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

186 michael detlefsenbe deduced from them than if they remain confused’ (Bolzano (1804), 172).It improved efficiency because learning is easier when concepts are ‘clear,correct, and connected in the most perfect order’ (loc. cit.). We must therefore... regard the endeavour of unfolding all truths of mathematics down to theirultimate grounds, and thereby providing all concepts of this science with thegreatest possible clarity, correctness, and order, as an endeavour which will notonly promote the thoroughness of education but also make it easier.Bolzano (1804, 172)The discipline of purity thus promised to increase the extent of ourknowledge while at the same time increasing our capacity to extend it further.We are thus to imitate Thales. That is, we are to look ‘inside’ theses, to analyzetheir concepts into constituent concepts and to excavate the basic truths thatconcern these sub-concepts. We are not to make use of conceptual resourcesoutside those of the given thesis. Bolzano’s basic recommendation was thusthat we should not rest content with a proof...if it is not ... derived from concepts which the thesis to be proved contains, butrather makes use of some fortuitous alien, instrumental concept (Mittelbegriff)¹⁴,which is always an erroneous metabasis eis allo genos.Bolzano (1804, 173)All in all, then, Bolzano saw purity as serving a variety of epistemic ends. First,by forcing us to look behind the ‘obvious’, it revealed hitherto unnoticed truths,the revelation of which increased the extent of our knowledge. Secondly, itpromised to improve efficiency through the clarification of the concepts thatappear in a thesis. Concepts are easier to grasp (and therefore easier to develop)the clearer they and their connections with other concepts are. Thirdly, purityprotected against the circularities, and the attendant epistemic futility, thatmarked so many of the proofs of 18th and 19th century analysis and traditionalelementary geometry.Later 19th century mathematicians also acknowledged the need, or at leastthe virtue of purity. They acknowledged, in particular, the importance ofpurging geometrical reasoning from analysis. Moreover, their reasons were,if anything, even more pressing than Bolzano’s. For while Bolzano acceptedthe reliability of ordinary geometrical reasoning (at least in his early writings),later thinkers did not, or at least not generally. Weierstrass’ example of aneverywhere continuous but nowhere differentiable function cast doubt on¹⁴ The translation in the Ewald volume by Stephen Russ renders ‘Mittelbegriff’ as ‘intermediateconcept’. I think it is more in keeping with Bolzano’s ideas to take ‘Mittel’ in the sense of a ‘means’ or‘instrument’.