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

144 paolo mancosuvarious conceptual components of the model. In addition, further discussionof Steiner’s account, aimed at improving the account, is provided in Weberand Verhoeven (2002).Kitcher’s model will be described at length in the research paper followingthis introduction. Criticisms of unification theories of explanation asinsufficient for mathematical explanation have also been raised forcefully inTappenden (2005). In the next section, I would like to point out some aspectsof Kitcher’s position that bring us back to the issue of generalization andabstraction. This will also be instrumental in introducing aspects of Kitcher’sthought relevant to the research paper to follow.5.4 Kitcher on explanation and generalizationI will start with a striking quote concerning generalization and its relation toexplanation within mathematics. It is taken from Cournot:Generalizations which are fruitful because they reveal in a single general principlethe rationale of a great many particular truths, the connection and commonorigins of which had not previously been seen, are found in all the sciences, andparticularly in mathematics. Such generalizations are the most important of all, andtheir discovery is the work of genius. There are also sterile generalizations whichconsist in extending to unimportant cases what inventive persons were satisfied toestablish for important cases, leaving the rest to the easily discernible indicationsof analogy. In such cases, further steps toward abstraction and generalization donot mean an improvement in the explanation of the order of mathematical truthsand their relations, for this is not the way the mind proceeds from a subordinatefact to one which goes beyond it and explains it. (Cournot, 1851, sect.16, Engl.trans. 1956, p.24, my emphasis)The central opposition in this text is that between fruitful generalizations vs.sterile generalizations. What distinguishes the two of them is that the formerare explanatory while the latter are not. Genius consists, according to Cournot,not in generalization tout court but in those generalizations that are able to revealthe explanatory order according to which mathematical truths are structured.A remarkably similar statement is found in an article by the mathematicianS. Mandelbrojt who claims that ‘La généralité est belle lorsqu’elle possédeun caractére explicatif’ and ‘l’abstraction est belle et grande lorsqu’elle estexplicative’ (Mandelbrojt, 1952, pp. 427–428). Also Mandelbrojt pointed outthat generalization can be cheap and boring. The generalization is informativewhen it is explanatory. Such explanatory generalizations can be obtained bythe right degree of abstraction and should show the object being studied in its