7 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) |
| 468 | Musical performance can reveal a range of virtues [Damon of Ath.] |
| Full Idea: In singing and playing the lyre, a boy will be likely to reveal not only courage and moderation, but also justice. | |
| From: Damon (fragments/reports [c.460 BCE], B4), quoted by (who?) - where? |
| 6853 | Consequentialism wrongly assumes a clear line between an act and its consequences [Crisp,R] |
| Full Idea: What is wrong with consequentialism is that it assumes that there is agreement about when an act ends and when its consequences begin. | |
| From: Roger Crisp (Interview with Baggini and Stangroom [2001], p.31) | |
| A reaction: I certainly agree that this is a mistake in consequentialism, which has a crude idea of what an action is, though I am not convinced that this is the key fault in the theory. The theory doesn't distinguish acts by people from those by machines. |
| 6854 | Does the environment have value in itself? [Crisp,R] |
| Full Idea: The debate which dominates environmental ethics is whether the environment has value in itself. | |
| From: Roger Crisp (Interview with Baggini and Stangroom [2001], p.37) | |
| A reaction: I say the answer has to be 'yes'. If it only has value in relation to conscious or human life, it will be difficult to demonstrate that they have value. Pleasure and pain, or wisdom, can't imply value on their own; a bigger picture is needed. |