more from this thinker | more from this text
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.
Gist of Idea
If x changes by less and less, it must approach a limit
Source
Richard Dedekind (Continuity and Irrational Numbers [1872], p.27), quoted by Philip Kitcher - The Nature of Mathematical Knowledge 10.7
Book Ref
Kitcher,Philip: 'The Nature of Mathematical Knowledge' [OUP 1984], p.263
A Reaction
[Kitcher says he 'showed' this, rather than just stating it]
18082 | Quantities and ratios which continually converge will eventually become equal [Newton] |
18084 | When successive variable values approach a fixed value, that is its 'limit' [Cauchy] |
18092 | Weierstrass made limits central, but the existence of limits still needed to be proved [Weierstrass, by Bostock] |
18087 | If x changes by less and less, it must approach a limit [Dedekind] |
18187 | Theorems about limits could only be proved once the real numbers were understood [Maddy] |
22886 | The modern idea of 'limit' allows infinite quantities to have a finite sum [Bardon] |