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

188 michael detlefsenPurity was thus widely accepted as an ideal of proof in 19th centurymathematics. This notwithstanding, it was not without its detractors. Theseincluded such eminences as Hilbert and Klein. Hilbert viewed purity as a‘subjective’ ideal (cf. Hilbert, 1899, 106–107).¹⁸ The important thing, he said,is that we learn as much as we can about the different means by which a giventheorem or body of theorems can be proved. Pure proofs—that is, proofsthat concern themselves only with the concepts contained in the theoremsproved—are therefore part of what we want to know. They are, however,only part and are not inherently more interesting or valuable than proofs thatdraw upon other conceptual resources.Klein believed that much of mathematics, including much of analysis,ought in many instances to be confirmed by geometrical intuition. In addition,he believed that without ‘constant use’ of geometrical intuition theproof of some of the most interesting and important analytic results concerningcontinuity would be impossible, at least practically speaking (cf. Klein,1894, 45).He also challenged purity in the other direction. That is, he proposed asystem of axioms for geometry that made use of analytic ideas and methods,and, generally speaking, he recommended this system for its efficiency. At thesame time, he decried the resistance of geometers to this simple approach, aresistance he also attributed to traditional ideas of purity, in this case thoseembodied in the retrograde refusal to use numbers in geometry (cf. Klein,1925, 160).Promoting efficiency as a more compelling ideal than purity was hardly a newattitude, however. It was, indeed, at the core of 17th, 18th, and 19th centurydebates concerning the use of algebra in geometry (cf. Wallis, 1685, 117–118,305–306;MacLaurin,1742, 37–50) and, relatedly, the division of mathematicalreasoning into methods of discovery and methods of demonstration.7.3 Purity as a contemporary idealPurity has thus come into conflict with other ideals of mathematical reasoning,conflicts which, at best, have been only partially resolved. This unresolved¹⁸ The subdivision of axioms in Hilbert’s Grundlagen bespeaks a moderate concern for purity, andone that’s reflected in the larger organization of the book. Thus, for example, Ch. I, §4 is entitled‘Consequences of the axioms of incidence and order’ and Ch. I, §6 ‘Consequences of the axioms ofcongruence’. His intention, he said, was to deduce the most important theorems in such a way as tobring to light ‘the meaning of the various groups of axioms, as well as the significance of the conclusionsthat can be drawn from the individual axioms’ (Hilbert, 1899, 2).