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The discussion of purity of method usually emphasizes the study of particular proofs of mathematical practice. A critique of this position questions the value of purity by claiming that "mathematical progress" depends essentially on the use of impure methods. This article shows that a more holistic perspective, focusing on how purity claims can operate in the construction of autonomous theories, allows us to identify a context where purity of method acquires value in mathematical practice. In particular, the relevance of such demands is shown in the case of Kronecker's and Dedekind's algebraic number theories. [ABSTRACT FROM AUTHOR]

The aim of this paper is to reject the structuralist pretension to conceive of theoretical models as isomorphic to real structures of the world. By means of a series of examples drawn from physics, I show that a phenomenon can be characterized in different ways, complementary and equivalent with each other, and such that none of them is more real or more fictive than the other ones. I defend that the isomorphism relation can be held only among algebraic structures, and, secondly, that nature cannot be defined extensionally, as it were a set of fixed and immutable elements. [ABSTRACT FROM AUTHOR]