8 ideas
| 17879 | Axiomatising set theory makes it all relative [Skolem] |
| Full Idea: Axiomatising set theory leads to a relativity of set-theoretic notions, and this relativity is inseparably bound up with every thoroughgoing axiomatisation. | |
| From: Thoralf Skolem (Remarks on axiomatised set theory [1922], p.296) |
| 17878 | If a 1st-order proposition is satisfied, it is satisfied in a denumerably infinite domain [Skolem] |
| Full Idea: Löwenheim's theorem reads as follows: If a first-order proposition is satisfied in any domain at all, it is already satisfied in a denumerably infinite domain. | |
| From: Thoralf Skolem (Remarks on axiomatised set theory [1922], p.293) |
| 17880 | Integers and induction are clear as foundations, but set-theory axioms certainly aren't [Skolem] |
| Full Idea: The initial foundations should be immediately clear, natural and not open to question. This is satisfied by the notion of integer and by inductive inference, by it is not satisfied by the axioms of Zermelo, or anything else of that kind. | |
| From: Thoralf Skolem (Remarks on axiomatised set theory [1922], p.299) | |
| A reaction: This is a plea (endorsed by Almog) that the integers themselves should be taken as primitive and foundational. I would say that the idea of successor is more primitive than the integers. |
| 17881 | Mathematician want performable operations, not propositions about objects [Skolem] |
| Full Idea: Most mathematicians want mathematics to deal, ultimately, with performable computing operations, and not to consist of formal propositions about objects called this or that. | |
| From: Thoralf Skolem (Remarks on axiomatised set theory [1922], p.300) |
| 8386 | Events are picked out by descriptions, and facts by whole sentences [Crane] |
| Full Idea: Events are picked out using descriptions ('The death of Caesar'), while facts are picked out using whole sentences ('Caesar died'). | |
| From: Tim Crane (Causation [1995], 1.4.2) | |
| A reaction: Useful, and interesting. He mentions that Kim's usage doesn't agree with this. For analysis purposes, this means that an event is a more minimal item than a fact, and many facts will contain events as components. |
| 3016 | Even the gods cannot strive against necessity [Pittacus, by Diog. Laertius] |
| Full Idea: Even the gods cannot strive against necessity. | |
| From: report of Pittacus (reports [c.610 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 01.5.4 |
| 8387 | A cause has its effects in virtue of its properties [Crane] |
| Full Idea: Some philosophers say that a cause has its effects in virtue of its properties. | |
| From: Tim Crane (Causation [1995], 1.4.2) | |
| A reaction: The trouble with this approach, I think, is that it encourages us to invent dubious properties, because every explanation of an effect will require one. Dormative properties, for example, are ascribed to sleeping pills. |
| 8384 | The regularity theory explains a causal event by other items than the two that are involved [Crane] |
| Full Idea: An unsatisfactory aspect of the regularity thesis is that it explains why this A caused this B in terms of facts about things other that this A and this B. But we want to know what it is about this A and this B that makes one the cause of the other? | |
| From: Tim Crane (Causation [1995], 1.3) | |
| A reaction: Well said. This is the failing of any attempt to define things by their relationships (e.g. functional definitions). Hume, of course, was only relying on regularity because when he focused on the actual A and B, they had no helpful experiences to offer. |