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

cognition of structure 492.3 Extending the approach2.3.1 More complicated structuresThe structured sets considered so far have all been very small finite sets undera single binary relation. You may reasonably harbour the suspicion that setsstructured by a plurality of relations or operations lie beyond any visual meansof cognizing structure. I will now try to allay that suspicion. Figure 2.3 is alabelled visual template for the structure of the power set of a three-memberedset {a, b, c} under inclusion.As before, this is a very small set structured by one binary relation. Theconfiguration of Fig. 2.3 provides a template, namely, the set H of nodes underthe relation ‘n = m, or, there is an upward path from m to n’, which we canshorten to ‘n m’. This structured set is easily seen to be isomorphic to thepower set of S under inclusion by means of the labelling in the figure. Insymbols,〈H; 〉 ∼ = 〈P(S); ⊆〉However, there is another way in which the set of nodes of Fig. 2.3 isstructured. In place of the binary relation there are operations and constantsdefined as follows:(∧) x ∧ y = the highest node n such that n x and n y.(∨) x ∨ y = the lowest node n such that x n and y n.S{a,b} {a,c} {b,c}{a}{b}{c}∅Fig. 2.3.