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

366 colin mclarty1930s (Washington, 1997, p.101). This could eliminate Galois cohomologyfrom the proof but Washington takes it the other way around as introducingthat cohomology to the reader. The elimination would be routine and wouldreplace conceptual arguments by opaque calculations. Of course, to shortcutWiles’s use of cohomology by genuinely simpler new insights would bewonderful—like shortening any proof by any simpler new insight.13.3.3 LanguagesSome philosophical theories of structuralism allow ‘no intelligible attempt—ofthe kind that Zermelo and Fraenkel undertook—to unify mathematical practice’(MacBride, 2005, p.572). Structural methods in practice, though, standout for making the unity of mathematics work more deeply and powerfullythan Zermelo and Fraenkel ever did. Barry Mazur well expressed this inaccepting the 2000 Steele Prize. He names the founders of category theoryEilenberg, Mac Lane, and Grothendieck, not for category theory but also notby coincidence. He names them for what they did with the theory:I came to number theory through the route of algebraic geometry and beforethat, topology. The unifying spirit at work in those subjects gave all the newideas a resonance and buoyancy which allowed them to instantly echo elsewhere,inspiring analogies in other branches and inspiring more ideas. One has only tothink of how the work of Eilenberg, Mac Lane, Steenrod, and Thom in topologyand the early work in class field theory as further developed in the hands ofEmil Artin, Tate, and Iwasawa was unified and amplified in the point of viewof algebraic geometry adopted by Grothendieck, a point of view inspired bythe Weil conjectures, which presaged the inextricable bond between topologyand arithmetic. One has only to think of the work of Serre or of Tate.¹⁶But mathematics is one subject, and surely every part of mathematics has beenenriched by ideas from other parts. (NAMS, 2000, p.479)These methods cultivate each branch of mathematics in its own terms, andhighlight how each one precisely in its own terms has bonds to others.Poincaré defended geometry and specifically his beloved analysis situs nowcalled topology against those who would reduce all mathematics to differentialanalysis. We must not ‘fail to recognize the importance of well constructedlanguage’:The problems of analysis situs might not have suggested themselves if the analyticlanguage alone had been spoken; or, rather, I am mistaken, they would haveoccurred surely, since their solution is essential to a crowd of questions in analysis.¹⁶ See references to Artin, Iwasawa, Tate, and Thom in Monastyrsky (1998). Serre and the Weilconjectures are central to the accompanying case study.