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

what structuralism achieves 357that the tools used in textbooks and research are more specific than mostphilosophical proposals and are daily tested for rigor in principle and feasibilityin practice.Section 13.3 gives sample achievements and looks at intuition, purity, andunification. While some philosophical structuralisms may make any unity ofmathematics unintelligible, structuralist practice makes that unity ever moreproductive.13.1 Beginning structuralism13.1.1 Separate structuresFollowing the imagery of Euclid’s Elements, Resnik calls mathematical objectspositions in patterns and says they have no properties except relations toothers in the same pattern. But this ‘threatens to conflict with mathematicalpractice’ (MacBride, 2005 p. 568). If, for example, real numbers have onlyrelations to each other, how do they gain relations to complex numbers? Theanswerisgiveninpractice.Innumerable textbooks say something like:We suppose you understand the real numbers! The complex numbers are formalexpressions x 0 + x 1 i with x 0 , x 1 real, combined by(x 0 + x 1 i) + (y 0 + y 1 i) = (x 0 + y 0 ) + (x 1 + y 1 )i(x 0 + x 1 i)(y 0 + y 1 i) = (x 0 y 0 − x 1 y 1 ) + (x 0 y 1 + x 1 y 0 )i(Conway and Smith, 2003, p.1)If Conway and Smith were Zermelo–Fraenkel set theorists this would meanthey do not embed the real numbers R in the complex numbers C. Arealnumber defined in ZF is not a formal expression made from two real numbersand a letter i. But Conway and Smith are not set theorists. They freelyequate each real number x with the complex number x + 0i. Virtually allmathematicians do. This extends to all uses of the numbers, so real polynomialsare considered a kind of complex polynomial as mentioned in (Carter, 2005,p. 298). Elementary textbooks typically define complex numbers in someway which would imply in ZF set theory that none are real numbers, thenadd explicitly: ‘we identify R with its image in C’ (Lang, 2005, p. 347).For some philosophers ‘the notion of ‘‘identity by fiat’’ makes dubioussense’ (MacBride, 2005, p.578). But mathematics thrives on violating thisdictum.