The Logic of Liberty 1
The Dimensions of Political Space
Two points determine a line. Politics in the US under our duopoly two party system is one dimensional. No matter what your personal opinions on the constellation of issues of personal and social life, all are projected onto the single Republican  Democrat axis. But that's not political reality.
The Libertarian Party represents literally a new dimension in political thought. It is, perhaps, one of the first products evidencing "Political Science" to actually be a science. And I think it no accident that the message of that product is a profound defense of individual liberty and property versus collectivized force and ownership. This dimension of liberty versus authority was "unfolded" by David Nolan, one of the founders of the Libertarian Party back around 1970. (Coincidentally, another term for dimension, often used in statistics, is "degrees of freedom" .) David was then, very appropriately, a recent graduate in political science from MIT. Nolan noticed that the right tended emphasize economic freedom but moralistically want to restrict personal freedoms, while the left tended to have the opposite emphasis. Both want to control some aspects of your life. Nolan plotted the two dimensions against each other like this:
( theadvocates.org/images/OriginalNolanChart.jpg )
Nolan was not the first to make such an unfolding. According to the Wikipedia article on the Nolan Chart, Jerry Pournelle produced a similar chart in 1963. The diagram might have remained just a hypothetical framework, but in the mid 1980s Marshall Fritz, founder of the Advocates for SelfGovernment, boiled down a set of five questions for each axis which succinctly and reliably place people in this 2 dimensional space. These questions have been modified somewhat over the years. Here is the current Advocate's list. World's Smallest Political Quiz (Choose A if you agree, M for Maybe, D if you disagree.) Personal Issues
Economic Issues
A final modification was to rotate the chart so left and right are left and right, and libertarian is at the top and authoritarian at the bottom. Here is the current version of the Advocates chart.
There are now quite a few variations of the quiz and chart on the web. A number of references are given below.
The "MulticandidateElection Simulator", referenced, shows the interesting phenomenon that when you have a duopoly like ours vote seeking politicians tend to converge to almost indistinguishable positions offering little real choice. Only if you are involved in third party or issue politics are you likely to appreciate the horrendous antifirstamendment barriers the duopoly has set up, and is pressing hard to extend (notably McCain in the senate and Obey in the house) under the name of "taking money out of politics", by which they mean freely donated money rather than taken tax money which they weirdly label "clean", In order to keep things this way.
Now I'm going to get more spacey. I'm going to talk some actual mathematics and even present a couple of thousand year old formula. Probably not suppose to do that for an American street publication, but hopefully the flavor, if not the details will come across.
The Nolan chart clearly shows that a libertarian  authoritarian dimension exists. But, it presents an a priori pair of axes and distributes positions simply by adding up scores of questions selected to tease out that particular second dimension.
Dimension is a mathematical concept which, while we spend our existences inescapably in this exactly 3 dimensional physical world, is often misunderstood and misused almost mystically by those who have never had the math explained to them. Look up at the corner of the room you are in. If it is a normal rectangular room you see three edges coming out from it. And, of course they are at right angles to each other. In math terms, they are perpendicular or orthogonal to each other. (Note this is exactly the opposite of being parallel to each other. "Parallel dimensions" is the silliest phrase used in dumb scifi. Lines and plane can be parallel, not dimensions.) You could take a ruler and mark off units along each edge. How far are your eyeballs from the corner of the room? Well, you can go out along one edge to the point on it closest to you (that is, look straight at the edge so the line from your eyes to it is perpendicular to it). Note how far along that axis you are. Let's call that d1. The name for this "dropping" a perpendicular to a plane or a line is called "projecting". Now, how far are you from that point? Well, first let's consider the point right over your head. Let's call the distance from that point to that first point d2. What is the distance from the corner of the room to that point over your head? Here's the true mystical fact which the Pythagoreans formed a religion around. The square of that distance is the sum of the squares of the distances d1 and d2! Now, to get the total distance from the corner of the room to your eyeball? Let's stick with the squares. We just apply Pythagoras's truth again and add the square of the distance, let's call it d3, from that point on the ceiling above your head to the squares you've already added. Thus, the total distance from the corner of the room to your eyeball is the squareRoot of d1^2 + d2^2 + d3^2 , which you hopefully learned in high school. Of course there is a bunch more geometry where when edges are not "orthogonal". You get terms like d1*d2 and d2*d3, etc which go into computing sines and cosines of angles, etc, but it's all based on this fundamental several thousand year old observation. (For those familiar with the term "correlation", the correlation between two sets of data is the cosine of the angle between them. Thus it is 0 if they are perpendicular to each other abd 1 if they are parallel in the same direction.) Note that you could make a set of coordinates for your room by kind of rotating one of the edges of the room around to the line between your eyes and the corner and rotating the other edges off (they would be buried in the walls and ceiling) to keep them perpendicular.
So, how about more dimensions? The math doesn't care if we can add on more squared distances, d4^2, d5^2, d6^2, â€¦. All that matters is that they are independent of each other (not parallel). If they are not perpendicular, then we have to subtract off a bunch of those d2*d5, etc, terms for the portions that are parallel, but it would take a semester or two to get into the details. Anybody who has taken any statistics knows you are adding up sums of squares all the time. What you are doing is adding up squared distances in as many dimensions as you have data points. In the Nolan Chart, for instance, we have 10 questions. That generates potentially a 10 dimensional space. What the Nolan chart shows is that people's responses to these questions don't fall on an orderly straight line from conservative to liberal. Some people want more state control of all aspects of their (and your) lives and some people have a more "live and let live, I'll run my own life, thank you" attitude.
There is one other aspect which you might have noticed makes the Nolan Chart somewhat cruder than it might be: You simply add up the scores on each question; you don't square them. This is what is called the "cityblock" metric. I won't go into it, but any power from 1 (the cityblock metric) to infinity (the "max" metric where the only possible scores would be the center and the corners of the chart) can be used in place of 2. These are what are known as "nonEuclidean" metrics. However, only with exactly 2 can you form a consistent geometry with a notion of angles between points and so forth.
Millions have seen the friendly face of Dr. Neil Clark Warren describe the "29 key dimensions that predict deep compatibility and happier, more lasting relationships" which make his eHarmony.com "America's #1 relationship site". He is using the term accurately. What he has done is ask hundreds of  lets just say 200  questions to thousands of people. Then, he has a cloud of points representing each person's answers in the 200 dimensional space generated by those 200 questions. Here's a picture I made years ago trying to show what such clouds of data points might look like, albeit in just 3 dimensions, ie, people's answers on some three questions..
Looking at this, you can get an idea of a couple of the major ways of analyzing such data. One method is called cluster analysis. If for instance the Republicans and Democrats really had core unifying principles around which they clustered, you might see distributions of answers like these two clouds. The other method is called factor or principal component analysis. It finds the axes along which there is the greatest variation. In this picture, there is one sort of up and down thru the two clouds, and one kind of diagonally along the length of the clouds. Like rotating the axes of your room so one axis is thru your eyes, these axes can be expressed by adding up varying proportions of the original axes  which in this case are the individual questions When eHarmony says it's found 29 key dimensions, it is saying it has found 29 combinations of weightings of its questions which determine axes which pretty much capture the variation between good relationships and bad.
Note that with this sort of analysis, the important axes are not determined in advance like they are with the Nolan chart; they emerge from the data as long a the set of questions is rich enough to include the important factors. The Wikipedia "Political_spectrum" page list a number of potential factors, for instance: Role of the church, Urban vs. rural, Foreign policy, Trade freedom, and Openness to change. So, if you do a broad analysis which includes all these potential factors, what factors emerge as the most dominant?
All of this math has been standard fare in the social as well as physical sciences for at least the last 40 or 50 years. So I have been quite surprised that I have not found an analysis of US political space. I found some for the EU, and a very good website for the UK. (Perhaps a reflection on the quality of our quantitative education?)
I quite recommend that if you want to get a better understanding of what I've been saying, and see such an analysis in the context of British (3 party) politics, check out the PowerPoint presentation at http://www.politicalsurvey2005.com/themap.ppt , and the sets of questions making up the 2 major axes which are listed at http://politics.beasts.org/scripts/eigenvectors . That page has the excellent caution that "The names [of the axes] should not be taken too seriously; they are only labels. If you want to know what each axis 'means', read the list of statements which make it up."
Most interestingly, while the names and some details are different in Britain, the two major axes which emerge are generally a leftright and a freeMarketsocialist axis very comparable to the LibertarianAuthoritarian axis revealed by the "World's Smallest Political Quiz.
An irrelevant but interesting footnote: The metric of Einstein's Special Relativity is simply x^2 + y^2 + z^2  t^2 where the units are adjusted to make the "speed" of light equal to 1. That's it. That's his whole theory from which everything else is derived. How do you get a  t^2? Well, that means time must be imaginary.
References: http://en.wikipedia.org/wiki/Nolan_chart http://www.theadvocates.org/quiz.html Enhanced Political Quiz : http://www.quiz2d.com/ http://en.wikipedia.org/wiki/Pournelle_chart 3D version : http://www.friesian.com/quiz.htm Multicandidate Election Simulator : http://www.jmarshall.com/polisim/ http://en.wikipedia.org/wiki/Political_spectrum http://www.politicalsurvey2005.com/ http://www.politicalsurvey2005.com/themap.ppt http://politics.beasts.org/scripts/eigenvectors
