Notes for CMTHU201 History of Mathematics in This Course We Will
Total Page：16
File Type：pdf, Size：1020Kb
Notes For CMTHU201 History of Mathematics In this course we will study five early centers of civilization and track the development of mathematics in those five areas. There are other ancient centers, but the records for these is not as complete. Our five centers are the Tigris-Euphrates region, the Nile River Valley, the Aegean Sea area, the Yellow River Valley in China and the Indus River in what is now Pakistan and the culture of India that grew from this. We will then go on to study the first civilization that benefited from the mathematical and scientific discoveries of all five of these centers, the combined Persian and Islamic societies. I Prehistory The origins of counting in a broad sense are earlier in the evolutionary process than the emergence of man. Many animal species can tell the difference between the presence of a predator and the absence of the same. This carries forward to the difference between 1 and 2 objects. Wolves and lions know well enough not to attack 2 or 3 of their kind when they are alone. Experiments have shown that crows and jackdaws can keep track of up to six items. (In the Company of Crows and Ravens, Marzluff and Angell) “Pigeons can, under some circumstances, estimate the number of times they have pecked at a target and can discriminate, for instance, between 45 and 50 pecks.” (Stanislas Dehaene, The Number Sense) Human infants can discriminate between 2 and 3 objects at the age of 2 or 3 days from birth. This hard-wired and primitive sort of counting is known as digitizing. It extends to 3 and perhaps 4 objects, but not beyond that. Babies at the age of 6 months can correctly associate a number of sounds to a number of objects near them, a sort of grasp of abstract integer counting. Five month old babies also have been shown to grasp that 1 + 1 is not 1, nor 3, but exactly 2. (Cf. Dehaene, ibid.) With regard to the earliest records of human representation of numbers, one of the oldest is a piece of baboon fibula found in a cave in the Lebombo Mountains in Swaziland, the Lebombo bone. It has 29 tallying notches on it and has been dated to about 35,000 B.C.E. Another is a piece of shinbone from a young wolf and was found in 1937 in Pekarna, Moravia, Czechoslovakia. This bone has 57 notches, arranged in groups of 5 and is dated from about 30,000 B.C.E. A third example is known as the Ishango bone. This tiny 10 centimeter curved bone was found by a Professor Heinzelin on the shore of the Semliki River in Zaire. It was on the Ugandan side of the river, but in Zaire. This is just north of Lake Edwards. It has been dated to about 20,000 years of age. This bone has notched columns engraved on it, yielding patterned numbers. One interpretation is of certain periods of the moon. Certainly the association of early mathematics and early astronomy is a fact. We see that early humans had the ability to count, and that it had importance to them, enough to warrant the keeping of records. II Examples of Prehistoric Civilizations One very old example of social organization is found in south-central Turkey at a site called Chatal − Ho˙˙ yu˙˙ k. The town here had up to 10,000 inhabitants at some times in its 1,300 years of use, from about 7500 B.C.E. to 6200 B.C.E. It had no streets, as the people had no wheels yet. The mud houses were clustered together and entrances were through the roof. There is no sign of king or ruling classes, as the dwellings are pretty even€ in their appearance of owners’ status. There also seems to have been equality of sexes, without more elaborate burials of one or the other. There are no fortifications or walls around this site. Why the site was deserted is not known. (The Goddess and The Bull, Michael Balter) Jericho contrasts with this site in being one of the first walled cities. The ruins at Jericho include some dating back to 9000 B.C.E. The city was deserted about 700 years after the building of a tower and a wall around the city, about 7300 B.C.E. The earliest construction here is believed to belong to the ancient Natufian culture. We have no records of any mathematics from these old cultures and sites, but they are large enough to require organization and planning for harvesting crops and schedules for planting and hunting. We turn to Mesopotamia, the land between the rivers Tigris and Euphrates, for the emergence of a culture which left us evidence we can decipher. The people known to us as Ubaid people settled along these rivers, and the people in cities in the southern area did not leave, but began to create even more complex cities and societies. Why we do not know. The Ubaid period is dated from 5600 to 3900 B.C.E. The early Ubaid people began making pottery on a slow wheel. The town of Eridu is the largest of their sites, having a population of perhaps 5,000 people by 4500 B.C.E. A large mud-brick temple here was used at least from 4500 to 2000 B.C.E. There is evidence that the temple was for Enki, the god of water and city god of Eridu. We thus see a shared religious practice, as opposed to the household ritual in Chatal − Ho˙˙ yu˙˙ k. Larger, fancier houses are associated with the temple. The agriculture included growing of wheat, barley and lentils together with the husbandry of sheep, goats and cattle. Towards the end of the Ubaid period, many towns associated with the Ubaid culture began building walls around the towns. € At the same time the pre-Elamite city of Susa, in the watershed of the Karun River, east of Mesopotamia, was growing. The city was founded about 4,000 B.C.E. and the earliest pottery is not of Mesopotamian style. There is a period of Sumerian influence, followed by the proto-Elamite culture from 3200 B.C.E. to 2700 B.C.E. The Elamite language has not been translated and is an isolated language, like Sumerian, not related to any of our remaining language groups. The control of Susa and Mesopotamia shifted back and forth between Elamite and Sumerian control in the years 2500-2400 B.C.E. The Elamite culture is poorly understood, but remained strong enough to become part of the basis for the later Achaemenid Persian Empire. It thus forms part of the cultural roots of such mathematicians as Omar Khayyam and Nasir al-Din al-Tusi, both of whom we will study later. The Elamites originally came from the Iranian plateau. The Uruk period extended from 3900 to 3100 B.C.E. and included the innovations of the fast pottery wheel and of the wheeled cart. The largest early Sumerian site is at Uruk, which had a population of 10,000 to 20,000 by 3100 B.C.E. Early writing occurred in a pre-cuneiform style by 3400 B.C.E., before the Sumerian ascendancy. It was in the form of pictographs, simplified pictures of a fish, a bird or a man, pressed into soft clay and baked. It was principally used to account for goods, and evolved to include letters between the elite and also the recording of myth. We find larger population centers in this era, with the formation of city-states, and conflict between these city states. The Sumerians made real advances in writing up until 2300 B.C.E. They ended up using lots of sound-based syllables in symbolic form. By 2500 B.C.E. they had scribal schools for the elite, who also had to practice solving mathematical problems in addition to learning to write. In order to keep such things as farming organized they had a purely lunar calendar at first. Ultimately they had to combine the sun and the moon to keep better track of the seasons. Since 29.5 days is the approximate period of the moon, they used 12 times this to create a 354 day year. They were then short by about 11 days and had to add a 13th month every 3 years. Here is the symbiosis of astronomy and early mathematics in a fairly structured social setting. One notices the organized, long-term astronomical observations in conjunction with the arithmetic of the calendar. In particular the number 12 has acquired importance. The Sumerian hours varied with the seasons. There were six daytime hours and six nighttime hours. Thus there were 12 hours in each day. Each hour was divided into 60 minutes of varying length, and each minute into 60 seconds. III Mesopotamian Mathematics There are three main achievements of Mesopotamian mathematics which we will focus on, and which we will experience first-hand in worksheets #1 and #2 and homework I. These are: the sexagesimal, or base 60, number system, the use of abstract symbols for numbers and a positional system for writing numbers, including fractions. The first of these accomplishments was in place in Sumer by 2350 B.C.E. It may have been a compromise between various different ways of counting in societies with whom the Sumerians traded. We shall come to know one of these, the ancient Harappan society of the Indus valley, in short order. This society was probably as old and civilized as the Mesopotamian societies, Some ancient cultures counted by 2’s at first.