ly considered from such construction; and which doth accompany it though otherwise constructed than is supposed.7 Wallis thus saw geometrical knowledge as more concerned with construction-invariant features of geometrical figures than with features deriving from their construction. In addition, he saw algebraic 5. Idem, p. 230 6. Knowing via minimal or simplest cause would have been better still, of course, since it could have been expected to distill a kind of “pure” cause by separating what is essential to the construction of a figure from what is accidental. 7. John Wallis, A Treatise of Algebra, both historical and practical: shewing the original, progress, and advancement thereof, from time to time, and by what steps it hath attained to the height at which it now is (London: John Playford, 1685), p. 291 methods as offering striking gains in simplification over their classical counterparts. Here his views seem directly opposed to Newton’s. In particular, Wallis was not persuaded that the only type of simplicity relevant to geometrical reasoning was what Newton described as “the more simple drawing of Lines” (loc. cit.). As he saw it, the use of algebraic methods commonly afforded more efficient discovery of convincing reasons for (as distinct from proper proofs or demonstrations of) geometrical truths, and in his view, the “official” preferences of traditional geometers had not properly reflected the value of this efficiency.8 Newton’s mathematical interpreter, Colin MacLaurin, followed the lead of his (Newton’s) philosophical interpreter Locke in cautioning against the uncritical use of Wallis’ infinitistic algebraic methods. He did this, however, while acknowledging their efficiency.9 The appeal to the simplicity of classical methods thus seems to have been weaker in him than in Newton. Mr. Lock . . . observes, “that whilst men talk and dispute of infinite magnitudes [where MacLaurin has ‘magnitudes’, Locke had ‘space or duration’], as if they had as compleat and positive ideas of them as they have of the names they use for them, or as they have of a yard, or an hour, or any other determinate quantity, it is no wonder if the incomprehensible nature of the thing they discourse of, or reason about, leads them into perplexities and contradictions . . . ” Mathematicians indeed abridge their computations by the supposition of infinites; but when they pretend to treat them on a level with finite quantities, they are sometimes led into such doctrines as verify the observation of 8. Wallis’ preference is therefore not well described as simply a preference for the simpler over the less simple. Rather, it was a preference for a particular type of simplicity—discovermental simplicity—that he believed had been traditionally undervalued. 9. MacLaurin expressly cited Wallis’ arguments in John Wallis, Arithmetica Infinitorum (Oxford: Tho. Robinson, 1656) as examples of the type he had in mind (cf. Colin MacLaurin, A Treatise of Fluxions (Ruddimans, 1742), p. 48). philosophers’ imprint 3 vol. 11, no. 2 (january 2011) detlefsen, arana Purity Of Methods this judicious author. . . . These suppositions [suppositions concerning the infinite] however may be of use, when employed with caution, for abridging computations in the investigation of theorems, or even of proving them where a scrupulous exactness is not required . . . Geometricians cannot be too scrupulous in admitting of infinites, of which our ideas are so imperfect.10 MacLaurin thus repeated the common observation that algebraic methods did not have the same value for demonstrating truths that traditional, constructional methods had. Wallis believed this to be compatible with their (i.e. algebraic methods’) having great value as instruments of discovery or investigation. MacLaurin conceded and even repeated this point. Properly controlled, algebraic methods had distinct value as methods of investigation. It was important, however, not to conflate methods of investigation with methods of demonstration and thus to overlook the limitations of algebraic methods as methods of demonstration. Such at any rate were MacLaurin’s views.11 Despite Newton’s reservations concerning algebraic methods, mathematicians of the eighteenth century and later generally followed Descartes and Wallis in sanctioning the relatively free use of “impure” algebraic methods in geometry.12 This should not be taken to suggest, 10. MacLaurin, op. cit., pp. 45–47 11. A reviewer rightly noted that despite the fact that MacLaurin was Newton’s interpreter, the opposition between his (MacLaurin’s) and Wallis’ views is not the same as that between Newton’s and Wallis’. Newton’s opposition to the use of algebraic methods in geometry was on grounds of purity. MacLaurin’s was primarily on grounds of confidence or security. In fact, the fuller truth is more complicated still. As we will see shortly, Wallis’ preference for algebraic methods can be seen as partially based on considerations of purity. He believed that geometry was (or ought to be) about certain construction-transcendant invariances, and that algebraic methods more purely reflect these invariances than do traditional geometrical methods. In addition, it is important to bear in mind that though Newton “officially” preferred geometrical to algebraic methods in geometrical investigations, in practice he made extensive use of the latter. 12. This was truer of mathematicians on the continent than of those in England. For more on the reception of algebraic methods in England, see Helena M. Pycior, Symbols, impossible numbers, and geometric entanglements (Cambridge: however, a general decline in the importance of purity as an ideal of mathematical reasoning. Even (perhaps especially) views like Wallis’ encouraged retention of purity as an ideal of proof. What set these views apart was a different conception of the subject-matter of geometry. It remained geometrical figures, but these were not conceived in the usual way. In particular, they were not seen as essentially tied to characteristic means of construction. Rather, their essential traits were taken to be those which were invariant with regard to the method(s) of construction. Or so it may be argued.13 The distinctive feature of the algebraists was therefore not a move away from purity as an ideal of proof, but a move away from the traditional conception of geometrical objects. Specifically, it was a move away from a view which saw geometrical objects as being given or determined by their (classical) methods of construction, and a move towards a view which saw the essential properties of geometrical figures to be those which were invariant with respect to their means of construction. For Wallis, these were mainly arithmetical or algebraic features. Purity was also an expressly avowed ideal of such later figures as Lagrange, Gauss, Bolzano, von Staudt and Frege. It figured particularly prominently in Bolzano’s search for a purely analytic (i.e. nongeometric) proof of the intermediate value theorem.14, the chief motive behind which he described as follows: [I]t is . . . an intolerable offense against correct method to derive truths of pure (or general) mathematics (i.e. arithmetic, algebra, Cambridge University Press, 1997) who discusses the reception of Newton’s views among British mathematicians, and its effects in slowing the adoption of the new algebraic methods for a century and a half. 13. For more on what was meant by invariance in early modern practice, cf. Michael Detlefsen, “Formalism,” in Stewart Shapiro (ed.), Handbook of the Philosophy of Mathematics and Logic (Oxford University Press, 2005), Section 4.2.1. 14. The intermediate value theorem states that if a real function f is continuous on a closed bounded interval [a, b], and c is between f (a) and f (b), there is an x in the interval [a, b] such that f (x) = c. philosophers’ imprint 4 vol. 11, no. 2 (january 2011) detlefsen, arana Purity Of Methods analysis) from considerations which belong to a merely applied (or special) part, namely, geometry. Indeed, have we not felt and recognized for a long time the incongruity of such metabasis eis allo genos? Have we not already avoided this whenever possible in hundreds of other cases, and regarded this avoidance as a merit? . . . [I]f one considers that the proofs of the science should not merely be certainty-makers [Gewissmachungen], but rather groundings [Begründungen], i.e. presentations of the objective reason for the truth concerned, then it is self-evident that the strictly scientific proof, or the objective reason, of a truth which holds equally for all quantities, whether in space or not, cannot possibly lie in a truth which holds merely for quantities which are in space.15 For Bolzano, then, truly scientific proof was demonstration from objective grounds and this was in keeping with the demands of purity. Frege’s logicist program also called for purity—specifically, the purification of arithmetic from geometry. Even in pre-scientific times, because of the needs of everyday life, positive whole numbers as well as fractional numbers had come to be recognized. Irrational as well as negative numbers were also accepted, albeit with some reluctance—and it was with even greater reluctance that complex numbers were finally introduced. The overcoming of this reluctance was facilitated by geometrical interpretations; but with these, something foreign was introduced into arithmetic. Inevitably there arose the desire of once again extruding these geometrical aspects. It appeared contrary to all reason that purely arithmetical theorems should rest on geometrical axioms; and it was inevitable that proofs 15. Bernard Bolzano, “Purely analytic proof of the theorem that between any two values which give results of opposite sign there lies at least one real root of the equation,” in William Ewald (ed.), From Kant to Hilbert (Oxford University Press, 1999), p. 228; our translation is slightly differ
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