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

cognition of structure 57infinite completed durations, or infinite speeds. Finiteness is in this sense ‘builtin’.Under these background constraints the category specification determinesthat the number marks are well-ordered by their relation of precedence. Thissuffices to determine a unique structure. So we can grasp the structure of thenatural number system as the structure of the set of number marks of themental number line under their order of precedence.Knowing the natural number structure in this way is much less direct thanthe kind of knowledge of finite structures discussed earlier. In this case wecannot experience an entire instance of the structure. So this knowledge ofstructure does not consist in the cognitive basis of an ability to recognizeinstances and to distinguish them from non-instances. We have to gather thenature of a number line from our inclinations to answer certain questions aboutit; although visual experience plays some role in this process, our answers arenot simply reports of experience. In becoming aware in this indirect way of thecontent of a visual category specification for a mental number line, we acquirea grasp of a type of structured set, and we can then know the structure N as thestructure of structured sets of this type. While this kind of knowledge is quitedifferent from the experiential knowledge of small structures discussed earlier,it does have an experiential element that distinguishes it from knowledge bya description of the form the structure of models of such-&-such axioms, which ofcourse requires knowing that the axiom set is categorical. To help appreciatehow significant this difference is, it is worth examining a contrasting case.2.4.1 An infinite structure beyond visual graspA contrasting case is the structure of the set of real numbers in the closed unitinterval [0, 1], under the ‘less than’ relation. I will call this structure U for‘unit’. We do have visual ways of representing U, but I claim that they do notgive us knowledge of it. I will now try to substantiate this claim.An obvious thought is that we can think of U as the structure of the setof points on a straight line segment with left and right endpoints, when eachpoint corresponds to a unique distance from one end, and the order of points isdetermined by the corresponding distance. We can certainly visualize a finitehorizontal line segment, taking its points to be the locations of intersection ofthe horizontal line segment with (potential) vertical line segments; and we canvisually grasp what it is for one such location to lie to the left of another. Whydoes not this give us a visual grasp of the structure U?One reason concerns points. If the points on a line constitute a set withstructure U they must be not merely too small to be seen by us, but absolutelyinvisible, having zero extension. Neither vision nor visualization gives us anyacquaintance with even a single point of this kind, let alone uncountably many