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

introduction 7concerns. On the contrary, nothing is further from the truth. Developing aformal language, such as Frege did, which aimed at capturing formally allvalid forms of reasoning occurring in mathematics, required a keen understandingof the reasoning patterns to be found in mathematical practice.³Central to Hilbert’s program was, among other things, the distinction betweenreal and ideal elements that also originates in mathematical practice. Delicateattention to certain aspects of mathematical practice informs contemporaryproof theory and, in particular, programs such as reverse mathematics. Finally,Brouwer’s intuitionism takes its origin from the distinction betweenconstructive vs. non-constructive procedures, once again a prominent distinctionin, just to name one area, the debates in algebraic number theoryin the late 19th century (Kronecker vs. Dedekind). Moreover, the analyticaldevelopments in philosophy of mathematics are also, to various extents,concerned with certain aspects of mathematical practice. For instance, nominalisticprograms force those engaged in reconstructing parts of mathematicsand natural science to pay special attention to those branches of mathematicsin order to understand whether a nominalistic reconstruction can beobtained.This will not be challenged by those working in the Lakatos tradition orby Maddy or by the authors in this collection. But in each case the appealto mathematical practice is different from that made by the foundationalisttradition as well as by most traditional analytic philosophers of mathematics inthat the latter were limited to a central, but ultimately narrow, aspect of thevariety of activities in which mathematicians engage. This will be addressed inthe following sections.My strategy for the rest of the introduction will be to discuss in broad outlinethe contributions of Corfield and Maddy, taken as representative philosophersof mathematics deeply engaged with mathematical practice, yet who comefrom different sides of the foundational/maverick divide. I will begin withCorfield, who follows in the Lakatos lineage, and then move to Maddy, takenas an exemplar of certain developments in analytic philosophy. It is withinthis background, and by contrast with it, that I will present, in Section 4, thecontributions contained in this volume and articulate, in Section 5, how theydiffer from, and relate to, the traditions being currently described. Regretfully,I will have to refrain from treating many other contributions that woulddeserve extensive discussion, most notably Kitcher (1984), but completeness isnot what I am aiming at here.³ For a reading of Frege which stresses the connection to mathematical practice, see Tappenden(2008).