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

396 colin mclartyIn short the objects of any topos are continuously variable sets as for examplethe topos Sh M contains the sets varying continuously over the space M.²⁴Mathematics in a topos is like classical mathematics with some differencesreflecting the variation. Specifically mathematics in Sh M differs from classicalby reflecting the topology of M. The cohomology of M measures the differencebetween classical Abelian groups and Abelian groups varying continuously overM. This measure also works in other toposes to give other cohomology theoriesand it was the key to producing étale cohomology. For one expert view ofhow far toposes can be eliminated from étale cohomology, and yet how theyhelp in understanding it, see Deligne (1977, 1998).Categorical thinking is also central to understanding schemes. At the basiclevel of the spaces to be studied, on either definition of schemes, the geometryof a scheme is poorly revealed by looking at it as a set of points with geometricrelations among them. It may even be ‘bizarre’ on that level:[In many schemes] the points ... have no ready to hand geometric sense. ... Whenone needs to construct a scheme one generally does not begin by constructingthe set of points. ... [While] the decision to let every commutative ring definea scheme gives standing to bizarre schemes, allowing it gives a category of schemeswith nice properties. (Deligne, 1998, pp. 12–13)One generally constructs a scheme from its relations to other schemes. Therelations are geometrically meaningful. The category of schemes capturesthose relations and incorporates Noether’s insight that the simplest arithmeticdefinitions often relate a desired ring to arbitrary commutative rings as inSection 14.1.4.14.5.2 Several reasons morphisms are not functionsFunctions are examples of morphisms, but overreliance on this example hasbeen a major obstacle to understanding category theory (McLarty, 1990, §7).Indeed schemes typify the failure of Bourbaki’s structure theory based onstructure preserving functions and so they witness the inadequacy of currentphilosophical theories of ‘structure’. It is not a problem with set theory. Settheory handles schemes easily enough. We handle schemes set-theoreticallyhere. But even on set theoretic foundations scheme maps are not ‘structurepreservingfunctions’ in the set theoretic sense—that is in Bourbaki’s sense orin the closely related sense of the model-theoretic structuralist philosophies ofmathematics.²⁵²⁴ William Lawvere has made this much more explicit but Grothendieck already found each toposis ‘like’ the universe of sets (McLarty, 1990, p.358).²⁵ Bourbaki (1958); Hellman (1989); Shapiro (1997).