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Early Sequence of Numerical Knowledge

Probably soon after humans could speak they could also count, at least up to ten, by using their fingers. It is possible that Neanderthals or Cro-Magnons could count as early as 35,000 years ago, based on parallel incised scratches on both a wolf bone in Czechoslovakia from about 33,000 years ago and a baboon bone in Africa from about 35,000 years ago.

Whether the scratches recorded the passage of days, numbers of objects, or were just scratched as a way to pass time is not known. The wolf bone is the most interesting due to having 55 scratches grouped into sets of five. This raises the probability that the scratches were used to count either objects or time.

An even older mastodon tusk from about 50,000 years ago had 16 holes drilled into it, of unknown purpose. Because Neanderthals and Cro-Magnons overlapped from about 43,000 BCE to 30,000 BCE, these artifacts could have come from either group or from other contemporaneous groups that are now extinct.

It is interesting that the cranial capacity and brain sizes of both Neanderthals and Cro-Magnons appear to be slightly larger than modern homo sapiens, although modern frontal lobes are larger. Brain size does not translate directly into intelligence, but it does indicate that some form of abstract reasoning might have occurred very early. Cave paintings date back more than 40,000 years, so at least some form of abstraction did exist.

In addition to counting objects and possessions, it was also important to be able to keep at least approximate track of the passage of time. Probably the length of a year was known at least subjectively more than 10,000 years ago. With the arrival of agriculture, also about 10,000 years ago, knowing when to plant certain crops and when to harvest them would have aided in food production.

One of the first known settlements was Catal Huyuk in Turkey, dating from around 7,000 BCE. This village, constructed of mud bricks, probably held several hundred people. Archaeological findings indicate agriculture of wheat, barley, and peas. Meat came from cattle and wild animals.

Findings of arrowheads, mace heads, pottery, copper, and lead indicate that probably some forms of trading took place at Catul Huyuk. Trading is not easily accomplished without some method of keeping track of objects. There were also many images painted on walls and this may indicate artistic interests.

The probable early sequence of humans acquiring numerical knowledge may have started with several key topics:

  • Prehistoric numeric and mathematical knowledge:

    • Counting objects to record ownership
    • Understanding the two basic operations of addition and subtraction
    • Measuring angles, such as due east or west, to keep from getting lost
    • Counting the passage of time during a year to aid agriculture
    • Counting the passage of daily time to coordinate group actions
  • Numeric and mathematical knowledge from early civilizations:

    • Counting physical length, width, and height in order to build structures
    • Measuring weights and volumes for trade purposes
    • Measuring long distances such as those between cities
    • Measuring the heights of mountains and the position of the sun above the horizon
    • Understanding the mathematical operations of multiplication and division
  • Numeric and mathematical knowledge probably derived from priests or shamans:

    • Counting astronomical time such as eclipses and positions of stars
    • Measuring the speed or velocity of moving objects
    • Measuring curves, circles, and irregular shapes
    • Measuring rates of change such as acceleration
    • Measuring invisible phenomena such as the speed of sound and light
  • Numeric and mathematical knowledge developed by mathematicians:

    • Analyzing probabilities for games and gambling
    • Understanding abstract topics such as zero and negative numbers
    • Understanding complex topics such as compound interest
    • Understanding very complex topics such as infinity and uncertainty
    • Understanding abstract topics such as irrational numbers and quantum uncertainty

Prehistoric numeric and mathematical knowledge probably could have been handled with careful observation assisted by nothing more than tokens such as stones or scratches, plus sticks for measuring length. Addition and subtraction are clearly demonstrated by just adding or removing stones from a pile.

Numeric and mathematical knowledge from early civilizations would have needed a combination of abstract reasoning aided by physical devices. Obviously, some kind of balance scale is needed to measure weight. Some kind of angle calculator is needed to measure the heights of mountains. Some kind of recording method is needed to keep track of events, such as star positions over long time periods.

Numeric and mathematical knowledge probably derived from priests or shamans would need a combination of abstract reasoning; accurate time keeping; accurate physical measures; and awareness that mathematics could represent intangible topics that cannot be seen, touched, or measured directly. This probably required time devoted to intellectual studies rather than to farming or hunting.

Numeric and mathematical knowledge developed by mathematicians is perhaps among the main incentives leading to calculating devices and eventually to computers and software. This required sophisticated knowledge of the previous topics, combined with fairly accurate measurements and intellectual curiosity in minds that have a bent for mathematical reasoning. These probably originated with people who had been educated in mathematical concepts and were inventive enough to extend earlier mathematical concepts in new directions.

One of the earliest cities, Mohenjo-Daro, which was built in Northern India about 3,700 years ago, shows signs of sophisticated mathematics. In fact, balance scales and weights have been excavated from Mohenjo-Daro.

This city may have held a population of 35,000 at its peak. The streets are laid out in a careful grid pattern; bricks and construction showed signs of standard dimensions and reusable pieces. These things require measurements.

Both Mohenjo-Daro and another city in Northern India, Harappa, show signs of some kind of central authority because they are built in similar styles. Both cities produced large numbers of clay seals incised both with images of animals and with symbols thought to be writing, although these remain undeciphered. Some of these clay seals date as far back as 3,300 BCE.

Other ancient civilizations also developed counting, arithmetic, measures of length, and weights and scales. Egypt and Babylonia had arithmetic from before 2,000 BCE.

As cities became settled and larger, increased leisure time permitted occupations to begin that were not concerned with physical labor or hunting. These occupations did not depend on physical effort and no doubt included priests and shamans. With time freed from survival and food gathering, additional forms of mathematical understanding began to appear.

Keeping track of the positions of the stars over long periods, measuring longer distances such as property boundaries and distances between villages, and measuring the headings and distances traveled by boats required more complex forms of mathematics and also precise measurements of angles and time periods. The advent of boat building also required an increase in mathematical knowledge. Boat hulls are of necessity curved, so straight dimensional measurements were not enough.

Rowing or sailing a boat in fresh water or within sight of land can be done with little or no mathematical knowledge. But once boats began to venture onto the oceans, it became necessary to understand the positions of the stars to keep from getting lost.

Australia is remote from all other continents and was not connected by a land bridge to any other location since the continents broke up. Yet it was settled about 40,000 years ago, apparently by means of a long ocean voyage from one (or more) of the continents. The islands of Polynesia and Easter Island are also far from any mainland and yet were settled thousands of years ago. These things indicate early knowledge of star positions and some kind of math as well.

Many early civilizations in Egypt, Mesopotamia, China, India, and South America soon accumulated surprisingly sophisticated mathematical knowledge. This mathematical knowledge was often associated with specialists who received substantial training.

Many ancient civilizations, such as the ancient Chinese, Sumerians, Babylonians, Egyptians, and Greeks, invested substantial time and energy into providing training for children. Not so well known in the West are the similar efforts for training in India and among the people of Central and South America, such as the Olmecs, Mayans, Incans, and later the Aztecs.

Japan also had formal training. For the upper classes, Japanese training included both physical skills in weapons and also intellectual topics such as reading, writing, and mathematics. All of these ancient civilizations developed formal training for children and also methods of recording information.

The University of Nalanda in Northern India was founded circa 472 BC and lasted until about the 12th century, with a peak enrollment during around 500 AD. It was one of the largest in the ancient world, with more than 10,000 students from throughout Asia and more than 2,000 professors. It was among the first universities to provide training in mathematics, physics, medicine, astronomy, and foreign languages.

The University of Nalanda had an active group of translators who translated Sanskrit and Prakrit into a variety of other languages. In fact, much of the information about the University of Nalanda comes from Chinese translations preserved in China since the University of Nalanda library was destroyed during the Moslem invasion of India in the 12th century. It was reported to be so large that it burned for almost six weeks.

Indian scholars were quite advanced even when compared to Greece and Rome. Concepts such as zero and the awareness of numerous star systems were known in India prior to being known in Europe. (The Olmecs of Central America also used zero prior to the Greeks.)

In ancient times, out of a population of perhaps 1,000 people in a Neolithic village, probably more than 950 were illiterate or could only do basic counting of objects and handle simple dimensional measures. But at least a few people were able to learn more complex calculations, including those associated with astronomy, construction of buildings and bridges, navigation, and boat building.

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