We have talked of the system of gurruṯu as a 'grid' and as an 'orderly mesh'. In ITEM 4.5 we see how Yolngu represent this grid. This dhulaŋ (bark painting) is a representation of ganma. Here we see two sets of gurruṯu, which each constitute a set of related people and places – Dhuwa on the left and Yirritja on the right – work together and develop mutual understandings, or a depth of knowledge: the dark band in the centre of the bark.
4.5
Ganma; moiety - Yirrirta; clan - Gumatj; painter - Yalmay Yunupiŋu, 1989
ochre on bark, 20 x 36 cm.
This bark was painted specifically to represent the concept of ganma. We see two grids - formalised representations of two groups of people. Here we have a formal account of the working of the gurruṯu system. Perhaps we can think of the intersecting zigzag lines as the Yirritja and Dhuwa moieties, twisting around each other and linked at each generation - each bend and each intersection of the lines. The cross-hatching within the diamonds represents the connection of the present-day clan With the Ancestors of the Dreaming. This painting represents the abstract relation of land and people as well as being a formal account of 'ideal' relations between people. The graphic forms in this bark are Gumatj - different from the forms of other Yolngu clans (see ITEMS 5.5 and 5.10).
At a superficial level we may interpret this bark in the following way; but there are many other levels of interpretation. Concentrate first on the yellow lines zig-zagging across each other down the bark. Here we have the main lines of the gurruṯu system, the Yirritja and the Dhuwa moieties interacting with each other across the generations. Across the bark we see one complete cycle of the märi-gutharra cycle: three generations; three complete diamonds. In the cross-hatching on the bark we see the grid stretching both backwards and forwards across time symbolising Ancestral power through the many layered relationships reaching back to the Dreamtime. This design presents us with the patterning and order that was wrought by the Ancestors in their living. It is a metaphor which assures us that the land and its people have meaningful existence. Vast ideas have been conveyed in precise ways, but this bark can inform any particular person about where she or he fits in only in the most general sense. What is encoded here could not be used to inform a specific decision on a practical everyday matter. But it does enable a time never experienced to be known.
MATHEMATICS AND ROM
Gurruṯu and number variously mediate involvement with the world. We used graphs of geographic variables to illustrate a use of number, but it is not so easy to illustrate the workings of gurruṯu. The linguistic texts in which its workings are made clear are for the most pan oral texts, and the graphic texts which might be used to illustrate the way gurruṯu mediates involvement with the material world cannot be reproduced here.
As they occupy such a central place in social life, it should not surprise us to find that gurruṯu and number are celebrated in social institutions: number is part of the powerful abstract discourse of mathematics, and gurruṯu is integral to the functioning of the powerful and abstract Yolngu discourse rom. Rom is often translated as 'law', but rom can also usefully be considered as analogous to mathematics. Both rom and mathematics deal in abstract forms, taken as self-evident facts of the world. Using these facts, people proceed through deductive reasoning to knowledge. Of course gurruṯu is not the only abstract form in rom, Just as number is not the only abstract form in mathematics.
4.6
The ideal normal distribution curve.
A normal distribution curve is a mathematical device which depicts the distribution of a characteristic for a given population in terms of the area under a bell-shaped curve.
To appreciate this analogy it will be useful to unwrap something of the mystery which imbues both mathematics and rom. Thus we might ask: How do the abstract forms that are dealt with in rom and mathematics apply to the 'real' world? How can we usefully view their (rival) universal claims to truly represent the world? Latour considers the case for mathematics. In practice, he notes, the abstract forms of mathematics are never actually applied to the empirical world; it is much more subtle than that. In the action of using mathematics, visual images are generated-graphs, diagrams and so on. In conventional ways actual things are transformed and the transformations are then inscribed. Quite diverse bits of the world are rendered into identical shapes: bacteria and people become normal distributions (see ITEM 4.6).
The adequation of mathematics with the empirical world is a deep mystery [but the] superimposition of one mathematical form on paper [e.g. the ideal normal distribution] and of another mathematical form drawn on the printout of an instrument [the normal distribution seen in a particular set of data] is not a deep mystery, but it is quite an achievement all the same. B. Latour, Science in action, Open University Press, Milton Keynes, 1987, p. 244
We can adapt Latour's account of the workings of mathematics to illustrate the workings of rom:
The adequation of [rom] with the empirical world is a deep mystery but the superimposition of one [rom] form [wrought in sand (an ideal generalisation about the related world) and of another [rom] form drawn on la boy's body at circumcision (a particular manifestation of the ideal)] is not a deep mystery, but it is quite an achievement all the same.
Neither gurruṯu nor number are neutral; each conjures up a particular political system, a particular knowledge-power network. Through gurruṯu, notions of value mediate relations in the world in a particular way, and this is quite different from the way the notion of value is engaged through the use of number. We might say that the number system works on the basis that everything has value in some positive sense, whereas the gurruṯu system works on the basis that every relation involves beholdenness, or has something like negative value.
This juxtaposition of different ways of evoking notions of value gives us a new way of viewing the process of ganma. We can recognise that in the past European Australia and Aboriginal Australia have been negotiating with quite different views on how value enters human lives. Perhaps acknowledgment of this disparity helps us towards greater understanding of each other. With mutual recognition that each side has constructed a system of valuing which is rigorous and rational, and mutual acceptance that each system has particular advantages, we can get somewhere.
FURTHER READING
T. Danzig, Number, the language of science, Macmillan, New York, 1954. A history of number showing how practical aspects of tallying have come to constitute number.
G. Frege, The foundations of arithmetic, tr. J. L Austin, Blackwell, Oxford, 1980. Draws out the relation between grammatical categories of language use and the foundations of mathematics.
G. Ifrah, From one to zero, tr. L Bain, Viking Press, Sydney, 1985. Shows the evolution of recursive tallying in many cultures.
Harold W. Schemer, Australian kin classification, Cambridge University Press, Cambridge, 1978.
Don Williams, Exploring Aboriginal Kinship, Curriculum Development Centre, Canberra, 1981.
L. Wittgenstein, Remarks on the foundations of mathematics, ed. G. H. von Wright, R. Rhees & G. E. M. Anscombe, tr. G. E. M. Anscombe, MIT Press, Cambridge, Mass., 1967.