four extreme parts, i.e. of five components. The number five and especially the five-part body is identified with the human body consisting of the main body and the four limbs.

An even higher organization is, for example, obtained from the union of two oelis of five parts, whereby one is assumed to be male, the other female. Then a new oeli is created with nine parts, because now again the two centres merge into the middle of the new system. Unfortunately, there is not enough time to go into this interesting issue.

For natural man, a number is not an abstract concept, not a link in a logically constructed system, but it is a mysterious individual, filled with good or evil potential. Some numbers make him happy, others frighten him. Besides, even for many of us the numbers 7, 11 and 13 still have a very special meaning. Almost every natural people has its "holy" numbers. For example, four is such a number in the vast majority of Indian tribes.

Some peoples also find a reluctance to count people, animals, or valuables for fear of unleashing demonic powers. In the past, when someone at a meeting of the South African tribe of Thongas inquired about the number of people present, they were shocked: "What? You want to count us? Who is here who you want to plunge into destruction?" In a fable of the Avatime a spider easily makes a living. She builds seven heaps of earth and has them counted by different animals, which of course are marked down for death. All of this strongly recalls the Bible story about the disaster that was brought upon Israel when Satan had urged David to count the sons of Israel.

Number magic and mysticism flourish especially among the higher natural peoples. Moreover, the cultured peoples of antiquity were in no way inferior to them. Just listen to the following so-called prayer of the Pythagoreans, addressed to tetraktys, the holy foursome:
"Bless us, divine number, you who have begotten the gods and men! O holy, holy tetraktys, you who contain the origin and source of the eternal stream of creation! For the divine number begins with pure and deep unity and then attains the holy foursome; then it begets the mother of the all, who embraces everything, connects everything, the first-born, who never deviates, who never gets tired, the holy ten, who holds the key to all things."

In many cases the odd numbers were considered to bring happiness as opposed to the even numbers, the latter often being identified with the feminine principle in the cosmos. For example, at Vergil an abandoned mistress tries to reclaim the unfaithful:
"Draw, O my singing, Daphne away from the city to my house. First I wrap these nine threads of three different colours around your image and run it three times around this altar; the odd number pleases the deity."

In Shakespeare it reads: "... I hope good luck lies in odd numbers ...., they say there is divinity in odd numbers either in nativity, chance or death."

In addition to the number three, especially seven was highly regarded. With the Babylonians, seven was the number of parts of which all perfect things consisted. There were seven heavens and the underworld also consisted of seven departments, closed by seven gates. In the myth of Istar's journey to this abode, the goddess must discard one of her seven cloths at each gate. Seven thereby became a concept that indicated that something was complete. The Babylonian world map shows seven countries, i.e. all the countries. It is said of the enormous number of demons: "They are seven, they are seven, twice they are seven!" Seven is also the "perfect" number in the Bible. The creation took seven days to express that it was complete. Job's friends were speechless for seven days because they were completely defeated at the sight of their friend's misery.

The secret doctrine of the numbers was highly regarded by the ancient cultured peoples, especially by the Babylonians. A clay tablet teaches us that it is the goddess Nisaba who "knows the meaning of numbers" and there is also a list of gods with the corresponding numbers. The mathematicians of those days had to be well aware of the magical powers that work in the numbers. King Sargon II informs us that he had the wall encircling Sargon's castle made 16280 cubits long, because this number corresponded to his name. The "wise writers of numbers," as a cuneiform text calls mathematicians, were united in a kind of "Mathematical Centre": "the House of Moemmoe." The divine patron Moemmoe was the son and spokesperson of Apsu, the god of primeval water, and we must see in him the personification of the intelligent world. When a king could not find the original building charter of a temple, "he gathered", as a text says, "the elders of the city, the inhabitants of Babel and the wise mathematicians who live in the House of Moemmoe and the great secret keep of the gods ".

Such a mysterious doctrine of numbers is still known today among some of the highly developed peoples. The place Lilibooy on Ambon and the neighbouring island of Noesa Laoet, for example, must still include old people who are proficient in this art. On the other hand, men such as Plato, Dante and Kepler were strongly influenced by numerical mysticism, yes, even a modern revival of Pythagoreanism can be seen in modern natural science; I only recall Eddington's well-known reflections.

We have seen in the foregoing how inadequate the number concept among primitive peoples is. Some writers, in particular the school of Lévy-Bruhl, saw in it a proof of their proposition that the spirit of natural man is essentially different from ours. Later. Subsequent investigations, however, have shown the incorrectness of this claim. Several times in the foregoing we had the opportunity to note, how primitive forms of thought and methods are apparent to us in the area of the number. We must assume that the concept of number among our ancestors was initially on a plan that was as low as that of the most uncivilized peoples, and that this concept has grown over time.

I cannot go deeply into this issue here, but I would nevertheless like to quote a few examples which show how poorly understood in ancient times was the understanding of large numbers. We repeatedly meet in the Bible where we are assured that certain numbers were too large to be counted. That is what it is written in 1 Kings 8: 5: "Now King Solomon and all the congregation of Israel who were assembled with him were with him before the ark, sacrificing sheep and oxen, who could not be recorded nor counted because of the multitude" and in Genesis 41: 49 we read: "And Joseph stored up grain in great abundance, like the sand of the sea, until he ceased to measure it, for it could not be measured."

Counting large numbers in antiquity was not a matter of which people thought lightly. Herodotus tells us how Xerxes counted his army of millions in a plane on the coast of Thrace: "A group of ten thousand men was brought together in a certain place and these people were squeezed as closely as possible. Then a circle was drawn around them, on which these men were released. Then a fence was erected along the circle at the height of a man's waist. Then other men were driven into the space inside, until the entire army was counted in this way."

Even the Greeks, who by the way contributed a lot to clarifying the concept of numbers, had a certain aversion to working with large numbers.

Yet there was one people in ancient times, for which the foregoing did not apply and whose spirit was, as it were, wandering in an orgy of large numbers. They were the Indians. They liked to dream in their own world of fantastic dimensions and their imagination recognized no boundaries. Often it was religious representations that stimulated them to continue the sequence of numbers. The Indian knows no measure in the number of his deities. There is a mention of 330 million in a popular saying and even $24 \times 10^{15}$ in the epic Mahabharata. Before Buddha retreated to the solitude of the wilderness, his father seeks to bind him to earthly life and to entrap him - as the story says - "in a dungeon, where the small jailer, the joy was a doorkeeper". He gave his son a harem with only 84,000 women! In another place we read that Buddha must have had 600,000 million sons. Also listen to the so-called Buddha proclamation:
"At 32 main and 80 secondary marks, Buddha, to 32 his mother, to 8 the house, where he will be born, being recognizable. His mother, the queen Maya-Devi, is served by 10 million women. Hundreds of thousands of saints and one hundred thousand millions of exalted ones will pay tribute to Buddha. His throne is composed of the good works during one hundred thousand millions of world periods. The great lotus, however, which blooms on the night of Buddha's conception, opens its flower in a circumference of 68 million miles."

You can see how the number here is a means to strengthen the poetic force.

With this I want to end; I hope to have given you an idea of the difficulties that primitive man had to overcome in simple counting, and also that the path that led to the modern abstract number concept must have been a very long and difficult one for mankind. I hope, furthermore, that with all these numbers I did not give you too much - to speak with the Bakai - "kinarachu iwano".

Last Updated November 2019