It was not until the early 1400s that Ptolemy's Geographia arrived in Europe, the same period in which Brunelleschi developed perspective geometry and its application in architecture. The *Geographi*a mapped the entire world and presented all the known information in a standardised and consistent way with grid lines of latitude and longitude (see ITEM 7.1). This metrication meant that all points were commensurable: that is, distances and directions could be established between one place and any other. Further, unknown places could be given co-ordinates. It was the synthesis of perspective geometry and Ptolemy's work that enabled the imposition of a grid on the known world. Once that grid was imposed, the mathematican Toscanelli was able to argue plausibly that sailing westwards across the Atlantic was a shorter voyage to the Spice Islands than the traditional route around the Cape of Good Hope and on to the East. Thus Columbus 'discovered' America even though in 1492 he was convinced that Cuba and Japan were one and the same.
Chinese grid map drawn on stone, known as 'Map of the tracks of Yu the Great' (1137).
There is of course nothing in reality that corresponds to such a grid; it is a human construct, and hence arbitrary, conventional and culturally variable. Ptolemy (AD 90-168) located his grid by designating the Fortunate Isles (Canary Islands) as the prime meridian, because they were the western extremity of the known world. Spanish and Portuguese cartographers used the Tordesillas line (see ITEM 10.2). The use of grids originated in China, probably with the work of Chang Heng in the 1st century AD. Although none of his mapwork survives, his biographer, Tshai Yung, wrote that he 'cast a network about heaven and earth and reckoned on the basis of it' (R. Temple, The genius of China, 1989, p.30). Subsequently the grid was in continuous use in China, and one of the two early maps inscribed in stone at Sian in 1137 is covered by a well-defined grid (ITEM 4.12). Phei Hsiu, the 3rd-century Chinese cartographer, laid down the use of the grid as one of his six principles of scientific cartography, claiming that 'When the principle of the rectangular grid is properly applied, then the straight and the curved, the near and the far, can conceal nothing of their form from us' (in P. D. A. Harvey, The history of topographical maps, 1980, pp. 133-4). The Romans used a grid system called 'centuration' by which they came close to turning 'all Europe into one vast sheet of graph paper' (S. Y. Edgerton, 'From mental matrix to Mappamundi to Christian Empire', 1987, p. 22). The British use the National Grid system and in Australia the Australian Map Grid is used.
Even the system of lines of latitude and longitude are conventional: 'At an international conference held in Washington in 1884 it was agreed by many countries that subsequently 0° of longitude would be assumed to pass through Greenwich [England]. This meridian is now widely, though not universally, used for mapmaking' (A. G. Hodgkiss, Understanding maps, 1981, p. 30). For a grid system to work it has to be literally conventional. Grid systems require real conventions, negotiations and agreements. In order to bring the distant and the large to your table top you need perspective geometry, reproducible and combinable representations, a grid and the agreement of your fellows. The power of maps lies not merely in their accuracy or their correspondence with reality. It lies in their having incorporated a set of conventions that make them combinable in one central place, enabling the accumulation of both power and knowledge at that centre. The significance of Ptolemy's Geographia was not just its use of a grid: it was also an atlas which enabled the co-ordination of maps of individual lands into one map of the world. Similarly, the map of China (ITEM 4.12) was constructed as a pathwork of local maps drawn from itineraries.
In Exhibit 5 we shall be considering Aboriginal Australian bark paintings as maps. These have the appearance of being incapable of being combined in the European or Chinese way. Their maps appear to have no grid, no standardised mode of representation. Nonetheless it is possible for Aboriginal people to know about, and to travel across, unknown, even distant, territory. Their knowledge is in fact combinable because it is in the form of narratives of journeys across the landscape. Aborigines inculcate and invoke conventions just as we do, through conferences and agreement. They call them business meetings; anthropologists call them ceremonies and rituals. Songlines (which are accounts of journeys made by Ancestral Beings in the Dreamtime) connect myths right across the country. One individual will only 'know' or have responsibility for one section of the songline, but through exchange and negotiation, the travels of the Ancestors can be connected together to form a network of dreaming tracks. These may be constituted as bark paintings or song cycles. The strength of the distinction between indexical maps and non-indexical maps will seem even less cogent when we come to consider the use of maps as instruments for navigation in Exhibit 9. It will then be seen that in order to find our way about we need at least a mental map, or a cognitive schema, and an indexical image of the landscape, and that we can never navigate with non-indexical statements alone.
That maps consisting entirely of non-indexical statements cannot be used for practical purposes involving direct interaction with the material world such as navigation constitutes another similarity with scientific theories. Scientific theories consist of nonindexical universal statements about reality, and as such cannot be applied directly to a particular circumstance, or be confirmed or falsified by particular items of evidence. Scientific theories always need additional assumptions and qualifications, or specification of conditions in order to apply in practice. Strictly speaking, scientific theories could be said to be, on the one hand, useless, or, on the other hand, false, in the sense that they can never apply without modification to a particular circumstance. If, for example, you wish to calculate the orbit of a planet around the sun, Newton's laws of motion and gravitation, or even Einstein's laws, are not sufficient. You have to assume that there are no other forces at work, that there are no unobserved bodies in our solar system, that space is empty, that planets are effectively point masses, that there are no effects of the system acting on itself and so on. Then you have to live with the fact that in the case of Mercury, for example, the predicted orbit does not completely fit the observations. It seems that scientific theories gain their non-indexicality at the expense of their applicability.
Another way of capturing the notion of indexicality is to recognise its connections to 'forms of life'. All indexical statements are embedded more or less explicitly in a form of life. Non-indexical statements attempt to transcend or deny a form of life. In the attempt to maximise the objectivity of scientific theories and to display them as universal truths they are increasingly distanced from their forms of life and consequently lose their connection to the world. This, however, does not make them either useless or powerless. They gain great strength and efficacy in the social and cultural domain. Once again this point is illustrated by maps as we shall see in Exhibits 8, 9 and 10.