The origin of mathematics accompanied with the evolution of social systems. Many, many social systems needed calculation and numbers. Conversely, the calculation of numbers enables more complex relations and interactions between people. Numbers and calculations with them require a well-organized operational system. As we will see, not just one but several number systems come to us from antiquity. However, as interesting as the basic notions of counting may be, the origin of mathematics includes more than just enumeration, counting, and arithmetic. Thus, also there were some other issues of mathematics to be considered as well. Number provides a common link between societies and a basis for communication and trade. This illustrates various entries into our ancient knowledge of how our number-system began.
Examples range from prehistory to contemporary. The ancient evidence is easy to accept, but the modern examples yield more conclusions. It is remark- able that even though mathematics achieved its first zenith twenty three hundred years ago, more than a hundred generations, there are people today that still count with their fingers or with stones, that have no real language for numbers, and moreover don't have the general concept of numbers beyond specific examples. There is every reason to believe that in the future, near or distant, there will exist a far more credible theory for the origin of counting than there was before for the development of geometric proof, the Pythagorean Theorem in particular.
Human being needed that inspired mankind’s first efforts in mathematics, arithmetic in particular were
• Counting
• calculations
• measurements
For example,
The number of animals own by a herdsman cannot be known unless some basics of counting are not known.
HOW MANY??
A wealth cannot be distributed unless certain facts about division (fractions) are known. A temple cannot be built unless certain facts about triangles, squares, and volumes are known.
From practical needs such as these, mathematics was born. One view is that the core of early mathematics is based upon two simple questions.
• How many?
• How much?
Two possibilities for the origin of counting have been posted. One is that counting spontaneously arose throughout the world more or less independently from place to place, tribute to tribute.
The other is that counting was invented just once and it spreaded throughout the world from that source. The latter view, maintained by Abraham Seidenberg, is based upon a remarkable number of similarities of number systems throughout the world.
For example, that odd numbers are male and even numbers are female seems to be virtually universal. (Of course this distinction has been lost in modern times.) Seidenberg’s anthropological studies further suggest that counting
“was frequently the central feature of a rite, and that participants in ritual were numbered.”
HOW MUCH??
As society formed and organized, the need to express quantity emerged. Even at this early level, perhaps as early as 250,000 years ago, there must have begun a transition from sameness to similarity of numbers.
one wolf → one sheep
two dogs → wolf → two rabbits
five warriors→ five spears → five fingers
This abstraction of the concept of number was a major step toward modern mathematics. From artifacts even more than 5,000 years old, notches on bones have been noted. Were these to count seasons, kills, children? We don’t know. But the need to denote quantity must have been significant.
As ancient civilizations developed, the need for practical mathematics increased. They required numeration systems and arithmetic techniques for trade, measurement strategies for construction, and astronomical calculations to track the seasons and cosmic cycles.
Our first knowledge of mankind’s use of mathematics comes from the Egyptians and Babylonians. Both civilizations developed mathematics that was similar in scope but different in particulars. There can be no denying the fact that the totality of their mathematics was profoundly elementary2, but their astronomy of later times did achieve a level com-parable to the Greeks.
The Babylonian civilization has its roots dating to 4000BC with the Sumerians in Mesopotamia. Yet little is known about the Sumerians. Sumer2 was first settled between 4500 and 4000 BC by a non-semitic.
This mathematical tablet was recovered from an unknown place in the Iraqi desert. It was written originally sometime around 1800 BC. The tablet presents a list of Pythagorean triples written in Babylonian numerals. This numeration system uses only two symbols and
- Their mathematical notation was positional but sexagesimal1.
- They used no zero.
- More general fractions, though not all fractions, were admitted.
- They could extract square roots.
- They could solve linear systems.
- They worked with Pythagorean triples.
- They solved cubic equations with the help of tables.
- They studied circular measurement.
- Their geometry was sometimes incorrect.
For enumeration the Babylonians used symbols for 1, 10, 60, 600, 3,600, 36,000, and 216,000, similar to the earlier period. Below are four of the symbols. They did arithmetic in base 60, sexagesimal.
1 10 60 600
CUNIEFORM NUMERALS
The Babylonians choosing the base of 60, the values 2,3,5,10,12,15,20, and 30 all divide 60. For example The number of days, 360, in a year gave rise to the subdivision of the circle into 360 degrees, and that the chord of one sixth of a circle is equal to the radius gave rise to a natural division of the circle into six equal parts. This in turn made 60 a natural unit of counting. (Moritz Cantor, 1880).
The Babylonians used a 12 hour clock, with 60 minute hours. That is, two of our minutes is one minute for the Babylonians. (Lehmann-Haupt, 1889) Moreover, the (Mesopotamian) zodiac was divided into twelve equal sectors of 30 degrees each.
The base 60 provided a convenient way to express fractions from a variety of systems as may be needed in conversion of weights and measures. In the Egyptian system, we have seen the values 1/1, 1/2, 2/3, 1, 2, . . . , 10. Combining we see the factor of 6 needed in the denominator of fractions. This with the base 10 gives 60 as the base of the new system. (Neugebauer, 1927)
References- http://www.math.tamu.edu/~dallen/masters/hist_frame.html and http://www.math.tamu.edu/~dallen/masters/egypt_babylon/babylon.pdf
In next article…
Greek Period (600 B.C. to 450 A.D.).......
DANUKA GUNASEKARA
2011/2011
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