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

visualizing in mathematics 33discover the general truth of Euclidean geometry that a square on a diagonalof a given square has twice the area of the given square. The many visual waysof reaching Pythagoras’s Theorem provide similar examples.While it is easy to reach the theorem Plato presents by means of thevisual image, it is very difficult to show that the mode of belief-acquisition isreliable, because much of the process is fast, unconscious, and sub-personal.While the way of reaching the belief that there are twice as many smalltriangles in the square on the diagonal as in the square with the horizontalbase is clear and open, the way of reaching the belief that the small trianglesare congruent is hidden. For an initial speculative account of the process,see Giaquinto (2005). (The relevant passage in the Meno is examined inGiaquinto (1993a) and a similar example is discussed at length in Giaquinto(1992).)My hypothesis is that the hidden process involves the activation of dispositionsthat come with possession of certain geometrical concepts (e.g. forsquare, diagonal, congruent). What triggers the activation of these dispositionsis conscious, indeed attentive, visual experience; but the presence and operationof these dispositions is hidden from the subject. Because the process is fastand hidden, the resulting belief seems to the subject immediate and obvious.The feeling of obviousness occurs even in some more complicated cases. Aswe occasionally turn out to be fooled by these feelings, and because (at leastpart of) the process of belief-acquisition in visual discovery is hidden, we needin addition some transparent means of reaching the discovered proposition asa check: this is proof.But in other cases there is nothing hidden. One makes a discovery by meansof explicit visual thinking using background knowledge. Figure 1.4 illustratesan open way of discovering that the geometric mean of two numbers is lessthan or equal to their arithmetic mean (Eddy, 1985).(a+b)/2a(a–b)/2bFig. 1.4.ab