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

7Purity as an Ideal of ProofMICHAEL DETLEFSEN7.1 The Aristotelian ideal of purityThe topic of this chapter—‘purity’ of proof—received its classical treatmentin the writings of Aristotle. It was part of his theory of demonstration, whichwas set out in the Posterior Analytics.Aristotle presented purity as an ideal. Proofs lacking it were not necessarilyworthless, but they did not provide highest or best knowledge of theirconclusions. It was thus a quality of highest or best proof.Aristotle presented the ideal in the form of a prohibition against what hetermed metabasis ex allo genos, or crossing from one genus to another in thecourse of a proof.To sum up, then: demonstrative knowledge must be knowledge of a necessarynexus, and therefore must clearly be obtained through a necessary middle term;otherwise its possessor will know neither the cause nor the fact that his conclusionis a necessary connexion. ...It follows that we cannot in demonstrating pass from one genus to another. Wecannot, for instance, prove geometrical truths by arithmetic.¹Aristotle (anc1), 75a29–75b12The prohibition against crossing from one genus to another in the courseof an argument had various motives. Among them were Aristotle’s oppositionto (i) the Pythagoreans’ reduction of all things to number (cf. Aristotle (anc3),1036b8–21), (ii) the related view that all things ultimately have but a singlebasic Form (loc. cit.), and (iii) Plato’s conception of dialectic as a kind of ‘masterscience’ (cf. Plato (anc2), 55d–59d; Plato (anc3), 533b, c; Aristotle (anc1),¹ In a later passage (cf. Aristotle (anc1), 76a22–25), Aristotle sanctioned the use of metabasis betweena science and its subordinates. Thus, geometrical argument could be used in mechanics and optics, andarithmetical arguments in harmonics.