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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]

In this paper, we investigate the tensions and interferences between philosophy and mathematics in the context of the emergence of pure mathematics in Prussia in the 19th century. In historiography the occurrence of an epistemological turn in mathematics in that century is well known. Such occurrence came to restructure the internal conceptions of its objects, in detriment of a quantitative notion and instead from a relational conception. We argue here that this change was facilitated, precisely, by the Kantian understanding that mathematics, by dealing with synthetic a priori judgments, would conceive itself as the foundation of thinkability. Our approach resides upon investigating how these notions were addressed and conceived of in the works of two authors: Jakob Fries, philosopher of science and mathematics at the beginning of the 19th century, and Hermann Grassmann, mathematician whose work would influence new conceptions by the second half of that century. Our research identifies, in the perspectives and practices of Fries and Grassmann, a conceptual unity, regarding the foundational reflection of mathematics from abstract structures of thinking. This finding reveal, on the other hand, the intervention of philosophical practices and references within the mathematical practice, producing tensions and encompassing a relevant role in the epistemological change in question. [ABSTRACT FROM AUTHOR]