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] |
| 5052 | When Gentiles follow the law, they must have the law written in their hearts [Paul] |
| Full Idea: When the Gentiles which have not the law, do by nature the things contained in the law, these, having not the law, are a law unto themselves, which shew the works of the law written in their hearts, their conscience also bearing witness. | |
| From: St Paul (06: Epistle to the Romans [c.55], 02.15) | |
| A reaction: This passage was used by theologians as proof of innate ideas, which are, of course, divinely implanted (in the guise of doing things 'by nature'). It is quoted by Leibniz. Thus Christians annexed credit for pagan morality to God. |
| 19087 | The meaning or purport of a symbol is all the rational conduct it would lead to [Peirce] |
| Full Idea: The entire intellectual purport of any symbol consists in the total of all modes of rational conduct which, conditionally upon all the possible different circumstances and desires, would ensue upon the acceptance of the symbol. | |
| From: Charles Sanders Peirce (Issues of Pragmaticism [1905], EP ii.246), quoted by Danielle Macbeth - Pragmatism and Objective Truth p.169 n1 | |
| A reaction: Macbeth says pragmatism is founded on this theory of meaning, rather than on a theory of truth. I don't see why the causes of a symbol shouldn't be as much a part of its meaning as the consequences are. |
| 7572 | Power is ordained by God, so anyone who resists power resists God, and will be damned [Paul] |
| Full Idea: Let every soul be subject unto the higher powers. For there is no power but of God: the powers that be are ordained by God. Whosoever therefore resisteth the power resisteth the ordinance of God: and they that resist shall receive to themselves damnation. | |
| From: St Paul (06: Epistle to the Romans [c.55], 13:1-2) | |
| A reaction: This notorious passage was used to justify the Divine Right of Kings in England in the seventeenth century. It strikes me as being utterly preposterous, though you might say that violent resistance to an evil dictator only brings worse evil. |