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

362 colin mclartymorphism with vu(X) = X. But the identity 1 R[X] also takes X to X, souniqueness implies vu = 1 R[X] . Similarly uv = 1 R[X ′ ].□Lang deduces everything he needs about R[X] fromFact1 plus propertiesof the real numbers. A ZF set theoretic definition of ‘polynomial’ is ratherbeside the point—except that we still need to prove:Theorem 4. There are rings with the properties given in Fact 1.Here the foundations matter. This theorem can be proved by specifying a setR[X] with these properties, or from more general theorems. Either way, onZF foundations, it will come down to specifying a set by specifying which setsare its elements. On ZF foundations to specify R[X]istospecifywhichsetsarepolynomials. Proofs in ZF cannot all be structural because the axioms are notstructural. Proofs in categorical set theory are entirely structural. Categoricalset theory specifies any set, including any candidate for R[X], only up toisomorphism entirely by the pattern of functions between it and other sets(McLarty, 2004).Lang (1993) does not specify a set theory. In particular he nowhere sayswhat functions are. He only says they take values. He calls them ‘mappings’and his only account of them reads:If f : A → B is a mapping of one set into another, we writex ↦→ f (x)to denote the effect of f on an element x of A. (Lang,1993, p.ix)ZF set theory formalizes this by taking elementhood as primitive and defining afunction f as a suitable set of ordered pairs, and defining f (x) = y to mean theordered pair 〈x, y〉 is an element of the set f . Categorical set theory formalizesit by taking function composition as primitive and defining elements as suitablefunctions, and defining f (x) = y to mean y is the composite of f and x.InResnik’swords this set theory treats functions ‘as positions in the pattern generated by thecomposition relation’ (1997,p.218). Either approach will work for Lang.The point is that structural practice isolates foundations in a few proofs,namely existence proofs as for Theorem 4. Definitions, theorems, and the greatmajority of proofs proceed rigorously without ever specifying the elements ofany set at all. So it does not matter to them if elements are:• other sets, as in ZF set theory.• global elements x :1→ S as in categorical set theory.• entities lower in type by 1 than the entity S as in Russell’s type theory.• etc.