object -> actions -> statements -> relationships
Henri Poincaré distinguished between the memory for steps in a chain of reasoning and the appreciation of the links between propositions. The later was regarded as the more important ability for a mathematician.
There are symbol systems for encoding actions performed upon objects and symbol systems for relations between objects.
1 + 1 = 2 | 2 - 1 = 1 | 2 x 2 = 4 | 4/2 = 2