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

52 marcus giaquintoinitial element and for every non-terminal element exactly two immediateR-successors.⁶ I am restricting attention to binary trees such that, for somepositive finite number k, all and only kth generation elements are terminal.⁷The rule is this:x bears relation R to y if and only if there is an upward path from the node(branching point) representing x to the node representing y.This convention clearly respects the transitivity of the relation. The largesttree that we can easily visualize has very few generations. But we have auniform rule for obtaining the next tree from any given n-generation tree bya visualizable operation:Add to each of the 2 n terminal points an appropriately sized ‘v’.This visualizable operation, together with the diagrams or images of the firstfew trees, provides us with a visual awareness of a kind of template forfinite binary trees. But however we proportion the ‘v’s of each succeedinggeneration to their predecessors, for some n which is not very large the partsof a tree diagram from the nth generation onwards will be beyond the scopeor resolution of a visual percept or image in which the initial part of the treeis clearly represented. Yet we have a visual way of thinking of a binary treetoo large for its spatial representation to be completely visible at one scale.We can visualize first the result of scanning a tree diagram along any one ofits branches; we can then visualize the result of zooming in on ‘v’s only justwithin visual resolution, and we can iterate this process. This constitutes a kindof visual grasp of tree diagrams beyond those we perceive entirely and at once.This visual awareness, arising from the combination of our visual experienceof the first few tree diagrams and our knowledge of the visualizable operationfor extending them, gives us a grasp of the structural kind comprising the finitebinary trees.The same considerations apply with regard to ternary and perhaps evenquaternary trees. But trees whose nodes split into as many as seven branchesdefy visualization beyond the first generation, and we cannot visualize eventhe first generation of a tree with 29 branches from each node, except perhapsin a way in which it is indistinguishable from trees with 30-fold branching. In⁶ So the ordered sets represented earlier with the ‘parent-of ’ and ‘child-of ’ ordering relations arenot binary trees as neither of these is transitive. But those same sets ordered by ‘forbear-of ’ and‘descendant-of ’ respectively are binary trees.⁷ The initial element is zeroth generation, its two successors are first generation, and so on.Accordingly, a tree whose terminal elements are first generation is a 1-generation tree, and in general atree whose terminal elements are nth generation is an n-generation tree.