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

visualizing in mathematics 29thinking, a topic I will have to leave aside. In the following discussion I willassume that appropriate text is present.The conclusion drawn is that the sum of integers from 1 to n is [n×(n + 1)]/2 for any positive integer n, but the diagram presents the case only forn = 6. We can perhaps avoid representing a particular number of dots whenwe merely imagine a display of the relevant kind; or if a particular number isrepresented, our experience may not make us aware of the number—just as,when one imagines the sky on a starry night, for no particular number k arewe aware that exactly k stars are represented. Even so, there is likely to besome extra specificity. For example, in imagining an array of dots of the formjust illustrated, one is unlikely to imagine just two columns of three dots, therectangular array for n = 2. Typically the subject will be aware of imaginingan array with more than two columns. This entails that an image is likely tohave unintended exclusions. In this case it would exclude the three-by-twoarray. An image of a triangle representing all angles as acute would excludetriangles with an obtuse angle or a right angle. The danger is that the visualreasoning will not be valid for the cases that are unintentionally excluded bythe visual representation, with the result that the step to the conclusion is anunwarranted generalization.What should we make of this? First, let us note that in a few cases the imageor diagram will not be over-specific. When in geometry all instances of therelevant class are similar to one another, for instance all circles or all squares,the image or diagram will not be over-specific for a generalization about thatclass; so there will be no unintended exclusions and no danger of unwarrantedgeneralization. Here then are possibilities for non-superfluous visual thinkingin proving.To get clear about the other cases, where there is a danger of overgeneralization,it helps to look at generalization in ordinary non-visual reasoning.Schematically put, in reasoning about things of kind K, once we have shownthat from certain premisses it follows that such-and-such a condition is true ofarbitrary instance c, we can validly infer from those same premisses that thatcondition is true of all Ks, with the proviso that neither the condition norany premiss mentions c. (If a premiss or the condition does mention c, thereasoning may depend on a property of c that is not shared by all other Ks,and so the generalization would be unsafe.) A question we face is whether infollowing an argument involving generalization on an arbitrary instance,⁴ thethinking must include a conscious, explicit check that the proviso⁵ is met. It is⁴ Other terms for this are ‘universal generalization’ and ‘universal quantifier introduction’.⁵ The proviso is that neither the condition nor any premiss mentions c.