To grasp the prodigious variety of customary ways of measuring land, we would have to imagine literally scores of “maps” constructed along very different lines than mere surface area. I have in mind the sorts of maps devised to capture our attention with a kind of fun-house effect in which, say, the size of a country is made proportional to its population rather than its geographical size, with China and India looming menacingly over Russia, Brazil, and the United Sates, while Libya, Australia, and Greenland virtually disappear.
–James C. Scott, Seeing Like a State
This quote comes in the middle of a chapter about culturally-relative units of measure–for example the distance between villages in Malaysia might be expressed in terms of how many rice-cookings it would take to walk from one to the other, or the size of a farm in premodern Ireland might be expressed in terms of the number of cows it could support. Scott’s aim is to investigate the social impact of carving up the world in these locally salient terms instead of more objective units like hours, kilometers, and acres. The quote above illustrates the way in which the same real world object–a farm, say–might have radically different representations if “mapped” in terms of how many cows it supported, or how many people it could feed, or how rocky its soil was. Scott makes some fascinating points about relationship between measurement and power, but here I’m just interested in his map metaphor.
The metaphor is clear and compelling, but as I was reading it a mathematical one came to mind: projections of geometric objects onto basis vectors. The farm is an object in some n-dimensional space. The basis vectors are locally-salient social constructs (rice-cookings, cows and so on). The measurements are projections of the object onto these basis vectors. I like this metaphor because it can draw on finely articulated properties of coordinate systems that help to make Scott’s point. For example, the notion of orthogonality. How long it takes to plow a field is related to how many people it can support, perhaps in an indirect and complicated way. To say that premodern measurements were projections onto skew coordinate systems is to capture this complex notion of interrelatedness succinctly. Also in coordinate systems we often think in terms of projection onto groups of basis vectors instead of just single ones, and I could see how it would be illuminating to think of the same farm’s projection onto distinct physical, temporal, kinship, political and economic hyperplanes. Finally, there is the notion that coordinate systems taken as a whole are systems–invented items with internal structure that can be transformed from one into the other–which is precisely in line with the broader thrust of Scott’s argument.
The downside is that most people will just have to take my word for it when I say that basis vectors make a great metaphor for premodern measurement systems, because while maps are part of daily life, basis vectors are something you only encounter if you’ve taken a linear algebra class. This doesn’t change the fact that mathematics is good for more than just balancing checkbooks and making sure that bridges don’t fall down. It provides a wealth of metaphors that have application outside the mathematical domain, and it’d be a good thing if those of us with a bit of mathematical knowledge could find ways to popularize these metaphors to those who have none.