THE USE OF NUMBERS
THE USE OF NUMBERS IN THE ANCIENT NEAR EAST AND IN GENESIS
1) The Anthropological Evidence
In considering the use of numbers by people who are not immersed in the scientific age we can look to two sources.
1) Anthropological studies among primitive tribes. These will give a far more likely answer to questions as to how the ancients saw numbers than guesswork.
2) Archaeological evidence, interpreting it partly in the light of anthropological studies.
Let us therefore look briefly at some of the anthropological evidence.
The Anthropological Evidence
D E Smith in “The History of Mathematics” informs us that when the number words of thirty selected Australian languages were examined, they were ‘mainly restricted to “one” and “two”, and in no case extended beyond “four”.’ He points out that, in general, everything beyond “two” is called “much” or “many”. This was not universally true of all Australian aborigines, for in one linguistic region tribes had number names reaching up to as much as “fifteen”, and even in some cases “twenty”, but they were the exception.
This is very significant as it demonstrates that in spite of the fact that a few groups could count in a very limited fashion, it did not ‘catch on’ with the remainder, who continued to ignore the use of numbers. (We might have expected the opposite).
Furthermore it seems reasonable to assume that there is a good probability that the restriction to “twenty” of even the ‘advanced’ aboriginal tribes was the result of the number of fingers and toes available. They did not use figures naturally, but laboriously. Clearly trading by number was much restricted, and the counting of animals or other objects very limited. Once even the ‘advanced’ men had reached the end of their fingers and toes the idea of number faded into obscurity, while for the majority it had never even begun!
So for the large majority of Australian aborigines numbers and number words had no meaning. Try to imagine such a world. It is almost inconceivable to us. How could they keep a record of animals owned, or objects possessed? How be aware of the size of their own tribes? Clearly it was not a problem to them. In tribal life numbers were of limited use.
The Bay of Bengal
These facts are borne out among the Andamans in the Bay of Bengal, where counting was accomplished by the use of fingers. In order to count they touched their noses with one finger after another, and once they had gone through their ten fingers they put their hands together, saying “all”, signifying they had reached what we would call “ten”. This was the limit of their numbering ability. What is even more interesting is that they did not have number words for even this primitive ‘counting’. They could simply ‘count’ silently by using fingers in strict order, the idea never being verbalised. Numbers were thus never used in conversation or in tribal stories.
Menninger in “Number Words and Number Symbols” describes an example among the Abiponese Indians in South America. A missionary working among them discovered that they had no idea of number at all, and when he attempted to teach them he found them very resistant. Yet he noted how they were able to keep track of their dogs and livestock merely by the fact that they noted any gap in their total world picture. They did not consider that an awareness of numbering would have any advantages to them. In lifestyle they provide an interesting parallel to the semi-nomadic patriarchs.
Another interesting example comes from Malaysia where a Malaccan old man, when asked his age, proudly replied “I am three years old”. To the Malaccans age was counted by whether you were in childhood, of adult age or of mature years. Most died at “two” years old. It was a feat to reach “three”.
These examples can be parallelled repeatedly. When I mentioned these examples to my brother-in-law, who worked among head-hunting tribes in New Guinea, he commented that they also did not count beyond three, and this in the latter part of the twentieth century after contact with ‘civilised’ society.
These examples bring out the fact that, in general, those who have not been affected by the influence of educational systems do not use or comprehend numbers, and when confronted with them, have no wish to do so. It is something foreign to their way of thinking, in which they have no interest, and to which they have no wish to apply themselves. The Ancient World
In the ancient world, therefore, we have good reason to assume that a similar state of affairs largely held good. The vast majority of people did not need to be able to count, and if our examples show us anything, would resist attempts to teach them. They would leave it to the ‘clever ones’, and interpret number words in a general, numerically vague, fashion..
Three Biblical examples spring to mind. When in, 1 Kings 17, Elijah meets the woman of Zarephath she says (v.13) that she will gather “two” sticks in order to bake a cake. What she means is “a few”. Had she intended to gather a lot she would have said “three”. (In ancient Egyptian the hieroglyph for “three” also originally meant “many” before it began to be used as a number symbol as well). She clearly had no use for number words except for generalisations.
Language Changes and Develops
We must remember that when we look at ancient languages from a modern point of view, there have been considerable changes over the centuries. “Hebrew” did not commence as a sophisticated language and stay the same over two thousand years and more. It developed and grew from primitive beginnings, and the meanings of words changed over the ages. Canaanite examples from ancient Ugarit (canaanite was a very similar language to hebrew) have helped us to know long forgotten meanings of hebrew words, and have shown us that we were mistranslating because the ‘experts’ had assumed that the ‘modern’ meaning of the word must be the correct one. The same was true of number words. The Hebrew word for “a thousand” also means a “family”, “clan”, or “captain”. At what stage did the change take place when the word for “family” was also used for “a thousand”? When did the idea conveyed by “ten families” (or clans), and the number it conveyed, change to mean “ten thousand”? It may be that we are misrepresenting a large number of early Hebrew “numbers”.
Other Biblical Examples
The second example is found in 2 Kings 6.10. Here the king takes advantage of Elisha’s advice a number of times, and this is represented by “not once, not twice”, clearly the equivalent of “a number of times”.
The third example is even more interesting. The compilers of the books of Samuel and Kings drew on ancient records, which were maintained by the court ‘recorders’ from the time of David onwards, when the court of Israel first became sophisticated. Included in these records were details of the kings reigns. “So and so was so many years old when he began to reign, and he reigned for so many years”. This refrain comes again and again. But when they looked back to Saul they had a problem. He had not had a “recorder”. He was a primitive war-leader. So they took advantage of an ancient device. They knew that he had become king at an early age, and that he had died in his maturity, so they wrote “Saul was one year old when he began to reign, and he reigned two years in Jerusalem” (1 Samuel 13.1 literally). Translators constantly had difficulty with this verse because they did not understand the meaning that Saul began to reign while still a youth, and only reigned to adulthood, not reaching old age. It is especially significant that the writers did not try to invent figures. (They were not as casual with the truth as modern men like to make out.)
The Conclusion We Can Draw
Thus it is clear that while numbers were handled quite skilfully in court circles, at least among the recorders, the majority of the people of Israel, like the primitives of modern times, were much more limited, and writings aimed at being used at their festivals would have this in mind. It is probable, for example, that the creation narrative and other covenant narratives, would be read out at one of their major festivals, as happened among other nations. As a “covenant people” something would need to be recited at the festivals which would remind them of their covenant relationship with God. Even by around the time of Jesus “three days and three nights” (Matthew 12.40) could mean part of today, tomorrow and part of the third day, - only two nights and two part days plus one full day being indicated. (Compare Luke 24.21; Mark 8.31; Luke 9.22; Matthew 12.40 for contrasts). This usage is witnessed to in literature outside the Bible. It seems remarkable to us, but it was in fact general usage.
How Then Did They Trade?
How then did people trade? The answer is that they used “tally sticks” or stones. They knew their requirements and indicated them, or checked that they had received the required quantity, by means of notches in wood or bones, or by assessing against a number of stones. A businessman would have a slave with him to carry the quantity of stones required. They did not need to calculate, as they knew in their heads the quantity (not numerated) that they required. Our modern examples have confirmed that this did not necessarily require the ability to use numbers.
Once they got beyond barter they made payment in gold and silver, which was weighed out on the merchants scales, taking place item by item.It is probable that the actual use of numbers was developed because once men began to live in a developed civilisation (e.g. the Sumerians, who probably first invented numbers, as they first invented writing) they had to control estates, pay taxes and trade more widely and to a larger extent. But it must be remembered they still did these things for many centuries without numbers. In the end it was just a staggering new invention, which took a long time to catch on away from Sumer.
(2) The Evidence from Archaeology
In The Exact Sciences in Antiquity, Neugebauer describes the growth of numbers in ancient Sumer. (Kramer has ably demonstrated that ‘Sumer’ is probably synonymous with ‘Shem’ in Genesis 10).
He points out that the word used for “one” (as) is the same word as that for “man”, the word for “two” (min) is the same as that for “woman”, and the word for “three” is the same as that for “many”. This ties in with the Egyptian use of the hieroglyph for “many” for the number “three”. This appears to go back to a time when the use of ‘numbers’ was restricted to myself, my wife and the rest of the world. Only gradually did they come to have genuine numerical significance. This fits in well with the anthropological evidence.
Hints in Other Languages
In both ancient hebrew and ancient greek (to name but two) verbs had a singular, a dual and a plural. One of a thing was represented by the singular, two of a thing by the dual, and many things by the plural. To the ancient Egyptians the hieroglyph of a dash “-” meant one of a thing, two dashes one above the other “=” meant two of a thing and three dashes under each other meant “many”. This latter eventually came also to represent ‘three’.
This conclusion is further bolstered up by the fact that in many languages the words for “one”, “two” and “three” declined like adjectives, because they were seen as descriptive rather than quantitative. It was only when numbers extended beyond these that they ceased ‘declining’ them, because numbers were gradually becoming distinguished from adjectives. As we shall see, among the common people this process took some time.
A Further Look at Sumer
The use of “numbers” as adjectives comes out forcibly in the writings of Sumer. Over the thousand years before the time of Abraham the Sumerians, who were almost certainly the inventors of numbering (certainly in recorded form), developed numeracy from small beginings to an amazing extent. Numbers were developed for business purposes (c.3300 - 2000 BC), although utilised in the later stages for historical records.
But, of course, this was only the scholars and highly educated business men. The stories for common people were not affected. In literature for the common people, the myths and legends of ancient Sumer, only the numbers “three” and “seven” were ever used (apart from one stereoyped phrase which in fact emphasises the fact). Both these numbers were used constantly, and an analysis of the literature shows clearly that “three” signified a sense of “completeness” - three of a thing meant that there was all that there needed to be - looking back to the time when three was the totality beyond which men could not go numerically. “Seven” had come to mean especially completeness in what was divine.
For example seven gates were portrayed to Sheol (the world of the dead) because this demonstrated their absolute impregnability. There was no way back from the underworld! (Ininna - later Ishtar - was to be the exception that proved the rule, but this was in order to achieve good harvests. The fertility of the fields was supposed to arise from the death and subsequent coming back - a reviving, not a resurrection - from the dead of a god or goddess).When the common people heard that there were ‘seven’ they did not think of the number but of the quality of divine completeness.
Remember these stories were not just tales to be told for entertainment, they were recited and acted out at religious festivals for the purpose of producing certain effects in nature for the benefit of all. A ‘myth’ was strictly the spoken part of the drama, carried down century after century, not a vaguely remembered history of the past.
Creation and Flood Stories in Sumer
The Sumerians also saw ‘creation’ as taking place in “seven” days (not the six days and one of rest of Genesis) and the flood storms lasting for “seven” days, both signs of a divine period of time and of the completeness of the actions, again in accounts intended for public religious purposes. Genesis 1 probably had a similar purpose, although not necessarily in the same way. In this case it was to remind the people of God’s covenant with them in providing them with food and control over nature (1.28-30). It is probable that it too was first intended for recital at festivals, as a reminder of their Creator God and of His covenant with Adam. By taking part in the festival the people saw themselves as entering into the benefits of the covenant).
This did not apply so emphatically with historical lists which had no such religious purpose. Thus, for example, in Sumerian literature (translated by Kramer) in a vivid narrative there is the rare use of a different number in the counting of five piles of bones of the dead after a battle, but that was in a strictly historical account and was a kind of list. The purpose of piling up the bones was to assess the number who had died in battle. The method demonstrates that the ability to count is restricted. It is clear that the use of “three” and “seven” had become standard usage in “religious stories”, used in religious ceremonies, while numbers were developing, and the practise had not changed. They conveyed to the majority, who would be innumerate, the idea of completeness and divine certainty. Other numbers were a business tool and useful for recording lists, but they were still a mystery to the majority.
This use of “three” and “seven” can be witnessed to in large numbers of later accounts from other sources outside Sumer where the numbers “three” and “seven” continued to have special significance.
The Sumerian King Lists
Another interesting example of number usage was in the Sumerian king lists. In these were listed the kings “before the flood”. They are portrayed as having lived for large numbers of years. However, their ages are all in round numbers (on the sexagessimal system) varying in one case between 10,800 (60 x 60 x 3) years and 21,600 years (60 x 60 x 3 x 2), and it must be seen as questionable whether they were ever intended to be taken literally. They were rather a deliberate attempt to portray their mystical status, and possibly contained within them some remembrance of their unusual longevity. (It may even be that the number symbols used had a different meaning when the lists were first compiled).
The General Position
Outside the use for recording business and historical information exact numbers were almost certainly seen as having little importance, and were undoubtedly a mystery to the majority of people. Such numbers as were used were rather used with an adjectival significance. They denoted quality, rather than quantity. Quantity was incidental. It is indeed questionable whether outside businessmen and scholars very many used numbers numerically at all. The few number words that were used, were used to denote an idea. We can compare this with when we say ‘I had a hundred (or a thousand) and one things to do’. It sounds exact, but it simply means ‘a lot’. Or when we say ‘I’ve got dozens of them’, multiples of twelve do not readily spring to mind. This was their natural way of interpreting number words.
The Magic of Numbers
The awe with which someone who could use numbers was looked on is brought out in the Egyptian funerary texts. In the ‘spell for a obtaining a ferryboat’ (to take someone across the river of death into the after world) we have the account of a deceased king who is seeking to persuade the ferryman to take him across. The ferryman objects on the grounds that the gods will question his right to carry across someone who could not “number his fingers” i.e. use his fingers for calculating. Clearly the art of calculating was looked on as giving men powers beyond the ordinary.
Fortunately for the king he was able to remember a rhyme enabling him, superficially, to “number” his ten fingers (the rhyme was to be remembered by all who wished to deceive the gods and cross the river of death securely, and was intended to be used as a spell) and he is allowed across as ‘a great magician’. Thus even a king would not necessarily be able to calculate with numbers and had to use a gimmick , thus portraying himself as having super-normal powers.
A further example is found in the Papyrus Rhind, dated about 1800BC, but from earlier sources. Its introductory statement reveals the awe in which number crunchers were held for it promised to give “directions for obtaining knowledge of all obscure things” and then goes on to deal with number problems.
It should perhaps be noted for completeness of the picture that the ‘Old Babylonians’ who followed the Sumerians, developed numbering into a mathematical art. They calculated pi exactly (the Egyptians found it approximately by measurement), and compiled large numbers of lists of equations, including tables using the so-called theorem of Pythagoras. They developed mathematics to a height not attained for another thousand years, but were hindered by not having thought of the zero. However, these advances took place after the time of Abraham (c.1900 BC).
(3) The Development of Numbers
The Oldest Known Record
Perhaps at this stage we could look at the development of numbers. It is, of course, to the Sumerians that we must look for the first example of numbers. The numbers one, two and three are found on the oldest clay tablet known, found near Kish and dating back to 3300BC, five hundred years before anything similar is found in Egypt. It is the first known example of ‘writing’, but contains only a few brief symbols and is so brief that it is untranslatable. It is possibly a business record, and probably an indication of the very limited nature of the symbols used for recording at that time.
There are certainly numbers on a tablet found at Jemdet Nasr (near Kish) dating from 3100BC. It is far more detailed and includes numerical symbols for 15 and 40 in the form that was to become the regular pattern for centuries to come - circles made by digging the end of the stylus into the clay to denote tens, and half moons made by digging the end of the stylus in at an angle to denote units.. It appears to be a record of property.
From now on things develop rapidly. The standard forms of business document were clay tablets the size of a man’s hand, and by careful examination we can trace the growth of different number systems. This demonstrates that numbers developed along with the ability to record them. There was thus not an already established number system used in speech. This confirms that the use of numbers was still in a formative stage. Numbers had not developed in oral form before their use in recording.
As might be expected it was not an exact science. The different systems developed side by side so that we have examples of both decimal and sexagessimal systems, but interestingly enough it was the sexagessimal system (still used by us to measure seconds and minutes) that prevailed for later purposes.
Examination of Sumerian number names suggest how numbers developed. The first five numbers had names of their own, distinctive from each other e.g. one = as; two = min; three = es; four = limmu; five = ia.
Six, however, was a symbol representing ia-as, that is five plus one, while seven was i-min (ia-min) representing five plus two. This suggests that at one time five was the limit of counting (on the fingers, compare the use in Egypt of the hieroglyph of a hand for “five”), and six and seven were made possible by retaining them in the memory. We can compare with this our own number system where eleven = eleph en, that is “one over”, while twelve is two eleph, that is, two over. Once the number of fingers had come to an end, the memory was able to work to another two. That may be why thirteen became seen as unlucky.
We can see from this why the numbers three and seven were special. “Three” was originally the sum total of counting as we have seen, representing ‘everything’, as it still did to the large majority of people. “Seven” became the same once numbers had developed sufficiently. Once seven was reached there was nowhere else to go - at least for a time. It was a number of the gods. Anything beyond was a mystery. They had reached the furthest extent of knowledge.
Numbers gradually developed as they were used for business records, such as lists, and, later, other transactions, the development taking place over hundreds of years.
Then at some stage someone had the idea that this new business method for noting down business records could be used to convey more complicated messages, and the writing of narrative came into being, which enabled the writing down of the scripts for the religious festivals and the recording of historical events.
The Sumerian Number System
Meanwhile numbering was developing further. Eight was issu, nine was ilimmu, (that is ia-lim i.e. five plus four), and ten was u. From ten to twenty u plus a numeral signified the number e.g. u as (ten plus one), u min (ten plus two); and so on. Twenty was nis, thirty was usu, (that is es-u i.e. three times ten), and forty was nin (ni-min i.e. two times twenty). Fifty was ninu (ni-min-u i.e. two times twenty plus ten). Finally sixty was reached which had its own name ges, and became the basis of the system that prevailed, although it was not the only system for a long time.
So even at the lower level it was not a decimal system, but a system based on fives, tens, twenties and sixties, and indication of gradual build up.
The next major stage was the sar which equalled three thousand six hundred (sixty times sixty), followed by the sar-u equalling thirty six thousand (ten times three thousand six hundred and the sar gal (the great sar) equalling two hundred and sixteen thousand (sixty times sixty times sixty). This development took place over a thousand years. It did not all happen at once.
There were symbols for units, represented by so many small half-moons, tens represented by small circles, and sixties symbolised by a larger half-moon, made with the end of a bigger stylus. A hundred was symbolised by a large circle. Thesystem was thus a mixture of a system of tens, twenties and sixties, demonstrating the complexity of its development and the differing systems which were incorporated. As most people could not even read we can see why they began to impute magical powers to mathematicians.
A further big step was made when wedge shaped writing (cuneiform) was developed, and symbols representing units and tens could be used to express all numbers, depending on placement. With all their wisdom, however, they never developed the zero, which proved something of a restriction and could sometimes lead to confusion as to what number was being represented.
Chinese mathematics was almost certainly borrowed from the Old Babylonians coming to them via the people of Susiana, who passed it on to the Bak tribes, who migrated to China. There seems little doubt that before the Sumerians numbering was unknown.