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

184 michael detlefsencases, and regarded this avoidance a merit? ... if one considers that the proofsof the science should not merely be confirmations (Gewissmachungen), but ratherjustifications (Begründungen), i.e. presentation of the objective reason for the truthconcerned, then it is self-evident that the strictly scientific proof, or the objectivereason, of a truth which holds equally for all quantities, whether in space or not,cannot possibly lie in a truth which holds merely for quantities which are in space.Bolzano (1817a, 228)By eliminating such appeals to geometric truths, Bolzano’s proof was, as hesaw it, more than ‘a mere confirmation (blosse Gewissmachung)’ (Bolzano, 1810,233) of the intermediate value theorem. It not only made that theorem certain,it gave its ‘objective justification’ (objektive Begründung) (loc. cit.). In so doing, itachieved the ideal of ‘genuinely scientific’ (echt wissenschaftlich) (loc. cit.) proof.Bolzano’s determination to free analysis of impure geometrical influencesrecalls similar earlier themes of Descartes and Wallis. Wallis, for example,defended his use of algebraic methods in geometry on grounds of superiorobjectivity. This objectivity was, he believed, borne of the fact that some ofthe properties of geometrical figures (specifically, in Wallis’ view, propertiesconcerning their rectification, quadrature, and cubature) reflect properties thatthe figures have in themselves, so to say, and independently of how they mightbe constructed (cf. Wallis, 1685, 291, 292–293, 298–299). In Wallis’ view,then, objectivity in geometry actually required the use of algebraic methods.Bolzano held an in some ways similar view, arguing that the persistentmisuse of geometrical reasoning in analysis was due to a lingering confusionbetween theoretical and practical mathematics and the associated mistake oftaking a mathematical object to be real (wirklich) only to the extent that it wasconstructible by certain means (cf. Bolzano, 1810, 218). In this respect, then,he was more neo-Platonist than neo-Aristotelian.Bolzano also pointed out (cf. Bolzano, 1817a, 227, 232) similar misgivingsin Gauss, misgivings which Gauss directed at the use of geometrical reasoningin his first (1799) proof of the fundamental theorem of algebra.⁹ Gauss laterexpressed similar concerns regarding his geometrical interpretation of thecomplex numbers.The representation (Darstellung) of the imaginary quantities as relations of pointsin the plane is not so much their essence (Wesen) itself, which must be graspedin a higher and more general way, as it is for us humans the purest or perhaps auniquely and completely pure example of their application.Gauss (1870–1927, vol.X,106)¹⁰⁹ Gauss (1799). ¹⁰ Letter to Moritz Drobisch, 14 August 1834.