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Axiomatics, methodology, and Dedekind's theory of ideals

Summary: Axiomatics, methodology, and
Dedekind's theory of ideals
Doug White
June 26, 2004
Richard Dedekind has had an incredible influence on modern mathematics,
largely due to his methodological demands which are still valued by math-
ematicians today. Through an investigation of some of his works written
between 1854 and 1877, I reveal a connection between these methodological
demands and features of axiomatic reasoning that he employed. I discuss two
foundational/philosophic works (his Habilitationsrede and Stetigkeit und ir-
rationale Zahlen), and his first two versions of the theory of ideals. Dedekind
himself assists in the endeavor as he often expresses his reasoning for choos-
ing one method over another. This self-reflective feature of Dedekind's efforts
provides a unique opportunity to use his comments as a guide to reading both
the foundational and mathematical works. Furthermore, his methodological
preferences can often inform an interpretation of the chronological develop-
ment of his work. Distinctive changes occurring between his first two versions
of the theory of ideals are particularly relevant to such a discussion. I provide


Source: Avigad, Jeremy - Departments of Mathematical Sciences & Philosophy, Carnegie Mellon University


Collections: Multidisciplinary Databases and Resources; Mathematics