Prehistoric mathematics

Summary: Historians believe that even the earliest people used mathematics.

Many books on the history of mathematics begin with the ancient Egyptians and Babylonians, but those civilizations did not begin until about 5000 years ago. Although historians do not know many details, human life had been progressing for several millennia prior to that time. Even archeology offers little detail on the earliest mathematics, so most knowledge comes from speculation. However, from what is known about human beings in general, and especially about prehistoric life, even the earliest people must have known and used some mathematics.

The use of “mathematics” probably even precedes the development of modern human beings. Studies of animal behavior have shown that animals, and especially birds, seem to possess limited number sense, recognizing the difference between groups of two and three and even larger sets. Bees can recognize and even communicate information about the location of orchards and fields for pollination, displaying a sense of space that could be called “geometry.” Even more spectacular are the long migratory trips of herd animals, flocks of birds, and groups of butterflies, often traveling thousands of miles to return to the same fields every year. These examples certainly do not represent a sophisticated concept of mathematics and are instinctual, but they show a mathematical organization in the brain.

Language, Counting, and Quantities

The earliest humans (wherever the line is drawn between pre-human and human) continued the mathematical thinking shown in animals. As their brains developed, their mathematics also grew stronger and more sophisticated. This progression continued as early grunts become proto-languages, for a key part of mathematics is not only having the concepts in one’s head, but also representing and communicating the concepts to others. Hence, language was a key ingredient in prehistoric mathematics (as it remains today).

A concept of counting must have come early, as people began to distinguish quantity. Even if they did not have linguistic terms for numbers beyond three or four, they would at least be able to make rough comparisons of large quantities and much larger quantities—consider that even modern humans often need notations, pictures, or concrete examples to handle specific large quantities, but certainly can tell the difference between a dozen and a hundred and a million. Many aspects of life require at least limited counting—to make sure all one’s goats (or children) are present, to share items fairly in a group or to calculate the size of a load to be carried, and many other applications.

It is only a small jump of abstraction to begin to record quantities with tally marks. It is likely that people first collected stones or other small objects to represent quantities and later began to “write” them as tallies. Tally marks have been found in many parts of the world scratched on cave walls or carved onto wooden sticks and were also likely written in sand or clay, which shifted to destroy the writing. Probably the most famous prehistoric mathematical object is the Ishango bone, found in south-central Africa, and thought to be at least 15,000 years old. The bone has several sets of tallies scratched onto it—some have pointed out that they are mostly prime numbers, but that is probably a coincidence. Using tallies quickly leads to a problem: a long line of marks is hard to deal with, even if one had some limited counting words. Probably, many people around the world recognized that some structure helped handle large quantities of tally marks—especially collecting them into groups of the same size. Not only does this make counting more efficient but it also leads to the concept of multiplication. In nearly all modern languages—most derived from ancient or even prehistoric languages—the higher counting words use a system of groups and groups of groups, now called “place-value,” but they reach back to the prehistoric convenience of putting tally marks together.

Measurement and Geometry

Closely tied to counting was the use of comparative relationships—especially large and small, tall (or long) and short, and even old and young. These may have come when exact counts were difficult, but the comparisons were obvious and usually visual. A tall stack of blocks would easily be seen to have more items than a short stack; a long line of tally marks (grouped or ungrouped) was a greater quantity than a short line. As actual counting developed and numbers were applied to comparisons, the beginnings of measurement occurred—measurement is really just comparisons of quantities where one side of the comparison is a defined unit. To make comparisons easier, certain items of specific size or quantity became units, and as people reached farther to wider audiences, units became at least roughly standardized. Often, body parts were used both for counting tabulations and as “standard” units. For example, the distance from the elbow to the fingertips was approximately the same for most adults, so in the Middle East, this length became the “cubit.”

Geometry also has deep roots in the human story. Circles must have been recognized in the shape of the sun and full moon and the apparent edge of the horizon. Efficiency caused people to arrange objects to fit together well in patterns—often circular but sometimes rectangular. The first tools used sharp angles, heavy weights, and tall, thin cylinders. The beginnings of farms led to more organized geometrical arrangements in the shapes of fields and structures. Often, the “invention” of the wheel is considered one of the big milestones of the start of civilization, and this represents a practical understanding of the geometry of circles. As objects became more sophisticated—woven mats, farming tools, larger structures, and even bridges—many more geometrical relationships and properties were discovered. These might be considered the beginnings of engineering—using mathematical properties in practical applications.

Pure Mathematics

Archeologists have also noted some prehistoric mathematics that may have been closer to pure mathematics. Cave paintings, carved sculptures, and textile patterns show contemporary mathematical objects such as circles, triangles, parallel lines, quadrangles, symmetric patterns, and the crosshatch. However, no one has yet deciphered what the geometric signs meant to prehistoric peoples. Some symbols appeared repeatedly in various parts of the world. They may have served practical or religious values, but they also were art—perhaps art for its own sake, for beauty. Certain numbers may have had mystical meanings that were seemingly less useful for day-to-day activity but important for esthetics and spirituality.

The overlap between this pure mathematics and the practical needs of early farmers was the use of mathematics in astronomy and calendars. Could the gods show the times for planting and harvesting? Could humans discern the plans of these gods and use them in practice? Most of the spectacular prehistoric structures, from Stonehenge in England to the huge geometrical patterns of Nazca in Peru, have been linked to measures of the sun’s movement and the seasons. Mathematics led prehistoric peoples in solving their daily problems and to thinking of the universe and infinity. Mathematics still serves modern humans in the same ways.

Bibliography

Boyer, Carl. A History of Mathematics. Hoboken, NJ: Wiley, 1991.

Burton, David M. The History of Mathematics: An Introduction. New York: McGraw-Hill, 2007.

Eves, Howard. Introduction to the History of Mathematics. New York: Saunders College Publishing, 1990.

Ifrah, Georges. The Universal History of Numbers. Hoboken, NJ: Wiley, 1994.

Von Petzinger, Genevieve. “Geometric Signs in Rock Art & Cave Paintings.” http://www.bradshawfoundation.com/geometric‗signs/geometric‗signs.php.