Mancosu - Philosophy of Mathematical Practice (Oxford, 2008).pdf

cognition of structure 51Fig. 2.4.diagram of the power set algebra of a four-element set: it can be done, butthe result will not be easy to take in and memorize. Moreover, there seemsto be no iterative procedure for obtaining a diagram or spatial model for thepower set algebra of an n + 1-element set from a diagram for the power setalgebra of an n-element set. If that is right the structures of the power setalgebras of finite sets do not constitute a kind that we can grasp by means ofvisual configurations. Some kinds of structure, however, are sufficiently simplethat visual representations together with rules for extending them can provideawareness of the structural kind. For discrete linear orderings with endpointsthe representations might be horizontal configurations of stars and lines, givinga structured set of stars under the relation ‘x is left of y’. The first few of thesewe can see or visualize, as illustrated in Fig. 2.4.We have a uniform rule for obtaining, from any given diagram, the diagramrepresenting the next discrete linear ordering with endpoints: add to thestar at the right end a horizontal line segment with a star at its right end.This is a visualizable operation that together with the diagrams of the initialcases provides a grasp of a species of visual templates for finite discrete linearorderings with endpoints. Through awareness of this kind of template we haveknowledge of the kind of structure which they are templates for. Binary trees,and more generally n-ary trees for finite n, comprise a more interesting kind ofstructure. We have already seen configurations that can be used to representa binary tree in earlier figures. Taking elements to be indicated by points atwhich a line splits into two and by terminal points, the three binary trees afterthe first (which is just a ‘v’) can be visualized as in Fig. 2.5.As mentioned before, a configuration represents a structured set only givensome rules governing the representation of relations, functions, and constants.Binary trees are sets ordered by a single transitive relation R with oneFig. 2.5.