Formal stuff

actually doing formal operations is brain-dead (you're a computer);
however, the -study- of -systems- of formal operations is -logic-, or
-algebra- (the connection is very close).

that is, it's an -aspect- of math, an -aesthetic- of math

here's a lil' EG, showing baby model theory
(kinda: relating the algebra of $\Pow(X)$ to logic):

- what are the -subsets- of $\bR$ ($\bQ, \bZ$) that can be -described-
  (in -finite- sentences)
  using only basic logic (not, and, or) and $<$?
  (answer: finite unions of intervals)
  
  NB: good exercise: show that logic and $<$ yields $\leq$
  (like, duh: $x \leq y \iff \not(y < x)$; but do with sets
   and compliments)

NB: contrast with topology:
in logic, the basic -algebraic- notions of -closure- should be:
-finite- intersect, -finite- union, compliment
(contrast to -topology-: no compliment, but -arbitrary- union)

- show similarly that these are the sets you get taking the -closure-
  of (open) -rays- under compliment, intersect, union

(see?! logic = algebra, here...)
