Evolution Of Number Systems Research Paper

Symbolic activities such as counting, the use of tallies for representing quantities, or performing calculations by means of tokens or systems of written symbols, are usually considered practices dealing with ‘numbers.’ However, such symbolic activities historically precede an explicit number concept. Explicit concepts of number are the outgrowth of reflection upon such activities.

When first explicit concepts of abstract ‘numbers’ emerged in history, they were restricted to the number system which is called today ‘natural numbers,’ that is, to the system ‘1, 2, 3, 4, …’ The subsequent history of the term number is characterized by a series of extensions of the number concept to cover ‘integers,’ ‘rational numbers,’ ‘real numbers,’ ‘complex numbers,’ ‘algebraic numbers,’ and other mathematical objects designated, with lesser or greater justification, as a number system. These extensions usually included numbers of earlier stages as special kinds of numbers, but necessarily also dismissed certain properties which seemed to be essential in earlier stages. Consequently, any extension of number systems was accompanied by controversial discussions about the question whether the newly-considered mathematical objects were really numbers. These doubts are often reflected in the designation of such numbers as ‘irrational,’ ‘negative,’ ‘imaginary,’ etc. Finally, the variety of potential candidates for the term number became so great that the answer to the question of what is and what is not a ‘number’ turned into a merely conventional decision. The focus of attention shifted from the investigation of types of numbers as they were historically determined towards the study of their common structures.

2. Cultures Without Number Systems

Until recently, ‘prearithmetical’ cultures existed without any arithmetical techniques at all such as finger counting, counting notches, counting knots, or tokens. The best known examples are the Australian aborigines (Blake 1981, Dixon 1980) and certain South American natives (Closs 1986) whose languages did possess terms for quantities, yet of an exclusively qualitative nature, terms such as ‘many,’ ‘high,’ ‘big,’ or ‘wide.’ The quantitative aspects of an object of cognition were not yet distinguished from its specific physical appearance. Since from periods before the Late Neolithic no objects or signs have been identified that might have had some kind of arithmetical function, it is likely that up to that time human culture remained entirely prearithmetical. While artifacts with repeated signs and regular patterns from periods as early as the Late Paleolithic have occasionally been interpreted as representations of numbers (Marshack 1972), such an interpretation seems to be untenable since these sign repetitions lack the characteristic subdivision by counting levels which is present in all real counting systems (Damerow 1998).

3. Proto-Arithmetic

The occurrence of the simplest genuine arithmetical activities known from recent nonliterate cultures (see, e.g., Saxe 1982) date back to the Late Neolithic and the Early Bronze Age. These activities aiming at the identification and control of quantities are based on structured and standardized systems of symbols representing objects. Their emergence as counting and tallying techniques may have been a consequence of sedentariness. Symbols are the most simple tools for the construction of one-to-one correspondences in counting and tallying that can be transmitted from generation to generation. The organization of agricultural cultivation, animal husbandry and household administration led to social conditions that apparently made symbolic techniques useful, and their systematic transmission and development possible. Such techniques are ‘proto-arithmetical’ insofar as the symbols represent objects and not ‘numbers,’ and consequently are not used for symbolic transformations which correspond to such arithmetical operations as addition and multiplication.

Early explorers and travelers encountering indigenous cultures using proto-arithmetical techniques often anachronistically interpreted their activities from a modern numerical perspective and believed the limitations of proto-arithmetic resulted from deficient mental abilities of such peoples. It was only in the first half of the twentieth century that anthropologists and psychologists challenged these beliefs (e.g., see Klein, Melanie (1882–1960); Boas, Franz (1858–1942)) and began seriously to study culturally specific mental constructions connected with proto-arithmetical activities (Levy-Bruhl 1923, Wertheimer 1925).

4. From Proto-Arithmetic To Systems Of Numerical Symbols

A further step in the evolution of number systems resulted from the rise of early civilizations and the invention of writing. This transition is particularly well documented in the case of the Ancient Near East. A system of clay tokens possessing proto-arithmetical functions has been identified which may have been widely used already during the period from the beginning of sedentariness in the areas surrounding the Mesopotamian lowland plain and in the Nile valley around 8000 BC until the emergence of cities around 4000 BC (Schmandt-Besserat 1992).

In the fourth millennium BC this proto-arithmetic system of clay tokens became the central tool for the control of a locally-centralized economic administration. In order to serve this purpose, the protoarithmetical capabilities of the system were exploited to the limits of their capacity. The safekeeping of tokens in closed and sealed spherical clay envelopes demonstrates that they were used for the encoding of important information. The tokens were complemented and later completely replaced by markings that were impressed on the surface of such clay envelopes or on the surface of sealed clay tablets. Thus, the system was transformed into a more suitable medium by substituting written signs for tokens. Around 3200 BC, the numerical notations were supplemented with pictograms, and the tablets achieved a more complex structure. They now displayed several quantitative notations arranged according to their function, as we find in an administrative form. The earliest attestation of addition as an operation with symbols is provided by the notation of totals of the entries on such tablets, usually inscribed on the reverse of the tablet containing these entries.

Based on their origin, the oldest written numerical notation systems exhibited, for a short period, very unusual characteristics. Since they initially still represented units of counting and measurements, and since the numerical relations between these units were dependent on the counted or measured objects, the numerical signs had no uniquely determined values. Their values resulted from their metrological context, and changed with the respective areas of application without any apparent attempt to attain unambiguous numerical values for the signs. From the viewpoint of modern arithmetic, the signs thus represented in different contexts different numbers (Damerow 1995, Damerow and Englund 1987). Some 500 years later, however, the system had developed into a sort of numerical sign system as is known from other early civilizations. In particular, the signs had now attained fixed numerical values.

5. Systems Of Numerical Symbols In Early Civilizations

Most, if not all, advanced civilizations, in particular the Egyptian empire, the Mesopotamian city-states, the Mediterranean cultures, the Chinese empire, the Central American cultures, and the Inca culture, have, independent from each other or by adaptation from other cultures, developed systems of numerical symbols that exhibit similar semiotic characteristics. The basic symbols of these systems were the same as at the proto-arithmetical level, signs for units and not for numbers, but since they represented now complex metrologies they had necessarily also to deal with fractions. Furthermore, part of the systems were now complex symbol transformations, that is, arithmetical techniques such as Egyptian calculation using unit fractions (Chace 1927), sexagesimal arithmetical techniques of the Babylonians (Neugebauer 1934), transformations of rod numerals on the Chinese counting board (Li and Du 1987), calendrical calculations in the pre-Columbian culture of the Maya (Closs 1986, Thompson 1960), or techniques of the use of knotted cords (quipu) as administrative tools by the Inca (Ascher and Ascher 1971–1972, Locke 1923).

The development of these numerical techniques was historically closely related with the administrative problems that had arisen through the concentration of economic goods and services in the governmental centers of early state organization. The dramatic rise in the quantities of products had to be controlled, and the immense variety of decision-making implications had to be executed administratively (Høyrup 1994, Nissen et al. 1993).

The rules dictating the use of numerical signs reflected this function; they corresponded to their meaning in the social context. The numerical sign systems thus exhibited a great variety of structures with no internal differentiation between rules representing a universal number concept and rules depending on the specific kind of symbolic representation and its function to control quantities in a specific social setting. It was only the cultural exchange between advanced civilizations with developed systems of numerical notations that created the preconditions for the differentiation between universal numbers and specific notations.

6. The Euclidean Conceptualization Of Number

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.

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.

7. The Impact Of Algebra

The problems resulting from the restrictive Greek number concept were aggravated with the renaissance of ancient mathematics in the Early Modern Era. In particular, the development of algebraic notation techniques and of analytical methods applied to continuous processes in mechanics made the distinction between numbers and magnitudes obsolete. The use of variables in order to solve equations was indifferent to this conceptual distinction. Fractions, negative numbers, irrational numbers, infinitesimals, and infinity became more and more accepted as numbers. Although infinitesimals were again expelled in the nineteenth century after their missing logical foundation resulted in serious contradictions in the conceptual system of mathematics, they were finally reintroduced in a new form by the nonstandard analysis of the twentieth century.

The introduction of algebraic notations thus contributed to the transformation of existing arithmetical techniques into conceptualized number systems, but it also initiated the creation of completely new ones. A new type of number system with no immediately obvious meaning emerged, reflecting no longer primary arithmetical techniques but rather abstract operations with variables and equations. A well-known example is provided by ‘imaginary’ square roots of negative numbers which occurred as solutions to higher degree equations. Such solutions, although calculated correctly according to rules for numbers, seemed to make no sense since squaring a number never results in a negative number. Nevertheless, it turned out to be possible to operate consistently with such imaginary numbers as if indeed they were, in fact, numbers. They were, thus, finally accepted as ‘complex numbers.’

The justification of seemingly meaningless number systems was not the only problem posed by the development of mathematics in the nineteenth century. The term number became questionable for other reasons too. On the one hand, a productive new theory, even today called simply ‘theory of numbers,’ was created; it is, however, a discipline with a specific topic. The term number is in this theory restricted to (positive and negative) integers. On the other hand, a growing number of new types of mathematical systems emerged which, with greater or lesser emphasis, were called number systems, for example, ‘algebraic numbers,’ ‘transcendental numbers,’ ‘higher order complex numbers,’ ‘quaternions,’ ‘biquaternions,’ etc. In order to organize the hodgepodge of artificially-constructed number systems, a heuristic principle was formulated. The construction of number systems should follow the ‘principle of the permanence of formal laws’ of an arithmetica universalis (Hankel 1867), that is, they should be constructed in such a way that some of the rules of ordinary numbers are preserved. It was hoped, in particular, that ‘higher order complex numbers’ of a certain type might serve as the foundation for an infinite number and virtually all possible number systems, an expectation which later turned out to be misguided.

8. Formalism And The Decline Of Number

The formalistic approach, which in the nineteenth century led to the flourishing number systems and ascribed to them a key role in the foundation of mathematics, reduced their importance in the course of the structuralist rebuilding of mathematics in the twentieth century. In fact, the recombination of structures and operations of mathematical objects in order to create new ones with new properties made it more and more difficult to distinguish number systems from other systems of mathematical objects. The designation of an object as a number was increasingly seen as merely an historical convention. Number systems were downgraded to instances of structures abstracted from them, in particular, of algebraic structures resembling addition and multiplication and topological structures determining distance and continuity. Not numbers, but ‘sets’ equipped with such structures, which were designated artificially as ‘groups,’ ‘fields,’ ‘rings,’ ‘lattices,’ ‘compact spaces,’ etc., became the building blocks of modern mathematics.

Bibliography:

1. Ascher M, Ascher R 1971–1972 Numbers and relations from ancient Andean Quipus. Archive for History of Exact Sciences 8: 288–320
2. Blake B J 1981 Australian Aboriginal Languages. Angus and Robertson, London
3. Chace A B 1927 The Rhind Mathematical Papyrus. Mathematical Association of America, Oberlin, OH
4. Closs M P (ed.) 1986 Native American Mathematics. University of Texas Press, Austin, TX
5. Damerow P 1995 Abstraction and Representation: Essays on the Cultural Evolution of Thinking. Kluwer Academic Publishers, Dordrecht, The Netherlands
6. Damerow P 1998 Prehistory and cognitive development. In: Langer J, Killen M (eds.) Piaget, Evolution, and Development. Erlbaum, Mahwah, NJ
7. Damerow P, Englund R K 1987 Die Zahlzeichensysteme der Archaischen Texte aus Uruk. In: Green M W, Nissen H J (eds.) Zeichenliste der Archaischen Texte aus Uruk. Gebr. Mann, Berlin
8. Dixon R M W 1980 The Languages of Australia. Cambridge University Press, Cambridge, UK
9. Hankel H 1867 Theorie der complexen Zahlensysteme insbesondere der gemeinen imaginaren Zahlen und der Hamilton’schen Quaternionen nebst ihrer geometrischen Darstellung. Voss, Leipzig
10. Heath S T L (ed.) 1956 The Thirteen Books of Euclid’s Elements. Dover, New York
11. Høyrup J 1994 In Measure, Number, and Weight: Studies in Mathematics and Culture. State University of New York Press, Albany, NY
12. Levy-Bruhl L 1923 Primitive Mentality. Macmillan, New York
13. Li Y, Du S 1987 Chinese Mathematics: A Concise History. Clarendon, Oxford
14. Locke L L 1923 The Ancient Quipu or Peruvian Knot Record. The American Museum of Natural History, New York
15. Marshack A 1972 The Roots of Civilization: The Cognitive Beginnings of Man’s First Art, Symbol and Notation. Weidenfeld and Nicolson, London
16. Neugebauer O 1934 Vorgriechische Mathematik. Springer, Berlin
17. Nissen H J, Damerow P, Englund R K 1993 Archaic Bookkeeping: Early Writing and Techniques of Economic Administration in the Ancient Near East. Chicago University Press, Chicago, IL
18. Saxe G B 1982 Culture and the development of numerical cognition: Studies among the Oksapmin of Papua New Guinea. In: Brainerd C J (ed.) The Development of Logical and Mathematical Cognition. Springer, New York
19. Schmandt-Besserat D 1992 Before Writing. University of Texas Press, Austin, TX
20. Thompson J E S 1960 Maya Hieroglyphic Writing. University of Oklahoma Press, Norman, OK
21. Wertheimer M 1925 Uber das Denken der Natur olker: Zahlen und Zahlgebilde. In: Wertheimer, M (ed.) Drei Abhandlungen zur Gestalttheorie. Palm and Enke, Erlangen, Germany