Purity of Methods

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 different from the translation there. which apparently established such a dependence should seem to obscure the true state of affairs. The task of deriving what was arithmetical by purely arithmetical means, i.e. purely logically, could not be put off.16 In expressing his concern that proofs reveal “the true state of affairs", Frege echoed Bolzano. Both seem to have believed in some type of objective ordering of mathematical truths a distinct exposition (or at least reflection) of which was proof’s highest calling. Purity remained a guiding ideal of twentieth-century thinking as well. An example is the search for an “elementary" proof of the prime number theorem, which states that the prime number theorem states that the number of primes up to an integer n is approximately n log n . The prime number theorem was first proved by Hadamard and de la Vallée Poussin in 1896. Because their proofs used methods from complex analysis, however, they were widely regarded as imperfect.17 In 1949, Paul Erdös18 and Atle Selberg19 found proofs that avoided such methods. This discovery was believed to be of such importance as to contribute significantly to Selberg’s earning a Fields Medal. But though Selberg and Erdös pursued and valued purity, they offered neither a clear general characterization of it nor a statement of why it should be valued. The Bourbakiste Jean Dieudonné did better. . . . an aspect of modern mathematics which is in a way complementary to its unifying tendencies . . . concerns its capacity for 16. Gottlob Frege, “Formal Theories of Arithmetic,” in Brian McGuinness (ed.), Collected papers on mathematics, logic, and philosophy (Blackwell, 1984), pp. 116–117 17. Cf. A.E. Ingham, The distribution of prime numbers (Cambridge University Press, 1932), p. 5, and a remark from G.H. Hardy quoted in Melvyn B. Nathanson, Elementary methods in number theory (New York: SpringerVerlag, 2000), p. 320. 18. Paul Erdős, “On a new method in elementary number theory which leads to an elementary proof of the prime number theorem,” Proceedings of the National Academy of Sciences, USA 35 (1949) 19. Atle Selberg, “An elementary proof of the prime-number theorem,” Annals of Mathematics (2) 50 (1949) philosophers’ imprint 5 vol. 11, no. 2 (january 2011) detlefsen, arana Purity Of Methods sorting out features which have become unduly entangled.. . . It may well be that some will find this insistence on “purity" of the various lines of reasoning rather superfluous and pedantic; for my part, I feel that one must always try to understand what one is doing as well as one can and that it is good discipline for the mind to seek not only economy of means in working procedures but also to adapt hypotheses as closely to conclusions as is possible.20 Dieudonné evidently saw “closeness” or “proximity” of topic between the premises and conclusion of a proof as an important quality of it. He seems also to have believed that working to maximize this closeness typically furthers the intellectual development of the prover.21 Here we see appreciation not of the epistemic value of purity, but of what, following traditional usage, we might call its intervenient value. In taking such a view, Dieudonné joins a long tradition of thinkers who have praised the study of mathematics as a means of improving capacity for reasoning. A well-known exponent of this view, Francis Bacon, memorably compared the benefits of doing of pure mathematics to those of playing tennis. In the Mathematics I can report no deficience, except it be that men do not sufficiently understand the excellent use of the Pure Mathematics, in that they do remedy and cure many defects in the wit and faculties intellectual. For if the wit be too dull, they sharpen it; if too wandering, they fix it; if too inherent in the sense, they abstract it. So that as tennis is a game of no use in itself, but of great use in respect it maketh a quick eye and a body ready to put itself into all postures; so in the Mathematics, 20. Jean Dieudonné, Linear algebra and geometry (Boston, Mass.: Houghton Mifflin Co., 1969), p. 11 21. Hans Freudenthal, “Review of Dieudonné, Algèbre linéaire et géométrie élémentaire,” American Mathematical Monthly 74/6 (1967), pp. 744–748 criticizes Dieudonné on these points. that use which is collateral and intervenient is no less worthy than that which is principal and intended.22 Many other thinkers have made similar claims.23 Among these was Bolzano, who wrote: It is quite well known that in addition to the widespread use which its application to practical life yields, mathematics also has a second use which, while not so obvious, is no less useful. This is the exercise and sharpening of the mind: the beneficial development of a thorough way of thinking.24 We call attention to this view of purity as an intervenient or developmental virtue in order to distinguish it from the treatment of purity we offer here. Intervenient value is broadly pragmatic in character. We are interested, by contrast, in the distinctively epistemic or justificative value of purity, one variety of which we will discuss in §§3, 4. For us, then, the question is: “What is the epistemic value of purity?” Here there is relatively little to go on in the literature. The little there is we will now briefly survey. In his 1900 Problems address, Hilbert made a statement that suggested that he saw purity as an epistemic virtue. Specifically, he claimed that in solving a mathematical problem we ought to stay as close as possible to the conceptual resources used in stating, or, perhaps better, in understanding that problem. As he put it: It remains to discuss briefly what general requirements may be justly laid down for the solution of a mathematical problem. I should say first of all, this: that it shall be possible to establish 22. Francis Bacon, The two bookes of Francis Bacon. Of the proficience and advancement of learning, divine and humane (London: Henrie Tomes, 1605), Bk. 2, VIII, 2, emphasis added 23. See e.g. the anonymously written dedication to Nicholas Saunderson, The elements of algebra (Cambridge University Press, 1740), xiv–xv. 24. Bernard Bolzano, “Preface to Considerations on some Objects of Elementary Geometry,” in William Ewald (ed.), From Kant to Hilbert (Oxford University Press, 1999), p. 172 philosophers’ imprint 6 vol. 11, no. 2 (january 2011) detlefsen, arana Purity Of Methods the correctness of the solution by means of a finite number of steps based upon a finite number of hypotheses which lie in the presentation of the problem (in der Problemstellung liegen) and which must always be exactly formulated.25 Unfortunately, Hilbert did not say more clearly what he meant by an hypothesis’ “lying in” the definition or presentation of a problem. Nor did he indicate why it might be requisite, or at least gainful, to restrict solutions to a given problem to such hypotheses.26 Elsewhere he made it clear that his endorsement of purity was qualified.27 Indeed, in the conclusion of the Grundlagen der Geometrie, written at approximately the same time as the Problems address, he characterized purity as a subjective constraint—specifically, a subjective form of the so-called “Grundsatz” of the Grundlagen,28 which Hilbert presented as follows: . . . to consider each presented question in such a way as to examine . . . whether or not it is possible to answer it by following 25. David Hilbert, “Mathematische Probleme,” Archiv der Mathematik und Physik (3rd series) 1 (1901), p. 257, emphasis added 26. For more on Hilbert’s views concerning purity, cf. Michael Hallett, “Reflections on the Purity of Method in Hilbert’s Grundlagen der Geometrie,” in Paolo Mancosu (ed.), The Philosophy of Mathematical Practice (Oxford University Press, 2008) and Michael Detlefsen, “Purity as an Ideal of Proof,” in Paolo Mancosu (ed.), The Philosophy of Mathematical Practice (Oxford University Press, 2008), p. 188. 27. This is worth noting if only because of the contrary suggestions made by various of Hilbert’s interpreters, chiefly Kreisel; cf. Georg Kreisel, “Kurt Gödel,” Biographical Memoirs of Fellows of the Royal Society 26 (1980), 150, 163, 167; Georg Kreisel, “Luitzen Egbertus Jan Brouwer: 1881–1966,” Biographical Memoirs of Fellows of the Royal Society 18 (1969), 60; Georg Kreisel, “Frege’s Foundations and Intuitionistic Logic,” The Monist 67 (1984), 74–75. 28. In classifying preference for pure proof as subjective Hilbert seems to have had something like the following in mind: we generally stand to learn as much from impure proof as from pure; therefore, a systematic preference for pure proof is unjustified (and in that sense “subjective”). out a previously determined method and by employing certain limited means.29 The pre-determined or limited means mentioned might represent a restriction to pure methods, but it need not represent only that. Restriction to pure proof is only one type of restriction. And though it may offer something to learn, so do other types of restrictions. That there is more to be learned from a restriction to pure methods than from other types of restrictions is an unwarranted assumption. Accordingly, a systematic preference for pure proof is likewise unwarranted. This at any rate is what Hilbert seems to have been asserting in the passage from the Grundlagen just noted. Purity has retained a following as an ideal of proof to the present day. Gel’fond and Linnik, for example, see it as representing a “natural desire” to attain elementary solutions to elementary problems. There can be no question of renouncing transcendental methods in modern number theory. Yet the investigator feels a natural desire to seek more arithmetic approaches to the solution of problems which can be stated in elementary terms. Apart from the obvious methodological value of such an approach, it is important by reason of giving a simple and natural insight into the theorems obtained and the causes underlying their existence.30 There are other cases we might mention as well.31 We trust, though, that we have said enough to motivate our interest in purity. In the remaining sections, we will attempt to say more clearly what purity is and how it might function as an epistemic virtue. We will also present and discuss some examples. 29. David Hilbert, Foundations of Geometry, second edition (La Salle, IL: Open Court, 1950), p. 82 30. A. O. Gel’fond and Yu. V. Linnik, Elementary Methods in the Analytic Theory of Numbers (Cambridge, MA: MIT Press, 1966), p. ix, emphasis added 31. Cf. Jean Dieudonné, History of Algebraic Geometry (Monterey, CA: Wadsworth, 1985), p. 9, for another recent example. Additionally, a 2002 philosophers’ imprint 7 vol. 11, no. 2 (january 2011) detlefsen, arana Purity Of Methods 3. Specific Ignorance & Its Relief One important goal of epistemic development is to reduce ignorance of which an investigator is aware and relief of which she in some important sense pursues. Among the different types of ignorance investigators may seek to alleviate is one we call “specific ignorance.” This is ignorance of solutions to specific problems of which we are aware and whose solution we desire.32 How, exactly, relief of ignorance ought to be thought of, and how, so conceived, it and its relief ought best to be measured are subtle and complicated matters. In this paper we operate from a basic but only partially developed conception of ignorance and its relief, one which sees specific ignorance as capable of relief even in cases where global ignorance (measured, for example, by the number of problems we do not know how to solve, or by the proportion of problems we know to which we have solutions) may not be. We should at this point, say a few words concerning the terminology of “relief.” We say “relief” rather than “reduction”, “decrease” or issue of the American Mathematics Monthly endorses purity as a worthy pursuit of undergraduate instruction, as follows. Problem 10830. Proposed by Floor van Lamoen, Goes, The Netherlands. A triangle is divided by its three medians into 6 smaller triangles. Show that the circumcenters of these smaller triangles lie on a circle. Editorial comment. The submitted solutions used analytic geometry (or complex numbers) and involved lengthy computations (some done with Maple or Mathematica). The editors felt that a coordinate-free statement deserves a coordinate-free solution; such a solution may shed more light on why the result is true. (cf. Gerald A. Edgar, Doug Hensley and Douglas B. West, “Problem 10830,” American Mathematical Monthly 109/4 (2002), pp. 396–7) As with Dieudonné, the editors quoted here justify their preference of purity because of their explanatory value. We will not pursue this suggestion in this paper, not least because the concept of explanation in mathematics remains largely undeveloped (cf. Paolo Mancosu, “Mathematical Explanation: Why It Matters,” in Paolo Mancosu (ed.), The Philosophy of Mathematical Practice (Oxford University Press, 2008)). 32. We are not assuming any specific basis or type of basis for this desire. Specifically, we are not assuming that it has to represent what might be thought of as purely epistemic reasons or motives. something similar because we believe that alleviating a case of specific ignorance may be an epistemic good even if it does not, overall, result in a lasting reduction of the extent of our ignorance.33 At the same time, however, we do not believe that absolutely all ways of eliminating instances of specific ignorance should count as relieving it. Specifically, we would not count the elimination of a case of specific ignorance as relief if application of the means of elimination itself systematically produced further cases of specific ignorance not eliminated by such application. So, even though relief of specific ignorance does not itself necessarily entail reduction, it does require the absence of systematic effects of replenishment. Relief of specific ignorance and pure problem solution are thus joined at the hip in our account. Relating them in this way requires that we make certain assumptions, of course. Specifically, it requires that we assume that (i) knowledge-seekers, or what we will generally refer to as investigators, can be aware of cases of specific ignorance, and that (ii) they are within their rights (i.e. they operate within the purview of their roles as investigators) to seek to relieve it.34 On the view we will now present, the basic epistemic virtue of pure problem solutions is that they are particularly effective means of relieving spe-