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) |
| 18617 | Substances, unlike aggregates, can survive a change of parts [Mumford] |
| Full Idea: Substances can survive a change in their parts in a way that a mere aggregate of parts. | |
| From: Stephen Mumford (Metaphysics: a very short introduction [2012], 3) | |
| A reaction: A simple but very important idea. If we then distinguish between 'substances' and 'aggregates' we get a much clearer grip on things. Is the Ship of Theseus a substance or an aggregate? There is no factual answer to that. What do you want to explain? |
| 18618 | Maybe possibilities are recombinations of the existing elements of reality [Mumford] |
| Full Idea: It has been suggested that we could think of possibilities as recombinations of all the existing elements of reality. | |
| From: Stephen Mumford (Metaphysics: a very short introduction [2012], 8) | |
| A reaction: [Armstrong 1989 is the source] The obvious problem would be that the existence of an entirely different reality would be impossible, if this was all possibility could be. It seems to cramp the style of the possible too much. Are properties elements? |
| 18619 | Combinatorial possibility has to allow all elements to be combinable, which seems unlikely [Mumford] |
| Full Idea: The combinatorial account only works if you allow that the elements are recombinable. ...But could Lincoln really have been green? It seems possible that you could jump to the moon, unless we impose some restrictions. | |
| From: Stephen Mumford (Metaphysics: a very short introduction [2012], 8) | |
| A reaction: Mumford suggests different combination rules for logical and natural possibility. The general objection is that combinatorial possibility is too permissive - which it clearly is. |
| 18620 | Combinatorial possibility relies on what actually exists (even over time), but there could be more [Mumford] |
| Full Idea: Can combinatorial possibility deliver enough possibilities? It uses the existing elements, but there might have been one more particular or one more property. Even extended over time, the elements seem finite, yet there could have been more. | |
| From: Stephen Mumford (Metaphysics: a very short introduction [2012], 8) | |
| A reaction: [compressed] One objection is that the theory allows too much, and now the objection is that it allows too little. Both objections are correct, so that's the end of that. But I admire the attempt to base modality on actuality. |
| 19698 | Deviant causal chain: a reason causes an action, but isn't the reason for which it was performed [Davidson, by Neta] |
| Full Idea: A 'deviant causal chain' is when an agent has a reason for performing an action, and for the reason to cause the performance, without that being the reason for which the agent performed it. | |
| From: report of Donald Davidson (Freedom to Act [1973]) by Ram Neta - The Basing Relation II | |
| A reaction: Davidson's thesis is that 'reasons are causes'. This was a problem he faced. I think this discussion is now obscured by the complex and multi-layered account of action which is emerging from neuroscience. |