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

cognition of structure 63to merely using each position in an ω-cube to represent an ω-cube?²⁰ Not ifthis is visuo-spatial imagination, as opposed to supposition. This is becausearepresentationofanω-cube is a representation of something with infinitespatial extension, while each element of an ω-cube is represented as finitelyextended. If, however, ‘imagine’ is understood broadly to include fictionalsupposition, we can imagine each element of an ω-cube to be a benign blackhole into which we can dive; and once in there, we find ourselves in a newinfinite three-dimensional space containing another ω-cube (and of course onecould iterate the story). Although this does not amount to forming a visualcategory specification for an ω-cube of ω-cubes, it does give us a semi-visualway of thinking of higher powers of ω. But even this semi-visual thinking runsout before ɛ 0 .²¹There may be other kinds of infinite structures, non-ordinal structures,that are knowable with the kind of visual awareness that we have for thestructure of ω and ω 2 under their standard ordering. That is a matter for furtherresearch. N, the structure of ω, is the cognitively simplest infinite structure.We have usable visual representations of instances of it that provide someawareness of the nature of the structure. But these representations are categoryspecifications, not images. The fact that we can never have a visual experienceencompassing the whole of an instance of this structure marks a qualitativedifference between our grasp of this simplest infinite structure and our graspof the finite structures discussed earlier, such as the two-generation binary treeor the power set algebra of a three-element set.2.5 ConclusionWe can have cognitive grasp of some structures by means of visual representations.For small simple finite structures we can know them through sensoryexperience of instances of them, somewhat as we can know the butterfly shapefrom seeing butterflies. This is a kind of knowledge by acquaintance. Thoughwe cannot have knowledge by acquaintance of any infinite structure, somesimple infinite structures can be known by visual means, and not merely as thestructure of models of this or that theory. The crucial representations in these²⁰ This question was put to me by Stewart Shapiro.²¹ Apparently Georg Kreisel used to say that ordinals less than ɛ 0 are visualizable (Paolo Mancosu,personal communication), but it is not clear why he was confident that visualizability did not stop wellbefore. There is an interesting earlier discussion of the matter by Oskar Becker in a letter to Weyl in1926 and in Becker (1927) in his attempt to provide phenomenological foundations for the transfiniteordinals reported in Mancosu and Ryckman (2002).