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) |
| 13536 | Skolem did not believe in the existence of uncountable sets [Skolem] |
| Full Idea: Skolem did not believe in the existence of uncountable sets. | |
| From: Thoralf Skolem (works [1920], 5.3) | |
| A reaction: Kit Fine refers somewhere to 'unrepentent Skolemites' who still hold this view. |
| 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) |
| 3061 | Anaxarchus said that he was not even sure that he knew nothing [Anaxarchus, by Diog. Laertius] |
| Full Idea: Anaxarchus said that he was not even sure that he knew nothing. | |
| From: report of Anaxarchus (fragments/reports [c.340 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 09.10.1 |
| 3364 | All mental phenomena contain an object [Brentano] |
| Full Idea: Every mental phenomenon contains something as object within itself. | |
| From: Franz Brentano (Psychology from an empirical standpoint [1874], p. 88), quoted by Jaegwon Kim - Philosophy of Mind p.21 | |
| A reaction: This gives rise to the slogan that 'intentionality is the mark of the mental', which notoriously seems to miss out the phenomenal aspect of mental life. We note now, though, that even emotions have objects. |
| 3225 | Mental unity suggests that qualia and intentionality must connect [Brentano, by Rey] |
| Full Idea: Brentano's thesis is that all mental phenomena are intentional i.e. representational. Support for this view is that assimilating phenomenal experience to attitudes we explain the essential unity of the mind. | |
| From: report of Franz Brentano (Psychology from an empirical standpoint [1874]) by Georges Rey - Contemporary Philosophy of Mind 11.5 | |
| A reaction: Unifying intentionality and qualia in a single theory looks like a good move, but which one has priority? Evolutionary theory says priority goes to whatever produces behaviour. My intuition is that qualia are more basic - in tiny insects, say. |