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

what structuralism achieves 361This says nothing to specify what polynomials are beyond the paradigmaticallyuninformative assertion that one of them is called X. ItdoessayR[X]isaring,so there are other elements, like X 2 = X · X,and3X 2 + X, described by theiralgebraic relation to X. Intuitively u a evaluates these at X = a. That is, sinceu a is a ring morphism:u a (X 2 ) = a 2 and u a (3X 2 + X) = 3a 2 + a and so on.But nothing in Fact 1 specifies the elements of R[X] as sets. It specifies whereR[X] fits in the pattern of rings and morphisms.Lang states Fact 1 in italics without calling it a definition or theorem oranything else (Lang, 1993, p. 99). If pressed, he would probably call it atheorem. Indeed either one of his official definitions of polynomials can bemade rigorous and used to prove the Fact. But Lang does not like using themin proofs. Fact 1 itself gives a rigorous definition of R[X] up to isomorphismwhich he does like to use.Philosophers usually define an isomorphism of structured sets as a one-tooneonto function preserving structure in each direction, as in (Bourbaki,1939). Mathematicians today use a simpler and more general definition, whichhowever agrees with Bourbaki’s in this case (Lang, 1993, p.54). For the sakeof generality it speaks of morphisms instead of structure preserving functions: Everystructure S 1 has an identity morphism 1 S1 : S 1 → S 1 .Amorphismf : S 1 → S 2is an isomorphism if it has an inverse, thatisamorphismg : S 2 → S 1 whichcomposes with f to give identities:2fg1 g f1 12 21 11 2The need for generality will appear in Section 13.3.1 and the accompanyingcase study. For now, note how directly this concept of isomorphism suits Fact 1.Theorem 3. Suppose a ring R[X] and function c satisfy Fact 1, and so do anotherR[X ′ ] and c ′ . Then there is a unique ring isomorphism u : R[X] → R[X ′ ] such thatu(X) = X ′ and uc = c ′ .c[X]Xc ′[X′] X′Proof. By assumption there are unique morphisms u with u(X) = X ′ and uc = c ′and u : R[X ′ ] → R[X]withv(X ′ ) = X and vc ′ = c. The composite vu is a ring