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

46 marcus giaquintoorder-preserving correlation (an isomorphism) is a mechanical-perceptual task.In this way we have epistemic access to a structure.Here I have been supposing that a visual template is one particular configurationof actual physical marks, e.g. those in Fig. 2.1 in the copy of the bookyou are now reading! Just as you can recognize different physical inscriptions asinstances of an upper-case letter A, so you can recognize different configurationsof physical marks as instances of the same type. You would have no difficultyin recognizing, as an instance of that type, the instance of Fig. 2.1 in anothercopy of the book. Moreover, you could recognize as instances of that typeother configurations that are geometrically similar but differing in size andorientation.To account for this recognitional capacity it is thought that the visual systemstores representations of types of visible configurations. A representation of avisual type is not itself a visual image; but(i) a visual stimulus can activate a representation of a type. This is requiredfor recognition as opposed to mere perception.(ii) Also, we can activate a type representation to produce an image of aninstance of the type by voluntary visual imagining.In this situation it seems reasonable to take the configuration type to have thestructure-fixing role that we had been assigning to a particular physical instanceof the type. This is still within the ambit of the template idea. In place of asingle physical template, however, we allow as templates any configuration ofmarks that we can recognize as instances of the type.2.2.2 Beyond visual templatesWe can have an awareness of structure that is more direct than the visual templatemode, at least in such very simple cases as this, two generations of binarysplitting. The awareness I have in mind is not tied to a particular configurationtype. Although a particular configuration of marks, viewed as a set of elementsvisibly related in a particular way, may serve a person as an initial instance ofa structure, one may later think of the structure without thinking of it as thestructure of configurations of just that visual type. Once one has perceived aconfiguration of marks as a structured set, one can acquire the ability to perceiveconfigurations of other visual types as structured in the same way. For example,one has no difficulty in seeing the configurations of Fig. 2.2 as structured inthe same way, even though they appear geometrically quite dissimilar.It is not clear how we do this. In these cases there is a visualizable spatiotemporaltransformation of one configuration into another that preservesnumber of members and relevant relations between them. (Relevant relations