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

8 paolo mancosu2 Corfield’s Towards a Philosophy of Real Mathematics(2003)A good starting point is Corfield’s recent book Towards a Philosophy of RealMathematics (2003). Corfield’s work fits perfectly within the frame of thedebate between foundationalists and ‘maverick’ philosophers of mathematics Idescribed at the outset. Corfield attributes his desire to move into philosophyof mathematics to the discovery of Lakatos’ Proofs and Refutations (1976)and he takes as the motto for his introduction Lakatos’ famous paraphrasingof Kant:The history of mathematics, lacking the guidance of philosophy, has becomeblind, while the philosophy of mathematics, turning its back on the mostintriguing phenomena in the history of mathematics, has become empty.(Lakatos,1976, p.2)Corfield’s proposal for moving out of the impasse is to follow in Lakatos’footsteps, and he proposes a philosophy of ‘real’ mathematics. A succinctdescription of what this is supposed to encompass is given in the introduction:What then is a philosophy of real mathematics? The intention of this term is todraw a line between work informed by the concerns of mathematicians past andpresent and that done on the basis of at best token contact with its history orpractice. (Corfield, 2003, p.3)Thus, according to Corfield, neo-logicism is not a philosophy of realmathematics, as its practitioners ignore most of ‘real’ 20th century mathematicsand most historical developments in mathematics with the exception of thefoundational debates. In addition, the issues raised by such philosophers arenot of concern to mathematicians. For Corfield, contemporary philosophyof mathematics is guilty of not availing itself of the rich trove of the historyof the subject, simply dismissed as ‘history’ (you have to say that with theright disdainful tone!) in the analytic literature, not to mention a first-handknowledge of its actual practice. Moreover,By far the larger part of activity in what goes by the name philosophy of mathematicsis dead to what mathematicians think and have thought, aside from an unbalancedinterest in the ‘foundational’ ideas of the 1880–1930 period, yielding too often adistorted picture of that time. (Corfield, 2003, p.5)It is this ‘foundationalist filter’, as Corfield calls it, which he claims isresponsible for the poverty of contemporary philosophy of mathematics.There are two major parts to Corfield’s enterprise. The first, the pars destruens,