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

cognition of structure 53these cases we operate by description and where possible by analogy with thetrees we can visualize, just as we sometimes use three-dimensional objects asanalogues of objects in higher dimensions. However, the sort of acquaintancewecanhavewiththebinarytreekindwecanalsohavewithmanyotherkinds of structure, if we can give a recursive specification of the kind. What isrequired is just that templates of the smallest two or three structures are easyto visualize and we have a rule for obtaining a template of the ‘next’ structurefrom a template of a given structure by a visualizable process. A structural kindof this sort comprises an infinite sequence of structures that are often nested,in the sense that every structure in the sequence is a substructure of all laterstructures. The union of such a sequence is an infinite structure. What kind ofvisual grasp of infinite structures is possible, if any at all? That is our next topic.2.4 Knowledge of infinite structuresIf there can be visual cognition of any infinite structure, the simple andfamiliar structure of the natural numbers will be the prime example. Here I amtalking about the structure of the finite cardinals under their natural ‘less than’ordering. There is significant evidence that people with a standard educationmentally represent positive integers as aligned in space. This is an active topicof cognitive research that goes under the title of ‘mental number lines’.⁸ Inour culture the standard mental number line is a horizontal alignment with aleft-to-right ordering. Details may vary. One possibility is a set of evenly spacedvertical marks on a horizontal line, with a single leftmost mark, continuingendlessly to the right such that every mark, however far to the right, is reachableby constant rate scanning from the leftmost mark (Fig. 2.6). One mark is takento precede another if it is to the left of it; with respect to this ordering theleftmost mark is the only initial mark and there is no terminal mark.⁹An obvious problem with the idea that a mental number line provides agrasp of the natural number structure is that we cannot see or visualize morethan a finite part of any such line. When it comes to actual images (or percepts),something like Fig. 2.6 will be the best we can do. So the idea that we have aFig. 2.6.⁸ This is the subject of Chapter 6 of Giaquinto (2007).⁹ An initial element is one that has no immediate predecessor; a terminal element is one that has noimmediate successor.