Philosophical Richness of Liouville's Theorem
We need singularities
I keep relating mathematical metaphors to social mobility. 1
Liouville’s Theorem is very interesting wrt social mobility. I often talk about tall poppy syndrome. There must be somewhere to escape to. We shouldn’t stymy growth. And here’s why I’m sceptical of absorbing states — too entropic:
If there are no singularities, i.e. nowhere for people to escape to — f is entire — then f will be constant. It’s like a kettle without a spout. Nothing grows.
The presence of singularities is what allows f to be non-constant elsewhere.
This is relevant re March for Billionaires. “If people can’t capture the surplus, they won’t bother innovating” (contentious, but true up to a point).
People reject pyramid base roles (Bostrom, Deep Utopia), but solution is simple — multidimensional — “I’ll lead on Mondays; you lead on Tuesdays, …”
The existence of any bound predicts Malthusian futures, repugnant conclusions, Hansonian post-dreamtime dynamics. We must fight these.
Will a singularity consume all our resources? Is it the utility monster of yore?
What’s more of a threat to you? Utility monsters, or deadened sensitivities?
If f is entire and bounded, then f must be constant. We can take contrapositive — if f non-constant / fluctuates, then f is either non-entire or non-bounded. Which might you prefer? Presence of singularities or more prosaic unboundedness? In the complex plane, pole-singularities are pretty simple: |f(z)| must run to ∞ as we approach the pole2.
Essential singularities are much weirder. “In any neighborhood of an essential singularity, f takes every complex value (with at most one exception) infinitely often.” This seems less ‘utility monster’ in nature than pole-singularities.
It’s escape to infinity / weirdness that makes all other variation possible. So thank those who widen the Overton window. What you can see is your ceiling.
In the context of Markov chains, I wrote about how even in the case of an irreducible, aperiodic chain (you can reach any state from any other…you can stagnate, yielding GCD pii = 1), two identically-chained agents may fail to converge to shared stationary distribution because IRL we do not have enough steps.
initially, i wanted to write ‘singularities can be positive or negative’, but then i remembered only |f(z)| is real; f(z) is complex and non-ordered