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This suggests that some mathematics was used in the centuries before B.

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There are two ways of learning about the mind and culture of prehistoric humans. We have learned about them through the discovery of ancient artifacts, which were found and interpreted by archaeologists. We have also learned about prehistoric civilization by observing primitive cultures in the modern world and by making inferences as to how prehistoric thought and customs developed. In our study of the development of ideas and understandings, both approaches are useful.

One of the most exciting archaeological discoveries was reported in by Karl Absolom as a result of excavations in central Czechoslovakia. Absolom found a prehistoric wolf bone dating back 30, years. Several views are shown in Figure Fifty-five notches, in groups of five, are cut into the bone.

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The first 25 are separated from the remaining notches by one of double length. Although we do not know how this bone was notched, the most plausible explanation is that some prehistoric man deliberately cut it. Perhaps he was recording the number of a collection, possibly of skins, of relatives, or of days since an event.

It is reasonable to assume that he made a notch for each object in the collection that he was counting. If this interpretation is correct, then we can recognize in this prehistoric record rudimentary versions of two important mathematical concepts. One is the idea of a one-to-one correspondence between the elements of two different sets of objects, in this case between the set of notches on the bone and the set of whatever the prehistoric man was counting.

The other is the idea of a base for a system of numeration. The arrangement of the notches in groups of 5 and of 25 indicates a rudimentary understanding of a base 5 system of numeration. Anthropological studies reinforce our belief in the existence of prehistoric number ideas. A study of the western tribes of the Torres Straits, reported by A. Haddon in , describes a tribe that had no written language which counted as follows: 1, urapun; 2, okosa; 3, okosaurapun; 4, okosa-okosa; 5, okosa-okosa-urapun; 6, okosaokosa-okosa.

Everything greater than 6 they called ras. A student of modern mathematics would recognize in this system of counting the beginnings of a base 2 numeration system.

The Historical Roots of Elementary Mathematics by Lucas N.H. Bunt

If a Torres Strait native had recognized this idea, however, he would have used a different word for 4 and would not have said ras for numbers greater than 6. Seidenberg has recently published a theory of the origin of counting see reference 12 at the end of the chapter. He believes that counting was invented for use in early religious rituals.

Many studies of primitive tribes as well as early Babylonian religious writings are cited which indicate that participants in religious rituals were called into the ritual in a definite order and that counting developed in connection with specifying this order. In his studies, Seidenberg found 2- counting to be the earliest counting that he could detect.

This seems to indicate that the counting of the Torres Straits natives is consistent with a method of counting that was in use thousands of years earlier. These two types of prehistoric number ideas, matching and counting, correspond to two different approaches to number which are common both in modern life and in modern education. One of these is the approach through the ideas of set and one-to-one correspondence between sets that have developed since the work of Georg Cantor in the latter part of the nineteenth century. This treatment is sometimes referred to as a cardinal approach to number.

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At about the time that Cantor was developing the beginnings of modern set theory, Giuseppe Peano was attempting to axiomatize the natural numbers and their arithmetic. To do this, he stated a set of five axioms. One of these axioms is that every natural number has a successor. Such a treatment is called an ordinal approach to number. It emphasizes the counting idea, in contrast to the matching idea stressed in the cardinal approach to number. These two approaches can be shown to be equivalent to each other, but our purpose here is merely to point out the antiquity of the underlying ideas which have recently been organized into important modern mathematical systems.

Evidences of other prehistoric mathematical ideas are not hard to find. One can read into primitive cave paintings some ideas of proportion and symmetry as skilled artists produced remarkably realistic drawings of animals and hunters. Ideas of number and one-to-one correspondence appear in connection with stickmen and four-legged animals. Elaborate geometric designs can be found on prehistoric pottery.

Prehistoric drawings showing different views of a wagon and horses have been found in Europe. Sketches from the time of ancient Babylonia that might be plans of a building, perhaps a temple, have been unearthed. What appears to be a decimally divided ruler has been unearthed at Mohenjo-Daro in Pakistan. More information about this seller Contact this seller 7. About this Item: Dover Publications , Condition: Fair. Presumed 1st as edit. Still very serviceable copy. More information about this seller Contact this seller 8.

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The Historical Roots of Elementary Mathematics.

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