| 18082 | Quantities and ratios which continually converge will eventually become equal [Newton] |
| Full Idea: Quantities and the ratios of quantities, which in any finite time converge continually to equality, and, before the end of that time approach nearer to one another by any given difference become ultimately equal. | |
| From: Isaac Newton (Principia Mathematica [1687], Lemma 1), quoted by Philip Kitcher - The Nature of Mathematical Knowledge 10.2 | |
| A reaction: Kitcher observes that, although Newton relies on infinitesimals, this quotation expresses something close to the later idea of a 'limit'. |
| 18084 | When successive variable values approach a fixed value, that is its 'limit' [Cauchy] |
| Full Idea: When the values successively attributed to the same variable approach indefinitely a fixed value, eventually differing from it by as little as one could wish, that fixed value is called the 'limit' of all the others. | |
| From: Augustin-Louis Cauchy (Cours d'Analyse [1821], p.19), quoted by Philip Kitcher - The Nature of Mathematical Knowledge 10.4 | |
| A reaction: This seems to be a highly significan proposal, because you can now treat that limit as a number, and adds things to it. It opens the door to Cantor's infinities. Is the 'limit' just a fiction? |
| 18092 | Weierstrass made limits central, but the existence of limits still needed to be proved [Weierstrass, by Bostock] |
| Full Idea: After Weierstrass had stressed the importance of limits, one now needed to be able to prove the existence of such limits. | |
| From: report of Karl Weierstrass (works [1855]) by David Bostock - Philosophy of Mathematics 4.4 | |
| A reaction: The solution to this is found in work on series (going back to Cauchy), and on Dedekind's cuts. |
| 18087 | If x changes by less and less, it must approach a limit [Dedekind] |
| 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] |
| 18187 | Theorems about limits could only be proved once the real numbers were understood [Maddy] |
| Full Idea: Even the fundamental theorems about limits could not [at first] be proved because the reals themselves were not well understood. | |
| From: Penelope Maddy (Naturalism in Mathematics [1997], I.2) | |
| A reaction: This refers to the period of about 1850 (Weierstrass) to 1880 (Dedekind and Cantor). |
| 22886 | The modern idea of 'limit' allows infinite quantities to have a finite sum [Bardon] |
| Full Idea: The concept of a 'limit' allows for an infinite number of finite quantities to add up to a finite sum. | |
| From: Adrian Bardon (Brief History of the Philosophy of Time [2013], 1 'Aristotle's') | |
| A reaction: This is only if the terms 'converge' on some end point. Limits are convenient fictions. |