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

4 paolo mancosuYet it is pertinent to ask whether there are not also other tasks for the philosophyof mathematics, tasks that arise either from the current practice of mathematics orfrom the history of the subject. A small number of philosophers (including one ofus) believe that the answer is yes. Despite large disagreements among the membersof this group, proponents of the minority tradition share the view that philosophyof mathematics ought to concern itself with the kinds of issues that occupy thosewho study the other branches of human knowledge (most obviously the naturalsciences). Philosophers should pose such questions as: How does mathematicalknowledge grow? What is mathematical progress? What makes some mathematicalideas (or theories) better than others? What is mathematical explanation? (p. 17)They concluded the introduction by claiming that the current state of thephilosophy of mathematics reveals two general programs, one centered onthe foundations of mathematics and the other centered on articulating themethodology of mathematics.Kitcher (1984) had already put forward an account of the growth ofmathematical knowledge that is one of the earliest, and still one of the mostimpressive, studies in the methodology of mathematics in the analytic literature.Starting from the notion of a mathematical practice,² Kitcher’s aim was toaccount for the rationality of the growth of mathematics in terms of transitionsbetween mathematical practices. Among the patterns of mathematical change,Kitcher discussed generalization, rigorization, and systematization.One of the features of the ‘maverick’ tradition was the polemic againstthe ambitions of mathematical logic as a canon for philosophy of mathematics.Mathematical logic, which had been essential in the development of thefoundationalist programs, was seen as ineffective in dealing with the questionsconcerning the dynamics of mathematical discovery and the historical developmentof mathematics itself. Of course, this did not mean that philosophyof mathematics in this new approach was reduced to the pure description ofmathematical theories and their growth. It is enough to think that Lakatos’Proofs and Refutations rests on the interplay between the ‘rational reconstruction’given in the main text and the ‘historical development’ provided in the notes.The relation between these two aspects is very problematic and remains oneof the central issues for Lakatos scholars and for the formulation of a dialecticalphilosophy of mathematics (see Larvor (1998)). Moreover, in addition toproviding an empiricist philosophy of mathematics, Kitcher proposed a theoryof mathematical change that was based on a rather idealized model (see Kitcher1984, Chapters7–10).² A quintuple consisting of five components: ‘a language, a set of accepted statements, a set ofaccepted reasonings, a set of questions selected as important, and a set of metamathematical views’(Kitcher, 1984).