Ideas from 'Continuity and Irrational Numbers' by Richard Dedekind [1872], by Theme Structure

[found in 'Essays on the Theory of Numbers' by Dedekind,Richard [Dover 1963,0-486-21010-3]].

green numbers give full details    |     back to texts     |     unexpand these ideas

---

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers

17611

We want the essence of continuity, by showing its origin in arithmetic

			Full Idea: It then only remained to discover its true origin in the elements of arithmetic and thus at the same time to secure a real definition of the essence of continuity.
			From: Richard Dedekind (Continuity and Irrational Numbers [1872], Intro)
			A reaction: [He seeks the origin of the theorem that differential calculus deals with continuous magnitude, and he wants an arithmetical rather than geometrical demonstration; the result is his famous 'cut'].

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / i. Reals from cuts

10572

A cut between rational numbers creates and defines an irrational number

			Full Idea: Whenever we have to do a cut produced by no rational number, we create a new, an irrational number, which we regard as completely defined by this cut.
			From: Richard Dedekind (Continuity and Irrational Numbers [1872], §4)
			A reaction: Fine quotes this to show that the Dedekind Cut creates the irrational numbers, rather than hitting them. A consequence is that the irrational numbers depend on the rational numbers, and so can never be identical with any of them. See Idea 10573.

6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / f. Arithmetic

17612

Arithmetic is just the consequence of counting, which is the successor operation

			Full Idea: I regard the whole of arithmetic as a necessary, or at least natural, consequence of the simplest arithmetic act, that of counting, and counting itself is nothing else than the successive creation of the infinite series of positive integers.
			From: Richard Dedekind (Continuity and Irrational Numbers [1872], §1)
			A reaction: Thus counting roots arithmetic in the world, the successor operation is the essence of counting, and the Dedekind-Peano axioms are built around successors, and give the essence of arithmetic. Unfashionable now, but I love it. Intransitive counting?

6. Mathematics / A. Nature of Mathematics / 5. The Infinite / l. Limits

18087

If x changes by less and less, it must approach a limit

			Full Idea: If in the variation of a magnitude x we can for every positive magnitude δ assign a corresponding position from and after which x changes by less than δ then x approaches a limiting value.
			From: Richard Dedekind (Continuity and Irrational Numbers [1872], p.27), quoted by Philip Kitcher - The Nature of Mathematical Knowledge 10.7
			A reaction: [Kitcher says he 'showed' this, rather than just stating it]