Ideal Types, Vector Bases, and Convex Combinations

Last update: 07 Dec 2024 23:56
First version: 18 September 2024

Attention conservation notice: Answering a question which probably no one else will find interesting, let alone puzzling. Accidentally posted a version with half the text deleted in mid-September 2024...

When I was an undergrad, I learned about two concepts in (relatively) close succession, and together they left me puzzled:

All the vectors in a vector space can be expressed as linear combinations of basis vectors, and there are infinitely many bases for any vector space, all equally good, if not perhaps equally convenient for every problem. In particular, any one vector can always be made into a basis vector (and you can then build the rest of the basis around it). If you have a $ d $-dimensional vector space, $ d < \infty $, then any set of $ d $ linearly independent vectors could be used as a basis. (Of course one often wants to have orthonormal bases, but there are straightforward prescriptions for building such a basis, starting from your favorite collection of $ d $ linearly independent vectors.)
Sociologists like to describe social forms (or even individual people) can be described in terms of "ideal types", often as combinations of ideal types. (Or at least some sociologists liked to do this...) So they go and identify, perhaps, three ideal types of authority, and then describe what "traditional" authority would look like, ideally, or "charistmatic" authority, or "rational-legal" authority, and might say that the authority of a corporate CEO is mostly of the rational-legal type though with, in some cases, touches of charisma.

Maybe you can already see what puzzled me from the way I've set this up: ideal types sounded a lot like basis vectors. Why choose that basis, rather than any other? Couldn't we declare any social form or individual we liked to be an ideal type, and start defining further types by contrast to Fred (or whoever)?

This is something I actually worried at, in my reading and in my offline notebooks, for more --- much more --- than a decade. A resolution did eventually occur to me, but only after I learned a lot about convexity. In fact I think this only occurred to me after convexity had figured crucially in a couple of papers...

The key is that a linear combination of two vectors is just their weighted sum, but if I am taking convex combinations, I am restricted to positive weights (which add up to one). If $\vec{u}$, $\vec{v}$, $\vec{w}$ are my basis vectors, and $\vec{x}$, $\vec{y}$, $\vec{z}$ are three (linearly independent) vectors I can write in terms of the bases, then I can indeed reverse my perspective and express $\vec{u}$, $\vec{v}$, $\vec{w}$ as linear combinations of $\vec{x}$, $\vec{y}$, and $\vec{z}$. But if $\vec{x}$, $\vec{y}$, $\vec{z}$ are convex combinations of $\vec{u}, \vec{v}, \vec{w}$, then I cannot do this reversal. Bounded convex sets have "extreme points", which cannot be generated by taking convex combinations of other points in the set. (Unbounded convex sets, like the whole Euclidean plane, need not have extreme points.) Whether a vector is an extreme point of a convex set is not affected by the basis used to represent the vectors. (Think of the corners of a regular polygon.) In fact, closed, bounded convex sets are precisely the ones which are generated by taking all the possible convex combinations of the extreme points. (Again, think of the corners of a regular polygon.)

So there you have the resolution to my perplexity: just think of "ideal type" as how a sociologist says "extreme point", and insist that when we describe actual phenomena in terms of ideal types, we are always looking at a convex combination of the ideal types. If taken seriously, this has an interesting implication: we can never put negative weight on an ideal type. Even if we want to say "the way this institution works is the polar opposite of charismatic authority", we would have to find one (or more) ideal types which we could positively mention in our description, and which point in that direction.

(Why, in all that time, did it not occur to me to go the other way, and ask what bits of math work like the way sociologists use ideal types? Because, in my mind, math and physics were obviously more real and more correct than social science or the humanities. That says something unfortunate about how I thought as an undergrad, and I doubt I have actually improved.)