The Roman numerals are undoubtedly one of the better-known numerical notation systems, and have received a tremendous amount of scholarly attention. Nevertheless, they constitute only a part of a larger phylogeny of numerical notation systems that originated, not among Romans, but among Etruscans and Greeks on the Italian peninsula around 600–500 bc. The name “Italic” refers only to this geographical origin, and thus does not reflect any shared linguistic or cultural affiliation. Italic systems flourished between 500 bc and 500 ad throughout the Mediterranean region, Western Europe, and North Africa, under conditions of Greek and Roman cultural hegemony and political domination. Ironically enough, however, the collapse of the Roman Empire brought about the greatest expansion of one particular system - the Roman numerals - in medieval Europe, and ultimately throughout the modern Western world. The most common variants of the Italic numeral-signs are shown in Table 4.1.

ETRUSCAN

The Etruscans were a non-Indo-European people whose civilization had its center in north central Italy, in the region of modern Tuscany (whose name is taken from the Latin Tusci, meaning Etruscan). The origins and civilization of the Etruscans are poorly understood, and large parts of their language remain undeciphered. Yet Etruscan civilization was the most potent political force on the Italian peninsula between around 800 and 300 bc, and significantly influenced Roman culture throughout the Republic and even later. The Etruscan alphabet, developed in the early seventh century bc on the model of the archaic Euboean Greek alphabet, usually runs from right to left (Bonfante 1996).

The systems discussed in the previous three chapters are primarily cumulative, repeating signs within each power of the base to indicate addition. In contrast, the next two families – the Alphabetic and South Asian systems – consist mainly of ciphered systems, which use, at most, a single sign for any power to indicate its different multiples: 1 through 9, 10 through 90, 100 through 900, and so on, in the case of decimal systems. Ciphered numeral-phrases are thus much shorter than cumulative ones, but require their users to be familiar with many more signs. Alphabetic numerical notation systems generally use phonetic script-signs, in a specified order, to express numerical values, and thus mitigate the effort needed to memorize both script-signs and numeral-signs. Despite the name, the scripts in question are not always alphabets; some, such as the Hebrew and early Arabic, are abjads or consonantaries, expressing primarily consonantal phonemes, and one, the Ethiopic Ge'ez script, is an alphasyllabary or abugida, expressing consonant + vowel clusters.

Alphabetic systems were used as far north as England, Germany, and Russia and as far south as Ethiopia, and throughout Africa and the Middle East from Morocco eastward to Iran. Their history spans over two thousand years, from the development of the Greek numerals around 600 bc to the present, but in some cases important historical questions remain unresolved. While they are mostly ciphered-additive, they are not structurally identical. We can learn much more from these structural diff erences than from the paleographic curiosities of the signs of various systems.

The Western world is a world of written numbers. One can hardly imagine an industrial civilization functioning without the digits 0 through 9 or a similar system. Yet while these digits have pervasive social and cognitive effects, many unanswered questions remain concerning how humans use numerals. Why do societies enumerate? How does the representation of numbers today differ from their representation in the past? Why does the visual representation of number figure so prominently in complex states? What cognitive and social functions are served by numerical notation systems? How do numeral systems spread from society to society, and how do they change when they do so? And, despite their present ubiquity, why have the vast majority of human societies not possessed them at all?

If you look up from this page and examine your surroundings, I am certain that you will encounter at least one instance of numerical notation, probably more. Moreover, unless you have a Roman numeral clock nearby, I am nearly certain that all of the numerals you encounter are those of the Hindu-Arabic or Western system. Numerals serve a wide variety of functions: denotation – “Call George, 876–5000”; computation – “21.00 × 1.15 = 24.15”; valuation – “25 cents”; ordination –”. Wash dishes, 2. Sweep fl oor, 3. Finish manuscript”; and so on. Most of the thousands of numerals we see each day barely register on our conscious minds; regardless, we encounter far more written numbers in our lifetime than we do sunsets, songs, or smiles. Until the past few centuries, the opposite was true for most people.

The primary function of numerical notation is to communicate numerical values. One cannot even lie effectively about how many enemies were killed in battle if the numerals being used are incomprehensible to the intended audience. Any attempt to explain the history of numerals without reference to the cognitive features underlying their structure is doomed to failure. Nevertheless, considerations of efficiency are not the sole or even the primary factor in the cultural evolution of numerical notation. While synchronic regularities may be explainable without reference to social context, diachronic regularities are not. Every cognitive advantage associated with a system is associated with disadvantages. The role of various social factors in explaining the history and development of numerical notation systems differs from case to case, depending on historical context, but they are always there. We cannot explain the replacement of Maya numerals by Western ones without consideration of the enormous social, political, and technological upheavals that were associated with the Spanish conquest of Mesoamerica. Numerical notation systems never exist as objects in isolation; their utility is not merely a function of their structure. By exploring the social contexts in which the transformation and replacement of numerical notation systems occur, it will be possible to evaluate the impact of social factors relative to purely cognitive and structural ones.

I have identified seventeen factors that influenced the changes in numerical notation systems examined throughout this study, any of which may apply to a particular historical event.

Around twenty systems do not fit neatly into the phylogenetic classification presented in Chapters 2 through 9. A few, such as the Inka khipu numerals, the Indus (Harappan) numerals, and the enigmatic Bambara and Naxi numerals, apparently arose independently of any other system, but gave rise to no descendant systems. Others are cryptographic or limited-purpose systems used in the medieval or early modern manuscript traditions of Europe and the Middle East. The majority of this chapter, however, deals with systems that emerged in colonial settings under the influence of the Western or Arabic ciphered-positional numerals, in conjunction with the development of indigenous scripts. Most of these systems were developed in sub-Saharan Africa, but Asian (Pahawh Hmong, Varang Kshiti) and North American (Cherokee, Iñupiaq) indigenous groups have also developed their own numerical notation systems. Finally, a few systems are probably members of other phylogenies, but their exact affiliations remain inscrutable enough that no definite conclusions can be reached.

INKA

The Inka civilization was an enormous state on the Pacific coast of South America that reached its pinnacle between 1438 and 1532. While writing is often (and mistakenly) seen as a sign of civilization, or at least as a necessity for large-scale bureaucracy, the pre-colonial Inka state operated in the apparent absence of any writing system capable of expressing phonetic values. Instead, the primary means of encoding information was a system of knotted cords of different colors, known as khipus, whose main purpose was to record numerical information to aid in the administration of the Inka state.

In Chapters 2 through 10, I described over 100 different numerical notation systems spanning over 5,000 years and every inhabited continent. While there are historically determined similarities among the systems of each phylogeny, the same structures and principles emerge independently multiple times. This situation creates a paradox only if we cling to the dichotomous assumption that historical explanations stand in stark contrast to universalizing ones. A set of interrelated cognitive factors help explain why systems are the way they are and why they change in the ways that they do. There are some domains of human experience for which the role of contingency is so great, or the functional constraints so minimal, that we cannot speak meaningfully of regularities or laws. Numerical notation is not one of them. In Chapter 1, I outlined the case that the study of cross-cultural regularities and universals is of equal importance to the study of unique or particular phenomena (Brown 1991). Here I will outline around thirty regularities that apply to numerical notation systems, while in the following chapter I will inject theoretical issues relating to social and historical context into this analysis.

Numerical notation systems exhibit both synchronic regularities, which apply to numerical notation systems considered as static structures, and diachronic regularities, which apply to relations between systems over time. Synchronic or diachronic regularities can be either universals (for which there are no exceptions) or statistical regularities (which hold true only for a preponderance of cases). While true universals are interesting, statistical regularities are also important, and may in fact be caused by cognitive factors similar to those that produce universals.