When preparing for my research proposal, I read a lot of articles in economics and political science. In the social sciences one often encounters statements such as "the institutions of country X and Y are close" (or not so close, as the case may be). The problem is, what does 'close' mean? For me, this kind of statement seems to assume that the institutions of country X and Y live in a space equipped with at least a metric (right?). However, as far as I can see, there is nothing that hints that this should be true a priori (i.e., without an inquiry specifically designed to elucidate the geometry or topology of the space in question), and even if it were true, I see no reason that the metric should be the euclidean metric, an assumption which seems to be the norm if I'm not mistaken.
It seems that before we start comparing objects in the real world, we should first ask questions about the structure of the set of objects that we wish to compare, although I am aware that this would probably be a very difficult task, the results of which possibly being subjective or contingent on the particular definitions of whatever we are interested in.
I know the topic will strike some as a bit diffuse or too broad, but is there any literature that deals with this kind of questions from a mathematical (or applied-mathematical) point of view?
I am aware of one such article:
Eguia, J. X. (2013). Challenges to the standard Euclidean spatial model. In Advances in Political Economy (pp. 169-180). Springer, Berlin, Heidelberg.
Here a standard assumption of so-called spatial models is discussed and challenged. In spatial models, the preferences of individuals are represented as points $x \in \mathbb{R}^n$ where the projections on the axes are preferences on individual issues (e.g., if $x = (x_1, x_2, \ldots)$ is a preference of some individual, then $x_1$ could be the preferred defense budget, $x_2$ might be the preferred tax rate, and so on). Spatial models can be used to generate predictions such as the median voter theorem and proofs usually (as far as I have seen, at least) rely on geometric or topological arguments.
In these models, it is generally assumed that the distance between preferences can be compared using the euclidean metric in $\mathbb{R}^n$, an assumption that is discussed among others in the article. Another assumption that is discussed is the "convexity" of preferences.
Moreover, this book:
Goertz, G. (2012). Social science concepts: A user's guide. Princeton University Press.
Contains some arguments that I would classify as being related to this realm. However, as far as I can tell, it seems researchers are not in general interested in asking questions about the topology or geometry of the objects that are being investigated, nor the spaces in which they "live". Although I can certainly see how the assumption that if we are studying X, then X must live in euclidean space, can arise, I also believe that we must ask the question is it really the case that X lives in euclidean space? To me it is not at all clear a priori that most things that are being compared in the social sciences should conform to this assumption (except perhaps "locally" whatever that means at this level of abstraction).
If you know of any similar literature that deals with this kind of questions, not just for spatial models, I would be very grateful!