(see notes-math/philosophy for more thoughts)
There are -many- distinct skills required to do math;
it's worth -pointing out- how different ones are required.

Some of these are more important/higher level than others, but the
lower ones are often -necessary- for higher level ones.
EG, you need -basic- spelling in order to write novels, just as you
need -basic- symbol shuffling to do group theory -- even though the
most interesting parts of both are -not- the basics.

(my favorite base distinction is:
 math reasoning vs. writing up;
 there's a skill in understanding -why- something is true,
 and there's a skill in explaining it/writing it rigorously,
 and these are -different- (though related) skills)

These include:
- mathematical reasoning

- writing up

- reading comprehension
  Math writing is -dense-, and the words are -technical- and
  -abstract-. Being able to make sense of them

- abstract thinking ++ formal axioms
  If you've managed to -contextualize- knowledge, put it in some
  -framework- that you understand, then it's much easier to work with;
  this is a key technique in learning anything (and eventually every
  system "creates its own context"). However, when doing something
  -new-, there's frequently little or no -context-: you must work with
  things abstractly, axiomatically.
  
  This skill is "abstract thinking"
  This has at least 2 sub-divisions:
   (1) "symbol shuffling"
     Can you work with symbols out of context, w/o them having any
     meaning?
     (e.g., being able to do implicit differentiation is valuable, 
     even if it's meaningless -- as you're able to follow certain
     rules)
   -- students are quite good at this, as most math means nothing to
   them; e.g., derivative rules: they're just working formally anyways --
   (2) "mental flexibility"
     how easily can you work in a new and unfamiliar field?
     make it your own?
     This is -more- valuable, but harder.
  
  (EG of made-up language, in IF competition: Gorsat? something like
  that. I may not know what a frible is, but I know it's something you
  can flobot. (i.e., axioms, formal symbols).)
  (this sometimes involves rapidly "creating context")
