Lecture Two The Dawn of Numbers
Scope:
Humans have an innate capacity to accurately compare small quantities. We will examine some early counting tools as a means to determine how humanity's understanding of numbers initially developed. Humankind has been counting for at least 30.000 year but are humans the only creatures to possess a number sense? In this lecture. we will see that even some animals appear to have the capacity for numerical concepts. The human concept of number may have developed in the same way it does in children. To compare large quantities, however, early civilizations used the idea of a one-to-one pairing. Notched bones, knotted strings, and piles pebbles allowed people to keep track of animals and conduct commerce. Although the human hand is one of the most fundamental counting tools studies of primitive cultures reveal the subtle use of the entire body in counting practices. Next, we will turn to the development of the abstract notion of number; when, for example, did the adjective three (e.g., "three" apples) become the noun three? Although this event did not occur at a precise moment evidence of abstract numbers in Mesopotamia dates back somewhere between 3500 and 3200 B.C.E. (dates range considerably).
Outline
I. What motivated humans to count?
A. Thousands of years before there were writing, literacy, or even numeral symbols, shepherds tending flocks had to keep track of their sheep.
B. As agricultural societies developed, people needed to measure and divide land, keep track of livestock, record harvests, and take census data.
C. With growing populations and clashing cultures came conflict requiring armies to face the logistics of arming and feeding their soldiers.
D. Bountiful agricultural fruits of labor required counting days and
lunar cycles as part of calendars to better predict the change in
seasons, annual floods, or dry spells.
II. Human beings have an innate number sense.
A. This innate number sense allows us to instantly compare small collections of objects.
1. If a Sumerian shepherd has a very small number of animals, he can keep track of them without the need for counting.
2. It is easy to see the difference between a herd of four sheep and a herd of three without actually counting.
3. With a larger herd, we are unable to determine (by simply looking) whether the collection of sheep we have after grazing is the same size as the collection with which we started. This limitation is referred to as the limit of four.
4. This limit of four might underlie the barred-gate system of counting we still employ today.
B. Other creatures also sense numbers.
1. Studies with goldfinches reveal that when presented with two small piles of seeds, they usually pick the larger of the two piles; crows have also been known to distinguish between collections of different sizes.
2. Evidence suggests that animals do not, however, have a notion of number as an abstract object.
C. The human concept of number may have developed in a manner similar to the way in which it develops in children.
1. Ordination comes first; that is, the ability to see that one set of objects is larger than another. We learn to order objects according to size before we learn to count them.
2. Learning ordered lists is a classic component of early education; children are taught to recite the alphabet, numbers, and even the days of the week, often well before they understand the meaning of these sequences.
3. Children next begin to grasp the idea of natural numbers (e.g., 1, 2, 3, 4 ...).
4. Finally, children master cardination (or true counting), in which the objects in one collection can be counted or paired up with objects from another collection.
III. Many societies used various forms of sticks as counting tools to record one-to-one correspondences.
A. Notched bones from as long ago as 30.000 B.C.E. have been found in Western Europe.
B. Notched sticks called tally sticks have been used for millennia and may have inspired the development of Roman numerals. A wooden tally stick could be marked and then split lengthwise so that two
parties could keep track of a transaction.
C. In order to make the one-to-one correspondence physical, the Incas and cultures along the Pacific Rim and in Africa used knotted strings.
1. In the 5th century B.C.E., Herodotus of Greece wrote in his History that Darius, the king of Persia, used a knotted cord as a calendar.
2. Catholic, Muslim, and Buddhist rosaries and prayer beads allow the devout to recite the appropriate number of litanies without the need for an abstract counting system.
D. As early as 3500 to 3200 B.C.E.. our Sumerian shepherd most likely used a pile of pebbles to "count" his sheep through a one-to-one pairing.
IV. The human hand is a natural counting tool.
A. The limit of four in humans made the five-digit hand particularly useful for counting and led to the use of five as a basic grouping for counting.
1. The continued popularity of the barred-gate tally system may be due to its basis in counting by 5s.
2. The 10 digits on our two hands may have led to the modern-day dominance of a base- 10 numeral system.
B. Toes are the obvious extension of the hand as a counting tool.
1. Given the hand's convenience, some cultures extended their counting to include the joints of fingers.
2. Other body parts have been incorporated into counting systems in many cultures.
V. Sumerian methods of counting have been studied extensively.
A. Sumerians created different clay tokens, called calculi. to represent quantities of different items.
1. To record a quantity of goods, such as measures of grain in a storehouse, Sumerians would seal the appropriate collection of tokens in a clay jar. Evidence for this method of counting can be found as early as 3200 B.C.E.
2. The tokens were used to make impressions on the outside of the jar before the clay hardened, indicating the quantity within.
3. The token markings later came to represent the numbers themselves, eliminating the need for the tokens in recordkeeping.
B. Sumerian markings led to one of the first numeral systems and, consequently, to what may have been the first form of written language: cuneiform.
Questions to Consider:
1. Without counting, determine whether the collection of @ signs below or the collection of & signs is larger.
&&&&&&&&&&&&&&&&&&&
$$$$$$$$$$$$$$$$$
How were you able to perform this task without counting?
Now determine without counting whether the collection of @ signs
below or the collection of backslashes is larger.
&&&&&&&&&&&&&&&&&&&
......................
Discuss why one comparison was easy and the other more difficult.
2. Find examples in your everyday life where you use a one-to-one pairing
to compare the sizes of two collections without actually counting.
Taught by Edward B. Burger
Williams College | Ph.D., The University of Texas at Austin
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memo :
・抽象的に数の概念が獲得される以前は単に形容詞としてそれが使用されていた。 『three apples』
・四つまでは数えることなしに判断可能であるかもしれないということ。
・大きさというものが数の概念の発達の初期にはとても重要な役割を果たしているということ。
