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

visualizing in mathematics 31given kind can fall short of universal applicability that if one had been presentedwith a schema that did fall short, one would have detected the failure.In the example at hand, the schema of visual reasoning involves at the starttaking a number k to be represented by a column of k dots, thence taking thetriangular array of n columns to represent the sum of the first n positive integers,thence taking that array combined with an inverted copy to make a rectangulararray of n columns of n + 1 dots. For a schema starting this way to be universallyapplicable, it must be possible, given any positive integer n, for the sum of thefirst n positive integers to be represented in the form of a triangular array, so thatcombined with an inverted copy one gets a rectangular array. This actually failsat the extreme case: n = 1. The formula [n × (n + 1)]/2 holds for this case; butthat is something we know by substituting ‘1’ for the variable in the formula,not by the visual method indicated. That method cannot be applied to n = 1,because a single dot does not form a triangular array, and combined with acopy it does not form a rectangular array. The fact that this is frequently missedby commentators suggests that the required sensitivity is often absent. This andsimilar ‘dot’ arguments are discussed in more detail in Giaquinto (1993b).Missing an untypical case is a common hazard in attempts at visual proving.A well-known example is the proof of Euler’s formula V − E + F = 2for polyhedra by ‘removing triangles’ of a triangulated planar projectionof a polyhedron. One is easily convinced by the thinking, but only becausethe polyhedra we normally think of are convex, while the exceptions are notconvex. But it is also easy to miss a case which is not untypical or extremewhen thinking visually. An example is Cauchy’s attempted proof (Cauchy,1813) of the claim that if a convex polygon is transformed into another polygonkeeping all but one of the sides constant, then if some or all of the internalangles at the vertices increase, the remaining side increases, while if some orall of the internal angles at the vertices decrease, the remaining side decreases.The argument proceeds by considering what happens when one transforms apolygon by increasing (or decreasing) angles, angle by angle. But in a trapezoid,changing a single angle can turn a convex polygon into a concave polygon,and this invalidates the argument (Lyusternik, 1963).The frequency of such mistakes indicates that visual arguments often lack thetransparency required for proof; even when a visual argument is in fact sound,its soundness may not be clear, in which case the argument is better thoughtof as a way of discovering rather than proving the truth of the conclusion. Butthis is consistent with the claim that visual thinking can be and often is a way ofproving something. That is, visual thinking can form a non-superfluous part ofa process of thinking through a proof that is not replaceable by some non-visualthinking (without changing the proof.) Euclid’s proof of the proposition that