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

purity as an ideal of proof 183Analysis, or the general theory of quantity, was more basic than geometry,the science of spatial quantities. In many instances, then, a proper (i.e. aproperly pure) proof of a geometrical theorem θ 2 would derive from a morebasic, general theorem of quantity θ 1 . In such cases, an impure proof of themore general θ 1 from the more particular θ 2 would, in effect, amount to acircular grounding of θ 2 on itself.Bolzano was thus convinced that, in analysis, ‘geometrical proof is, ... inmost cases, really circular’ (Bolzano, 1817a, 228). He therefore set out topurge analysis of its geometrical contaminations. His best known work in thisconnection is his purely analytic proof (rein analytischer Beweis)oftheintermediatevalue theorem—the theorem that for a real function f continuous on a closedbounded interval [a, b], for every µ, f (a)