Essays on the Theory of Numbers - Internet Archive

Not infrequently this opposition is debated in terms of which approachis “the right one”, with the implication that only one ofthem but not the other is legitimate. (This started with Dedekind andKronecker. We already noted that Dedekind explicitly rejectedconstructivist strictures on his work, although he did not rule outcorresponding projects as illegitimate in themselves. Kronecker, withhis strong opposition to the use of set-theoretic and infinitarytechniques, went further in the other direction.) But another questionis more basic: Can the contrast between the two approaches toalgebraic number theory, or to mathematics more generally, be capturedmore sharply and revealingly; and in particular, what is itsepistemological significance?

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Essays on the Theory of Numbers by Richard Dedekind

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Essays in the theory of numbers, 1

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The third main part of Dedekind's approach connects and complementsthe first two. Not only does he study systems of objects or wholeclasses of such systems; and not only does he attempt to identifybasic concepts applicable to them. He also tends to do both, often inconjunction, by considering mappings on the systems studied,especially structure-preserving mappings (homomorphisms etc.). Thisimplies that what is crucial about a mathematical phenomenon may notlie on the surface (concrete features of examples, particularsymbolisms, etc.) but go deeper. And while the deeper features areoften captured set-theoretically (Dedekind cuts, ideals, quotientstructures, etc.), this points beyond set theory in the end, towardscategory theory (Corry 2004, McLarty 2006, as well as Ferreirós2016a).

Essays on the Theory of Numbers: I

Essays on the Theory of Numbers - I

Dedekind's published this account of the real numbers only in 1872,fourteen years after developing the basic ideas on which it relies. Itwas not the only account proposed at the time; indeed, variousmathematicians addressed this issue, including: Weierstrass, Thomae,Méray, Heine, Hankel, Cantor, and somewhat later, Frege(Dieudonné 1985, ch. 6, Boyer & Merzbach 1991, ch. 25,Jahnke 2003, ch. 10). Most familiar among their alternative treatmentsis probably Cantor's, also published in 1872. Instead of using“Dedekind cuts”, Cantor works with (equivalence classesof) Cauchy sequences of rational numbers. The system of such (classesof) sequences can also be shown to have the desired properties,including continuity. Like Dedekind, Cantor starts with the infiniteset of rational numbers; and Cantor's construction again reliesessentially on the full power set of the rational numbers, here in theform of arbitrary Cauchy sequences. In such set-theoretic respects thetwo treatments are thus equivalent. What sets apart Dedekind'streatment of the real numbers, from Cantor's and all the others, isthe clarity he achieves with respect to the central notion ofcontinuity. His treatment is also more maturely and elegantlystructuralist, in a sense to be spelled out further below.

Essays on the theory of numbers (Book, 2010) …

Essays On The Theory Of Numbers, I

The first part is closely tied to Dedekind's employment ofset-theoretic tools and techniques. He uses these to construct newmathematical objects (the natural and real numbers, ideals, modules,etc.) or whole classes of such objects (various algebraic numberfields, rings, lattices, etc.). Even more important andcharacteristic, both in foundational and other contexts, is anotheraspect. Not only are infinite sets used by Dedekind; they are alsoendowed with general structural features (order relations, arithmeticand other operations, etc.); and the resulting systems are studied interms of higher-level properties (continuity for the real numbersystem, unique factorization in algebraic number fields, etc.).