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

364 colin mclartyto each other. Some were already known in the 1920s tobeinequivalenttoeach other when applied to some more arcane spaces. Topologists had all toomany nuts and bolts definitions of homology but no structural characterization.They had no homological ‘Fact 1’.The Fact, when it was found by Eilenberg and Steenrod (1945), turned outto characterize homology (up to isomorphism) as a pattern of functors from acertain category of topological spaces and continuous maps to the category ofAbelian groups. This characterization enabled massive progress in homology.It also suggested useful variants such as K-theory which began by solving theclassical problem of independent vector fields on an n-dimensional sphere S n .Itshowed how many everywhere different ways there are to ‘comb the hair flat’on S n . There is only one way on the circle S 1 : combing all clockwise is notdifferent in this sense from combing all counterclockwise. There is no way tocomb it all flat on the sphere S 2 or any even-dimensional S 2n . The matter iscomplicated in higher odd dimensions. Carter (2004) looks at K-theory andphilosophical structuralism.Noether’s number theory was simplified and extended by similar functorsapplied to groups in place of topological spaces. This group cohomology lies behindthe Langlands Program, and so behind Wiles on Fermat’s Last Theorem, andmuch more. One earlier classical result was Gerd Faltings’s proof of theMordell conjecture, a problem that had drawn top number theorists for oversixty years: Any algebraic curve above a certain low level of complexity has atmost finitely many rational points (Monastyrsky, 1998, p.71). For philosophicdiscussion of these ideas see Krieger (2003), especially on Robert Langlands andAndré Weil.All these proofs ignore nuts and bolts as far as possible—which is veryfar indeed by the conventional, functorial methods. That made the proofsfeasible. Even when the morphisms of a category are structure preservingfunctions, these methods abstract away from the ‘internal’ nature of functionsin favor of the patterns they form. The categorical definition of isomorphismgiven in Section 13.2 is a textbook example. In other important categories themorphisms are not functions but the useful pattern-theoretic properties stillhold for them. The accompanying case study gives examples, and see Corfield(2003, esp.ch.10).13.3.2. Intuition and purityOlga Taussky-Todd joked that she had trouble with Noether’s courses onnumber theory because ‘my own training was in ... the kind of number theorythat involves numbers’ (Taussky-Todd, 1981, p.79). Noether created radicalnew methods on the board as she lectured and everyone needed time to