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Sunday, June 26, 2011

Inane Mathematical Metaphors

I haven't abused any mathematical metaphors on this blog for some time. To atone for this deficiency, today I will introduce some mathematical concepts that beg to be misapplied to political discourse.

Let's start with geometry. The first three geometries one usually encounters are
Euclidean, Spherical, and Hyperbolic. Euclidean is the geometry we all learn in grade school. It is the geometry of flat space. Spherical geometry is the geometry of the surface of a ball. We learn a few aspects of spherical geometry in grade school. Given the reasonable approximation of the surface of the Earth as a sphere, spherical geometry enters into many planetary scale computations. Unless you major in mathematics or physics, you are unlikely to encounter the third classical geometry: hyperbolic space. One feature of hyperbolic space which makes it useful for metaphor abuse is the fact that in hyperbolic n-space, the volume of a ball of radius R grows like e^{(n-1)R} instead of like R^n as in Euclidean space. (Volume growth of balls in spheres for R greater than half the circumference of the sphere is obviously not particularly interesting.) Equivalently, the volume of a sphere of radius R grows like e^{(n-1)R} in hyperbolic space as opposed to growing like R^{n-1} in Euclidean space. As we often do, we turn to Hayek to develop our metaphors.

In Hayek's Road to Serfdom, Hayek discusses at great length the undirected emergence of complex economic organizations from the individual actions of billions of people. In The Constitution of Liberty, Hayek extends his reflections on the spontaneous emergence of complex phenomena from the realm of economic organization to social. He notes that complex social conventions, like complex economic arrangements, arise undirected; they are the product of the experiences and the experiments - both successful and failed - of all our predecessors. He implies that we should therefore be wary of challenging established custom because we are not simply opposing our reason and experience to that of our neighbors, but also to the accumulated knowledge and experience of all of our predecessors. On the other hand, his analysis contains the implicit assumption that people will continue to experiment and modify social conventions, contributing incremental improvements.

Hayek's fundamentally conservative viewpoint on challenging social norms is based on the assumption that the accumulated experience of our predecessors is greater in magnitude than the new experience of the current generation. So, for Hayek, a Euclidean ball of radius T is a better model for the accumulation of experience in time T than a hyperbolic ball. In Euclidean space, the sphere contains only a small fraction (1/T) of the total knowledge of the ball; in hyperbolic space the sphere and the ball contain comparable data. Therefore, those who believe the experience of their generation outweighs that of all their predecessors might prefer a hyperbolic model of accumulation of experience. I suppose the more extreme conservative position would be one from spherical geometry: after a fixed finite time (presumably already passed) no new knowledge is gained.

So, on social issues, then, we can divide people into spherical, flat, and hyperbolic. We will have to work harder, however, to tie these models to Hayek's more fundamental concern - the spontaneous rise of structure from the independent actions of millions of people (as opposed to the directing hand of government). Actually that seems to be too tall an order, if we wish these structures to differentiate scenarios supporting or opposing the spontaneous rise of structure. In geometry, almost all interesting natural structures arise from optimizing some local energy condition. In two dimensions, spherical, Euclidean, and hyperbolic geometries can all be created from uglier geometries by allowing the geometric structure to flow locally, subject to the instruction to minimize some locally defined energy. So the spontaneous rise of order is hardwired into much of geometry, as I have discussed elsewhere.

There is a subtler (and even sillier) link between these geometric structures and the political spectrum. If you traverse a small circle counterclockwise while holding a small ruler pointing in a fixed direction as you travel (modelled by a covariant constant vectorfield along the curve, for those of you who want precision), then in Euclidean geometry, when you have returned to your starting point, the ruler will be in its original position. In hyperbolic geometry it will have shifted to the left. In spherical, it will have shifted to the right. On these grounds, we will award hyperbolic geometry to the liberal side of the political spectrum and spherical to the conservative side.

If this assignment does not match your politics with your preferred geometry, simply replace counterclockwise travel with clockwise travel to reverse the political spectrum.

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