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

purity as an ideal of proof 187even the reliability of geometrical thinking as regards continuity phenomena.¹⁵Analysts of the latter half of the 19th century thus had, if anything, evenstronger reasons to try to rid analysis of geometrical thinking.An important example is Dedekind, who expressed his commitment topurity this way:In discussing the notion of the approach of a variable magnitude to a fixed limitingvalue ... I had recourse to geometric evidences. Even now such resort to geometricintuition ... I regard as exceedingly useful, from the didactic standpoint ... . Butthat this form of introduction into the differential calculus can make no claim tobeing scientific, no one will deny. For myself this feeling of dissatisfaction was sooverpowering that I made the fixed resolve to ... find a purely arithmetical andperfectly rigorous foundation for the principles of infinitesimal analysis.Dedekind (1872, 1–2)Elsewhere Dedekind stated as the first procedural demand on his theory ofalgebraic integers that ‘arithmetic remain free from intermingling with foreignelements (mélange d’éléments étrangers)’ (Dedekind, 1877, 269). This meant forhim that ‘the definition ... of irrational number ought to be based on phenomenaone can already define clearly in the domain R of rational numbers’ (loc. cit.).Dedekind’s fellow logicist, Frege, also accepted purity as an ideal. Indeed,he described it as a principal motive for his logicist program, one which set itapart from the misbegotten attempt by Gauss to found complex arithmetic ongeometry.¹⁶The overcoming of this reluctance [to accept complex numbers, MD] wasfacilitated by geometrical interpretations; but with these, something foreign wasintroduced into arithmetic. Inevitably there arose the desire of ... extruding thesegeometrical aspects. It appeared contrary to all reason that purely arithmeticaltheorems should rest on geometrical axioms; and it was inevitable that proofswhich apparently established such a dependence should seem to obscure the truestate of affairs. The task of deriving what was arithmetical by purely arithmeticalmeans, i.e. purely logically, could not be put off.Frege (1885, 116–117)¹⁷¹⁵ Bolzano himself discovered a function of this type in 1834. The work in which this was done(namely, Bolzano, 1834, §75), however, was not published until 1930.¹⁶ As noted above, Gauss himself conceded the impurity of his geometrical interpretation in hislater writings. In his 1831 essay, however, he credited it with providing a sensible representation ofthe complexes ‘that leaves nothing to be desired’ (Gauss, 1832, 311). ‘[O]ne needs’, he said, ‘nothingfurther to bring this quantity [i.e. √ −1, MD] into the domain of the objects of arithmetic’ (Gauss,1832, 313). It should perhaps also be noted here that, towards the end of his life, Frege returned to theidea of a geometrical foundation for arithmetic.¹⁷ See Frege (1884, §§103–104) for similar statements.