There are infinite isms

11 Jun, 2025

If not neocapitalism, then what?

Let

be a countable or perhaps larger set of atomic features; yes/no statements such as:

• $f_1$: "private property is protected"
• $f_2$: "banks may lend at interest"
• $f_3$: "the state plans heavy industry"

So that each ideology can flip those switches on or off.

We can then define an ism as a subset $I\subseteq \mathcal{F}$ purely based on the membership of features (the behaviours it embraces). But perhaps a better way to define it would be as some deductively closed set $T$ of propositions over $\mathcal{F}$ so that it captures the implicit rules that hold the membership. The latter lets us say things like if $f_1$ and $f_2$ hold, then $f_4$ must hold.

Now, if $|\mathcal{F}|={\aleph }_0$ then its powerset has cardinality $|\mathcal{P}(\mathcal{F})|=2^{{\aleph }_0}$. Which means that even after we throw out logically inconsistent collections, we still get uncountably many consistent theories. We could say that we are living in a Boolean algebra of subsets, or its Lindenbaum–Tarski algebra (all theories), whose points form a compact totally disconnected Stone space.

Therefore, the slogan of "it’s either capitalism or communism" is effectively flattening a 3D city map onto a single street. Continuing on our naive mathematical formulation, we can explore some properties of the ideology space as (1) Order where inclusion $I\subseteq J$ says J contains every principle of I, (2) Distance where Hamming or Jaccard distances on $\{0,1{\}}^{|\mathcal{F}|}$ count perhaps how many switches differ, and (3) Lattice moves where intersection lets us find common ground and join (union + closure) fuses programs.

A toy example

Take just four features:

Then $2^{\mathcal{F}}$ has $2^4=16$ possible ideologies.

Already 14 of the 16 points are neither pure capitalism nor pure communism. Because the space is a full-blown Boolean cube (and, with logic, a Stone space), there is (1) continuum of coherent isms, (2) natural neighbourhoods (market-friendly socialism), and (3) structured ways to travel with reform paths, compromises, hybrid programs.

This was just a simple exploration. However, I think this sets ground for many open questions to ask. Should atomic features be degrees rather than binaries, vectors in $[0,1{]}^n$? How can we better capture reactive effects? What about temporal properties?

#social-science