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

‘there is no ontology here’ 397In the plainest technical sense neither of our definitions of schemes hasset theoretic functions as maps. On the structure sheaf definition a map from(X, O X ) to (Y, O Y ) consists of a continuous function f : X → Y plus manyring morphisms in the opposite direction (in general, infinitely many). For thevariable set definition each map is a natural transformation and thus a structurepreservingproper class of functions. Either way a map is more complex than aset theoretic function.Topology gives another kind of morphism more complex than functions.Continuous functions f , g : S → S ′ between topological spaces are calledhomotopic if each one can be continuously deformed into the other (Mac Lane,1986, p.323). Write f for the homotopy class of f , that is the set of functionsS → S ′ homotopic to f .Thehomotopy category Toph has topological spacesS as objects, and homotopy classes f : S → S ′ as morphisms. The morphismsof Toph are not functions but equivalence classes of functions. By the globalaxiom of choice we can select one representative function from each class.But there is provably no way to select representatives so that the composite ofevery pair of selected representatives is also selected. The categorical structure ofToph cannot be defined by composition of functions but requires compositionof equivalence classes of functions (Freyd, 1970).A case study by Leng (2002) includes an example from functional analysis.C ∗ -algebras originated in quantum mechanics and the paradigms are algebrasof operators on function spaces. The mathematician George Elliott sought atheorem characterizing certain ones. The theorem used inductive limits, wherethe inductive limit of an infinite sequence of C ∗ -algebras and morphismsA 1 A 2 … A n…is a single C ∗ -algebra A ∞ combining all of the A n in a way compatible withthe morphisms. The inductive limit is an actual C ∗ -algebra. On the way tothe theorem, though, Elliott first treated each infinite sequence of C ∗ -algebrasas a single new object, and defined morphisms between sequences in just sucha way as to make the sequences act like their own inductive limits.²⁶ As Lengsays: ‘Defining the morphisms between these sequences turned out to requiresome ingenuity’ (2002, p.21). But it worked. Elliott’s formal inductive limitssupported the proofs needed to imply the theorem for the actual C ∗ -algebrainductive limits. It would be triply useless to think of the ingeniously definedmorphisms of sequences as ‘structure-preserving functions’ in the logician’s²⁶ Compare Grothendieck’s ind-objects and pro-objects in (Artin et al., 1972, ExposéI).