R Hansen mentions: >...abstract analytical (non visual) thinking.
This just reminds me of something. A famous computer scientist (not mathematician, unfortunately) once confessed he could never find a relation "More Abstract Than" on which to base a systems architecture on, even though CS people had talked about "levels of abstraction" for years prior. (Parnas went on to write on of the classic papers in the field after clearing away some built up B.S.)
The above strikes me as somewhat the same. Abstraction means to focus on some properties of something while disregarding others. So Euclidean lines and planes are already abstract. Much visual thinking used in mathematics would be abstract by the same token. Some thinking might be guided by algebraic manipulation of expressions -- most people would not call that visual thinking though the expression on paper are visual objects.
Ultimately, mathematics often strives towards the "proof" as the gold standard. A proof, in the most rigorous sense, is a formal object in a formal system, which is abstract in one sense (if its tokens and primitives are thought to *refer* to something else -- the "abstraction" involved is distilling just those properties of that something else one wants to encode in the formal system.) On the other hand, if the system is thought of as simply a game of symbols and rules with no external referents, its a bit of a misnomer to call it abstract, as it abstracts from *nothing* in that case -- its just a game. The same could be said for algebraic systems, if we really view them as purely formal games. I know in colloquial speech people would ordinarily refer to any symbolic game of this type to be "abstract". But from a different viewpoint, they are really *concrete*, in that the symbols and rules are rigidly fixed.
Most real life proofs fall are simply reviewed arguments that enough qualified people think could be turned into a formal proof if push comes to shove. Any kind of thinking, visual or otherwise, that leads to a proof (formal or not) is mathematical, abstract thinking, in the usual (colloquial) sense.
Some people, I think, would regard expression manipulation as "more abstract than" thinking about lines and slopes and planes etc. I think I'll go with Parnas on that issue though.