9 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) |
| 12354 | A 'categorial' property is had by virtue of being or having an item from a category [Wedin] |
| Full Idea: A 'categorial' property is a property something has by virtue of being or having an item from one of the categories. | |
| From: Michael V. Wedin (Aristotle's Theory of Substance [2000], V.5) | |
| A reaction: I deny that these are 'properties'. A thing is categorised according to its properties. To denote the category as a further property is the route to madness (well, to a regress). |
| 12358 | Substance is a principle and a kind of cause [Wedin] |
| Full Idea: Substance [ousia] is a principle [arché] and a kind of cause [aitia]. | |
| From: Michael V. Wedin (Aristotle's Theory of Substance [2000], 1041a09) | |
| A reaction: The fact that substance is a cause is also the reason why substance is the ultimate explanation. It is here that I take the word 'power' to capture best what Aristotle has in mind. |
| 12346 | Form explains why some matter is of a certain kind, and that is explanatory bedrock [Wedin] |
| Full Idea: The form of a thing (of a given kind) explains why certain matter constitutes a thing of that kind, and with this, Aristotle holds, we have reached explanatory bedrock. | |
| From: Michael V. Wedin (Aristotle's Theory of Substance [2000], Intro) | |
| A reaction: We must explain an individual tiger which is unusually docile. It must have an individual form which makes it a tiger, but also an individual form which makes it docile. |
| 1748 | Archelaus was the first person to say that the universe is boundless [Archelaus, by Diog. Laertius] |
| Full Idea: Archelaus was the first person to say that the universe is boundless. | |
| From: report of Archelaus (fragments/reports [c.450 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 02.Ar.3 |
| 5989 | Archelaus said life began in a primeval slime [Archelaus, by Schofield] |
| Full Idea: Archelaus wrote that life on Earth began in a primeval slime. | |
| From: report of Archelaus (fragments/reports [c.450 BCE]) by Malcolm Schofield - Archelaus | |
| A reaction: This sounds like a fairly clearcut assertion of the production of life by evolution. Darwin's contribution was to propose the mechanism for achieving it. We should honour the name of Archelaus for this idea. |