What is exchange of cultural feature?
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The emergence of an abstract concept of number as a consequence of cultural exchange is particularly well known from the ancient Greek culture which played a crucial role in the historical process of transforming the heritage of early Near Eastern civilizations into the intellectual achievements of the hegemonic cultures of the Hellenistic and Roman world. The reflective restructuring of divergent bodies of knowledge brought about new kinds of general concepts and, in particular, an abstract concept of number (see Reflexivity in Anthropology).
The oldest known examples of general propositions about abstract numbers, for instance, the statement that the number of prime numbers is infinite, are handed down to us through the definitions and theorems of Euclid's Elements (Heath 1956). The relevance of these definitions and theorems was no longer based on their practical applicability, but only on their role within a closed system of mental operations. These operations reflected certain arithmetical activities in the medium of written language. In the case of the number concept these operations were the techniques of counting and tallying. Euclid defined a number as a ‘multitude composed of units’ and hence restricted the abstract concept of number to natural numbers. Platonism, in particular, dogmatically excluded from theoretical arithmetic all numerical structures that did not come under the Euclidian definition. Fractions of a number, for instance, and even more so irrational numbers such as the square roots of natural numbers that are not squares were not accepted themselves as numbers.
In order to circumvent the problems of incommensurability, a theory of proportions was developed, not for numbers, but for entities designated as ‘magnitudes’ (Heath 1956). This theory served at the same time as a substitute for an expanded concept of number that could cover all the arithmetical activities and symbolic operations existing at that time.
At a practical level, however, operations with notations for other types of number systems had been developed and were used long before they were theoretically reflected in an extended number concept. Any exchange of goods by merchants or by administrators of a centralized economy implicitly involves fractions. The balancing of debits and credits in bookkeeping similarly involves negative numbers. Accordingly, arithmetical techniques for dealing implicitly or even explicitly with fractions or with negative and irrational numbers had thus been developed already in early civilizations long before they were reflected in an explicit number concept.