Little progress was made in understanding numerical values and space relations until the transition occurred from the mere gathering of food to its actual production, from hunting and fishing to agriculture. With this fundamental change, a revolution in which the passive attitude of man toward nature turned into an active one, we enter the New Stone Age, the Neolithic.

This great event in the history of mankind occurred perhaps ten thousand years ago, after the ice sheet that covered Europe and Asia had melted and made room for forests and deserts. Here nomadic wandering in search of food came slowly to an end. Fishermen and hunters were in large part replaced by simple farmers. Such farmers, remaining in one place as long as the soil stayed fertile, began to build more permanent dwellings; villages emerged as protection against the climate and against predatory enemies. Many such neolithic settlements have been excavated. The remains show how gradually elementary crafts such as pottery, carpentry, and weaving developed. There were granaries, so that the inhabitants were able to provide against winter and hard times by establishing a surplus. Bread was baked, beer was brewed, and in late neolithic times copper and bronze were smelted and prepared. Inventions appeared, notably the potter’s wheel and the wagon wheel; boats and shelters were improved. All these remarkable innovations occurred only within limited areas and did not always spread to other localities. The American Indian, for example, did not learn much about the technical use of the wagon wheel until the coming of the European. Nevertheless, as compared with paleolithic times, the tempo of technical improvement was enormously accelerated.

Between the villages a considerable trade existed, which so expanded that connections can be traced between places hundreds of miles apart. The discovery of the arts of smelting and manufacturing, first copper, then bronze tools and weapons, strongly stimulated this commercial activity. This again promoted the further formation of languages. The words of these languages expressed very concrete things and very few abstractions, but already there was room for some simple numerical terms and for some form relations. Many Australian, American, and African tribes were in this stage at the period of their first contact with Europeans: some tribes are still living in these conditions, so that it is possible to study their habits and forms of expression, and to some extent to understand them if we can strip ourselves of preconceived notions.

Here is an example from some Australian peoples:

The development of the crafts of commerce stimulated this crystallization of the number concept. Numbers were arranged and bundled into larger units, usually by the use of the fingers of the hand or of both hands, a natural procedure in trading. This led to numeration first with five, later with ten as a base, completed by addition and sometimes by subtraction, so that 12 was conceived as 10 + 2, or 9 as 10–1. Sometimes 20, the number of fingers and toes, was selected as a base. Of 307 number systems of primitive American peoples investigated by W. C. Eels, 146 were decimal, 106 quinary and quinary decimal, vigesimal and quinary vigesimal. ³ The vigesimal system in its most characteristic form occurred among the Mayas of Mexico and the Celts in Europe.

Numerical records were kept by means of bundling: strokes on a stick, knots on a string, pebbles or shells arranged in heaps of fives—devices very much like those of the old-time innkeeper with his tally stick. From this method to the introduction of special symbols for 5, 10, 20, etc., was only a step, and we find exactly such symbols in use at the beginning of written history, at the so-called dawn of civilization.

One of the oldest examples of the use of a tally stick dates back to paleolithic times and was found in 1937 in V

stonice (Moravia). It is the bone of a young wolf, 7 inches long, engraved with 55 deeply incised notches, of which the first 25 are arranged in groups of 5. They are followed by a simple notch twice as long which terminates the series; then, starting from the next notch, also twice as long, a new series runs up to 30.⁴ Other such marked sticks have been found.

It is therefore clear that the old saying found in Jakob Grimm and often repeated, that “counting started as finger counting,” is incorrect. Counting by fingers, that is, counting by fives and tens, came only at a certain stage of social development. Once it was reached, numbers could be expressed with reference to a base, with the aid of which large numbers could be formed; thus a primitive type of arithmetic originated. Fourteen was expressed as 10 + 4, sometimes as 15 − 1. Multiplication began where 20 was expressed not as 10 + 10, but as 2 x 10. Such dyadic operations were used for millennia as a kind of middle road between addition and multiplication, notably in Egypt and in the pre-Aryan civilization of Mohenjo-Daro on the Indus. Division began where 10 was expressed as “half of a body,” although conscious formation of fractions remained extremely rare. Among North American tribes, for instance, only a few instances of such formations are known, and this is in almost all cases only of 1/2, although sometimes also of 1/3 or 1/4.⁵ A curious phenomenon was the love of very large numbers, a love perhaps stimulated by the all-too-human desire to exaggerate the extent of herds of enemies slain; remnants of this tendency appear in the Bible and in other sacred and not-so-sacred writings.

Neolithic man also developed a keen feeling for geometrical patterns. The baking and coloring of pottery, the plaiting of rushes, the weaving of baskets and textiles, and later the working of metals led to the cultivation of plane and spatial relationships. Dance patterns must also have played a role. Neolithic ornamentation rejoiced in the revelation of congruence, symmetry, and similarity. Numerical relationships might enter into these figures, as in certain prehistoric patterns which represent triangular numbers; others display “sacred” numbers.

Figures 1-Figures 4 below give examples of some interesting geometrical patterns occurring in pottery, weaving, and basketry. The design in Fig. 1 can be found on neolithic pottery in Bosnia and on objects of art in the Mesopotamian Ur period.⁸ The motif in Fig. 2 exists on Egyptian pottery of the Predynastic period (c. 4000-3500 B.C.).⁹ Fig. 3 shows patterns which were used by pole-house dwellers near Ljubljana (Yugoslavia) in the Hallstatt period (Central Europe, c. 1000–500 B.C.).¹⁰ The designs in Fig. 4, rectangles filled with trian...