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.

Frogs, Cigarettes, and Libertarians

I often ask my friends of the Libertarian persuasion,
Question: 'What do you call a Libertarian who wants to get elected?'
Answer: 'A Republican.'
Of course most Republicans are not Libertarians, but the GOP is far and away the natural home for any rational Libertarian. The Republican Liberty Caucus is the home of one subgroup of self identified libertarian Republicans. I have detailed aspects of both my philosophical differences and my agreements with the Libertarians in numerous posts such as Libertarians vs Conservatives, Steepest Descent, Vouchers, Libertarians, and Our Failing Public Schools. Despite my differences with my libertarian colleagues, I am probably viewed by them as a fellow traveler or an ally and am occasionally invited to Libertarian functions. Last week, I attended a state Republican Liberty Caucus (RLC) function and left the event with a burning question: are Libertarians disproportionately smokers? In (American!) Mathematics circles, I do not encounter many smokers and was under the impression that the habit was dying among professionals and, perhaps even more generally, among the college educated. In fact, a quick Googling asserts that smoking has dropped to around 9% of college grads (vs. 27% for high school grads). Yet, at the state RLC function, interesting conversations were frequently interrupted by the departure of one or more participants, heading outside to smoke. (No. They were not simply fleeing me.) At the one district GOP convention I have attended, RLC delegates pushed resolutions attacking smoking bans. The erosion of our liberties at the hands of the nanny state, although upsetting, is still somewhat abstract to me. This week I finally realized the obvious: the erosion of liberty is very real to smokers. Laws have been passed greatly restricting their rights. I greatly enjoy the consequences of these laws, no longer encountering smoking in public places, but here, I use the term public loosely, including privately owned restaurants which are not allowed to permit smoking on their premises (and which I am free to avoid should smoking be permitted there).

Seeing how strongly smokers feel the curtailment of their freedom, I wondered what abridgement of liberty would pinch me similarly. What current painful abridgements can I identify? According to another quick Google, I work from 1/3 to 1/2 or more of the year for the federal government. Since I love my work (and often work through vacations if I am not careful), I don't resent this loss of my time. If I translate this loss of time into a loss of income, the loss remains abstract unless I think about how I might have spent money lost to excessive taxes. The first thought that came to mind was how nice it would have been to send my children to excellent private schools with the taxes spent to maintain bad public schools. This example is not compelling, however, because (i) I could have always sacrificed some other good in order to send my children to a good private school, and (ii) there were not any obviously excellent private schools available. The first weakness illustrates the fact that infringement of liberty by means of taxation is the optimal means of infringement because it allows the victim/taxpayer to forfeit what he values least rather than having the government decide what material good (can a good education be called a material good since it costs money?) he must forfeit. The impending loss of my freedom to allocate as much of my income as I desire to pay for healthcare for my family will likely be my first exposure to a loss of liberty sharp enough to cause strong discomfort. I imagine small business owners have no end of more immediate examples.

Republican speakers endlessly repeat the metaphor of boiling a live frog in a pot to describe the gradual erosion of our liberties. They assert that if you put a frog in a pot of warm water and gradually turn up the heat, the frog will not jump out, whereas he will jump immediately if thrown into a pot of hot water. This sounds unlikely to me, and I don't know why Republican speakers spend so much time boiling live frogs (having steadfastly declined my French hosts' occasional offers of frog legs). Nonetheless, it is clear that if erosion of liberty occurs through taxation, the taxpayer will always eliminate his least valued good first, and is thus more easily lured into accepting ever increasing encroachments on his sphere of activity.