Can visual thinking lead to discovery of an idea for a proof in more advanced contexts? Yes. Carter (2010) gives an example from free probability theory. The case is about certain permutations (those denoted by “\(p\)” with a circumflex in Carter 2010) on a finite set of natural numbers. Using specific kinds of diagram, easily seen properties of the diagrams lead one naturally to certain properties of the permutations (crossing and non-crossing, having neighbouring pairs), and to a certain operation (cancellation of neighbouring pairs). All of these have algebraic definitions, but the ideas defined were noticed by thinking in terms of the diagrams. For the relevant permutations \(\sigma\), \(\sigma(\sigma(n)) = n\); so a permutation can be represented by a set of lines joining dots. The permutations represented on the left and right in Figure 8 are non-crossing and crossing respectively, the former with neighbouring pairs \(\{2, 3\}\) and \(\{6, 7\}\).