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

14‘There is No Ontology Here’:Visual and Structural Geometryin ArithmeticCOLIN MCLARTYIn Diophantine geometry one looks for solutions of polynomial equations whichlie either in the integers, or in the rationals, or in their analogues for numberfields. Such polynomial equations {F i (T 1 , ... T n )} define a subscheme of affinespace A n over the integers which can have points in an arbitrary commutativering R. (Faltings, 2001, p.449)Structuralists in philosophy of mathematics can learn from the current heritageof the ancient arithmetician Diophantus. A list of polynomial equations definesa kind of geometric space called a scheme. By one definition these schemesare countable sets built from integers in very much the way that LeopoldKronecker approached pure arithmetic. In another version every scheme is afunctor as big as the universe of sets. The two versions are often mixed togetherbecause they give precisely the same structural relations between schemes. Thepractice was vividly put by André Joyal in conversation: ‘There is no ontologyhere’. Mathematicians work rigorously with relations among schemes withoutchoosing between the definitions. The tools which enable this in principle andrequire it in practice grew from topology.The three great projects for 20th century mathematics were to absorbRichard Dedekind’s and David Hilbert’s algebra, to absorb Henri Poincaré’sand Luitzen Brouwer’s topology, and to create functional analysis.¹ Algebraand topology made explosive advances when Emmy Noether initiated a seriesof ever-deeper structural unifications. Her group theory became the method of¹ Functional analysis also joined the structural unification (Dieudonné, 1981). Leading workers in allthree projects contributed to mathematical logic.