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

12 paolo mancosusentiments appear in the writings of many philosophers of mathematics who holdthat the goal of philosophy of mathematics is to account for mathematics as it ispracticed, not to recommend reform. (Maddy, 1997, p.161)Naturalism, in the Maddian sense, recognizes the autonomy of mathematicsfrom natural science. Maddy applies her naturalism to a methodological study ofthe considerations leading the mathematical community to the acceptance orrejection of various (set-theoretical) axioms. She envisages the formulationof ‘a naturalized model of practice’ (p. 193) that will provide ‘an accuratepicture of the actual justificatory practice of contemporary set theory andthat this justificatory structure is fully rational’ (pp. 193–4). The method willproceed by identifying the goals of a certain practice and by evaluating themethodology employed in that branch of mathematics (set theory, in Maddy’scase) in relation to those goals (p. 194). The naturalized model of practiceis both purified and amplified. It is purified in that it eliminates seeminglyirrelevant (i.e. philosophical) considerations in the dynamics of justification;and it is amplified in that the relevant factors are subjected to more preciseanalysis than what is given in the practice itself and they are also applied tofurther situations:Our naturalist then claims that this model accurately reflects the underlyingjustificatory structure of the practice, that is, that the material excised is trulyirrelevant, that the goals identified are among the actual goals of the practice (andthat the various goals interact as portrayed), and that the means-ends reasoningemployed is sound. If these claims are true, then the practice, in so far as itapproximates the naturalist’s model, is rational. (Maddy, 1997, p.197)Thus, using the example of the continuum hypothesis and other independentquestions in descriptive set theory, she goes on to explain how the goal ofproviding ‘a complete theory of sets of real numbers’ gives rational support tothe investigation of CH (and other questions in descriptive set theory). Thetools for such investigations will be mathematical and not philosophical. Whilea rational case for or against CH cannot be built out of the methodology thatMaddy distils from the practice, she provides a case against V = L (an axiomthat Quine supported).We need not delve into the details of Maddy’s analysis of her case studies andthe identification of several methodological principles, such as maximize andunify, that in her final analysis direct the practice of set theorists and constitutethe core of her case against V = L. Rather, let us take stock.Comparing Maddy’s approach to that of the ‘maverick’ tradition, we canremark that just as in the ‘maverick’ tradition, there is a shift in whatproblems Maddy sets out to investigate. While not denying that ontological