What is the relation between the medium in which we think and the medium in which we talk? This is a question asked by Stephen C. Levinson (2003) in his path-breaking work on spatial cognition and language. The medium in which we think, so argues Levinson, is at heart a spatial one, defined by an intuitively recognized, inherently memorable, and intersubjectively shared coordinate system, composed of a mental model of a geometric object that is independent of one’s imagined position in it. It is the independence of the geometric object that in fact enables us to “hold multiple levels of action in view at once” (Riles 1998, 379).

Levinson’s work is of crucial significance to anthropology, because the discipline has largely unquestioningly embraced cognitive science’s dictum of an “anchored” mind. Constrained by a physical world “out there,” the psycho-biology of the human visual system, and upright posture, western traditions of scientific and philosophical thought assume the human body at the center of the universe to be the “natural” disposition, congruent to a child’s conception of the world. Anthropology’s generic adoption of the egocentric, relative, and anthropomorphic position ignores the theoretical and methodological challenges of its own ethnographic accounts that point to the contrary. Space, we now know, can be conceptualized in absolute terms by peoples who operate in the world on par with post-Newtonian scientists, independent of any objects that space might contain, while also attending to space egocentrically. To show that this is so relies on a methodology that traces how people think and talk about the nature of relation by attending to the shape of such relation, formed by points in space arranged in neighborhoods that are as much immanent within objects as they are intuited by people.

The anthropologist Claude Lévi-Strauss (1969) had famously drawn on the posthumously published work by Cambridge natural scientist turned missionary and ethnographer A. Bernard Deacon (1934) to decode the kinship system practiced on the island of Malekula in Vanuatu, an island chain south west of Papua New Guinea. The Malekulans, Deacon realized, utilize the topological and rotational capabilities of an algebraic system known as quaternion number system to envision the relations between six classes of kin as sequences of recursive actions rather than as classificatory types. For Lévi-Strauss this recognition of mathematical ideas informing an understanding of an order immanent to social relations was the breakthrough for a theoretical and methodological appraisal of social relations in the study of kinship. The anthropologist Jürg Wassmann (1994, 648–49) showed, leading on from this, that in Australian Aboriginal societies, as among the Yupno in New Guinea, an absolute system of four cardinal edges is used at the same time as an egocentric one, allowing manifold spatial relations to be held stable while contemplating them from different perspectives.

Research into the spatial cognition at work in these societies has shown that they deploy a topological conception of space, which approaches space not as flat, but as curved, comprised of points arranged in open sets of relations or neighborhoods (Glowczewski 1989; Hutchins 1995). Mathematically understandable via the tools of differential geometry, the idea of the curved surface is shaping thinking about the space between two points in ways that are radically distinct as the space now is heterogeneous and multiple, and yet globally integrated as well as precisely measured, made tangible in patchwork-like assemblages of points. The topology between points in curved surfaces is made understandable and knowable by representing it in objects such as stick charts used by Micronesian seafarers or in piecework coverlets of Eastern Polynesia used for the reckoning of genealogical relations (Küchler 2017). These objects evince the idea of a neighborhood of points in relation to one another in ways that allow for a simultaneous conception of their insideness and outsideness, disjunction and connection, and sequences of actions of folding, stretching, and squeezing. Via these objects, people move not through, but in space, both simultaneously and sequentially, rather than to fixed points along a trajectory.

The recognition of the use of an absolute conception of space alongside an egocentric one, one in which the idea of relation is inseparable from its shape, has far-reaching consequences for method in anthropology as it cannot rely on observation alone (Küchler 2017). In his classical study of gift exchange, Marcel Mauss (1954) set the tone for future work in anthropology and social science at large by setting out to explain how people are bound together through the agency of objects imbued with notions of personhood. By attending to objects in which the environment is no longer outside, but inside, invisible to the eye and yet conceptually accessible and predictable via a geometry that invites multiple views at once, we can begin to ask new questions about how these objects work and the difference they make to the societies that make and deploy them. The close-up analysis of such objects that enables an integration of egocentric and absolute conceptions of space, however, requires social science to rethink how it is to be taught and practiced alongside the natural sciences and computing.