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

purity as an ideal of proof 193is purity which, practically speaking, enforces a certain symmetry betweenthe conceptual resources used to prove a theorem and those needed for theclarification of its content. The basic idea is that the resources of proof oughtideally to be restricted to those which determine its content.There are two main views concerning the significance of such symmetry.For Aristotelians, neo-Aristotelians (e.g. Leibniz and, to some extent, Bolzano)and some neo-Platonists (e.g. Wallis and, to some extent, Bolzano), includingmany working mathematicians (those mentioned plus, for example, Rotaand Ingham), it represents (or has represented) an epistemic ideal, a viewconcerning the qualitative type of knowledge that proof ought, or at least mightideally provide. According to this view, proof is at its best when it providesknowledge of the most basic reasons why the proposition proved is true.For others, including many working mathematicians, it has been more astrategic or pragmatic ideal, albeit one serving epistemic ends. By this I meanthat it has not so much been prized for the knowledge it itself constitutes, asforthe knowledge it in some broadly pragmatic sense provides for.In this latter connection, it has been characterized as ‘good discipline forthe mind’ (cf. Dieudonné, 1969, 12), a mental training that increases theprover’s potential for future epistemic development. It has also been saidto improve efficiency in classroom learning by decreasing the time and theconceptual distance that separates the definition of terms needed to understanda proposition from the demonstration of that proposition (cf. fns. 22, 27, andthe quote from Bolzano (1804, 172) onp.186). Finally, it has been thought toimprove the epistemic efficiency of a community by making better use of theway(s) in which it divides labor. Pure proofs put theorems at the disposal ofthose who, in terms of their training and expertise, are in the best position touse them to develop further knowledge of the concepts involved.Even for those who take a basically pragmatic view of the value of purity,it is therefore generally true that they see it as serving epistemic ends, whetherof increased extent or of improved efficiency.BibliographyArchimedes (2002), The Method of Archimedes, in The Works of Archimedes, ed.Sir Thomas Heath (Mineola, NY: Dover).Aristotle (anc1), Posterior Analytics, Oxford Translations, ed. W. D. Ross and J. A.Smith, trans. G. R. G. Mure (Oxford: OUP) 1908–1954.(anc2), Posterior Analytics, The Complete Works of Aristotle, vol.I.RevisedOxfordTranslation, ed. J. Barnes (Oxford: OUP) 1984.