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

146 paolo mancosucomplex book and I will not attempt to give a general overview of its contents.However, one of the main questions Kitcher raises is: How does mathematicsgrow? What are the patterns of change which are typical of mathematics? Isthe process of growth a rational one? In Chapter 9 of his book he sets the goalas follows:Here I shall be concerned to isolate those constituent patterns of change andto illustrate them with brief examples. I shall attempt to explain how theactivities of question-answering, question-generation, generalization, rigorizationand systematization yield rational interpractice transitions. When these activitiesoccur in a sequence, the mathematical practice may be dramatically changedthrough a series of rational steps. (Kitcher, 1984, p.194)Let us then consider generalization:One of the most readily discernible patterns of mathematical change, one whichI have so far not explicitly discussed, is the extension of mathematical languageby generalization. (Kitcher, 1984, p.207)As examples Kitcher mentioned Riemann’s redefinition of the definiteintegral, Hamilton’s search for hypercomplex numbers, and Cantor’s generalizationof finite arithmetic. Kitcher’s goal is ‘to try to understand the processof generalization which figures in these episodes and to see how the searchfor generalization may be rational’ (Kitcher, 1984, p.207). However, not allgeneralizations are significant. In fact it is easy to concoct trivial generalizations.What distinguishes the trivial generalizations from the significant ones? This iswhere explanation comes in again:significant generalizations are explanatory. They explain by showing us exactlyhow, by modifying certain rules which are constitutive of the use of someexpressions of the language, we would obtain a language and a theory withinwhich results analogous to those we have already accepted would be forthcoming.From the perspective of the new generalization, we see our old theory as a specialcase, one member of a family of related theories. (Kitcher, 1984, pp. 208–209)Building upon such considerations, Kitcher tries to distinguish between therationally acceptable generalizations and those that are not so:Those ‘generalizing’ stipulations which fail to illuminate those areas of mathematicswhich have already been developed are not rationally acceptable. (1984, p.209)In other words, to account for the rationality of processes of generalizationin mathematics we need an account of mathematical explanation. Moreover,one of Kitcher’s envisaged benefits of his analysis in terms of explanatory