It would seem that all cultural traditions, most certainly the two which flow together in ganma, find connecting patterns and apparent symmetries throughout nature. Individually and collectively, we accept these patterns as part of the background of our lives: beyond question or doubt, all Australians agree that we live in an orderly, knowable universe. Yet, the patterns which organise, and the laws which govern, European knowledge and perception apparently have little in common with the patterns which make sense of the Aboriginal world. The differences between these two patterns are usually seen in terms of these contrasts: Western science as opposed to Aboriginal religion; mathematics as opposed to myth; rationality as opposed to totemism; in short, the 'Industrial Age' versus the 'Stone Age', To polarise these two systems in this way is not only misleading but entirely misinformed.
As we shall see, and have already seen, both kinds of patterning form complex and mathematically sophisticated systems, both have powerful ideological underpinnings, both are social constructions emerging from historically identifiable contexts, both attempt to account for natural as well as social phenomena, both are rational in principle and practical in application. And, finally, both systems involve re-occurring patterns and cycles which can be used, and indeed are used, as an organising and explanatory framework for understanding all things. What, then, are these two great systems, these extraordinary conceptual tools, by which the two Australian cultures make sense of the universe? We can call the system used by Aborigines a genealogical pattern. By this we mean ordered ways of naming and construing the relationships of natural things according to perceived ancestral or familial linkages. That used by European Australians we can call a number pattern: ordered ways of naming and construing natural relationships according to conventional techniques of counting and measuring. These two ways of patterning are of course very different in content and application, yet in a deep mathematical sense they are similar in kind. This was elegantly demonstrated a little over one hundred years ago by a philosophically inclined mathematician named Gottlob Frege, who showed how the recursive definition (a form of mathematical induction), which characterises both these systems, really just formalises the idea of 'and so on'. Both the number patterns that variously developed in Europe, Asia and Africa and the genealogical patterns of Aboriginal Australia are recursions in a strict mathematical sense. Of course, both kinds of patterning – genealogy and number – are found as systems of organisation, in all human societies. Number is not absent from Yolngu life; nor is the genealogical pattern absent from Western life. Indeed, in some spheres of Western life, ordering through genealogy remains strong: patriarchy, for example, dominates far more than the nuclear family. Yet, genealogical linkages recognised by European Australians are not generally applicable beyond certain family and social relationships. In Western life, kinship classifications are discontinuous and discrete. They are variously regarded as important or not important. They tell us little or nothing about the world of nature. If Westerners should lie about their genealogy, their world would not collapse. Kinship relations are not taken as objective, inevitable and deterministic except, sometimes, in the strict biological sense, and even then, genetic determinations are usually seen as a statistical potential, or a set of limits, rather than as a set of relationships and responsibilities. As we shall see, all this stands in great contrast to Aboriginal ordering through genealogv. For Aborigines, the genealogical pattern explains all relationships in both the social world and the world of nature. Furthermore, and this is often difficult for Westerners to understand, the number system is involved in Aboriginal life only secondarily. Little hangs on the functioning of number, which is to say number does not carry the deterministic weight nor the aura of objectivity and inevitability that it carries in non-Aboriginal Australia. In other words, in all Aboriginal-Australian communities the genealogical recursion, not the number system, patterns both knowledge and behaviour and carries the productive processes of the social order.