The topology of opinion space

The purpose of an analogy is to explain an idea by relating it to something simpler.  Unfortunately, what I consider to be simple is not the same thing as what most other people consider simple.  Because of my math and physics education, I often think of analogies relating to math and physics.  To most people, these "analogies" only end up making things more complicated.

Here's an analogy in that vein:

When we want to describe people's views and opinions, we usually don't simply list out their views and opinions.  Instead, we try to simplify by describing it with an axis.  For example, liberal vs conservative is an axis.  If this axis isn't enough to describe what we want to describe, then we add more axes.  Religious vs secular.  Socially liberal vs socially conservative.  Isolationist vs interventionist.  We also invent axes to describe various other things in life, like sexual orientation or personality.

We sure love our axes!  That's why we have such things as the Political Compass, the Klein Orientation Grid, and the Myers-Briggs Type Indicator.

But what is the mathematical structure of an axis?  Most people are implicitly thinking of the interval (0,1), or possibly (-inf,+inf).  All the above quizes basically use an orthotope in Rⁿ, where n is the number of axes.  The greater n is, the more sophisticated the model is, supposedly.

But speaking as a student of maths, why R?  R, the set of real numbers, is a mathematical space with an awful lot of unnecessary structure to it.  R has addition, multiplication, subtraction, and division.  What does it even mean to take the sum or product of two opinions?  0 is defined as the addition identity element (ie x+0=x for all x), but there is no need for this concept if there is no addition.  Likewise, 1 is defined as the multiplication identity element, and there is no need for that either.

If you take all that away, what remains is not R, but a special metric space, which we'll call S.  It's a metric space, because we're imagining that we have defined the distance between any two points in S.  A distance function is basically a special function which takes an unordered pair of points in S, and outputs a positive real number (unless the two points are the same, in which case it outputs the distance 0).  The distance function must also obey the triangle inequality.  S also has some additional properties beyond being a mere metric space.

But I find myself wondering, is it necessary to even have a metric space?  I feel like it doesn't really make sense to say that my political view is equally distant from two other political views.  In other words, I find myself puzzled by the idea that between any two political views, there is another one which is exactly in the "middle".  What does the "middle" even mean?  In my humble opinion, what we view as the "middle" is very subjective and reactionary.

We could also define distance according to the results of the questionnaire.  But I do not think this solves the problem of subjectivity.  Instead, it merely pushes aside the problem of subjectivity onto the person designing the questionnaire.

To try a more objective distance function, we could define the distance as the percentage of people between the two points.  For example, if I'm introverted, and you're more introverted, we can say that the distance between us is equal to the percentage of people in the world who are more introverted than me, and less introverted than you.  But this would give us weird results.  For example, we could never say that lots of people are gathered in one particular part of the axis, since by definition people are evenly distributed along the axis.

My preference is to not think of an opinion axis as a metric space at all.  Instead, I think of an opinion axis as a topology, one that is topologically equivalent to (0,1).  However, there is no distance, and there is no middle.  Or at least, there is no natural way to define a distance or middle.  Instead, we can only talk about whether one opinion is to the left or right of another.  And we can talk about whether a third opinion is between the first two opinions.  But we can't talk about whether the third opinion is closer to the first or the second.

This isn't much to go on, but we can add a little structure by adding landmarks.  For example, the Kinsey Scale defines 7 landmarks of sexual orientation with the integers between 0 and 6.  Calling the landmarks by numbers is misleading, because we can't talk about the distance between two landmarks.  We can't, for instance, say that the distance between 3 and 4 is the same as the distance between 0 and 1.  Nor can we say that there is anything "special" about these landmarks, any reason we couldn't have chosen different landmarks.  But we can talk about being to the left or right of various landmarks.  For example, a person could be between 4 and 5.  That tells us something.

What are the advantages of thinking of an opinion axis as a topology?  It seems like we actually get less information this way, since we simply can't talk about distances.  But I prefer it this way because I think it eliminates the bad information, or at least emphasizes that it is bad information.  The fact that we can't talk about distances is an advantage, because distances don't really make sense.