The concluding section of the book takes a specific look at theological conceptions of the relationships among the measurable finite, the immeasurable infinite, cosmology and worldview. These chapters represent reactions to the evidence and questions asked of it in the preceding sections and present additional perspectives on the relationships between measurement and our attempts to engage with the material world and elements beyond it.

LeRon Shults, from a background in theology and philosophy, focuses his commentary in particular on human engagement with the ‘beyond’ element of the title of the conference, ‘Measuring the World and Beyond’. With the ability to measure and quantify the world arises an awareness of the immeasurability of certain phenomena. Shults draws out the theme of ‘control’ as being central to many of the chapters within the volume and, in particular, how this often leads to attempts to control the uncontrollable – for example, the uncontrollability of time and the degeneration of our corporeal form and material goods. He says that the questioning of such issues as what lies at the ends of our personal time – birth and death – lies at the roots of spirituality. If spirituality can be considered to be (at least partly) the process of making meaning of the experience of finite temporality, then as archaeologists we can hope to identify attempts to engage in this process.

Introduction

Κατὰ δὲ Πρωταγόραν τὸν σοφώτατον πάντων χρημάτων ἄνθρωπον μέτρον εἶναι, κατὰ δὲ Θεαίτητον τούτων οὕτως ἐχόντων αἴσθησιν ἐπιστήμην γίγνεσθαι (the doctrine of the great philosopher Protagoras, that man is the measure of all things, and the doctrine of Theaetetus that, since these things are true, perception is knowledge) (Platon, Theaetetus 160 d.9). This quotation may perhaps bring to mind the Egyptian cubit, of the length of a human forearm, and its subdivisions into palms and digits. In architecture, ergonomic requirements, such as the height of a door or the measurements of a staircase, exhibit dimensions related to the human body, whose nature makes them universal and diachronic (cf. Palyvou 2005, 156 with reference to Modular Man by Le Corbusier). The technique of measurement by weight is very close to Theaetetus's doctrine that perception leads to knowledge, since everyone can feel which is the heavier of two objects held in the two hands. Thus the mechanism of the balance, which actually predates the invention of weights, in fact reproduces an action performed by the human body itself. In the tomb of Ka-irer, dating to the Old Kingdom of Egypt, there is a scene showing ingots being weighed in a curiously designed balance in the form of woman with arms stretched horizontally (Kisch 1965, 26; Lauer 1976, 77). Furthermore, the most widely occurring largest unit in various metric systems is the heaviest load that a man can comfortably carry on his shoulders, that is, about 30 kilos (Figure 7.1).

The Inca Empire of Andean South America emerged in the 15th century AD as one of the great political triumphs of world history. In a short 100 or so years, the Inca developed in the central Andean highlands out of a multitude of competing polities in the post–Middle Horizon period (Figure 17.1). The Middle Horizon, circa AD 500–1100, represented the apogee of the three first-generation states of the Andes – first Moche in the first centuries AD up to around AD 700, then Wari and Tiwanaku (Figure 17.2). The latter two states developed almost in tandem circa AD 500–1000 extending their reach over most of the Andean highlands, from northern Peru to north central Chile.

The collapse of the Wari and Tiwanaku states created a volatile political context throughout the region. In the north coast, several polities developed out of the remains of the Moche state, ultimately culminating in the first empire of Chimor in the beginning of the second millennium AD. Moche defined statecraft as it was practiced in the pre–European contact Andes, beginning at the end of the first millennium BCE and continuing into the middle of the first millennium AD. Moche culture most likely directly influenced Wari and Chimu and indirectly influenced Tiwanaku as these two cultures developed their models of statecraft in the immediate post-Moche periods.

The Quechua-speaking Inca peoples were one of a number of smaller polities that developed in this general cultural context of the post–Middle Horizon period.

Part I. Numeration and numerical cognition

Cognition is a system consisting of representations of knowledge together with the processes that operate on those representations. What is fundamental to numerical cognition is the representation of knowledge about number and numeration, and the operations that are performed on them. This section explores features of numeral systems that relate to several different perspectives on these matters within cognitive and linguistic anthropology. It begins with a selection from Greenberg's (1987) findings on cross-linguistic regularities in the structure of numeral systems in spoken languages.

Number words and phrases

One way to explore how number is conceived is through its representation in language. For many linguistic anthropologists, it seems almost inevitable that the grammatical structure of words relating to regularly relevant semantic categories and domains will help to shape and/or be shaped by the ways they process information involving those categories and domains. Whether or not this is so in the case of number, the relevance of its linguistic representation follows from two empirical observations. (1) The vast majority of languages have numerals – ­conventional terms used widely in a speech community to represent specific numbers. (2) Worldwide, numeral systems – the linguistic organizations of terms representing numbers – show great similarity in their basic structure. The major work relevant to this chapter is by Greenberg (1987; see also Stampe 1977, Hurford 1987), who provides 54 generalizations capturing the structures of a wide variety of documented numeral systems.

To date, the major corpus of published information on Indus architecture is still that from the early excavations in the 1920s and early 1930s of Mohenjo-Daro (Mackay 1938; Marshall 1931). With the later excavations by Wheeler (Wheeler 1953) and Dales (Dales & Kenoyer 1986) no major data of physical remains could be added, except the Granary and some fortification in the citadel area reported by Wheeler (excavated in 1947 and 1950, respectively), and a low-lying area with a slope situated at the western edge of the ‘lower City’, west of HR area (UPM area) (see Figure 10.1), excavated in 1964 by George Dales. In terms of the conception of the interpretation of the Indus Valley Civilization, both excavations were important, as Mortimer Wheeler discovered the “centralized economic Priest-King system” of this civilization, represented by the ‘State Granary’. He also foresaw the great importance of his deep dig (1950), which clearly indicated a gigantic substructure of the citadel. This surrounded, at least, the whole western part of the settlement, as was demonstrated later by the German Research Project at Mohenjo-Daro (1982–84). Today we know that the visible part of Mohenjo-Daro, covering approximately 100 hectares, only represents the ‘tip of the iceberg’; in 1989, within the UNESCO Campaign programme for its protection against the Indus floods and for the stabilisation of the Indus River banks, large spurs were built into the Indus bed.

In this chapter I will focus on the archaeology of Neolithic circular enclosures in order to understand better their potential for further analysis regarding early measurement of space and time. In particular I will focus on the Neolithic circular enclosure in Goseck and not only introduce the site, but also present fresh data that may help shift our understanding of the meanings and functions of the phenomenon of the Middle Neolithic enclosures in Europe. These enclosures are especially interesting because they emerged and were used over a period of approximately two centuries (ca. 4900–4700 BC) and then just as abruptly disappeared. I will also discuss evidence for the astronomical function of the Goseck enclosure. This evidence is mainly based on Wolfhard Schlosser's work on the Goseck material, and I am grateful to him for letting me use and present his astronomical results in my analysis of the Goseck excavation data. Furthermore, I will discuss how this evidence could be operationalized to address questions of an early measuring system as well as of cosmology, cult and ritual, and the complex roles of these regarding identity, memory and experience for these early agriculturalists in Europe.

Neolithic enclosures in Europe

Neolithic enclosures have been known in Europe for over 100 years and have fascinated archaeologists and the public alike. The image of monumental architecture made of stone, wood or earth constructed some 7,000 years ago was so powerful that it has long incited archaeologists to search for the origin of this European phenomenon and to investigate its spread and function.

Introduction

This chapter discusses a number of different aspects of measurement relevant to past societies, and the archaeological record. First, it explores some of the concepts underlying different types of measurement, the differences between them, and the implications of those differences for how measurement might be, and has been, conceived. Different types and concepts of measurement are labelled with specific terms, and the relationships between them considered. Whilst it is hoped that the concepts discussed and terms used might be useful in wider considerations of measurement, the chapter is written keeping in mind the nature and implications of the types of measurement activity that are likely to have been important in the context of hunter-gatherer subsistence, especially concepts of time, cycles and distance. In particular, this chapter seeks to highlight the fact that a great many of the activities that were carried out by past societies would have involved quantification in some form. In order to explore fully the use of measurement of the world and beyond in past societies we must explore the implications of the activities for which we do have archaeological evidence, in addition to looking for direct evidence of quantification.

The final sections of the chapter in particular relate some of these conceptualisations – especially of time and cycles of events – to supernatural and spiritual systems of belief for their explanation, and ritual systems of practice for their mediation.

The theme of the relationships between terrestrial and celestial aspects of the world is elaborated further in Section IV, whose chapters are united by a principal focus on the measurement of time. The opening section of David Brown's contribution focuses on the notions of concrete and abstract number in Mesopotamia, before going on to consider in detail the measurement of time and its relationship with terrestrial and cosmological measures. Brown initially argues that the existence of ‘concrete number’ systems in Mesopotamia (with both quantity and quality [commodity] being represented by a single symbol, repeated the appropriate number of times) does not represent a lacuna in human cognition (an inability to conceptualise abstract number) but is instead a product of administrative practice. When this administrative mechanism changes, the use of abstract number becomes conspicuously part of the arithmetical system. However, he argues, evidence for the cognition of the concept of abstract number exists before; that is, it is a transformation in practice, rather than human cognition, that is represented by this change.

In Mesopotamia the measurement of the passage of time became linked intimately with the measure of distance, via the distance between ‘rest stops’ (covered at walking speed on foot march) and the duration of a day. This formed part of an idealised system of relationships among distance, time and astronomical movements, including idealised durations for days, months and years, which, Brown suggests, accorded with a concept of the ideal creation.

Orientation in space is an important aspect of measurement most commonly associated with direction finding. Humans tend to divide geographical space in a horizontal plane into four primary directions or areas deriving from an embodied experience of the world: in front, behind, left and right (Tuan 1977, 34–37). As the body is mobile and positions are relative, establishing, remembering or communicating direction requires external reference. In the absence of a compass or equipment for measuring direction, reference can be made to landscape features or, over longer distances and in the absence of distinctive landscape features (for example, at sea), reference can be made to the Sun and stars, or, where they exist, prevailing winds (see also Morley, this volume).

In settled communities, patterning in orientation is sometimes found in architecture. In many regions this derives from environmental considerations: structures oriented towards or away from a prevailing wind or towards or away from the Sun, depending on the conditions of a particular area. Patterning is also frequently found in the orientation of structures toward landscape features such as the sea, a river, or a route or feature created by human intervention in the landscape. Sometimes, however, patterning in orientation may be found in contexts in which pragmatic explanations are lacking or seem unlikely. These may suggest underlying religious or cosmological considerations; examples include patterning in the orientation of tombs, the orientation of mosques toward Mecca or churches toward the east (replacing earlier orientation toward Jerusalem).

The second section of the book deals specifically with the archaeological identification of the application of measurement to systems of economy and trade. Anna Michailidou opens her consideration of weighing in the Bronze Age Aegean by giving an overview of the occurrence of relative and absolute measurement in Egypt and ancient Greece, where – with an explicit connection between measurement and religion – the invention of systematic weighing was credited to the god Thoth and the hero Palamedes (great-great-great-grandson of Poseidon), respectively. Weighing is attested archaeologically by balances and numerous balance weights, as well as in written sources. The weighing system and activities of the Aegean have to be understood in the context of the other weighing systems and trade activities of the time. Some weights in use in the Aegean and Near East seem to have been deliberately manufactured to act as ‘nodal points’ for other weighing systems of the period. For example, the Aegean unit of ca. 65.5 grams could have represented 5 Egyptian deben, 8 Egyptian shaty, and 10 Near Eastern shekel.

It seems that many foodstuffs (e.g., grain, flour, pulses, oil, wine) were measured by volume rather than weight (excepting bread, fish, and meat), but that commodities such as metals, yarns, textiles, some wood and ivory, and condiments, dyes, and spices were measured by weight.

It was a profoundly significant step when, in the remote past, a human being, in undertaking an act of measurement, formulated the notion of measure. For to measure – whether in the dimensionality of weight, or of distance or of time – is to develop a new kind of material engagement with the world that is at once practical and conceptual. It is an act of cognition – a cognitive act. Such an act has philosophical implications, for measurement allows us to transcend the limitations of the here and the now. It involves observation, and it facilitates construction. It encapsulates the seeds of mathematics and of science. It makes possible architecture and design. It is the basis for systematic observation and prediction. It leads on towards astronomy and cosmology. It is the basis for any complex economic system. It is one of the foundations of all urban civilisations.

This volume, arising from the Roots of Spirituality project conducted at the McDonald Institute for Archaeological Research, sets out to explore the new and creative relationships with the world implied by the first deliberate development of measurement and of systems of measure in the early days of the human story.

The theme was chosen as a means of investigating, at a global level, some fundamental issues in the origins of human cognition in the early days of the different trajectories of cultural development. These issues bear upon the very process of becoming fully human in an increasingly complex world.

Introduction

The objective of this chapter is to explore various ­features of the encoding of information in the khipu (Quechua: ‘knot’), the knotted-string device used for record keeping in the Inka empire of pre-Columbian South America. Specifically, I will discuss what the testimony concerning khipus contained in Spanish colonial documents, as well as study of museum samples of khipus, can teach us about the types and standards of measurements used by local and state administrators in the Inka empire. Given these objectives, we should be clear from the beginning about the range of concepts and practices connected with ‘measure’ and ‘measurement’, at least as these are understood in English. To measure means “to compute, estimate, or ascertain the extent, quantity, dimensions, or capacity of [something], especially by a certain rule or standard” (Webster's New Twentieth Century Dictionary 1978). In addition to this dauntingly wide range and variety of activities, measurement involves more abstract concepts and forms of evaluation, including “to estimate by reference to any standard; to judge of the value, extent, magnitude, or greatness of [something/someone]” (Webster's New Twentieth Century Dictionary 1978).

Clearly, if our objective here is to discuss the recording of measure(ment)s in the Inka khipus, we will have to find some way to limit the domains of reference, as space will not permit a full consideration of the many and varied principles and activities evoked previously.

This chapter deals with a system of counters – clay tokens – used for over 4,000 years in the prehistoric Near East (7500–3100 BC). Relying on a database of some 8,000 tokens from Turkey, Syria, Jordan, Israel, Iraq and Iran (Schmandt-Besserat 1992, I & II), I discuss the evolution of the token system, the method of counting it implies and how it led to writing and abstract numbers (Butterworth 1999, 29–32; Rogers 2005, 81–84). Lastly, in the light of the token system, I address the relation of counting and measurements to the economy and to cognition.

Tokens and pictographic writing

Before starting my discussion I explain how the Mesopotamian pictographic and cuneiform scripts are critical to understanding the token system (Schmandt-Besserat 1996).

During the first 500 years following its invention about 3200 BC, writing in Mesopotamia was used exclusively for accounting (Cooper 2004, 72). The tablets served a city state administration scrupulously to record entries and expenditures of goods in the temple and palace. The first Mesopotamian script featured two kinds of signs: impressed signs stood for numerals and signs traced with a stylus represented the goods accounted (Figure 3.1). As is explained later, both of these types of signs, impressed and traced, were images or ‘pictographs’ of small counters, that is, tokens previously used for record keeping. Some of the pictographs can be understood by matching them to the cuneiform signs that derived from them. The pictographs therefore constitute a ‘Rosetta Stone’ to decipher the age-old token system.