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

48 marcus giaquintoA category specification is a nexus of related feature specifications that cannotbe ‘read off’ the subjectively accessible features of the visual experience of aninstance of the category. Just as with face recognition, the subject may haveno way of knowing precisely which congregations of features and relationsleads to recognition of the category when presented with an instance. A visualcategory specification is a kind of representation in the visual system, but it isvery unlike a visual image or percept. An image or percept is a transient item ofexperience of specific phenomenological type, whereas a category specificationis a relatively enduring representation that is not an item of experience, thoughits activation affects experience.We can recognize a perceived configuration of marks as an instance ofa certain structure, by activation of an appropriate visual category specification.Thus, I suggest, we can have a kind of visual grasp of structurethat does not depend on the particular configuration we first used as atemplate for the structure. We may well have forgotten that configurationaltogether. Once we have stored a visual category specification for a structure,we have no need to remember any particular configuration as a meansof fixing the structure in mind. We can know it without thinking of itas ‘the structure of this or that configuration’. There is no need to makean association. So this is more direct than grasp of structure by a visualtemplate.What about identifying the structure of a non-visual structured set givenby verbal description? Recall the method mentioned earlier using a visualtemplate: (1) wefirstnamethemembersoftheset,(2) then label elementsof the template with those names, and (3) then check that the labellingprovides an isomorphism. Though this can happen, we often do not needany naming and labelling. Consider the following structured set: {Mozart, hisparents, his grandparents} under the relation ‘x is a child of y’. Surely onecan tell without naming and labelling that it is isomorphic to the structuredsets given earlier. We know this for any set consisting of a person, her parentsand grandparents under the ‘child-of’ relation (assuming no incest). It is asif our grasp of these sets as structured sets already involves activation of avisual category specification for two generations of binary splitting. Exactly thesame applies to the set obtained from the first two stages in the constructionof the Cantor set by excluding open middle thirds, starting from the closedunit interval, under set inclusion. Naming and labelling are not needed torecognize this as a case of two generations of binary splitting. I offer as atentative hypothesis that we can recognize the structure in this case as a resultof activation of a visual category specification for two generations of binarysplitting.