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

purity as an ideal of proof 191advantage is enhanced if a solution is given in terms of concepts most familiarto her, and that (iii) the purer the proof the more it conforms to this type ofsolution. Viewed this way, purity is a pragmatic virtue, albeit one which servesepistemic ends (namely, the effective utilization of knowledge to produce moreknowledge). It does not itself constitute an epistemic virtue; a pure proof isnot, sheerly by dint of its purity, a better justification of the theorem it proves.It is, however, a more effective instrument for gaining further knowledge.We see here, then, a suggestion of the idea that purity generally increases theeffectiveness of divisions of epistemic labor based on specialization.²²Purity is thus a common concern among combinatorialists. It is not, however,unique to them. Indeed, it extends across a wide range of mathematical fields,and has been of concern both to researchers and educators throughout the20th century.A simple illustration is the search for a pure proof of the Erdös–Mordelltheorem. (Let ABC be a triangle and P be a point in its interior. Constructperpendiculars from P to each of the sides, letting the points of intersectionbe A ′ , B ′ , C ′ , respectively. The Erdös–Mordell theorem says that PA + PB +PC ≥ 2(PA ′ + PB ′ + PC ′ ).) Finding an elementary proof of this theorem wasa going concern in the middle decades of the 20th century, one that ended insuccess in the late 1950s (cf. Kazarinoff, 1957; Bankoff, 1958).Still other examples include Formanek (1973), Edmonds (1986), Gilmer andMott (1971), and Woo (1971). The first gives a proof of a theorem of algebra(the Eakin–Nagata theorem) the virtue of which is said to be that it uses onlydefinitions of terms contained in the theorem.²³ The second presents a purelytopological proof of a topological theorem and, like the case of Stanton andZeilberger mentioned above, it expresses the hope that such a proof will makethe theorem more ‘accessible’ to specialists.The third concerns a theorem first proved by Abraham Robinson usingmodel-theoretic methods. This theorem, though algebraic in character, was saidby Robinson not to have an accessible purely mathematical proof independentof his model-theoretic approach. The authors challenge this by providing apurely algebraic proof.²⁴I mention this because it points to a larger concern regarding purity,namely, whether pure proofs always exist. This question, in turn, points to²² Recall here the remark of Bolzano’s (from Bolzano, 1804, 172) quoted in Section 2 concerningthe advantages of purity not only for purposes of making education more thorough but of making iteasier and more efficient.²³ The theorem states that if T is a commutative Noetherian ring finitely generated as a moduleover a subring R, thenR is also Noetherian.²⁴ See Friberg (1973, 421) for a different example concerning algebraic purity.