13 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] |
| 9169 | A statement can be metaphysically necessary and epistemologically contingent [Putnam] |
| Full Idea: A statement can be (metaphysically) necessary and epistemologically contingent. Human intuition has no privileged access to metaphysical necessity. | |
| From: Hilary Putnam (Meaning and Reference [1973], p.160) | |
| A reaction: The terminology here is dangerously confusing. 'Contingent' is a term which (as Kripke insists) refers to reality, not to our epistemological abilities. The locution of adding the phrase "for all I know" seems to handle the problem better. |
| 5819 | Conceivability is no proof of possibility [Putnam] |
| Full Idea: Conceivability is no proof of possibility. | |
| From: Hilary Putnam (Meaning and Reference [1973], p.159) | |
| A reaction: This strikes me as a really basic truth which all novice philosophers should digest. It led many philosophers, especially rationalists, into all sorts of ill-founded claims about what is possible or necessary. Zombies, for instance… |
| 9168 | I can't distinguish elm trees, but I mean by 'elm' the same set of trees as everybody else [Putnam] |
| Full Idea: My concept of an elm tree is exactly the same as my concept of a beech tree (I blush to confess). ..We still say that the extension of 'elm' in my idiolect is the same as the extension of 'elm' in anyone else's, viz. the set of all elm trees. | |
| From: Hilary Putnam (Meaning and Reference [1973], p.154) | |
| A reaction: This example is clearer and less open to hair-splitting than his water/XYZ example. You could, with Putnam, say that his meaning of 'elm' is outside his head, but you could also say that he doesn't understand the word very well. |
| 5820 | 'Water' has an unnoticed indexical component, referring to stuff around here [Putnam] |
| Full Idea: Our theory can be summarized as saying that words like 'water' have an unnoticed indexical component: "water" is stuff that bears a certain similarity relation to the water around here. | |
| From: Hilary Putnam (Meaning and Reference [1973], p.160) | |
| A reaction: This is the causal theory of reference, which leads to externalism about concepts, which leads to an externalist view of thought, which undermines internal accounts of the mind like functionalism, and leaves little room for scepticism… Etc. |
| 9170 | We need to recognise the contribution of society and of the world in determining reference [Putnam] |
| Full Idea: Traditional semantic theory leaves out two contributions to the determination of reference - the contribution of society and the contribution of the real world; a better semantic theory must encompass both. | |
| From: Hilary Putnam (Meaning and Reference [1973], p.161) | |
| A reaction: I strongly agree that there is a social aspect to reference-fixing, but I am much more dubious about the world 'determining' anything. The whole of his Twin Earth point could be mopped up by a social account, with 'experts' as the key idea. |
| 5817 | Language is more like a cooperative steamship than an individual hammer [Putnam] |
| Full Idea: There are tools like a hammer used by one person, and there are tools like a steamship which require cooperative activity; words have been thought of too much on the model of the first sort of tool. | |
| From: Hilary Putnam (Meaning and Reference [1973], p.156) | |
| A reaction: This clear thought strikes me as the most fruitful and sensible consequence of Wittgenstein's later ideas (as opposed to the relativistic 'language game' ideas). I am unconvinced that a private language is logically impossible, but it would be feeble. |
| 5818 | If water is H2O in the actual world, there is no possible world where it isn't H2O [Putnam] |
| Full Idea: Once we have discovered that water (in the actual world) is H2O, nothing counts as a possible world in which water isn't H2O. | |
| From: Hilary Putnam (Meaning and Reference [1973], p.159) | |
| A reaction: Presumably there could be a possible world in which water is a bit cloudy, so the fact that it is H2O is being judged as essential. Presumably the scientists in the possible world might discover that we are wrong about the chemistry of water? |
| 16713 | Philosophers are the forefathers of heretics [Tertullian] |
| Full Idea: Philosophers are the forefathers of heretics. | |
| From: Tertullian (works [c.200]), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 20.2 |
| 6610 | I believe because it is absurd [Tertullian] |
| Full Idea: I believe because it is absurd ('Credo quia absurdum est'). | |
| From: Tertullian (works [c.200]), quoted by Robert Fogelin - Walking the Tightrope of Reason n4.2 | |
| A reaction: This seems to be a rather desperate remark, in response to what must have been rather good hostile arguments. No one would abandon the support of reason if it was easy to acquire. You can't deny its engaging romantic defiance, though. |