10 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) |
| 19400 | Possibles demand existence, so as many of them as possible must actually exist [Leibniz] |
| Full Idea: From the conflict of all the possibles demanding existence, this at once follows, that there exists that series of things by which as many of them as possible exist. | |
| From: Gottfried Leibniz (Exigency to Exist in Essences [1690], p.91) | |
| A reaction: I'm in tune with a lot of Leibniz, but my head swims with this one. He seems to be a Lewisian about possible worlds - that they are concrete existing entities (with appetites!). Could Lewis include Leibniz's idea in his system? |
| 19401 | God's sufficient reason for choosing reality is in the fitness or perfection of possibilities [Leibniz] |
| Full Idea: The sufficient reason for God's choice can be found only in the fitness (convenance) or in the degree of perfection that the several worlds possess. | |
| From: Gottfried Leibniz (Exigency to Exist in Essences [1690], p.92) | |
| A reaction: The 'fitness' of a world and its 'perfection' seem very different things. A piece of a jigsaw can have wonderful fitness, without perfection. Occasionally you get that sinking feeling with metaphysicians that they just make it up. |
| 19402 | The actual universe is the richest composite of what is possible [Leibniz] |
| Full Idea: The actual universe is the collection of the possibles which forms the richest composite. | |
| From: Gottfried Leibniz (Exigency to Exist in Essences [1690], p.92) | |
| A reaction: 'Richest' for Leibniz means a maximum combination of existence, order and variety. It's rather like picking the best starting team from a squad of footballers. |
| 8872 | It is widely supposed that externalism cannot be reconciled with first-person authority [Davidson] |
| Full Idea: It has been widely supposed that externalism, which holds that the contents of a person's propositional attitudes are partly determined by factors of which the person may be ignorant, cannot be reconciled with first-person authority. | |
| From: Donald Davidson (Epistemology Externalized [1990], p.197) | |
| A reaction: It is certainly a bit puzzling if you go around saying 'Actually, people don't know their own thoughts'. Davidson aims to defend first-person authority. The full story is developed in Tyler Burge's views on 'anti-individualism'. |
| 8874 | It is hard to interpret a speaker's actions if we take a broad view of the content [Davidson] |
| Full Idea: It will explain a speaker's actions far better if we interpret him as he intended to be interpreted, than if we suppose he means and thinks what someone else might mean and think who used the same words 'correctly'. | |
| From: Donald Davidson (Epistemology Externalized [1990], p.199) | |
| A reaction: This comes down to the fact that our actions have to be motivated by internal meanings. If I defer to experts on the essence of gold, I still buy gold according to how I myself understand it. So meaning has two components? |
| 8873 | The cause of a usage determines meaning, but why is the microstructure of water relevant? [Davidson] |
| Full Idea: While I agree that the usual cause of the use of the word determines what it means, I do not see why sameness of microstructure is necessarily the relevant similarity that determines my reference of the word 'water'. | |
| From: Donald Davidson (Epistemology Externalized [1990], p.198) | |
| A reaction: This is a problem for essentialists who build their views on semantic considerations. But the stability of what causes 'water' thoughts is the microstructure of water. However, that is an explantion of meaning, not a definition of it. |