I think you could say that. Another way to look at it would be to say it's about organising your code in such a way that side-effects don't really matter. For instance, if only I have access to a reference, then I'm free to update it without risk of disrupting anyone else's world view. You could say that there is no side-effect from my update because no-one can observe the change.

This is nice for functional programming because it gives you the performance of mutable state and destructive update, but in a way that is 'pure', or side-effect free, from the outside.

I asked myself a bit of a related question. Is this theory related to https://en.m.wikipedia.org/wiki/Static_single_assignment_for... ?

I think SSA is related to CPS, but I'm not sure of the correspondence with Linearity. I think the key difference (but I'm not sure), is that SSA only asserts that a variable is assigned once, but might be consumed multiple times. Linearity says that things must be consumed exactly once. For instance, I think:

y <- x + 1

z <- x + 2

is a legal SSA expression, but it is not a legal linear expression because x appears twice.

(Corrections welcome)

FWIW:

https://www.cs.princeton.edu/~appel/papers/ssafun.ps