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

purity as an ideal of proof 185Later still, and in terms strongly reminiscent of Bolzano’s earlier assessment ofthe usual proofs of the intermediate value theorem, Gauss further observed that... the real content (eigentliche Inhalt) of the entire argumentation belongs to ahigher realm of the general, abstract theory of quantity (abstracten Grössenlehre),independent of the spatial. The object of this domain, which tracks the continuityof related combinations of quantities, is a domain about which little, to date, hasbeen established and in which one cannot maneuver much without leaning on alanguage of spatial pictures (räumlichen Bildern).Gauss (1870–1927, vol. III, 79)¹¹Like Bolzano, then, Gauss too was concerned for purity in analysis. Lessclear is whether he shared Bolzano’s views concerning the historical reasonsfor impurity and the circularities of reasoning he took it to represent.¹²In truth, though, avoidance of circularity was not Bolzano’s only reason foradvocating purity. Indeed, it was not his principal reason. More fundamentalwas his acceptance of the traditional distinction between two types of reasoningin mathematics and in science generally. These were (i) confirmatory reasoning,or reasoning which convinces that (what Bolzano termed Gewissmachung),and (ii) reasoning which reveals the objective reasons for truth (objektiveBegründung).¹³ The former, he observed, falls short of the latter: ‘the obviousness(Evidenz) of a proposition does not absolve me of the obligation to look for aproof of it’ (Bolzano, 1804, 172).Bolzano pointed to the earliest instances of proof as his inspiration. Thales,he said, did not settle for knowledge that the angles at the base of an isosceles triangleare equal, though this was doubtlessly evident to him. Rather, he pressedon to understand why. In so doing, he was rewarded by an extension of hisknowledge. Specifically, he obtained knowledge of those truths that implicitlyunderlay common-sense belief in the theorem. Since these were ‘new truthswhich were not clear to common sense’ (op. cit., 173), his knowledge wasthereby extended.In addition, it promoted the further extension of his knowledge, both byincreasing its reach and by improving its efficiency. It increased its reachbecause ‘if ... first ideas are clearly and correctly grasped then much more can¹¹ This is from the Jubiläumschrift of 1849, where Gauss was commenting on the use of geometricalmethods to prove the existence of roots of equations.¹² Also problematic is how to harmonize Gauss’ view of the impurity caused by the use of geometricalreasoning in analysis with the contrast he drew between the created character of number and the objectivecharacter of space (cf. 1817 letter to Olbers, Gauss (1976), 651–652; 1829 letter to Bessel, Gauss(1870–1927), vol. VIII, 200; Gauss(1832), 313, fn.1).¹³ See Cicero (anc., 459–460); Ramus (1574, 17, 71, 93–94); Viéte (1983, 28–29); Arnauld (1662,302); Wallis (1685, 3, 290, 305–306), among others, for earlier statements of this distinction.