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

360 colin mclartydefinitions of real and complex numbers neither imply nor preclude anyidentity relation between them until we stipulate one.⁹ But on this view thereare no ‘systems’ in Lang, or in Conway and Smith, or nearly anywhere outsideof textbooks on ZF set theory. It remains to understand structuralist practicein terms of ‘structures’ themselves.13.2 Working structuralismPatterns themselves are positionalized by being identified with positions of anotherpattern, which allows us to obtain results about patterns which were not evenpreviously statable. It is [this] sort of reduction which has significantly changedthe practice of mathematics. (Resnik, 1997, p.218).One good example is the structure, specifically a ring R[X], of all realpolynomials. Lang says there are ‘several devices’ for reducing polynomials tosets and suggests one for undergraduates and another for graduate students.¹⁰What does not change are the rules for adding, subtracting, and multiplyingpolynomials. To collect the real polynomials into one structure Lang turns torings and ring morphisms. A commutative ring is a structure with 0,1, addition,subtraction, and multiplication following the familiar formal rules.¹¹ A ringmorphism f : A → B is a function which preserves 0,1, and the ring operations:f (0) = 0 f (x − y) = f (x) − f (y) etc.We will say ‘ring’ to mean commutative ring. Lang characterizes R[X]:Fact 1. R[X] is a ring with element X ∈ R[X] and a morphism c : R → R[X] calledthe insertion of constants.¹² For each ring morphism f : R → A and element a ∈ A, thereis a unique morphism u a : R[X] → Ac[X]u aXf A awith u a (X) = a and agreeing with f on constants—that is u a c = f .⁹ Shapiro’s notion of offices does not bear on embeddings like this since it only connects isomorphicstructures (Shapiro, 1997, p.82).¹⁰ The quote is in both Lang (2005, p.106) and Lang (1993, p.97).¹¹ See Mac Lane (1986, p.98) or any modern algebra textbook.¹² Is a real number the same thing as a constant real polynomial? Lang identifies them, although aZF reading of either of his definitions of polynomials says no.