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

26 marcus giaquintoMiller himself has surely gone through exactly the kind of process describedabove. Of course the actual event would have been split up in time, and Millerwould have already known the conclusion to be true; still, the whole thingwould have been a case of thinking through a proof, a highly untypical case ofcourse, in which visual thinking occurred in a non-superfluous way.This is enough to refute claim (a), the claim that all diagrammatic thinkingin thinking through a proof is superfluous. What about Tennant’s claim that aproof is ‘a syntactic object consisting only of sentences’ as opposed to diagrams?A proof is never a syntactic object. A formal derivation on its own is a syntacticobject but not a proof. Without an interpretation of the language of theformal system the end-formula of the derivation says nothing; and so nothingis proved. Without a demonstration of the system’s soundness with respectto the interpretation, one lacks reason to believe that derived conclusions aretrue. A formal derivation plus an interpretation and soundness proof can be aproof of the derived conclusion. But one and the same soundness proof canbe given in syntactically different ways, so the whole proof, i.e. derivation+ interpretation + soundness proof, is not a syntactic object. Moreover, thepart of the proof which really is a syntactic object, the formal derivation,need not consist solely of sentences; it can consist of diagrams, as Miller’sexample shows.The visual thinking in this example consists in going through a sequence ofdiagrams and at each step seeing that the next diagram results from a permittedalteration of the previous diagram. It is a non-superfluous part of the processof thinking through a proof that on any straight line segment an equilateraltriangle is constructible. It is clear too that in a process that counts as thinkingthrough this proof, the visual thinking is not replaceable by non-diagrammaticthinking. That knocks out (b), leaving only (c): some thinking that involves adiagram in thinking through a proof is neither superfluous nor replaceable bynon-diagrammatic thinking.This is not an isolated example. In the 1990s Barwise led a programmeaimed at the development of formal systems of reasoning using diagrams andestablishing their soundness. There was renewed interested in Peirce’s graphicalsystems for propositional and quantifier logic, and systems employing Eulerdiagrams and Venn diagrams were developed and investigated, culminating inthe work of Sun-Joo Shin (1994). Barwise was interested in systems which bettermodel how we reason than these, and to this end he turned his attention toheterogeneous systems, systems deploying both formulas and diagrams: he andEtchemendy developed such a system for teaching logic, Hyperproof, andbeganto investigate its metalogical properties (Barwise and Etchemendy, 1996b). Thiswas part of a surge of research interest in the use of diagrams, encompassing