Dedekind and the Set-theoretical Approach to Algebra
The connections between the work of Dedekind and that of such figures as Noether or Bourbaki explain the close attention that historians have given him in the last 25 years. It is certainly true that Dedekind’s work is the outcome of a serious and deep attempt to reconceive and systematize classical mathematics, and that it prepared the ground in an essential way for modern abstract mathematics. But the idea that Dedekind (and Galois) gave modern algebra its structure, which can be found here and there,2 is too simplifying-the emergence of the structural viewpoint in algebra was a lengthy process, and there is reason to doubt that Dedekind ever viewed mathematics from a strictly structural perspective. For him, pure mathematics was the science of numbers in all its extension and derivations, number systems being more basic than any possible abstract structure. Nonetheless, the abstract-conceptual viewpoint, that he took from Riemann and pursued in a new direction (§1.4), led him to prefer a kind of approach and methods that would prove to be extremely fruitful in the context of 20th-century structural mathematics. This accounts for his influence on Noether and others, and makes it particularly interesting to explore the methodological and conceptual traits of Dedekind’s work that underlay his preferred mathematical style.
KeywordsIdeal Theory Number System Ideal Factor Algebraic Number Galois Theory
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