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

30 marcus giaquintoclearly not enough that the proviso is in fact met. For in that case it might justbe the thinker’s good luck that the proviso is met; hence the thinker would notknow that the generalization is valid and so would not have genuinely thoughtthrough the proof at that step. This leaves two options. The strict option isthat without a conscious, explicit check one has not really thought throughthe proof. The relaxed option is that one can properly think through the proofwithout checking that the proviso is met, but only if one is sensitive to thepotential error and would detect it in otherwise similar arguments. For thenone is not just lucky that the proviso is met. Being sensitive in this contextconsists in being alert to dependence on features of the arbitrary instance notshared by all members of the class of generalization, a state produced by acombination of past experience and current vigilance. Without a compellingreason to prefer one of these options, decisions on what is to count as provingmust be conditional.How does all this apply to generalizing from visual thinking about anarbitrary instance? Take the example of the visual route to the formula fortriangular numbers just indicated. The image used reveals that the formulaholds for the 6th triangular number. The generalization to all triangularnumbers is justified only if the visuo-spatial method used is applicable to thenth triangular number for all positive integers n, that is, provided that themethod used does not depend on a property not shared by all positive integers.A conscious, explicit check that this proviso is met requires making explicitthe method exemplified for 6 and proving that the method is applicablefor all positive integers in place of 6. For a similar idea in the context ofautomating visual arguments, see Jamnik (2001). This is not done in practicewhen thinking visually, so if we accept the strict option for thinking through aproof involving generalization, we would have to accept that the visual routeto the formula for triangular numbers does not amount to thinking througha proof of it; and the same would apply to the familiar visual routes to othergeneral positive integer formulas, such as that n 2 = the sum of the first n oddnumbers.But what if the strict option for proving by generalization on an arbitraryinstance is too strict, and the relaxed option is right? When arriving at theformula in the visual way indicated, one does not pay attention to the fact thatthe visual display represents the situation for the 6th triangular number; it is asif the mind had somehow extracted a general schema of visual reasoning fromexposure to the particular case, and had then proceeded to reason schematically,converting a schematic result into a universal proposition. What is required,on the relaxed option, is sensitivity to the possibility that the schema is notapplicable to all positive integers; one must be so alert to ways a schema of the