# ‘There is no Ontology Here’: Visual and Structural Geometry in Arithmetic

# ‘There is no Ontology Here’: Visual and Structural Geometry in Arithmetic

Today's number theory solves classical problems by structural tools that violate standard philosophical expectations in ontology. The far-reaching practical demands of this mathematics require on one hand that the tools be fully explicit and more rigorous than many philosophical theories are, and on the other hand that they relate as directly and as concisely as possible to guiding intuitions. The standard tools grew from a century-long trend of unifying algebra, topology, and arithmetic, notably in the Weil conjectures; and they rely on devices that Kronecker produced for his idea of a pure arithmetic. Very large functors serve to organize individually simple kinds of data that can themselves even be depicted in simple pictures. Mathematicians and philosophers have debated issues of individuation and identity raised by these tools.

*Keywords:*
diophantine equation, Kronecker, Weil conjectures, schemes, ontology, number theory, identity, functor, purity of method

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