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

28 marcus giaquintoTypically diagrams (and other non-verbal visual representations) do notrepresent their objects as having a property that is actually ruled out by theintention or specification of the object to be represented. But diagrams veryfrequently do represent their objects as having properties that, though notruled out by the specification, are not demanded by it. In fact this is oftenunavoidable. Verbal descriptions can be discrete, in that they supply no moreinformation than is needed.³ But visual representations are typically indiscrete,because for many properties or kinds F, a visual representation cannot representsomething as being F without representing it as being F in a particular way.Any diagram of a triangle, for instance, must represent it as having three acuteangles or as having just two acute angles, even if neither property is requiredby the specification, as would be the case if the specification were ‘Let ABCbe a triangle’. As a result there is a danger that in using a diagram (or othervisual representation) to reason about an arbitrary instance of class K, we willunwittingly rely on a feature represented in the diagram that is not commonto all instances of the class K. Thus the risk of errors of sort (2), unwarrantedgeneralization, is a danger inherent in the use of diagrams.The indiscretion of diagrams is not confined to geometrical figures. Thedot diagrams of ancient mathematics used to convince one of elementarytruths of number theory necessarily display particular numbers of dots, thoughthe truths are general. Figure 1.2 is an example, used to justify the formulafor the nth triangular number, i.e. the sum of the first n positiveintegers.Some people hold that the figure alone constitutes a proof (Brown, 1999);others would say that some accompanying text is required, to indicate howthe image is to be interpreted and used. If there is no text, some backgroundof conventions of interpretation must be assumed. But often more than oneset of conventions is available. In fact Grosholz (2005) shows how the multipleinterpretability of diagrams has sometimes been put to good use in mathematicaln+1Fig. 1.2.n³ This happy use of ‘discrete’ and its cognates is due to Norman, 2006.