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

what structuralism achieves 363Even existence proofs usually specify elements only ‘in principle.’ Lang neveractually spells out in ZF detail what a polynomial is.So far the pattern or category of rings and morphisms is defined by a priorset theoretic definition of rings and morphisms. But standard tools can alsodescribe the pattern simply as a pattern ignoring, as Resnik says, the ‘internal’nature of rings and morphisms.¹³ Then a ring A is just a place in the pattern sothat, depending on the choice of foundations, A either has no elements in a settheoretic sense or it has them but they are unspecified and irrelevant. Eitherway morphisms are not functions defined on set theoretic elements. They arejust morphisms or more pictorially arrows in the pattern. The pattern includes aplace or places identifiable up to isomorphism as the ring Z of integers, anda place or places identifiable up to isomorphism as the ring Z[X] of integerpolynomials. Fact 1 has an analogue with Z in place of the real numbers Rwhich says we can define ‘elements’ of any ring A in a pattern theoretic senseto be the morphisms Z[X] → A. These correspond exactly to the classical settheoretic elements so that all the classical axioms on elements are verified, andthus all the classical theorems.The pattern of rings and ring morphisms, i.e. the category Ring, canbedefined in turn as a position in the pattern of categories and functors i.e. in thecategory of categories. Each ring A appearsasafunctorA : 1 → Ring where1istheterminal category. Ring morphisms f appear as functors f : 2 → Ringwhere 2 is the arrow category (Lawvere, 1966). Ring elements can still be definedby morphisms as in the previous paragraph. All the classical theorems on rings,elements, and morphisms reappear at this level. There is no obstacle to takingany one of these levels as basic with the others derived from it.13.3 The unifying spirit13.3.1 A few uses of patternsThese structural methods have deep roots in practice. As a notable example,topology by the 1920s relied on a booming but chaotic notion of homology.There were many versions.¹⁴ Topologists assumed that results proved for oneversion would normally hold for all—but they could not say just what was‘normal’ here. Some versions of homology were later proved exactly equivalent¹³ Universal algebra defines the category up to isomorphism (Lawvere, 1963). Monads define it noteven up to isomorphism but up to equivalence (Mac Lane, 1998, Ch.6).¹⁴ e.g. the Čech, Vietoris, and singular theories of Chapter 8 in Hocking and Young (1961) allexisted before the unifying category theoretic language of Chapter 7.