6 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] |
7903 | The six perfections are giving, morality, patience, vigour, meditation, and wisdom [Nagarjuna] |
Full Idea: The six perfections are of giving, morality, patience, vigour, meditation, and wisdom. | |
From: Nagarjuna (Mahaprajnaparamitashastra [c.120], 88) | |
A reaction: What is 'morality', if giving is not part of it? I like patience and vigour being two of the virtues, which immediately implies an Aristotelian mean (which is always what is 'appropriate'). |
20558 | Your representative owes you his judgement, and betrays you if he gives your opinion instead [Burke] |
Full Idea: Your representative owes you, not his industry only, but his judgement; and he betrays instead of serving you if he sacrifices it to your opinion | |
From: Edmund Burke (Address to the Voters of Bristol [1774]), quoted by Adam Swift - Political Philosophy (3rd ed) | |
A reaction: Nice rhetoric, but I'm not sure about the logic of it. Do I betray you if I give my stupid judgement rather than your wise one? Am I so arrogant as to think my judgement is always preferable? His audience was entirely of property owners. |