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

182 michael detlefsenThe objective grounding relationship induced an objective (partial) orderingof truths, and it was this ordering that Leibniz was interested in.... we are not concerned here with the sequence of our discoveries, which differsfrom one man to another, but with the connection and natural order of truths,whichisalwaysthesame.Leibniz (1764), 412 (Bk. IV, ch. vii, §9)Such an ordering of truths suggests a parallel conception of purity: pureproof is proof which recapitulates a segment of the natural, objective orderingof truths concerning a given subject.In the early years of the 19th century, Bolzano also articulated suchan idea and applied it to the reformation of mathematics generally, andparticularly to analysis.⁵ It comprised, indeed, a prime motive of his earlyattempts to ‘arithmetize’ analysis. This arithmetization, or, perhaps better,this de-geometrization, was wanted in order to combat what Bolzano saw asa pervasive type of circularity in proofs in analysis—a circularity borne ofimpurity.⁶The impurity represented by the importation of geometrical considerationsinto the proofs of genuinely algebraic or analytic theorems had serious consequences.In particular, it inverted the objective ordering of truths. In sodoing, it introduced circularities of reasoning into analysis.⁷ More particularly,a geometrical proof presented a theorem of analysis θ 1 as depending on ageometrical theorem θ 2 when, in fact, the opposite is true.⁵ Bolzano also vigorously pursued reform in geometry, including, perhaps especially, elementarygeometry. In his view, not even a proper theory of triangles and parallels had been given. This wasbecause a proper theory—that is, a properly pure theory—would be based solely on a theory of thestraight line. Yet all past attempts had presupposed axioms of the plane, axioms the proper founding ofwhich would itself require a theory of triangles. Bolzano was thus convinced that... the first theorems of geometry have been proved only per petitionem principii; and even if thiswere not so, the probatio per aliena et remota [proof by alien and remote elements, MD] ... mustnot be allowed.Bolzano (1804, 174)Bolzano’s thinking here is reminiscent of the ancient division of geometrical problems into linear,planar, and solid that Pappus emphasized. Bolzano also opposed the long standing common practiceof appealing to motion in the proof of ‘purely geometrical truths’ (op. cit., 173). See Mancosu (1996,Ch. 1) for a useful discussion of the early modern controversy concerning such appeals.⁶ At approximately the same time, Lagrange pursued a different purification program, one whichurged the liberation of analysis from the kind of dependency on motion encouraged by Newton’sconception of fluxions. He claimed that to introduce motion into a calculus that had only algebraicquantities as its objects was to introduce an alien or extraneous idea (une idée étrangère). Cf. Lagrange(1797, 4).⁷ Not every ‘departure from’ an ordering need be an ‘inversion of’ certain of its elements. Bolzano’sargument seems, however, to require this latter, narrower conception of ‘departing’ from an ordering.