The move to advanced mathematical thinking, using a full range of personal mental
imagery to develop new theories formulated in terms of systematic proof is more than
just the appreciation of a formal development from definitions and axioms. It builds in
the kind of structure exhibited in figure 5, with the advanced mathematical thinker using
visuo-spatial ideas, symbol-sense and all kinds of intuitions to develop new theories that
can be woven into the Bourbaki-like systematic development that forms the solid
theoretical basis of the subject.