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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']. |
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. |
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? |
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] |