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

50 marcus giaquintoWe name the bottom and top nodes, our constants, Min and Max respectively.(−) −x = the node n such that x ∧ n is Min and x ∨ n is Max.The set of nodes structured by ∧, ∨, −, Max and Min is isomorphic tothe power set algebra of S, which is the power set of S under the operationsof intersection, union, and relative complement, and the constants S and theempty set. In symbols,〈H; ∧, ∨, −, Max, Min〉 ∼ = 〈P(S); ∩, ∪, ∼, S, ∅〉The isomorphism can be checked visually. A structured set of this kind isknown as a three-atom Boolean algebra. Thus we have a visual template for thestructure of a three-atom Boolean algebra, and this is a structure involvingthree operations, known as meet (∧), join (∨), and complement (−), and twoconstants.With a small amount of practice it is easy to acquire the visual ability tosee the meet, join, and complement of nodes in Fig. 2.3 right away. Theleast straightforward is complement, but here is how it is done. Viewing theconfiguration as a cube, we see that every node is at one end of a uniquediagonal of the cube; the complement of a node is the node at the other endthe diagonal. So, although it is an exaggeration to say that we can simply seethe configuration of Fig. 2.3 as the structured set 〈H; ∧, ∨, −, Max, Min〉, itis strictly correct to say that we can have a perceptual grasp of that structuredset. But how do we grasp its structure, the structure of a three-atom Booleanalgebra? First, given an instance of this structure we can map it isomorphicallyonto a configuration like the one shown here, construed as a structured set inthe way described. This is the visual template method. But practice may takeus beyond it. We may eventually acquire an ability to tell, given an instance,that it can be mapped isomorphically onto the configuration without actuallyhaving carried out the mapping, even in thought. Provided that this ability isdiscriminating, in that it would not also lead us to think, of a non-instance, thatit too can be mapped isomorphically onto the configuration, our grasp of thestructure may consist in our having this ability, or, if that is too behaviourist,in our having the cognitive basis of the ability.2.3.2 Structural kindsThe cognitive abilities involved in discerning the structures of the last examplecanalsobeusedtodiscernkinds of structure. Representing a structure by aspatial pattern can be useful when the structure is very small or very simple,but moderate size and complexity usually nullifies any gain. Try drawing a