- How does one get to know a space after all?
As is often the case, the way in which a question is posed dictates its possible answers. Therefore, to ask how to know a space suggests that there exists a plurality of spaces. One of the possible responses to this question lies in interrogating that plurality.
The multiplicity of spaces translates into a larger set of problems. What types of spaces are there? Where do spaces exist? What space do they occupy? Is the plurality of spaces to be found in an all-embedded global Space? If yes, then are spaces mere sub-divisions of that same Space? In other words, are they homogeneous? If not, what differentiates them and, as heterogeneous divisions of Space, what can they tell us in return about It, if anything at all?
- Throughout history, the accumulation and distribution of space has also been the driving force of our knowledge of Space. Space first became metricized—or better, geo-metricized—for territorial reasons: to restore land ownership markings that were washed away by periodic river floods. In a sense, what necessitated geometry was its own impermanency, which partially explains subsequent efforts to abstract it from its earthbound application. One could argue that the history of Space over the last two millennia has been one of
parameterizing and de-metricizing those first territorial markings.1
Much has been hypothesized about Space, and the debunking of those assumptions consistently created new spatial fields and practices. The term “geometry” was gradually rendered inadequate to describe the spatial determinations (metrics) used. This explains the necessity to introduce a new spatial practice, namely, geodetics that literally translates into “the division of earth.”
- Colonialist expansion and its own redistribution of space called for greater precision in its representations of newly acquired territories. The application of geodesy provided just that by taking into account the topographyof a region and the intrinsic curvature of a surface without alluding to a presupposed transcendent reference system. However, mathematically speaking, since there is no isometric correspondence between spheres and planes, there is no transformation that preserves distances from one space to another. Earth’s surface cannot possibly be mapped onto a flat two-dimensional plane without tearing it apart or distorting portions of it. In this
sense, the history of cartography is predestined to be one of mistranslation.
Being able to determine curvature without referring to a presupposed external coordinate system (x, y, z), geodesy highlighted that locality on a surface (analysis situs) was an important property for an immanent analysis of space. Since metric relations are not necessarily preserved when moving through different spaces and across dimensions, how is dimensionality constructed?
R1: Let a point travel infinitely along a straight line, back and forth but not left and right.
R2: Let points along the constructed straight line travel left and right, but not up and down, to generate a plane.
R3: Let points along the plane travel up and down to generate a three-dimensional space.
In this construction, which can be taken up to Rn, the assumption was that the first dimension is modeled after the externalized straight line. Any other structures of multi-dimensional space would carry this prefiguration. In order to address this assumption, Bernard Riemann constructed a new concept of space: manifolds. Withinmanifolds, one does not uncritically assume the transcendent construction of dimensions and, hence, the determinability of its metrics. Instead, to determine the spatial properties of manifolds one begins from a state of absolute determinability and the process of determination involves zooming in on a specific region within Space and carefully comparing it topologically to its neighboring regions.
This method is, in many ways, far more modest. It does not make sweeping claims about the nature of Space and it acknowledges that different regions of Space involve different rules to be determined within their corresponding regions alone. There is no Universal Ruler to measure all space. This, in turn, pluralizes spaces such that they share fewer commonalities than what we have assumed so far. Going beyond experienced space or mathematical space, this could include other abstractions of space: color, sound, art, as well as economic, urban and political space.
In terms of the determinability of metrics and dimensionality, getting to know any space becomes a practice of carefully observing and caressing the intricacies of one’s own occupied region, whatever that may be. Without assuming the externally imposed de facto metrics, knowing a space involves constructing or identifying the right metrics for the currently occupied space. This might lead to a more sophisticated form of relativism. It would then take imaginative ways to overcome the red herring of relativist thought and successfully regain the openness that a variable Space has to offer.
1 In mathematics, a metric is a function that defines distances between each pair of elements in a set. Metricizing involves expressing a space in metric terms, that is, finding a metric that can describe that space.Parameterization involves defining all the parameters needed for a complete model of a space.
essay published in Marina Kassianidou's publication, for her exhibition: How to Know: A Space.