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

visualizing in mathematics 27computer science, artificial intelligence, formal logic, and philosophy. For arepresentative sample see the papers in Blackwell (2001).¹All that is for the record. Mathematical practice almost never proceeds byway of formal systems. For most purposes there is no need to master a formalsystem and work through a derivation in the system. In fact there is reason toavoid going formal: in a formalized version of a proof, the original intuitiveline of thought is liable to be obscured by a multitude of minute steps. Whileformal systems may eventually prove useful for modelling actual reasoning withdiagrams in mathematics, much more investigation of actual practice is neededbefore we can develop formal systems that come close to real mathematicalreasoning. This prior investigation has two branches: a close look at practicesin the history of mathematics, such as Manders’s work on the use of diagramsin Euclid’s Elements (see his contributions in this volume), and cognitive studyof individual thinking using diagrams in mathematics. Cognitive scientists havenot yet paid much attention to this; but there is a large literature now on visualperception and visual imagery that epistemologists of mathematics can drawon. A very useful short discussion of both the Barwise programme and therelevance of cognitive sudies (plus bibliography) can be found in (Mancosu,2005).² So let us set aside proofs by means of formal systems and restrictattention to normal cases.Outside the context of formal diagrammatic systems, the use of diagrams iswidely felt to be unreliable. There are two major sorts of error:1. relevant mismatch between diagrams and captions;2. unwarranted generalization from diagrams.In errors of sort (1), a diagram is unfaithful to the described construction:it represents something with a property that is ruled out by the description,or without a property that is entailed by the description. This is exemplifiedby diagrams in the famous argument for the proposition that all trianglesare isosceles: the meeting point of an angle bisector and the perpendicularbisector of the opposite side is represented as falling inside the triangle,when it has to be outside—see Rouse Ball (1939). Errors of sort (1) arecomparatively rare, usually avoidable with a modicum of care, and notinherent in the nature of diagrams; so they do not warrant a general charge ofunreliability.¹ The most philosophically interesting questions concern the essential differences between sententialand diagrammatic representation and the properties of diagrams that explain their special utility andpitfalls. Especially illuminating in this regard is the work of Atsushi Shimojima: see the slides for hisconference presentation, of which Shimojima (2004) is the abstract, on his web page.² Two other pertinent references are Pylyshyn (2003) andGrialouet al. (2005).