I am trying to follow G. Spencer-Brown's argumentation regarding 'Equations of the Second Degree' in the Laws of Form.
He shows how infinitely growing self-referential forms can be created by using a pattern of transformation steps, e.g.
$((a) b) \rightarrow ((((a) b) a) b)$
and so forth, such that
$f = (((( ... a) b) a) b) = ((fa) b)$
There are two elementary values, marked $m$ and unmarked $n$, and hence—using his principal algebra—the four permutations of possible values for $a$ and $b$ yield three determinate solutions for $f$, and one indeterminate:
- $((fm) m) = n$
- $((fm) n) = m$
- $((fn) m) = n$
- $((fn) n) = m$ or $n$
So the last equation has two solutions. The following claim does not convince me, though:
It is evident, then, that, by an unlimited number of steps from a given expression $e$, we can reach an expression $e'$ which is not equivalent to $e$.
I do not understand this claim. I am not a professional mathematician, so maybe there is the problem. For me, the fact that the expression is grown by an infinite number steps does not change the fact that they are equivalent to each other (do not both $e$ and $e'$ have two solutions? Maybe $e$ being constructed from a finite number of steps is determinate?)