ARITHMETICS – ΑΡΙΘΜΗΤΙΚΑ – АРИТМИТИКА


The origin of numbers. Counting without numbers. “Natural” number systems.
Introduction

It is often said that the essential tools for life are reading, writing and arithmetic. While this may not have been true for most people until very recently in history, it is certainly true for science. A moment’s reflection will also show that the most basic of the three tools is arithmetic: Writing may allow the science-priests to hand on their knowledge to future generations, who can absorb it through reading; but without the means of quantification the best science does not make much progress. The need to be quantitative varies of course from discipline to discipline, but even an essentially descriptive science such as taxonomy has to make quantitative statements about sizes and life spans. A history of science therefore has to begin with the question: Why did people invent numbers? To find the answer we follow George Ifrah’s Universal History of Numbers (Ifrah, 2000).

The invention of the modern number system is such a stroke of human genius that it is worth a detailed discussion. Its development. can be described as a passage through four stages:

  1. Grasping quantities
  2. Counting without numbers
  3. Using absolute number systems to count and calculate
  4. Using place-value number systems to calculate with pen and paper or calculator
Grasping quantities

We begin with the question: Do you need numbers to count? Framed in a different way, we may ask: If we are shown a group of identical objects, can we tell how many objects we see without counting?

It is known from observation and experiment that animals cannot count, but some animals can grasp quantities to a limited extent. An interesting anecdote about a French nobleman reports that he tried to rid himself of a raven that lived in the tower of his castle, but every time he approached the tower the raven flew to a nearby tree and waited until the nobleman had left. The nobleman sent two of his servants out and told them to enter the tower together; one of them should then leave, while the second servant should wait in the tower for the raven’s return. The raven was not fooled and waited until two persons had left the tower. The nobleman repeated the exercise with three of his men, then with four. The raven always waited until all men had left. But when five men entered the tower and four left, the raven could no longer grasp the difference and returned. (Dantzig, 1931)

This observation shows that the ability of animals to grasp the number of objects in a group without counting is quite limited. That the same is true for humans can easily be demonstrated with a simple test, which you should do at this point before reading on.

Compare your own data with the correct sequence(2 4 9 1 7 5 9 2 8 3 1 6 4 7 6 3 8 5)

You will find that the largest number the human brain can comprehend without counting or guessing is 4. Beyond that most people can identify 5 elements in a group by quickly counting them; everything beyond 5 can only be a guess, unless there is enough time for a count.

We can make the statement about the test result more scientific by giving it to a group of people and looking at the aggregate performance. I gave the test to two groups of “University of the Third Age” students, where the average age was well over 65, and to two groups of secondary school students and plotted the results. The graph shows (

test results



Here are the results from four classes that took the test. The curves show the percentage of students that recognized the quantity shown on the slide correctly. Blue curves are from classes of people 65 years of age and older, red curves are from classes of high school students. The n values indicate the sample size (people in each class).

The results show that virtually all students recognize quantities up to 4 or 5 correctly. (The high school students were less organised than the senior students; some missed the first few slides, which explains their performance slightly below 100% for small quantities.) For larger quantities the success rate falls off rapidly irrespective of age.

) that the accuracy of the result deteriorates quickly for numbers larger than 5 and that this is true regardless of age.

At the hunter-gatherer stage people had no need to count and invented words only for quantities that could be grasped without counting. Languages of such a society therefore had names only for the quantities one, two, three and four; everything beyond that was “many.” Examples for such a situation can still be found today where the hunter-gatherer society survives; they are (among others) the Aranda of Central Australia, the people of the Murray Islands in Torres Strait north of Australia, the Indians of Brazil and in Tierra del Fuego, the Abipón of Paraguay and the Bushmen of Africa. The Australian examples may stand here for all:

image

Such languages are often called primitive languages. This is correct in the sense that they represent a development of human society before the onset of civilization. It would, however, be extremely misguided to think that the people of today are intellectually superior to the people of the Neolithic age. The species Homo sapiens did not evolve significantly since the stone age, and as our test showed our ability to grasp quantities beyond 4 is the same as the ability of the hunter-gatherers of old. As our investigation progresses we shall see that the average Neolithic man or woman would not have any difficulty at all to catch up with our civilization within less than one generation. Every society develops the tools needed for its everyday activities. Egalitarian societies in which no person owns anything much and land is communally held have no need to count. If today’s hunter-gatherers do not count beyond 4 that only proves that they have no need for it.

Counting without numbers

If primitive societies have no need to count and therefore do not develop names for numbers beyond 4, when did the need to count arise? It could not have been the large numbers of animals in a herd – the Australian Aborigines also encountered many kangaroos during their hunting trips and were satisfied to say that there were many kangaroos, enough to satisfy their need for food and clothing. The need to establish the exact number of animals in a herd arises as soon as this herd becomes private property and can be traded.

Does the owner of the herd need words for numbers beyond 4 to be able to count his animals? Consider this problem: The owner of the herd asks you to deliver 40 head of sheep to a friend of his and return proof of delivery. He hands you a clay container, sealed with his personal stamp, containing 40 pebbles.

His friend will open the container and count the sheep by picking up one pebble for every sheep. He will then give you a receipt, for example a clay tablet with his personal stamp. By handing over the clay tablet to the vendor you can offer proof that you delivered 40 head of sheep as asked. Nobody involved in the transaction has to know the names of any numbers or be able to count. Evidence that this method of “counting” was widespread through several millennia is found in excavationsfrom Mesopotamia and retained in the English language: Calculus is Latin for “pebble”, so to calculate means “to move pebbles.”

This method, sometimes called counting by association, works fine as long as the number of sheep is not too large. How do you use pebbles to count very large numbers? You introduce pebbles of different shape. This system of counting was in use in many early societies of all continents.

Number systems

To associate pebbles of different shape with different values requires a number system. Our modern number system, known as the decimal system, uses the base 10, so we would use different pebbles for 1, 10, 100, 1000 etc.

Is there a “natural” number system? The usual response to that question is: “Of course; 10, because we have 10 fingers.” All evidence shows that when humans begin to count they use their fingers, so using body parts as a base for a number system does indeed come naturally. Nevertheless, the usual answer is a good example of cultural bias. When I visited Ocean University of Qingdao in China in the 1980s and 1990s I did not speak Chinese, but I quickly learnt how people in China indicate numbers with only one hand – they can indicate any number with five fingers by forming shapes with their fingers, rather than using the second hand to lift one finger at a time to indicate 6, 7, 8 etc.

The practice of counting on one hand was in use in Europe until at least 1600. Calculation tutorials from the time were not considered exhaustive if they did not include a description of the finger counting method. The method is superior to our way of counting with ten fingers in several respects. Anyone who has observed the activity in a noisy stock exchange has seen brokers signalling numbers across the room with two hands. The Chinese method allows the use of only one hand and leaves the other free for the mobile phone.

There are several ways to use your fingers for counting. A method still widely practiced in the Middle East uses the thumb to point to different parts of each finger. Each finger can be used to indicate three numbers, so the four fingers of one hand cover the numbers 1 – 12. This produces the number 12 as an alternative base for a “natural” number system, known as theduodecimal system.

A logical extension of the duodecimal system is the sexagesimal system, which uses the base 60. It uses one hand to count from 1 to 12 and the other hand to indicate the multiples of 12.

Yet another “natural” number system developed in regions of benign climate where people do not wear shoes and therefore can also use their toes for counting. This produces the vigesimal system, a system with the base 20.

We notice that the question what is “natural” does not have an easy answer. Even tday we can find evidence for a range of number systems. Some people were content to use only one hand for counting to five; their quinary system used the base 5. This system survived in the language of the Api from Vanuatu in the South Pacific. They use the decimal system today, but their names for the numbers 1 – 10 are

image

Evidence for the use of the duodecimal system during some developmental stage of science survives in most European languages. The normal construction of numbers larger than 20 in the English language is twenty-one, twenty-two, twenty-three etc. A similar system is used for the numbers 13 – 19: Three-ten (thirteen), four-ten (fourteen), five-ten (fifteen) etc. But the two numbers 11 and 12 have their own names, eleven and twelve, which are not composites such as twenty-eight or thirteen. The same “anomaly” occurs in the German, French and other languages.

Evidence for the use of the sexagesimal system survives in the French language. French names for numbers follow the normal decimal construction up to sixty (soixante) and continue on in the normal way sixty-and-one (soixante-et-un), sixty-two (soixante-deux), sixty-three (soixante-trois) and so on up to sixty-nine (soixante-neuf). But the numbers then continue as sixty-ten (soixante-dix), sixty-eleven (soixante-onze), sixty-twelve (soixante-douze) etc. The system returns to normal decimal counting at 100.

The following table shows which number systems have been in use in different parts of the world. It is evident that today’s decimal system is only one of several options. It has certain advantages over the sexagesimal system, which will become evident in the next lecture, but is not superior to any of the other systems. Its use today is more the result of historical accident than inherent advantage.

number systems used by different societies.

image

 

Summary

The next lecture will discuss how different civilizations developed writing systems for numbers. It will build on the outcome from this lecture, so let us summarize the essential points:

  • The ability of the human brain to comprehend quantities without counting is limited to 4 or 5 items.
  • This ability is the same in any human society, from hunter-gatherer to modern urban populations.
  • The need to develop methods for counting did not exist until some individuals acquired large personal wealth in early husbandry societies.
  • The invention of numbers is not the result of general curiosity but the answer to a problem that occurred with the accumulation of wealth during the development of society.
  • Different civilizations used different number systems. All were based on the use of the human body for counting.
Reference

Dantzig, T. (1930) Number, the language of science. London.

Ifrah, G. (2000) Universal History of Numbers. John Wiley & Sons. Translated from Ifrah, G. (1981) Histoire Universelle des Chiffres, (Seghers).

 

SOURCE  http://www.es.flinders.edu.au/

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