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

visualizing in mathematics 23Pasch said, and he was echoed by Hilbert and Russell (Pasch, 1882; Hilbert,1894; Russell, 1901).In some quarters this turn led to a general disdain for visual thinking inmathematics: ‘In the best books’ Russell pronounced ‘there are no figures atall.’ (Russell, 1901). Although this attitude was opposed by some prominentmathematicians, others took it to heart. Landau, for example, wrote a calculustextbook without a single diagram (Landau, 1934). But the predominant viewwasnotsoextreme:thinkingintermsoffigureswasvaluedasameansoffacilitating grasp of formulae and linguistic text, but only reasoning expressedby means of formulae and text could bear any epistemological weight.By the late 20th century the mood had swung back in favour of visualization:Mancosu (2005) provides an excellent survey. We find books that advertisetheir defiance of anti-visual puritanism in their titles, such as Visual Geometryand Topology (Fomenko, 1994) andVisual Complex Analysis (Needham, 1997);mathematics educators turn their attention to pedagogical uses of visualization(Zimmermann and Cunningham, 1991); the use of computer-generatedimagery begins to bear fruit at research level (Hoffman, 1987; Palais, 1999);and diagrams find their way into research papers in abstract fields: see forexample the papers on higher dimensional category theory by Joyal et al.(1996), Leinster (2004), and Lauda (2005). But attitudes to the epistemologyof visual thinking remain mixed. The discussion is almost entirely confined tothe role of diagrams in proofs. In some cases, it is claimed, a picture alone isa proof (Brown, 1999, Ch.3). But that view is rare. Even the editor of Proofswithout Words: Exercises in Visual Thinking, writes ‘Of course, ‘‘proofs withoutwords’’ are not really proofs’ (Nelsen, 1993, p. vi). At the other extreme is thestolid attitude of Tennant (1986, p.304):[the diagram] has no proper place in the proof as such. For the proof is a syntacticobject consisting only of sentences arranged in a finite and inspectable array.Between the extremes, others hold that, even if no picture alone is a proof,visual images can have a non-superfluous role in reasoning that constitutes aproof (Barwise and Etchemendy, 1996a; Norman, 2006). Visual representations,such as geometric diagrams, graphs, and maps, all carry information. Takingvalid deductive reasoning to be the reliable extraction of information frominformation already obtained, Barwise and Etchemendy (1996a) pose thefollowing question: Why cannot the representations composing a proof bevisual as well as linguistic? The sole reason for denying this role to visualrepresentations is the thought that, with the possible exception of veryrestricted cases, visual thinking is unreliable, hence cannot contribute to proof.In the next section I probe this matter by considering visualization in proving,