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

mathematical concepts: fruitfulness and naturalness 299between choices of basic categories and the success of predictions (where thepredictions may be clearly stated conjectures or less specific expectations of‘fruitfulness’).) Finally, I sketched the Riemann–Dedekind account of ‘innercharacteristic properties’ as a promising example of a method, and a methodology,incorporating this insight. This methodology has the additional attractivefeature that in its details it resonates with the Port Royal Principle, sinceidentifiable mathematical cases drive it and give it substance. Of course, so farI’ve given only the roughest sketch of the Riemann–Dedekind account andthe mathematics informing it, and I’ve given only unargued hints about its relationshipto contemporary mathematical practice. In part this has been a functionof space, but in part it is a necessity arising from the texts themselves. Riemannand Dedekind doled their methodological dicta out frugally. As mathematiciansare prone to do, they let their mathematics do most of the talking, whichleaves the philosopher/scribe a lot of detail to spell out. But this shouldn’t bea surprise: Arnauld and Nicole warned us that this is what to expect.BibliographyArmstrong, David M. (1986), ‘In Defence of Structural Universals’, Australasian Journalof Philosophy, 1, 85–88.(1989), Universals: An Opinionated Introduction (Boulder: Westview Press).Avigad, Jeremy (2006), ‘Methodology and Metaphysics in the Development ofDedekind’s Theory of Ideals’, in José Ferreirós and Jeremy Gray (eds.), The Architectureof Modern Mathematics (Oxford: Oxford University Press), pp. 159–186.Benacerraf, Paul(1965), ‘What Numbers Could Not Be’, The Philosophical Review,74(1), 47–73.Corfield, David (2003), The Philosophy of Real Mathematics (Cambridge: CambridgeUniversity Press).Cox, David (1988), ‘Quadratic Reciprocity: its Conjecture and Application’, TheAmerican Mathematical Monthly, 95(5), 442–448.(1989), Primes of the Form x 2 + ny 2 (New York: Wiley).Darrigol, Olivier(2005), Worlds of Flow: A History of Hydrodynamics from the Bernoullisto Prandtl (Oxford: Oxford University Press).Dedekind, Richard (1854), Über die Einfürung neuer Funktionen in der Mathematik,Habilitation lecture, Göttingen. Reprinted in Werke, III, pp. 428–438.(1877), Theory of Algebraic Integers, ed. and trans. John Stillwell (Cambridge MathematicalLibrary) Translation published 1996; originally published 1877.(1895), ‘Über die Begründung der Idealtheorie’, Nachrichten der KöniglichenGesellschaft der Wissenschaften zu Göttingen, 106–113. Reprinted in Werke, IIpp. 50–58.