7 ideas
| 17611 | We want the essence of continuity, by showing its origin in arithmetic [Dedekind] |
| 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 [Dedekind] |
| 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 [Dedekind] |
| 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 [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] |
| 14961 | Clearly a pipe can survive being taken apart [Cartwright,R] |
| Full Idea: There is at the moment a pipe on my desk. Its stem has been removed but it remains a pipe for all that; otherwise no pipe could survive a thorough cleaning. | |
| From: Richard Cartwright (Scattered Objects [1974], p.175) | |
| A reaction: To say that the pipe survives dismantling is not to say that it is fully a pipe during its dismantled phase. He gives a further example of a book in two volumes. |
| 14962 | Bodies don't becomes scattered by losing small or minor parts [Cartwright,R] |
| Full Idea: If a branch falls from a tree, the tree does not thereby become scattered, and a human body does not become scattered upon loss of a bit of fingernail. | |
| From: Richard Cartwright (Scattered Objects [1974], p.184) | |
| A reaction: This sort of observation draws me towards essentialism. A body is scattered if you divide it in a major way, but not if you separate off a minor part. It isn't just a matter of size, or even function. We have broader idea of what is essential. |
| 14470 | Explanatory exclusion: there cannot be two separate complete explanations of a single event [Kim] |
| Full Idea: The general principle of explanatory exclusion states that two or more complete and independent explanations of the same event or phenomenon cannot coexist. | |
| From: Jaegwon Kim (Mechanism, purpose and explan. exclusion [1989], 3) | |
| A reaction: This is a rather optimistic view of explanations, with a strong element of reality involved. I would have thought there were complete explanations at different 'levels', which were complementary to one another. |